Karl Franzens University of Graz. Institute of PhysicsDepartment of Theoretical Physics

Dissertation submitted for the degree of Doctor in Natural Sciences

Symmetry and Functional Aspects inQuantum Field Theory

Candidate: Selym Villalba Chavez

Supervisor: Dr. Reinhard Alkofer

Graz, Austria2010

Ich versichere dass meine Dissertation eine eigenstandige Originalarbeit darstellt, dievon mir selbstandig angefertigt und abgefaßt worden ist, andere als die angegebenenQuellen und Hilfsmittel nicht benutzt und mich auch sonst keiner unerlaubten Hilfe bedi-ent habe.

to my beloved wife

Abstract

In this thesis, the effects of the Lorentz symmetry break down in the gauge sector of QED inpresence of a supercritical magnetic field are analyzed. We show that a photon anomalousmagnetic moment is dynamically induced by a superstrong magnetic field. The possibleinteraction between the virtual electromagnetic radiation and the external magnetic fieldsuggests that a background of virtual photons is a source of magnetization to the wholevacuum. The corresponding contribution to the vacuum magnetization density is deter-mined by considering the individual contribution of each vacuum polarization eigenmodein the Euler-Heisenberg Lagrangian. Additional issues concerning the transverse pres-sures are analyzed. We also determined the connection between the Hamilton and thestandard Lagrange formalism for a generic Quantum Field Theory. By means of func-tional methods a systematic procedure is presented to identify the full correlation func-tions, which depend on the momentum fields, as functionals of those usually appearingin the standard Lagrange formalism. Within the same context, the Poincare generatorsare obtained and the connection between the form factors and the boost operator is estab-lished. Considering the latter, we note that what are usually called the pion square chargeradii, defined from derivatives of the pion form factor at zero squared momentum transfer,is completely blurred out by relativistic and interaction corrections, so that it is not clear atall how to interpret these quantities in terms of the pion charge distribution.

i

Acknowledgments

It has been three years from my arrival to Graz. A catastrophic and stressful situationplagued the starting of my PhD. Fortunately, everything concerning to that stage is just akeepsake. Due to this reason, it is a pleasure for me to write this humble recognition tothose persons which made this work a reality.

I would like to expresses my deep gratitude to Reinhard Alkofer, my supervisor, foraccepting me as his student three years ago and for his support and useful advice duringthis research. I am particularly grateful to Felipe J. Llanes-Estrada and Willibald Plessasfor valuable discussions and very helpful conversations.

I wish to express my gratitude to A. Shabad and Florian Hebenstreit for their criticalassessment, important remarks and comments on the first part of this dissertation, whichhas strengthened the current presentation.

A special thank goes to Kai Schwenzer, Klaus Lichtenegger and Erek Bilgici for manydiscussions regarding our research and for the essential help beyond the framework ofthe Doktoratskolleg. I am also grateful to Claudia Spidla, our doctoral assistant, for herconstant and valuable help from my arrival to Graz. I want to mention as well the namesof Christof Gattringer, the speaker of the Doctoral Program, Elmar Biernat, Markus Huberand Helios Sanchis Alepuz.

I have not words enough to thank my beloved wife for her patience, spirit and love,and in particular for reminding me that live is more than Theoretical Physics.

A final word of gratitude goes to my small family in Cuba for being a strong motivationin my life and for paying my absence for three long years.

to all of them, thanks a lot

This thesis has been supported by the Doktoratskolleg ``Hadrons in Vacuum, Nuclei andStars” of the Austrian Science Fund (FWF) under contract W1203-N08.

ii

Introduction

Symmetries and functional approaches have played an important role from the early timesof Quantum Field Theory (QFT). Whilst fundamental symmetries of nature classify the el-ementary particles, the functional methods allow to determine the Green’s functions nec-essary to describe the dynamics of interacting fields. There is, however, some physical sce-narios where the quantum processes break an invariance which is originally consideredas fundamental. Historically, this fact has been linked with the well known Higgs mech-anism [1], in which the break down of the vacuum symmetry of an intermediate scalarboson generates masses in the gauge fields sector of the Standard Electroweak Theory.

QFT in presence of an external classical electromagnetic field is another subject inwhich a symmetry breaking occurs, but this time, of the underlying spacetime symmetriesthemselves. A usual treatment of these theories requires a relativistic-invariant Lagrangianwith fields transforming as irreducible representations of the Poincare group (ISO(3, 1))[2, 3, 4, 5, 6, 7]. In this context, the external field (Fµν) is considered as a classical non-dymanical entity whose consequences emerge as soon as its interaction with charged par-ticles is taken into account. This procedure provides the theory with a dependence on theFµν−structure and therefore on the reference frame. Consequently, the Minkowski spaceis no longer isotropic nor homogeneous and a break down of the usual Poincare symmetryarises.

The most elementary consequence associated with this issue emerged from the Diracrelativistic theory and concerns to the fact that an electron with charge e, mass m, andspin one-half s has a magnetic moment m = g e

2ms with a Lande factor g = 2. Experi-ment measurements revealed, however, that g exceeds the value predicted by the the Diractheory. The explanation of this deviation constitutes one of the most important achieve-ments of Quantum Electrodynamics (QED). According to this, an electron acquires ananomalous magnetic moment (m′ = α

πe

2ms) [9] proportional to the fine-structure constantα = e2/4π = 1/137 due to the interactions between the background of virtual electron-positron pairs with the external magnetic field B. Consequently, the vacuum acquires anon-trivial energy associated with the Euler-Heisenberg Lagrangian [10, 11, 12, 13] andhigher order corrections to m have given rise to a series in power of α/π. Furthermore, ithas been possible to generate an electron mass and extend the concept of the anomalous

iii

magnetic moment m for the case of massless QED [16, 17] inducing a chiral symmetrybreaking.

Similar to an electron, a photon could carry a magnetic moment mγ [18]. From a classi-cal point of view, there is not reason to introduce this quantity. However, QED in an exter-nal field and the Maxwell-Einstein theory predict processes like photon splitting [19, 20],photon capture [21, 22, 23, 24, 25] and photon-graviton mixing [26, 27, 28], which pro-vide evidences that electromagnetic radiation interacts with the external magnetic field bymeans of the vacuum polarization tensors. Throughout this work we will show that forlow energies and superstrong magnetic fields |B| À Bc, Bc = m2/e = 4.42 · 1013G, theMaxwell equation in the ``medium”1 seems to describe a neutral massless particle witha genuine magnetic moment. We will point out that mγ is dynamically induced by Bthrough its interaction with the virtual electron-positron pair whose dynamics tend to bequasi confined in 1+1 dimensions [29, 30, 31, 32]. This fact is manifest as soon as the effectsof the vacuum polarization tensor on the Coulomb-potential of a point-like static chargeare considered. In this context, the attraction force between two charges acquires a string-like form which is directed parallel to B. The electric lines of force produced by the chargesare gathered inside a narrow tube, whereas the potential grows linearly with distance inbetween the Larmour l = (e|B|)−1/2 and Compton λC = m−1 lengths and hence provides``confinement” [33, 34, 35, 36]. As a consequence, the electron-positron interaction turnsout to be stronger favoring the particle-antiparticle pairing.

The interesting point associated with this phenomenology is that, despite the absenceof asymptotic freedom, QED in a superstrong magnetic field has some features inherentto the theory of strong interacting particles, namely Quantum Chromodynamics (QCD)[37, 38]. In this framework, the confinament of a color singlet particle in a color neutralobject and the dynamical chiral symmetry breaking constitute fundamental low-energyproperties which are presumably related to each other. Both phenomena seem to be en-rolled within the non-trivial structure of hadrons and therefore in their spectrum. How-ever, after four decades of intense effort, a genuine analytical solution to the latter problemremains a challenge in QFT. This fact is partially connected with the quark-confinementproblem which has not a concluding analytical picture as well.

Dyson-Schwinger Equations (DSEs) and Renormalization Group Equations constitutepowerful analytical functional methods used to describe the infrared behavior of QCD.These have been successfully applied in Landau gauge Yang-Mills theory, for a reviewsee [39]. However their implementation in Coulomb gauge Yang-Mills theory remain elu-sive. In contrast to Landau gauge, Coulomb gauge Yang-Mills theory has two advantageswhich makes it rather special. First, the number of variables within the theory is reducedto the physical degrees of freedom. Second, the so-called ``Gribov-Zwanziger” scenarioof confinement [40, 41] becomes especially relevant because the temporal component of

1The use of this terminology will be explained later on.

iv

the gluon propagator provides a long-range confining force while the transverse spatialcomponents are suppressed in the infrared. However, this theory is plagued by severalnon-trivial problems that emerge in a perturbative treatment within the standard Lagrangeformalism [42, 43, 44, 45]. The most relevant concerns to the so-called energy divergenceswhich cannot be regularized using any of the standard procedures. Due to this reason, theproblem of renormalizing Coulomb gauge Yang-Mills theory is still unsolved.

One way to avoid the energy divergences is to work within the Hamiltonian formalism,where formal arguments show that such divergences cancel exactly [41]. This framework,however demands the introduction of extra field-variables and in consequence DSEs be-come more cumbersome [46, 47]. Due to the dramatically complicated form of the func-tional equations in the Hamiltonian formalism, an explicit non-perturbative study is far tooinvolved to be computationally feasible. Therefore, an analysis in the Lagrange formalismwould be highly desirable. To this end we provide general connections between the Greensfunctions in the two formulations that should help to perform the renormalization in theLagrange framework utilizing the insight in the renormalization procedure obtained in theHamilton framework. In order to establish general relations between dressed correlationfunctions in the different formulations we exploit the following properties. At vanish-ing sources associated to the momentum fields the Effective Action (i.e. the GeneratingFunctional of one-particle-irreducible (1PI) Green’s functions) of the Hamilton formalismreduces to the one of the Lagrangean approach. This allows to reduce the set of Dyson-Schwinger equations (DSEs) [48, 49] in the first order formalism to the corresponding setderived from the standard path integral representation.

To expose our results we have structured this thesis in three parts. Part I concerns to theeffect of the photon radiative correction on the vacuum energy in a superstrong magneticfield. A precessing photon anomalous magnetic moment is derived, and its connectionwith the classical angular momentum of the radiation field is established. We shall showthat in the infrared domain the magnetic moment turns out to be a vector with two or-thogonal components in correspondence with the cylindrical symmetry imposed by theexternal field. We also explore the possibility to define such a quantity in the high energylimit. In addition we show that the vacuum energy is slightly modified by the electro-magnetic radiation. The corresponding contribution to the vacuum magnetization den-sity is determined by considering the individual contribution of each vacuum polarizationeigenmodes in the Euler-Heisenberg Lagrangian. A paramagnetic response is found inone of them, whereas the remaining ones are diamagnetic. Additional issues concerningthe transverse pressures are analyzed.

In the Part II we establish the connections between the Hamilton and the standard La-grange formalism for a generic Quantum Field Theory with vanishing vacuum expectationvalues of the fundamental fields. By means of functional methods a systematic procedureis presented to identify the full correlation functions, which depend on the momentumfields, as functionals of those usually appearing in the standard Lagrange formalism. We

v

show that Lagrange correlation functions can be decomposed into tree diagrams whereasthe decomposition of Hamilton correlation functions involves loop corrections similar tothose arising in n-particle effective actions. To demonstrate the method we derive for the-ories with linearized interactions the propagators of composite auxiliary fields and theones of the fundamental degrees of freedom. The formalism is then utilized in the case ofCoulomb gauge Yang-Mills theory for which the relations between the two-point correla-tion functions of the transversal and longitudinal components of the conjugate momentumto the ones of the gauge field are given.

The final aspects of our work are presented in Part III, which contains the derivationof the Poincare generators in Coulomb-Gauge Yang-Mills theory employing some resultsfrom the path-integral formalism. In particular, we give the complete operators for QCD.We immediately apply the boost operator to note that what are usually called the pionsquare, quartic... charge radii, defined from derivatives of the pion form factor at zerosquared momentum transfer, is completely blurred out by relativistic and interaction cor-rections, so that it is not clear at all how to interpret these quantities in terms of the pioncharge distribution. The form factor therefore measures matrix elements of powers of theQCD boost and Moller operators, weighted by the charge density in the target’s rest frame.In addition we remark that the decomposition of the η′ wavefunction in quarkonium, glu-onium, ... components attempted by the KLOE collaboration combining data from φ ra-diative decays, requires corrections due to the velocity of the final state meson recoilingagainst a photon. This will be especially important if such decompositions are to be at-tempted with data from J/ψ decays.

Further discussions are given in the conclusions while essential steps of many calcula-tions have been deferred to several appendices.

vi

Notations

Generals: We will use the Heaviside-Lorentz System with natural units, i.e., we set ~ = c = 1.The the fine structure constant reads α = e2/(4π).

The tensorial four-dimensional indices are designed by Greek indices µ, ν . . . , andrun over the four spacetime coordinate labels 0, 1, 2, 3.

The tensorial three-dimensional indices are designed by Latin indices i, j, ` . . . , andrun over the three spatial coordinate labels, usually taken 1, 2, 3, equivalent to x, y, z.

Repeated indices indicate summation.

The three-dimensional vectors are denoted by bold characters.

In Part I: The metric tensor is

ηµν = ηµν =

1 0 0 00 1 0 00 0 1 00 0 0 −1

.

The total antisymmetric tensor is identified with εµνρσ such that ε0123 = +1. The fieldstrength tensor and its dual can be written as

Fµν =

0 Bz −By −Ex−Bz 0 Bx −EyBy −Bx 0 −EzEx Ey Ez 0

, Fµν =

12εµν%σF%σ =

0 −Ez Ey −BxEz 0 −Ex −By−Ey Ex 0 −BzBx By Bz 0

The position four-vector is xµ = (x, x0), xµ = ηµνxν = (x,−x0), with x0 = ct.

The volume element is d4x = dxdydzdx0. The scalar product of the four-vectors iswritten as

ab = aµbµ = ηµνaνbµ.

The derivative operator with respect to the four-coordinate is: ∂µ = ∂/∂xµ =(∇, ∂/∂x0

).

vii

The d′Alambert’s operator is given by ¤ ≡ ηµν∂2/∂xµ∂xν = ∇2 − ∂2/∂t2, where ∇2

is Laplace’s operator ∇2 = ∂2/∂xi∂xi.

In the Fourier transforms the factors 1/2π in the momentum integral is absorbed intothe momentum element. For instance, in D−dimensions,

f(x) =∫

dkf(k) exp(ik · x),

f(k) =∫

dDxf(x) exp(−ik · x),

with dk ≡ dDk/(2π)D. Here kµ is the conjugate variable of the four-coordinate xµ,ck0 ≡ ω, k2 ≡ kµkµ = k2 − ω2, k · x ≡ kµxµ = kx− k0x0. The other important factorsof 2π to remember appear in

∫dDxf(x) exp(ik · x) = (2π)D δ(D)(k)

In Part II: In contrast to part I, in this part the metric tensor is gµν = gµν = −ηµν . Consequentlythe field strength tensor reads

Fµν =

0 −Ex −Ey −EzEx 0 −Bz ByEy Bz 0 −BxEz −By Bz 0

, Fµν =

12εµν%σF%σ =

0 Bx By Bz−Bx 0 −Ez Ey−By Ez 0 −Ex−Bz −Ey Ex 0

The position four-vector is xµ = (x0,x), xµ = gµνxν = (x0,−x), with x0 = ct.

The volume d4x = dxdydzdx0. The scalar product of the four-vectors is written as

ab = aµbµ = gµνaνbµ.

The derivative operator respect to the four-coordinate: ∂µ = ∂/∂xµ =(∂/∂x0,∇)

.

The d′Alambert’s operator is given by ¤ ≡ gµν∂2/∂xµ∂xν = ∂2/∂t2 −∇2.

In the Fourier transforms the factors 1/2π must be in the momentum integral. Forinstance, in D−dimensions,

f(x) =∫

dkf(k) exp(−ik · x),

f(k) =∫

dDxf(x) exp(ik · x),

viii

with dk ≡ dDk/(2π)D. Here kµ is the conjugate variable of the four-coordinate xµ,ck0 ≡ ω, k2 ≡ kµkµ = ω2 − k2, k · x ≡ kµxµ = k0x0 − kx. The other important factorsof 2π to remember appear in

∫dDxf(x) exp(ik · x) = (2π)D δ(D)(k)

In Part III: we use the convention of Part II.

ix

Contents

Abstract i

Acknowledgments ii

Introduction iii

Notations vii

I Photon Magnetic Moment and Vacuum Magnetization 1

1 Introduction 2

2 Vacuum polarization and reduction of the Lorentz symmetry. 52.1 Diagonal decomposition of the vacuum polarization tensor . . . . . . . . . . 52.2 The effective action and the anisotropy of the spacetime . . . . . . . . . . . . 82.3 Group theoretical analysis of the Lorentz symmetry breaking . . . . . . . . 102.4 The low energy effective action . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Noether Charges 153.1 The translation generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 The Lorentz generators and their conservation . . . . . . . . . . . . . . . . . 173.3 Spin and Helicity for purely parallel propagation . . . . . . . . . . . . . . . . 203.4 Spin density for perpendicular propagation . . . . . . . . . . . . . . . . . . . 22

4 Photon anomalous magnetic moment 244.1 The current distribution and the photon magnetic moment . . . . . . . . . . 244.2 On-shell approximation to the photon magnetic moment. . . . . . . . . . . . 274.3 Precession and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.4 The magnetic moment density for an ultraviolet photon . . . . . . . . . . . . 32

x

5 Two loop term of the Euler-Heisenberg Lagrangian 335.1 The unrenornamalized contributions . . . . . . . . . . . . . . . . . . . . . . . 335.2 Renormalized contributions due to the photon polarization modes . . . . . 365.3 Asymptotic behavior at large magnetic field strength . . . . . . . . . . . . . 39

6 Magnetic properties of the vacuum in a superstrong magnetic field 416.1 Role of the photon polarization modes on the vacuum magnetization . . . 416.2 Transverse pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

7 Summary 46

II Connection between Hamilton and Lagrange formalism in QFT 47

8 Introduction 48

9 Functional equations 519.1 First order Dyson-Schwinger Equations . . . . . . . . . . . . . . . . . . . . . 519.2 The connection between Lagrange and Hamilton formalism . . . . . . . . . 559.3 Pure and mixed momentum correlation functions . . . . . . . . . . . . . . . 589.4 Connecting the Symmetry-Related identities . . . . . . . . . . . . . . . . . . 61

10 Decomposition of proper Lagrange correlation functions 6210.1 Relations between the bare elements . . . . . . . . . . . . . . . . . . . . . . . 6210.2 Diagrammatic decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 6410.3 Inclusion of Grassmannian fields . . . . . . . . . . . . . . . . . . . . . . . . . 69

11 Decomposition of connected Hamilton correlation functions 7111.1 The mixed, connected 2-point function . . . . . . . . . . . . . . . . . . . . . . 7111.2 The momentum propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7311.3 The inverse propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7511.4 Recovering the first order DSEs . . . . . . . . . . . . . . . . . . . . . . . . . . 76

12 Applications 8012.1 Theories with auxiliary fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 8012.2 Coulomb Gauge Yang-Mills theory . . . . . . . . . . . . . . . . . . . . . . . 8212.3 Relating the renormalizability in Hamilton and Lagrange Coulomb gauge

QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8812.4 First order formalism and BRS-invariance . . . . . . . . . . . . . . . . . . . . 90

13 Summary 92

xi

III The boost operators in Coulomb-gauge QCD and the relativistic formfactor 94

14 Introduction and overview 95

15 The relativistic Pion form factor 9915.1 Relation between the Pion form factor and the boost operator . . . . . . . . 9915.2 No square radius interpretation for the pion . . . . . . . . . . . . . . . . . . . 10115.3 The form factor and the Breit frame . . . . . . . . . . . . . . . . . . . . . . . . 102

16 Poincare generators of QCD 10416.1 The Translation Generators of a pure Yang-Mills theory . . . . . . . . . . . . 10416.2 The Lorentz Generators of a pure Yang-Mills theory . . . . . . . . . . . . . . 10916.3 Inclusion of the Quark sector . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

17 φ-radiative decays 11517.1 The boosted wavefunctions of decay products . . . . . . . . . . . . . . . . . 11517.2 The gluonium content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

18 Summary 121

Conclusions and outlook 122

A 124A.1 Asymptotic behavior of L

(2)iR at large magnetic field strength . . . . . . . . . 124

A.1.1 Leading behavior of L(2)1iR in an asymptotically large magnetic field . 125

A.1.2 Leading behavior of L(2)2iR in an asymptotically large magnetic field . 127

B 132B.1 p−integration of Z[J ]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132B.2 Proof of the general form of the decomposition of proper Lagrange correla-

tion functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133B.3 First order Dyson-Schwinger equations . . . . . . . . . . . . . . . . . . . . . 134B.4 Explicit projection on the individual momentum components . . . . . . . . 135

C 138C.1 The Fadeev-Popov Determinant and the Hermiticity of Π

a(x). . . . . . . . . 138

xii

Part I

Photon Magnetic Moment andVacuum Magnetization

1

Chapter 1

Introduction

Large magnetic fields |B| À Bc, Bc = m2/e = 4.42 · 1013G in the surface of stellar objectsidentified as neutron stars [50, 51, 52] might provide physical scenarios where quantumprocesses predicted in such a regime could become relevant for astrophysics and cos-mology. According to QED in strong background fields, the most important effects arelikely to be pair creation [9, 10, 53, 54, 55], photon splitting [19, 20] and photon capture[21, 22, 23, 24, 25]. The last two essentially depend on the drastic departure of the photondispersion relation from the light cone curve, due to the vacuum polarization tensor Πµν

[56] which depends on both the Landau levels of virtual electron-positron pair, as well ason the external magnetic field. As a result, the issue of light propagation in empty space,in the presence of a magnetic field B, is similar to the dispersion of light in an anisotropic``medium”, with the preferred direction corresponding to the external field axis. The phe-nomenological aspects associated with this issue have been studied for a long time andrecently, the effects of the vacuum polarization tensor on the Coulomb potential in a su-perstrong magnetic field have been considered as well [33, 34, 36]. However, the problemconcerning the magnetism carried by a photon has not attracted sufficient attention. Thispart of the dissertation is focused on this topic. Our first purposes is to show that both ob-servable and virtual photons carry a magnetic moment, whereas the second goal concernsto the magnetization generated by the virtual radiation.

The possible existence of a photon anomalous magnetic moment was pointed out in arecent volume of Physical Review D [57]. The authors of this reference derived this quan-tity in two different regimes of the vacuum polarization tensor. On the one hand for lowenergies in weak fields (|B| ¿ Bc) and on the other hand (originally studied in reference[58]) near the first pair creation threshold and for a moderate fields (|B| ∼ Bc). How-ever, they ignored that the photon magnetic moment is actually a vector, consequently,the connection between this quantity and its angular momentum was not presented and,furthermore, they did not comment about its precession around the external field axis.

2

In the present paper we reveal the asymptotic conditions where the concept of a pho-ton anomalous moment mγ may be adequate. We will analyze the cases of superstrongmagnetic fields (|B| À Bc) for both the low and the high energy limit of the vacuum po-larization tensor. In these domains, Πµν depends linearly on the external field. In the highenergy limit this dependence suggests a photon behavior similar to a neutral vector bosonwith non-vanishing mass [59]. However, for lower energies, we shall see that the Maxwellequations in the ``medium” seem to describe a neutral massless particle with a genuinemagnetic moment, whose intrinsic properties are very similar to those associated with theelectron magnetic moment [9, 16, 17]. The latter arises due to the infrared structure of aΠµν−eigenvalue, which can be decomposed covariantly showing a linear dependence onboth the external field tensor and on the polarization mode. This behavior is also mani-fest in the corresponding dispersion law and provides evidence concerning the connectionbetween the photon anomalous magnetic moment and the classical angular momentumof the electromagnetic radiation. In addition, we will show that mγ turns out to be avector with two orthogonal components in correspondence with the cylindrical symme-try imposed by the external field. Our derivation will point out that mγ is dynamicallyinduced in the presence of the magnetic field B through its interaction with the virtualelectron-positron pair whose dynamics tend to be reduced from 3 + 1 to 1 + 1 dimensions[29, 30, 31, 32]. Moreover, we shall see that a non-vanishing torque exerted by B gener-ates a Larmor type precession of mγ . This precession is a manifestation of the reductionof the rotation symmetry and, consequently Lorentz symmetry. This fact corroborates andextends a recent result presented in Ref. [60] which claims that a magnetic dipole momentof truly-elementary massive neutral particles is a signal of Lorentz symmetry violation.Because the photon anomalous magnetic moment is a coefficient of the linear expansion inthe external magnetic field it may become a characteristic quantity since the break-down ofLorentz invariance leads to a non-conservation of the helicity, which is the standard pho-ton label. This fact attracts our attention because it seems that a magnetic moment maybe ascribed to a photon propagation mode in a magnetized vacuum. In consequence onemay attribute a spin to it.

As we already mentioned in this introduction, we want also to address the questionin which way the virtual photons contribute to a measurement of the vacuum magnetiza-tion and therefore to increase the external field strength. This aspect might be importantfor astrophysics since the origin and evolution of magnetic fields in compact stellar ob-jects remains poorly understood [61]. Some investigations in this area provide theoreticalevidences that |B| is self-consistent due to the Bose-Einstein condensation of charged andneutral boson gases in a superstrong magnetic field [62, 63, 64, 65]. In this context, the non-linear QED-vacuum possesses the properties of a paramagnetic medium and seems to playan important role within the process of magnetization in the stars. Its properties have beenstudied also in [70, 71, 72, 73] for weak (|B| ¿ Bc) and moderate fields (|B| ∼ Bc) in oneloop approximation of the Euler-Heisenberg Lagrangian [10]. New corrections emerge

3

by considering the two-loop term of this effective Lagrangian [11, 12, 13, 14, 15] whichcontains the contribution of virtual photons created and annihilated spontaneously in thevacuum and interacting with the external field through the vacuum polarization tensor.

The two-loop term of the Euler-Heisenberg Lagragian was computed many years agoby Ritus [11, 12]. A few years latter, Dittrich and Reuter [13] obtained a simpler integralrepresentation of this term and showed that their results agreed with those determined byRitus in the strong magnetic field limit. In the last few years, it has been recalculated byseveral authors using the worldline formalism [14, 15] and it has been extended to the caseof finite temperature as well [74]. Nevertheless, in all these works it is really cumbersometo discern the individual contributions given by each virtual photon propagation modeto the Euler-Heisenberg Lagrangian which should allow to determine the global magneticcharacter generated by each of them. In this thesis we compute these contributions sep-arately. In addition we derive the individual vacuum magnetizations provided by eachvirtual photon propagation mode for very large magnetic fields (|B| À Bc). Whilst thecomplete two-loop contribution is purely paramagnetic we find a diamagnetic responsecoming from that Πµν−eigenvalue which is used to obtain the photon anomalous mag-netic moment.

In order to expose our results we have structured this part of our dissertation as fol-lows:

In Chapter 2: we recall some basic features of the photon propagation in the external mag-netic field. In addition a group theoretical analysis of the Lorentz symmetry breakdown is developed within the gauge sector of QED.

In Chapter 3: we evaluate the effects of the Lorentz symmetry break down on the Noethercharges associated with the spacetime symmetries. These computations are devel-oped for the radiation field using the basis that diagonalizes the polarization tensor.

In Chapter 4: the photon anomalous magnetic moment is found and its precession aroundB is studied.

In Chapter 5: we perform the calculation of the two-loop contributions to the Euler-Heisen-berg Lagrangian given by each virtual propagation mode. In addition we obtain theirasymptotic behavior at very large magnetic field.

In Chapter 6: the corresponding contribution to the vacuum magnetization density is de-termined by considering the individual contribution of each vacuum polarizationeigenmodes. Additional issues concerning the transverse pressures are analyzed aswell.

A short conclusion of this part is presented in Chapter 7

4

Chapter 2

Vacuum polarization and reduction ofthe Lorentz symmetry.

2.1 Diagonal decomposition of the vacuum polarization tensor

We start our analysis remembering that in the presence of an external field the averagedcurrent jµ(x) = ieTr [γµG (x, x|A )] , does not vanish. Here G (x, x|A ) denotes the electronGreen’s function, A represents the external constant electromagnetic field

Aµ(x) = −12Fµνxν =

(0,−1

2B× x

), (2.1)

and Fµν = ∂µAν − ∂νAµ is the external electromagnetic field tensor. Under these condi-tions, the effective action which describes small-amplitude electromagnetic waves aµ(x)over the constant background field reads:

Γ = −12

∫d4xd4x′aµ(x)D−1

µν (x, x′|A )aν(x′) (2.2)

where

D−1µν (x, x′|A ) ≡ − δ2Γ

δaµ(x)δaν(x′)

∣∣∣∣a=0

,

δ2Γδaµ(x)δaν(x′)

∣∣∣∣a=0

= [¤ηµν − (1− ζ)∂µ∂ν ] δ(4)(x′ − x) + Πµν(x, x′|A )

(2.3)

is the inverse photon Green’s function of our problem with ζ being a gauge parameter.According to Eq. (2.2) a photon can interact with the external field through the vacuum

polarization tensor Πµν(x, x′|A ) which is the coefficient of the linear expansion of jµ(x) in

5

aµ(x) i.e. δjµ(x)/δaν(x′)|a=0 = Πµν(x, x′|A ). In this context, the QED Schwinger-Dysonequation for the photon field aµ(x) is given by

[¤ηµν − (1− ζ)∂µ∂ν ] aν(x) +∫

d4x′Πµν(x, x′|A )aν(x′) = 0. (2.4)

The latter constitutes the Maxwell equation in a neutral polarized medium. The first termis the classical part, while the second term is responsible for the quantum corrections that,for small amplitude waves, are described by the vacuum polarization tensor Πµν(x, x′|A ).The terms whose power in a(x) is higher than one are not considered. Therefore, Eq. (2.4)describes the wave propagation as a linear process, which means that they are consideredas non-interacting with themselves.

The external field strength is independent of the spacetime coordinates, therefore, thepolarization tensor, as a gauge invariant quantity, should correspond to a spatially ho-mogeneous optical medium, whose properties do not change with the time. This is pro-vided by the translational invariance of Πµν : it depends only on the coordinate differenceΠµν(x, x′|A ) = Πµν(x − x′|A ). In this case a Fourier transform converts Eq. (2.4) into alinear homogeneous algebraic equation given by

[k2ηµν − (1− ζ)kµkν −Πµν(k|A )

]aν(k) = 0 (2.5)

withΠµν(k|A ) =

∫Πµν(x− x′|A )e−ik(x−x′)d4(x− x′). (2.6)

To understand what follows it is necessary to recall some basic results developed inRef. [21, 56]. The presence of a constant magnetic or electric field creates, in addition to thephoton momentum four-vector [(4)

µ = kµ, three other orthogonal four-vectors which wewrite as four-dimensional transverse kµ[(i)µ = 0 for i = 1, 2, 3. These are [(1)

µ = k2F 2µλk

λ −kµ(kF 2k), [(2)

µ = Fµλkλ, [(3)µ = Fµλkλ. Here Fµν = 1/2εµνρσFρσ represents the dual of

Fµν , F = 1/4FµνFµν is one of the external field invariants, whereas the remaining oneΦ = 1/4FµνFµν = 0, vanishes identically. We remark that [(i)µ fulfills both the orthogonal-ity condition: [(i)µ [µ(j) = δij

([(i)

)2and the completeness relation:

ηµν − kµkνk2

=3∑

i=1

[(i)µ [

(i)ν(

[(i))2 . (2.7)

Note that from [(i), one obtains the fundamental scalars

k2 = z1 + z2,

z1 =kF 2k

2F, and z2 = −kF 2k

2F.

(2.8)

6

The diagonalization of Πµν defines the energy spectrum of the electromagnetic waves.For reference frames in which F > 0 or F < 0 with Φ = 0 this diagonalization is written interms of [(i)

Πµν =4∑

i=0

κi(z1, z2,F)[(i)µ [

(i)ν(

[(i))2 (2.9)

with κi being the eigenvalues of the vacuum polarization tensor. Owing to the transversal-ity property (kµΠµν = 0), the eigenvalue corresponding to the fourth eigenvector vanishesidentically (κ(4) = 0).

Substituting Eq. (2.9) into Eq. (2.5) and using the orthogonality condition, we find itssolutions in the form of a superposition of eigenmodes given by

aµ(k) =3∑

i=1

δ(k2 − κi)[(i)µ . (2.10)

In addition, the Green’s function obtained by inversion of Eq. (2.3) can be decomposed as

Dµν(k|A ) =3∑

i=1

1k2 − κi

[(i)µ [

(i)ν(

[(i))2 +

ζ

k2

kµkνk2

. (2.11)

According to Eq. (2.10) and Eq. (2.11) three non-trivial dispersion relations arise

k2 = κi (z2, z1,F) for i = 1, 2, 3. (2.12)

We want to point out that, z2 and z1 acquire simple forms in reference frames whichare either at rest or moving parallel to the external field. For a magnetic-like backgroundF > 0, z2 = k2

⊥ and z1 = k2‖ − ω2 where k⊥ and k‖ are the components of k perpendicular

and along the external field respectively, with k2 = k2⊥ + k2

‖ − ω2. Solving Eq. (2.12) fork2 = k2 − ω2 in terms of z2 yields

ω2i = k2 + m2

i (z2,F), (2.13)

with the term mi arising as a sort of dynamical mass. Note that mi vanishes for z2 = 0due to the gauge invariance condition κi(0, 0,F) = 0. Obviously, the dispersion law givenin Eq. (2.13) differs from the usual light cone equation. This difference increases near thefree pair creation thresholds reflecting the quasi-particle feature of a photon in an externalmagnetic field. For more details we refer the reader to Ref. [21].

Moreover, by considering aµ ∼ [(i)µ as the electromagnetic four vector describing the

eigenmodes, we obtain the corresponding electric and magnetic fields of each mode

e(i) ' i(ω(i)[(i) − k[(i)0 ) and b(i) ' −ik× [(i). (2.14)

7

Up to a non-essential proportionality factor, they are explicitly given by:

e(1) ' −in⊥ω, b(1) ' ik‖ × n⊥,

e(2)⊥ ' ik⊥k‖, e

(2)‖ ' in‖(k

2‖ − ω2),

b(2) ' iω(k⊥ × n‖),

e(3) ' iω(n⊥ × n‖),b

(3)‖ ' in‖k⊥, b

(3)⊥ ' −in⊥k‖.

(2.15)

Here, n‖ = k‖/|k‖| and n⊥ = k⊥/|k⊥| are the unit vectors associated with the parallel andperpendicular direction with respect to B.

2.2 The effective action and the anisotropy of the spacetime

In classical electrodynamics, the Maxwell action of a continuous medium with dielectrictensor εij(k, ω) and magnetic permeability tensor µij(k, ω), is given by

S =12

∫d4k

(2π)4d(−k,−ω) · e(k, ω)− h(−k,−ω) · b(k, ω) .

Here the inductions d,h and the fields e, b are connected by means of the relations

di(k, ω) = εij(k, ω)ej(k, ω), hi(k, ω) = µ−1ij (k, ω)bj(k, ω). (2.16)

The optical properties of anisotropic medium depend primarily on the symmetry ofits tensors εij and µij . In an uniaxial medium one of the principal axes of εij and µij co-incides with the optical axis. In what follows we shall take it as the z−axis, denoting thecorresponding principal value of εij and µij by ε‖ and µ‖, respectively. The direction ofthe remaining two axis, are arbitrary, and the corresponding principal values, which wedenote ε⊥, (µ⊥) are equal. Having the latter in mind, the Maxwell action acquires thefollowing form

S =12

∫d4k

(2π)4

ε⊥(k)|e⊥(k)|2 − 1

µ⊥(k)|b⊥(k)|2 + ε‖(k)|e‖(k)|2 − 1

µ‖(k)|b‖(k)|2

(2.17)

where we have decomposed e = (e⊥,e‖) and b = (b⊥, b‖). Here the symbol⊥ and ‖ refersto the optical axis as well.

Let us consider now the quadratic effective action corresponding to the dynamicalgauge field sector of QED in an external field. Substituting Eq. (2.9) into Eq. (2.2) and

8

making use of Eq. (2.7), we find in momentum space

Γ =∫

dkLeff ,

Leff = L0

(1 +

κ1

k2

)+

18κ1 − κ2

kFk

(FµνFµν

)2+

18κ1 − κ3

kF 2k(FµνFµν)

2

(2.18)

where L0 = −14FµνFµν = 1

2(|e(k)|2 − |b(k)|2) denotes the Maxwell Lagrangian. Heree(x) = ∇a0(x) − ∂0a(x) and b = ∇ × a are the averaged electric and magnetic fieldassociated with the electromagnetic wave.

For a magnetized vacuum F > 0 the following relations arise

−14FµνFµν = e ·B, 1

4FµνFµν =

12b ·B. (2.19)

Having the latter in mind, the effective Lagrangian reads

Leff =12

(1− κ1

k2

) (|e|2 − |b|2) + 2κ1 − κ2

kF 2k(e ·B)2 +

12κ1 − κ3

kF 2k(b ·B)2

. (2.20)

For further convenience, we express Leff in term of (e⊥, e‖) and (b⊥, b‖) relative to thedirection of B

Γ =∫

dk

12

(1− κ1

k2

) (|e⊥|2 − |b⊥|2)

+12

(1− κ1

k2+ 8F

κ1 − κ2

kF 2k

)|e‖|2 (2.21)

− 12

(1− κ1

k2− 2F

κ1 − κ3

kF 2k

)|b‖|2

.

A comparison with Eq. (2.17) shows that the Minkowski space occupied by an external mag-netic field seems to behave like an uniaxial anisotropic material whose dielectric constant andmagnetic permeability are given by

ε⊥ = µ−1⊥ = 1− κ1

k2, ε‖ = 1− κ1

k2+ 8F

κ1 − κ2

kF 2k,

µ−1‖ = 1− κ1

k2− 2F

κ1 − κ3

kF 2k.

(2.22)

Moreover, by using Eq. (2.21), we obtain that the electric and magnetic induction can beexpressed as

d =∂Leff

∂e= d⊥ + d‖ and h = −∂Leff

∂b= h⊥ + h‖ (2.23)

9

withd⊥ = ε⊥e⊥, d‖ = ε‖e‖,

h⊥ = µ−1⊥ b⊥, h‖ = µ−1

‖ b‖.(2.24)

Note that the procedure shown in this section is independent of any approximation madein the calculation of the vacuum polarization tensor.

Actually the anisotropic character of Γ arises due to a Lorentz symmetry breakdownwhich is not manifest in the Maxwell Lagrangian. In order to analyze the latter we shouldkeep in mind that z2 and z1 depend on the F−structure. Consequently, the explicit formof these scalars as well as κi will depend on the reference frame. This Lagrangian, there-fore, is not Lorentz invariant. However, it turns out to be invariant under those Lorentztransformations which leaves the external field invariant Fµν = ΛρµΛσνFρσ.

2.3 Group theoretical analysis of the Lorentz symmetry breaking

Let us suppose an external constant electromagnetic field that is embedded in MinkowskispaceM3+1 as no longer removable quality of it. In consequence Fµν induces an anisotropywithinM3+1 and only those Lorentz transformations which leave the external field invari-ant provide a symmetry group of an anisotropic spacetime. Thus, the proper inhomoge-neous orthochronous Lorentz transformations required to describe the kinematics in thepresence of an external classical field must fulfill the conditions

xµ → Λµνxν + εµ, ηλρ = ηµνΛµλΛ

νρ,

Fµν = ΛρµΛσνFρσ, detΛ = 1, Λ00 > 0.

(2.25)

The set of pairs ε,Λ satisfying Eq. (2.25) form a subgroup of ISO(3, 1) which will bereferred as the ``Amputated Poincare Group” ISOA(3, 1). Clearly, Λ ∈ SOA(3, 1) whereSOA(3, 1) is the ``Amputated Lorentz Group”. Due to Eq. (2.25) the Lorentz transforma-tion associated with our problem can be written as [75]

Λµν = δµν + ωµν with ωµν = ϕFµν + ζFµ

ν . (2.26)

where ϕ and ζ are real infinitesimal parameters. The corresponding unitary operatorU[ε,Λ] acting on the Hilbert space reads

U[ε,Λ] = 1− iεµPµ

+ iϕΞ + iζ ˆΞ (2.27)

10

with ε denoting the infinitesimal parameter associated with the translations1. This expres-sion involves the generators

Ξ =12Fµν J

µν , and ˆΞ =12Fµν J

µν . (2.28)

Here Pµ and Jµν are the ordinary generators of the ISO(3, 1)−Lie algebra:[Pµ, Pν

]= 0, i

[Pµ, Jρσ

]= ηµρPν − ηµσPρ,

i[Jµν , Jρσ

]= Jµσηνρ − Jνσηµρ − Jρµησν + Jρµησν .

Consequently, the Lie algebra associated with the ISOA(3, 1) is given by[Pµ, Pν

]= 0,

[Ξ, ˆΞ

]= 0,

[Ξ, Pµ

]= −iFµ

νPν ,[ˆΞ, Pµ

]= −iFµ

νPν ,

(2.29)

whereas the Casimir invariants are

P2 = Z1 + Z2,

Z1 =PF 2P

2Fand Z2 = − PF 2P

2F.

(2.30)

Note, that the scalars involved in Eq. (2.8) are, therefore, eigenvalues of P2, Z1, and Z2

with P replaced by the photon momentum K.Now, finite dimensional representations of the Lorentz group SO(3, 1) classify the dif-

ferent Lorentz covariant fields, local symmetry currents and conserved charges that arisein the Quantum Field Theory. Let us suppose a wavefunction f`(x) transforming under afinite dimensional SO(3, 1) representation

f`(x) → f′`(x) =∑

¯

DSO(3,1)

`,¯

[Λ−1

]f¯(Λ−1x), (2.31)

1Since Fµν is no longer removable part ofM3+1 the spacetime might interact with charged particles. Con-sequently, it turns out to be inhomogeneous for such particles. Thereby not all of the eigenvalues of the usual

translation generators are conserved quantities and instead Πµ = Pµ

+ 12

eFµνxν with

hΠµ, Πν

i= −ieFµν

seems to be the right one to describe the kinematic of the elementary particle states. The replacement ofPµ → Πµ provides generators which map the translational invariance of the ``modified” spacetime.

11

where ` denotes all possible Poincare indices that it might carry. Here

DSO(3,1)

`,¯

[Λ−1

]= exp

i

2ωµν (J µν)`¯

is some unspecified and non-trivial representation of Λ ∈ SO(3, 1) and (J µν)`¯ are a setof matrices satisfying the algebra of the ordinary Lorentz group. The finite dimensionalrepresentation of SOA(3, 1) is determined by inserting Eq. (2.26) into the previous equation

DSOA(3,1)

`,¯

[Λ−1

]= exp

i

2

(ϕFµν + ζFµν

)(J µν)`¯

. (2.32)

Supposing that our wave function is connected to a field operator Ψ`(x) by means ofthe relation f`(x) = 〈0|Ψ`(x)|f〉. By considering the relation U[ε,Λ]|f〉 = |f′〉 and Eq. (2.31)the field transformation law reads

U[ε,Λ]Ψ`(x)U−1

[ε,Λ] = DSOA(3,1)

`,¯

[Λ−1

]Ψ¯(Λx + ε). (2.33)

The infinitesimal version of Eq. (2.33) allows to express

δϕΨ`(x) = iϕ[Ξ,Ψ`(x)

]= iϑO`,¯

[Ξ

]Ψ¯(x),

δζΨ`(x) = iζ[ˆΞ,Ψ`(x)

]= iζO`,¯

[ˆΞ]Ψ¯(x),

δεΨ`(x) = −iεµ[Pµ,Ψ`(x)

]= εµOµ

`,¯[P]Ψ¯(x),

(2.34)

with the differential operator representation O`,¯ of the algebra given by

Oµ`,¯

[P] = ∂µδ`,¯,

O`,¯[Ξ] = 12Fµν

[(J µν)`,¯+ xµOν

`,¯[P]− xνOµ

`,¯[P]

],

O`,¯[ ˆΞ] = 12Fµν

[(J µν)`,¯+ xµOν

`,¯[P]− xνOµ

`,¯[P]

].

(2.35)

In a reference frame where the field is purely magnetic F > 0 or purely electric F < 0,the Lie algebra is

[Px, Py

]= 0,

[Jz, Px

]= iPy,

[Jz, Py

]= −iPx

[Pz, P0

]= 0,

[Kz, Pz

]= −iP0,

[Kz,P0

]= −iPz

(2.36)

Here J = (J23, J31, J12) denotes the rotation generators whereas K = (K10, K20, K30) arethe boost components.

12

Each row in Eq. (2.36) manifest a subalgebra: the first one corresponding to 2−dimensionalEuclidean group the ISO(2),whereas the second one corresponds to the (1+1)−dimensionalpseudoeuclidean group ISO(1, 1). Therefore, in a reference frame where the field is purelymagnetic, the symmetry breaking for neutral particles has the following pattern:

ISO(3, 1) + B → ISO(2)× ISO(1, 1). (2.37)

The effective physical configuration of Minkowski space manifests itself as the direct prod-uct of ISO(2) and ISO(1, 1). The two resulting symmetry groups are associated with thetransversal and pseudoparallel planes with respect to the B−direction.

2.4 The low energy effective action

In one loop approximation the eigenvalues of Πµν are given by [22]

κ1 = −12k2I1,

κ2 = −12(z1I2 + z2I1), κ3 = −1

2(z1I1 + z2I3),

(2.38)

with

Ii =4απ

∫ ∞

0dτe−m2τ

∫ 1

0dη

eBσi(η, s)sinh(s)

exp−z2

M(s, η)eB

− z1N(s, η)

eB

− 4απ

∫ ∞

0

dττ

e−m2τ

∫ 1

0dη

1− η2

4. (2.39)

Here and in the following s ≡ eBτ,

σ1(s, η) =14

sinh(s) cosh(sη)− η sinh(sη) cosh(s)sinh(s)

, σ2(s, η) =1− η2

4cosh(s),

σ3(s, η) =M(s, η)sinh(s)

, M(s, η) =cosh(s)− cosh(sη)

2 sinh(s), N(s, η) =

1− η2

4s.

(2.40)

Note that the substitution of Eq. (2.40) into Eq. (2.18) leads to a non-local effectiveaction, since it depends on momentum in a complicated way. However, Leff becomeslocal if we restrict our selves to the infrared limit and superstrong magnetic fields. In fact,for m2b À m2 À ω2 − k2

‖ with m2b À k2⊥, and b = |B|/Bc the above eigenvalues are

reduced to [25]

κ1 =α

3πk2

[ln

(bγπ

)− 0.068

],

κ2 = κ1 − α

3π(b− 1.90)z1, κ3 = κ1 +

α

3πz2.

(2.41)

13

with ln(γ) = 0.577 . . . being the Euler constant whereas α = e2/4π = 1/137 is the fine-structure constant.. Inserting Eq. (2.41) into Eq. (2.18), we find

Γ =∫

d4xLeff ,

Leff = ςL0 +β

16F

(FµνFµν

)2+

λ

16F(FµνFµν)

2 .(2.42)

Here the numerical coefficients are ς ≡ 1− α3π

[ln

(bγπ

)− 0.068

], β ≡ α

3π (b− 1.90), λ ≡ α3π .

It follows from Eq. (2.41) and Eq. (2.22) that

ε⊥ = µ−1⊥ = ς, ε‖ = ς + 4β, µ−1

‖ = ς − λ. (2.43)

The effective action given by Eq. (2.42) has been derived by Shabad and Usov [78]. Fromit, the causality and unitarity principles were analyzed. In what follows, we will focus onthe Noether charges associated with Leff .

14

Chapter 3

Noether Charges

3.1 The translation generators

The symmetry reduction by the external magnetic field does not alter the translationalgroup of ISO(3, 1). Therefore, for an uncharged particle, the spacetime configuration withan external classical field is translation invariant. The associated Noether current of theelectromagnetic radiation aµ(x) provides a conserved canonical stress-energy tensor:

T µν = ηµνLeff − ∂Leff

∂ (∂µaλ)∂ν aλ. (3.1)

Here

∂Leff

∂ (∂µaν)= − ςFµν +

β

4F

(F%σF%σ

)Fµν +

λ

4F(F%σF%σ) Fµν . (3.2)

According to this expression, the canonical momentum π` = ∂Leff/∂ (∂0a`) . For a magnetic-like background reads

π(can) = −ςe− 2βe‖,π

(can)⊥ = −ςe⊥, π

(can)‖ = −(ς + 2β)e‖.

(3.3)

It is remarkable that π(can) is different from the averaged momentum π = −e [81]. In whatfollows we will express the Noether charges in terms of π.

Substituting Eq. (3.2) and Eq. (2.42) into Eq. (3.1) we obtain

T µν = ςFµλ∂ν aλ − β

4F

(F%σF%σ

)Fµλ∂ν aλ − λ

4F(F%σF%σ) Fµλ∂ν aλ (3.4)

+ ηµνςL0 +

β

16F

(Fσ%Fσ%

)2+

λ

16F(Fσ%Fσ%)

2

.

15

We exploit the known ambiguity in the definition of the stress-energy tensor to intro-duce the tensor

Θµν = T µν + ∂λKλµν (3.5)

where Kλµν = ∂Leff∂(∂µaλ) a

ν is antisymmetric in its first two indices. Inserting the latter ex-pression into Eq. (3.5) and using the equation of motion we find that

Θµν = ηµνLeff − ∂Leff

∂ (∂µaλ)Fνλ. (3.6)

Note that Θµν is divergence-free (∂µΘµν = 0) so it satisfies the same conservation law asT µν . Substituting Eq. (3.2) and Eq. (2.42) into Eq. (3.6)

Θµν = ςFµλFνλ −β

4F

(F%σF%σ

)FµλFνλ −

λ

4F(F%σF%σ) FµλFνλ (3.7)

+ ηµνςL0 +

β

16F

(Fσ%Fσ%

)2+

λ

16F(Fσ%Fσ%)

2

.

The conserved Hamiltonian and Poynting vector are defined according to

H =∫

d3xΘ00 and P =∫

d3xΘ0i. (3.8)

In terms of the averaged momentum and for F > 0 the latter read

H =∫

d3x

12ς(π2 + b

2)− 1

2λb

2‖

,

P = −∫

d3xς(π × b

)+ 2β

(π‖ × b

).

(3.9)

To derive the above expressions we have used Eq. (2.19). We remark that for each eigen-mode the Poynting vector reads p(i) ' ς

(e(i) × b(i)

)− 2β

(e

(i)‖ × b(i)

)with e and b given

in Eq. (2.15). In particular

p(1) ' ςk‖,

p(2)‖ ' ςk2

⊥k‖, p(2)⊥ ' (ς + 2β)(ω2 − k2

‖)k⊥,

p(2)‖ ' ςk‖, p

(3)⊥ ' ςk⊥.

(3.10)

Different photon degrees of freedom contribute in the presence of B, depending on thedirection of wave propagation: for a pure longitudinal propagation k ‖ B, the Poynt-ing vector p(2) = 0. As a consequence, eigenmode 2 is a pure longitudinal non-physical

16

electric wave and does not carry energy. On the other hand, the first and third modehave well-defined Poynting vectors along the external field p(1) = p(3). In this case, eachset

e(1), b(1),p(1)

and

e(3), b(3),p(3)

forms an orthogonal set of vectors and represent

waves polarized in the transverse plane to B. Consequently, for pure parallel propagation[(1)µ and [(3)

µ represent physical waves.Now, if the photon propagation involves a non-vanishing transversal momentum com-

ponent k⊥ 6= 0, we are allowed to perform the analysis in a Lorentz frame that does notchange the value k⊥, but gives k‖ = 0 and does not introduce an external electric field. Asa consequence, the energy flux of the first eigenmode p(1) = 0 and becomes purely electriclongitudinal and a non-physical mode. In the same context each set

e(2), b(2),p(2)

and

e(3), b(3),p(3)

forms an orthogonal set of vectors and represent waves polarized in thetransverse plane to B. Hence, for a photon whose three-momentum is directed at any non-zero angle with the external magnetic field, the two orthogonal polarization states [(2)

µ and[(3)µ propagate. Note that mode 3 represents a physical wave independent of the direction

of propagation.

3.2 The Lorentz generators and their conservation

The Noether current associated with Lorentz invariance is usually split into two pieces

J λµν ≡ Lλµν + S λµν (3.11)

the orbital part Lλµν and intrinsic spin S λµν1. For an electromagnetic field described bymeans of Leff , we obtain

Lλµν = xµT λν − xνT λµ,

S λµν = −i ∂Leff

∂(∂λaρ(x))(J µν)ρσ a

σ(x).(3.12)

By considering (Jαβ)σρ = i

(δσβηαρ − δσαηβρ

)we can express the spin contribution as

S λµν = ς[Fλµaν(x)− Fλν aµ(x)

]− β

4F

(F%σF%σ

) [F λµaν(x)− Fλν aµ(x)

]

− λ

4F(F%σF%σ)

[Fλµaν(x)−Fλν aµ(x)

]. (3.13)

1The appropriateness of this division may be seen from the well-known result that in the case of a scalarfield S λµν is absent. However, this division is neither unique, nor physically meaningful, unless it is possibleto refer the particle to a ``rest” system.

17

Thus, the integral of the time-components of these ``currents” then provide the angularmomentum tensor

Jµν ≡ Lµν + Sµν ,

Lµν ≡∫

d3xL0µν , Sµν ≡∫

d3xS 0µν(3.14)

The spatial components defined from 12εijkJjk with ε123 = 1 define the classical angular

momentum of the electromagnetic field

J = L + S (3.15)

where L and S are the contribution coming from the orbital and spin part, respectively.For F > 0 the latter are give by

L =∫

d3xx×

[ςe`(x)− β

4F

(F%σF%σ

)B`

]∇a`(x)

=∫

d3x−ς

[x× π`(x)∇a`(x)

]− 2β

[x× π‖∇a‖(x)

](3.16)

and

S =∫

d3xς [e(x)× a(x)] +

β

4F

(F%σF%σ

)a(x)× B

=∫

d3x−ς [π(x)× a(x)]− 2β

[π‖(x)× a(x)

].

(3.17)

It is of interest to examine further the form of the total angular momentum J. In orderto do this we substitute Eq. (3.16) and Eq. (3.17) into Eq. (3.15)

J = −∫

d3xς [x×∇a (π · a) + π × a] + 2β

[x×∇a

(π‖ · a

)+ π‖ × a

](3.18)

where we have introduced the notation ∇a, which means that the subscripted gradientoperates only on the factor a. This expression may be written in the following form2

J = −∫

d3xς[x× (

π × b)− x× (∇ · π) a

]+ 2β

[x× (

π‖ × b)− x× (∇ · π) a

].(3.19)

2Using the identity ∇B(A ·B) = A× (∇×B) + (A ·∇)B

18

By virtue of Gauss law ∇ · π = 0 we find

J = J1 + J2,

J1 = −ς∫

d3x[x× (

π × b)], J2 = −2β

∫d3x

[x× (

π‖ × b)].

(3.20)

We want to remark that the total angular momentum derives from the Poynting vectordensity:

J =∫

d3xx× [−ς (

π × b)− 2β

(π‖ × b

)]. (3.21)

Let us consider the part concerning the boost transformation. In this case the compo-nents of the Noether charge are given by K` ≡ J`0. In consequence, the components of theorbital contribution read

K` = L`0 = −∫

d3x

x0Θ0` − x`Θ00. (3.22)

With Eq. (3.7) and Eq. (3.9) we find that

K = −tP +12

∫d3xx

ς(π2 + b

2)− λb

2‖. (3.23)

Within the same context, the contribution coming from the spin K` ≡ S`0 can be written as

K =∫

d3x−ςe(x)a0(x) +

β

4F

(F%σF%σ

)B a0(x)

=∫

d3xςπ(x)a0(x) + 2βπ‖a0(x)

. (3.24)

Obviously, the total boost operator is

K = K + K. (3.25)

In order to analyze the behavior of S and K in the presence of B we regard e(i) in Eq.(2.14) as the electric field associated with the four potential ([(i)). Up to a non-essentialfactor of proportionality we have

s(i) ' ςe(i) × [(i) = ς[(i)0 b(i)

κ(i) ' ς[(i)0 e(i) + 2βe

(i)‖ [

(i)0 δi2.

(3.26)

19

Manifestly Eq. (3.26) expresses the connection between the Noether charges and the dif-ferent polarization planes associated with each eigemode. In particular

s(1) ' ς(k‖ × k⊥), κ(1) ' ςk⊥,

s(2) ' ςω(k‖ × k⊥),κ

(2)⊥ ' ςk⊥k2

‖, κ(2)‖ ' (ς + 2β)k‖(k

2‖ − ω2),

s(3) ' 0, κ(3) ' 0.

(3.27)

According to the reduced Lorentz symmetry, i.e. Eq. (2.37), only the parallel compo-nents s

(i)‖ and κ

(i)‖ are related to conserved quantities and therefore, just the component

of the electric and magnetic field along B can generate conserved charges. However, fora purely parallel propagation (k⊥ = 0), this connection is rather obscure due to the ab-sence of e

(1,3)‖ and b

(1,3)‖ . In this case the first and third mode may be combined to form a

circularly polarized transversal wave which is allowed by the degeneracy property:

κ1(z1, 0,F) = κ3(z1, 0,F). (3.28)

In this context the photon is labeled by the helicity λ = J‖·n‖ which seems to be a conservedquantity. For non-vanishing transversal propagation k⊥ 6= 0, however, λ stops being a welldefined quantum number due to the non-conserved rotations transversal to B.

3.3 Spin and Helicity for purely parallel propagation

According to Eq. (2.15), only mode-2 has an electric field parallel to B. As a consequence,for a purely parallel propagation, Eq. (3.9) and Eq. (3.17) are reduced to

P = −ς∫

d3xπ × b

and S = −ς

∫d3x π(x)× a(x) , (3.29)

respectively. In order to establish the connection between helicity and classical spin of theelectromagnetic field we consider

S = S+ + S−

S± =12ς

∫d3x

(e± ib

)× a.

(3.30)

The complex fields 12

e± ib

transform irreducibly under spin (1, 0) and (0, 1) represen-

tation of SO(3, 1) ∼ SU(2)× SU(2), respectively [3, 5, 6, 7]. These field combinations fulfill

20

the free Maxwell equation for left- and right circularly polarized radiation

∇× (e± ib)∓ i∂

∂t(e± ib) = 0 (3.31)

corresponding to λ = ∓1.For propagation purely parallel to the external magnetic field we define the electric

field e(c) = e(1) + e(3) and magnetic field b(c) = −ik × [(c) = b(1) + b(3) associated with[(c) ' [(1) + [(3), respectively. At this point it is meaningful to analyze

s(c) ' ςe(c) × [(c) and p(c) ' ςe(c) × b(c). (3.32)

Inserting the explicit expression of e(c), b(c) and [(c) in Eq. (3.32) we get

s(c) ' s(1) + iςk⊥(n‖k⊥ − n⊥k‖

), p(c) ' p(1) + p(3) − ς(k⊥ × n‖). (3.33)

Note that

limk⊥→0

s(c) = 0 and limk⊥→0

p(c) = 2ςk‖. (3.34)

The resulting limits are expected for a circularly polarized wave.Further considering Eq. (3.30) we obtain

s± ' 12ς(e(c) ± ib(c)

)× [(c) =

12s(c) ± i

2ς[b(c) × [(c)

]. (3.35)

Note that[b(c) × [(c)

]= ik + ik2

[k‖k⊥ − (ω2 − k2

‖)k‖]− i(ω2 − k2

‖)[n‖ × k⊥

](3.36)

− iςk‖[k⊥ × k‖

].

The above relation was obtained by inserting the explicit form of e(c), b(c) and [(c). There-fore, the angular momentum associated with left and right circular polarization are repre-sented by

limk⊥→0

s± ' ∓12ςk‖. (3.37)

Regarding the normalized version of the above limit (s± ≡ ∓n‖) as the angular mo-mentum used to define the helicity, we can write λ = s± ·p(c) = ∓1, where p(c) = n‖ is thenormalized version of the second limit computed in Eq. (3.34).

21

3.4 Spin density for perpendicular propagation

Let us consider now a photon propagating perpendicular to the external magnetic field. Inthis case, the behavior of the photon spin reads

S(t) = −ς∫

d3x(

e(t) × a(t))

+ 2β(e

(t)‖ × a(t)

). (3.38)

Here e(t) = e(2)+e(3) and b(t) = b

(2)+b(3) denotes the electric and magnetic field associated

with a(t)µ = a

(2)µ + a

(3)µ , respectively. Considering the latter relations and the fact that only

mode-2 carries an electric field parallel to B we find

s(t) ' ς[(t)0 b(t) + 2β

(e

(2)‖ × b(t)

). (3.39)

In the present context s(t) describes the behavior of the photon spin density. The substitu-tion of Eq. (2.15) into the latter leads to

s(t) ' s(2) − kω× s(2) − 2β(ω2 − k2

‖)k⊥ (3.40)

where we have used Eq. (3.27) and the relations [(t)0 (k, ω) = −k‖, [(3)0 = 0. For α → 0 the

photon dispersion law is ω = |k|, as a consequence the density s(t) is reduced to

s(t) ' s(2)0 − n× s

(2)0 (3.41)

with n = k/|k| being the wave vector and s(2)0 ' ω(k‖ × k⊥).

The photon spin density s(t) must be an invariant vector under spatial inversion3.However, the second term in Eq. (3.41) −n × s(2) does not exhibit parity symmetry. Aparity invariant theory requests the existence of +n×s(2). Consequently, we can introducethe total spin density for a free photon as

sγ =

0 for k⊥ = 0s(2) ∓ n× s(2) for k⊥ 6= 0

. (3.42)

which itself depends on the direction of propagation. Here s(2) ≡ s(2)0 /|s(2)

0 | = (n‖ × n⊥).

3Note that [(2) is pseudovector. Its spatial part [(2) is parity invariant, whereas the temporal elementb(2)0 changes its sign. Consequently, the magnetic field b(2) = −ik × b(2) transforms as a pure vector and

s(2)0 = b

(2)0 b(2) is parity invariant as well.

22

Due to the fact that s(2) · k = 0, the total spin density of the electromagnetic field isorthogonal to the wave vector (sγ · n = 0). We remark that for transversal propagation sγhas a parallel component to B given by

s‖γ = ∓k⊥|k|n‖, lim

k‖→0s‖γ = ∓n‖

which is conserved. We shall see that these are related to the corresponding componentsof the photon anomalous magnetic moment.

23

Chapter 4

Photon anomalous magnetic moment

4.1 The current distribution and the photon magnetic moment

In this chapter we explore the possibility that a photon might carry a magnetic moment.From a classical point of view the electromagnetic radiation lacks a magnetic moment.However, in QED a photon interacts with an external field leading to a nonvanishing vac-uum polarization tensor which carries a non-zero magnetic moment since it involves vir-tual electron-positron pairs whose spins and charges are opposite in sign.

Let us start analyzing the Newton rotation law:

τ γ =dJ

dx0=

dJ1

dx0+

dJ2

dx0(4.1)

where τ γ is the total torque exerted over the photon, whereas J1 and J2 are given in Eq.(3.20). The first term in the right hand side vanishes identically since it corresponds, upto a coefficient, to the angular momentum of a free photon which is conserved. With thisdetail in mind and inserting the explicit value of β we find

τ γ =dJ2

dx0,

J2 =4α3π

e2m2

∫d3x

[(x× bπ‖

)×B]+

4α3π

e2m2

∫d3xπ‖

[(B× x)× b

]+ . . .

(4.2)

where we have used the Jacobi identity. The first term on the right hand side of Eq. (4.2) de-fines a torque exerted by the external field on a magnetic dipole associated with a photon.According to this result a photon seems to carry a magnetic moment and thus an electriccurrent. The latter can be obtained by differentiation of the effective action Eq. (2.42) withrespect the external field potential. Indeed,

δΓδAµ(x)

= −Jµ(x) ≈ 12β

2F∂ν

(FρσFρσFµν

), (4.3)

24

where we have neglected terms decreasing with b. Note that, in this approximation Jµ isa divergence-free vector ∂µJµ = 0. We insert Fµν = ∂µAν − ∂νAµ within the second termof the effective action Eq. (2.42)

Γ =∫

d4xςL0 − 1

4β

2F

(FρσFρσ

)Fµν∂νAµ +

18λ

2F(FρσFρσ)

2

. (4.4)

After an integration by parts, we are able to write

Γ =∫

d4xςL0 − 1

2Jµ(x)Aµ(x) +

18λ

2F(FρσFρσ)

2

(4.5)

where we have considered the definition of Jµ. The substitution Eq. (2.1) into the aboveexpression allows to write the second term of Eq. (4.5) as

Γ(2) =∫

dx0

[12µγ ·B

]with µγ =

12

∫d3x [x× J(x, t)] (4.6)

being the photon anomalous magnetic moment. Inserting J into Eq. (4.6) and making useof Eq. (2.19) we find

µγi =12εijk

∫d3x

[xjJk(x, t)

]=

4α3π

e2m2

12εijk

∫d3xxj

[∂l

(Fkl(x)e‖(x)

)+ ∂0

(Fk0(x)e‖(x)

)].

We integrate the first term by parts to express the magnetic moment in the form

µγ = µ(1)γ + µ(2)

γ ,

µ(1)γ = g

e2m

∫d3xπ‖(x)π(x)m−1, µ(2)

γ = ge

2m

∫d3x

[x× F[∂0]b(x)π‖(x)

] (4.7)

where g = 4α3π is a sort of Lande factor whereas F[∂0] = ∂0/m is a form factor.

The above expression deserves some comments. First of all, µγ is an off-shell expres-sion. This implies that both virtual and observable photons carry a magnetic moment.We want to remark that both µ

(1)γ and µ

(2)γ are gauge invariant quantities. Since µ

(1)γ is

independent of the spatial coordinates one can interpret it as the spin part. However, thisis a possible view and only further studies can tell us how far the interpretation can bestretched.

Let us now analyze the on-shell expression of the photon anomalous magnetic mo-ment. In order to do this we restrict ourselves to x0 = 0 and write µ

(1)γ in momentum

space

µ(1)onγ =

∫dkmγ , mγ = g

e2m

Son,

Son =1

4ω2

[e‖(k, ω) + e‖(k,−ω)

][e(−k, ω) + e(−k,−ω)]m−1,

(4.8)

25

with d = d3/(2π)3. To derive the above expression we have considered the followingdecomposition

e(i)(x) ≈ −∫

dk∫

dωe(i)(k, ω)δ(k2

)exp[ik · x]. (4.9)

Here the symbol≈means the neglect of first order and higher powers of the electric charge.Due to fact that only mode-2 has an electric field parallel to B, µ(1)on

γ is physically rel-evant for photons propagating with non-vanishing transversal momentum components.Consequently, e(−k, ω) and e(−k,−ω) turn out to be a superposition of electric fields cor-responding to mode−2 and mode−3. Therefore, by considering Eq. (2.15) and disregard-ing unessential factors, the integrand in Eq. (4.8) can be written as

Son ' f [k⊥](n× s(2)

)(4.10)

with s(2) given in Eq. (3.42) and f [k⊥] = k⊥/m being a dimensionless factor. According tothis result, the density of the photon magnetic moment can be expressed as

mγ ' ±ge

2mf [k⊥]

(n× s(2)

)(4.11)

where we have incorporated the consequence of charge conjugation by introducing ±1.The above expression shows that an infrared photon in a superstrong magnetic field be-haves as a quasi-particle having both a charge and a magnetic moment. This behaviorocurrs in some scenarios of condensate matter and has allowed to introduce the conceptof polariton. In our context, a positive (negative) polariton has a magnetic moment in thedirection which is the same as (opposite to) that of the external magnetic field.

So, mγ is the sum of two orthogonal components, as dictated by the cylindrical sym-metry of our problem: the first one along the external field direction

m‖γ = g

e2m

f [k⊥]n‖ sinφ, (4.12)

whereas the second one is perpendicular to B

m⊥γ = −g

e2m

f [k⊥]n⊥ cosφ. (4.13)

Here 0 ≤ φ ≤ π is the polar angle between the wave vector n and the external field (tanφ =k⊥/k‖). Furthermore, both components of mγ show opposite magnetic behavior: while

the parallel one is essentially paramagnetic (m‖γ > 0), the perpendicular one is purely

diamagnetic (m⊥γ < 0). They become non-magnetic for propagation along B, (k⊥ = 0), in

which case the radiation is, -like in the free Maxwell theory-, insensitive to the magneticfield.

26

We want to conclude this section pointing out that m‖γ is invariant under rotation

around the external field and thus conserved. In addition, for k⊥ → 0 the photon anoma-lous magnetic moment vanishes which is a direct consequence of the gauge invariance ofthe eigenvalues of Πµν .

4.2 On-shell approximation to the photon magnetic moment.

For an asymptotically large magnetic field b = |B|/Bc À 1 the constant and logarithmicterm in κ2 can be neglected compared to the term growing linearly with the external fieldstrength. Therefore we will consider

κIR2 ' −%(b)z1 = − α

3πe

m2Fµνkµ[ν(2), (4.14)

with %(b) = α3πb. Note that we have used the decomposition: z1 = kµF λ

µ F νλ kν/(2F) =

kµF λµ [

(2)λ /(2F). The expression for κIR

2 is analogous to that of the invariant interactionenergy of the electron [66]:

ε =e

2m2FµνpµSν . (4.15)

Here pµ is the electron four momentum and Sµ = γ5(γµ − Πµ

m

)is the electron spin with

Πµ = pµ + i2eFµ

νxν . In the electron rest frame, ε describes the interaction energy betweenthe electron magnetic moment and the external magnetic field. In the photon case thestructure of Eq. (4.14) shows that the electron spin Sµ is replaced by the photon polarization[(2)µ , describing thespin caused by the leptonic substructure of the photon in the plane

perpendicular to the external field. Note, however, that the absence of a photon rest frameprevents the definition of a photon spin, unlike the electron case. To avoid this problemand for further convenience we will analyze the photon propagation by investigating itsdispersion law Eq. (2.12), which reads

ω2 = k2 − %(b) (1 + %(b))−1 z2. (4.16)

For magnetic field strength 10 < b ¿ 3π/α one should treat %(b) as small. The expansionof ω up to first order in %(b) gives the dispersion law

ω = |k| − 12%(b)z2/|k|. (4.17)

Obviously, the first term in Eq. (4.17) corresponds to the light cone equation whereas thesecond arises due to the dipole moment contribution of the virtual electron-positron pair.In this approximation the dispersion law does not essentially deviate from its vacuumshape and its interacting term grows linearly with the external magnetic field. Precisely,

27

this linear dependence on |B| provides evidence for a possible connection between thephoton anomalous magnetic moment and its angular momentum. In other external fieldregimes (like those analyzed in reference [57]), this connection is rather obscure since κ2

depends on the external field in a very complicated way.Hereafter we will denote the magnetic interaction energy by

U(mag)γ = −1

2%(b)z2

|k| =12α

3πe

m2

1|k|Fµνkµ[ν(3) (4.18)

where the decomposition z2 = −kµF νµ F λ

ν kλ/(2F) = −kµF νµ [

(3)ν /(2F) has been used.

With this in mind, we find by symmetrization of Eq. (4.18)

U(mag)γ = −1

2MµνF

µν with Mµν ≡ −ig e2m

f [k⊥]Sµν . (4.19)

Here g = α/(3π) and Sµν ≡ F(3)µν /(|k||k⊥|) is a dimensionless tensor with F

(3)µν = −ikµ[(3)

ν +ikν[

(3)µ referring to the antisymmetric electromagnetic tensor generated by the third prop-

agation mode. Note that F(3)ij = εijkb

(3)k .

The spatial part of Mµν can be written as:

Mij = ge

2mb(3)k

|k| f [k⊥]J (k)ij . (4.20)

Here J(i)lm = −iεilm is the 3−dimensional representation of the SO(3)−generators, fulfill-

ing[J (i),J (j)

]= iεijkJ (k) and (J (i)

lm )2 = J(i)lu J

(i)um = 2δlm.

In a Lorentz frame where Fµν is purely magnetic the structure of Eq. (4.19) is expandedto

U(mag)γ = −g

e2m

f [k⊥][n‖ sinφ− n⊥ cosφ

] · B. (4.21)

Note that the expression inside the brackets is a unit vector orthogonal to the direction ofpropagation. We then write the interaction energy as

U(mag)γ = −mγ · B (4.22)

withmγ ≡ ±g

e2m

f [k⊥](n× s(2)

). (4.23)

The above expression is the potential energy of a magnetic moment mγ in an externalmagnetic field.

The dependence of the photon magnetic moment on k⊥ is displayed in Fig. 4.1 forpurely perpendicular propagation with respect to B. Note that in this case mγ · n = 0.

28

0.0002 0.0004 0.0006 0.0008 0.001f @k¦D

1´10-7

2´10-7

3´10-7

4´10-7

5´10-7

6´10-7

7´10-7

mΓΜB

Figure 4.1: Photon anomalous magnetic moment drawn against the transverse momentumcomponent. Here µB = e/(2m) = 9.274 · 10−21erg/G is the Bohr magneton.

Within the range of frequencies for which mγ is defined, it is four orders of magnitudesmaller than the electron anomalous magnetic moment (µ′ ' α

πe

2m) [9] but it is twelveorders of magnitude larger than the neutrino magnetic moment (µνe = 10−19µB) [125].However, due to the astonishing experimental precision with which the anomalous mag-netic moment of both the electron and muon are measured, there is some hope for anexperimental -probably astronomical- measurement of the photon anomalous magneticmoment.

4.3 Precession and discussions

We have seen that the external magnetic field leaves only the rotational symmetry aroundof B invariant. Consequently, m

‖γ is an invariant under the rotation around the axis of

B while m⊥γ does not. Note, in addition, that m⊥

γ does not contribute to the interactionenergy. However m⊥

γ turns out to be relevant in another case: the external field does exerta torque on the magnetic dipole which tends to line up mγ with B

τ γ = mγ × B = m⊥γ × B = g

e2m

f [k⊥]|B| cosφ s(2). (4.24)

Due to the fact that the torque is collinear to s(2), the transversal components of total angu-lar momentum in that direction is not conserved. This is a manifestation of the reductionof the rotation symmetry and, consequently Lorentz symmetry. Therefore mγ is a signalof a break-down of the Lorentz symmetry. This fact corroborates and extends the resultpresented in Ref. [60] which claims that a magnetic dipole moment of truly elementarymassive neutral particles is a signal of Lorentz symmetry violation.

29

Note that τ γ is orthogonal to n × s(2) which causes a precession of mγ around B inanalogy to the Larmor precession for electrons. This analogy suggests the existence of aprecession frequency which might be estimated by considering the equation for the mag-netic moment density:

∂sγ/∂t ≈ −Ωγ ×(n× s(2)

)= τ γ (4.25)

where Ωγ denotes the precession frequency. We insert Eq. (4.24) and Eq. (3.42) into Eq.(4.25) and multiply both members by s(2) to obtain

Ωγ ' ge

2mf [k⊥]|B| cosφ. (4.26)

Note that Ωγ is meaningful whenever the photon magnetic moment is directed at somefinite angle with respect to the external magnetic field direction. In the limit φ → 0 itapproaches Ωγ ' g e

2mf [k⊥]. On the other hand for purely transverse propagation withrespect to B, Ωγ vanishes identically pointing out that the magnetic dipole is aligned withthe external field.

Apart from being a signal of Lorentz symmetry break-down, we want to draw the at-tention to other possible implications of the photon anomalous magnetic moment. Perhapsone of the most interesting consequences of mγ arises directly from the dispersion law

ω = |k| −mγ ·B. (4.27)

This expression allows to determine the fundamental quantities that describe the opticalproperties of the ``medium” in terms of the photon magnetic moment. Indeed, the vacuumrefraction index for this mode can be written as

n =|k|ω' 1 +

mγ ·B|k| . (4.28)

The latter expression depends on the direction of the photon momentum and reaches itsmaximum for a purely transversal propagation (k‖ = 0) in which case

nmax ' 1 +α

3πe

2m2|B|.

For b ∼ 100, nmax ≈ 1.038 which exceeds the values of typical gases at atmospheric pres-sure in absence of B [25]. We want to remark that Eq. (4.28) might play an important rolein the evaluation of the magnetic effect on gravitational lenses in the vicinity of highlymagnetized stellar objects [67, 68].

As it was already mentioned in subsection 3.2 for non-vanishing transversal propa-gation k⊥ 6= 0, the helicity stops being a well defined quantum number due to the non-conserved rotations transversal to B. In addition the structure of λ is slightly modified due

30

to radiative corrections. In order to evaluate this change we write

λω =J · kω

=J · k|k| n = λn > 1 (4.29)

where we have used the definition of the refraction index. Due to the presence of λ in Eq.(4.29) we find that λω is not conserved as well. As a consequence, alternative conservedquantities are needed to identify a photon interacting with B. Within the current context,m‖γ seems to be the most elementary one. However, a realistic treatment of this problem

requires a group theoretical analysis including mγ similar to those developed by Wignerfor a free photon [2]. The latter is beyond the scope of this manuscript, but a study in thisdirection is in progress.

Let us consider now a system consisting of N independent photons described by thedispersion equation Eq. (4.27). In addition we will suppose that they propagate purelytransversal to the external magnetic field. According to these assumptions, the total inter-action energy of the system is

Utotal = −N∑

i=i

m‖(i)γ |B| = −N 〈m‖

γ〉|B| (4.30)

wherem(i)γ denotes the the magnetic moment of each photon. Here the average of the pho-

ton magnetic moment is 〈m‖γ〉 =

∑Ni=im

‖(i)γ /N ∼ 〈k⊥〉. For a uniform photon distribution

in the transversal plane to B, one expects that 〈k⊥〉 = 0 which implies that Utotal = 0. As aconsequence, the system does not carry a magnetization Mγ = −∂Utotal/∂B. If the systemis considered as a monochromatic beam 〈k⊥〉 6= 0, Utotal 6= 0 and

|Mγ | = −∂Utotal/∂|B| = N 〈m‖γ〉 = g

e2m

N 〈f [k⊥]〉, 〈f [k⊥]〉 = 〈k⊥〉/m (4.31)

In consequence the beam carries a non-zero magnetization which alters the external fieldB:

H = B + 4π(Mγ + Mvac).

Here Mvac > 0 is the vacuum magnetization (for details see Sect. 6.1). Within the mag-netic field interval for which mγ is defined, |Mvac| ∼ 1010 − 1012erg/(cm3G). Now, inthe frequency range of x-rays (〈k⊥〉 ≥ 150 eV) the averaged magnetic moment 〈mγ〉 ∼10−27erg/G. In order to produce a photon magnetization of the order of ∼ |Mvac|, a pho-ton density of order ργ = N /V ∼ 1036 − 1038 cm−3 would be necessary. Moreover, forργ & 1042cm−3 the magnetization carried by the beam is |Mγ | ∼ 1015erg/(cm3G) andthus larger than |Mvac| even more, it is one order of magnitude larger than the externalmagnetic field |B| ∼ 1014G/cm3.

31

4.4 The magnetic moment density for an ultraviolet photon

Next we consider the high-energy regime. In the limit m2b À ω2−k2‖ →∞ and m2b À k2

⊥,the second eigenvalue approaches

−κUV2 = m2

2 =2απ

e|B|. (4.32)

Here m2 is the photon effective mass [59] corresponding to the topological one in the2−dimensional Schwinger model [69]. Its derivation is closely related to the chiral limit inwhich the axial current is not conserved (for more details we refer the reader to Ref. [76]).As a consequence, an ultraviolet photon seems to behave like a neutral massive vectorboson whose movement is quasi-confined in 1 + 1 dimensions

ω ≈(k2‖ + m2

2

)1/2. (4.33)

In contrast to the previous case κUV2 does not depend explicitly on the polarization mode.

This fact is also manifest in the dispersion law and therefore a clear relationship betweenthe photon magnetic moment and the classical spin definition of the radiation field is notexpected. Indeed, the photon anomalous magnetic moment derived from Eq. (4.33) isgiven by

mγ = − ∂ω

∂B

∣∣∣∣B⊥→0

= −απ

eω

n‖. (4.34)

Note that in this regime the photon anomalous magnetic moment behaves diamagnet-ically mγ < 0 and depends on the external field strength. Charge conjugation leads tointroduce a ∓ instead of −1 in the above expression. In particular for k2

‖ ¿ m22

mγ ' ∓απ

em2

n‖ = ∓(

2απb

)1/2

µBn‖, (4.35)

which tends to vanish for |B| → ∞. Note the lack of precession around the external fieldsince mγ is a vector along the B direction.

32

Chapter 5

Two loop term of theEuler-Heisenberg Lagrangian

Virtual photons can interact with the external magnetic field by means of the vacuumpolarization tensor. As a consequence they might be a source of magnetization to thewhole vacuum. In what follows we compute this contribution for very large magneticfields (b À 1), which might exist in stellar objects like neutron stars.

5.1 The unrenornamalized contributions

We start our analysis with the Euler-Heisenberg Lagrangian

LEH = L(0)R + L

(1)R + . . . (5.1)

withL

(0)R = −1

2B2 and B = B0Z−1/2

3(1loop). (5.2)

Here L(0)R is the free renormalized Maxwell Lagrangian in a Lorentz frame where the elec-

tric field vanishes, E = 0. Hereafter B0, m0 and e0 will be referred as the ``bare magneticfield strength”, ``bare electron mass” and ``bare charge”, respectively. Without the index0 these quantities must be understood as renormalized. On the other hand, L

(1)R is the

regularized one loop contribution due to the virtual electron-positron pairs created andannihilated spontaneously in vacuum and interacting with the external field [10]:

L(1)R = − 1

8π2

∫ ∞

0

dττ3

e−m20τ

(s coth(s)− 1− s2

3

)(5.3)

33

with s = eBτ and e = e0Z1/23 . In this context, the one loop renormalization constant is

given by

Z−13(1loop) = lim

τ0→0

1 +

α

3πln

(1

γm20τ0

), (5.4)

where ln(γ) = 0.577 . . . is the Euler constant.

+ ...++LEH=

LH2LLH1LLH0L

Figure 5.1: Two loop expansion of the Euler-Heisenberg Lagrangian. The solid lines rep-resent the electron positron Green’s functions, whereas the wavy line refers to the photon.Here L(1) is represented by the one loop graph which gives the contribution of the virtualfree electron-positron pairs created and annihilated spontaneously in vacuum and inter-acting with the external field. The radiative corrections (involved in L(2)) emerge from thetwo-loop graph due to exchange of virtual photons.

The contribution due to virtual photons interacting with the external field by means ofthe vacuum polarization tensor is expressed as

L(2) =i

2

∫d4k

(2π)4Πµν(k|A )Dµν(k|A ). (5.5)

Substitution of Eq. (2.9) and Eq. (2.11) into the latter expression gives

L(2) =i

2

3∑

i=1

∫d4k

(2π)4κi

(k2 − κi). (5.6)

Manifestly, this expression shows that each photon propagation mode contributes inde-pendently. The quantity Πµν(k)Dµν(k) =

∑i κi(k2 − κi)−1 represents the intercation en-

ergy between the full photon propagator with the vacuum polarization tensor. In order toobtain the leading term in an expansion in powers of e2, i. e. the two-loop contributionas shown in Fig. (5.1), we just neglect κi in the denominator of the photon propagator.Hereafter we consider this approximation, in which case

L(2) =3∑

i=1

L(2)i with L

(2)i ≡ i

2

∫d4k

(2π)4κik2. (5.7)

34

Note that κi denotes the unrenormalized eigenvalue. Its expression is given by Eqs. (2.38-2.40) but considering only the first line of Eq. (2.39) where all physical parameters areunrenormalized.

Since the analytical properties of any κi differs from the others, each contribution L(2)i

will be different and therefore, the original 2−loop graph can be decomposed into threediagrams (see Fig. 5.2). Since we are interested in the magnetic properties generated byeach photon propagation mode, each term L

(2)i will be studied separately.

+ +

a3a2a1LH2L=

Figure 5.2: Diagrammatic decomposition of L(2) in terms of the vacuum polarization eigen-modes.

The symmetry of our problem suggests to perform the k integration with two sets ofpolar coordinates corresponding to the two planes (kx, ky) and (k‖, k4), with k4 = iω. How-ever, it is convenient to use the integration variables z2 = k2

⊥ and z1 ≡ k2‖ + k2

4 > 0, suchthat

L(2)i =

132π2

∫ ∞

0d`dz2dz1κi(z2, z1)e−`(z2+z1). (5.8)

Substitution of κi in Eq. (5.8) yields:

L(2)i = − α0

32π3

∫ ∞

0dτ exp(−m2

0τ)∫ 1

0dηQi(s, η) (5.9)

with s = e0B0τ and

Q1(s, η) =σ1

sinh sV(s, η) + W(s, η) , (5.10)

Q2(s, η) =σ2

sinh sV(s, η) +

σ1

sinh sW(s, η), (5.11)

Q3(s, η) =σ1

sinh sV(s, η) +

σ3

sinh sW(s, η). (5.12)

The functions V(s, η) and W(s, η) are given by the following integral representations

V(s, η) =∫ ∞

0d`dz2dz1z1e−z2( M

eB+`)−z1(N

B+`), (5.13)

W(s, η) =∫ ∞

0d`dz2dz1z2e−z2( M

eB+`)−z1( N

eB+`). (5.14)

35

These integrals can be explicitly calculated:

V(s, η) =2e3

0B30

(M−N)N− 2e3

0B30

(M−N)2ln

(MN

), (5.15)

W(s, η) =2e3

0B30

(N−M)M+

2e30B

30

(M−N)2ln

(MN

), (5.16)

such thatV(s, η) + W(s, η) = 2e3

0B30M

−1(s, η)N−1(s, η). (5.17)

5.2 Renormalized contributions due to the photon polarizationmodes

While the τ−integral in Eq. (5.9) does not diverge at τ → ∞, the integrand is singularfor τ → 0. In order to regularize L

(2)i we introduce a finite lower limit τ0 > 0 for the

proper time integral of the electron propagators contained in the polarization tensor. As aconsequence, we can write

L(2)i = − α0

32π3

∫ ∞

2τ0

dτe−m20τ

∫ η0

0dη Qi(s, η)−Qi0(τ, η) (5.18)

with η0 = 1− 2τ0/τ and

Qi0(s, η) =8(e0B0)3

(1− η2)s3. (5.19)

The substraction of this term guarantees that L(2)i = 0 for vanishing magnetic fields. Obvi-

ously, Eq. (5.18) differs from the original two-loop contributions in a term which is mag-netic field independent and therefore a constant.

We proceed by adding and subtracting the functions Qi2(s, η) in the integrand Eq.(5.18), such that

L(2)i = − α0

32π3

∫ ∞

2τ0

dτe−m20τ

∫ η0

0dηQi2(s, η) (5.20)

− α0

32π3

∫ ∞

2τ0

dτe−m20τ

∫ η0

0dη Qi(s, η)− Qi(s, η)

with

Q12(s, η) = −2(e0B0)3

3s, Q22(s, η) =

4(e0B0)3

3s1

1− η2,

Q32(s, η) = −(e0B0)3

3s− 4(e0B0)3

3s1

1− η2.

36

Note that the function Qi(s, η) ≡ Qi0(s, η) + Qi2(s, η) is the expansion of Qi(s, η) up toquadratic terms in the external field. Obviously, for τ0 → 0 the first term in Eq. (5.20) islogarithmically divergent (∼ ln τ−1

0 ). This divergence will be ``reabsorbed” in the courseof charge renormalization.

To regularize the remaining integration over η we express Eq. (5.20) as follows

L(2)i = − α0

32π3

∫ ∞

2τ0

dτe−m20τ

∫ η0

0dηQi2(s, η)− α0

32π3

∫ ∞

2τ0

dτe−m20τ

∫ η0

0dηHi(s, η)

− α0

32π3

∫ ∞

0dτe−m2

0τ

∫ 1

0dη Qi − Qi − Hi (5.21)

where Hi(s, η) is a function determined by the singular part of the Laurent series ofQi(s, η)−Qi(s, η) at η = 1. As a consequence, the integrations in the third line of Eq. (5.21) convergesfor τ0 → 0. In particular,

H1(s, η) =4(e0B0)3

s3(1− η2)

(s coth(s) +

s2

sinh2(s)− 2

), (5.22)

H2(s, η) =2(e0B0)3

s3(1− η2)

(3s coth(s) +

s2

sinh2(s)− 2s2

3− 4

), (5.23)

H3(s, η) =2(e0B0)3

s3(1− η2)

(s coth(s) +

3s2

sinh2(s)+

2s2

3− 4

). (5.24)

We consider the integration over η in the first and second line of Eq. (5.21) to express(after some algebraical manipulations)

L(2)1 = − α2

0

6π2L(0)

∫ ∞

2τ0

dττ

e−m20τ +

13δm2∂L

(1)R

∂m20

+ L(2)1R, (5.25)

L(2)2 = − α2

0

6π2L(0)

[ln

(γm2

0τ0) ∫ ∞

2τ0

dττ

e−m20τ −

∫ ∞

2τ0

dττ

e−m20τ ln

(γm2

0τ)]

(5.26)

− α0

2πln

(γm2

0τ0)L

(1)R +

16δm2∂L

(1)R

∂m20

+ L(2)2R,

L(2)3 =

α20

6π2L(0)

[ln

(γm2

0τ0) ∫ ∞

2τ0

dττ

e−m20τ −

∫ ∞

2τ0

dττ

e−m20τ ln

(γm2

0τ)]

− α20

12π2L(0)

∫ ∞

2τ0

dττ

e−m20τ +

α0

2πln

(γm2

0τ0)L

(1)R +

12δm2∂L

(1)R

∂m20

+ L(2)3R (5.27)

where L(0) = −1/2B20. Additionally,

δm2 =3α0m2

0

2π

[ln

(1

γm20τ0

)+

56

](5.28)

37

is the correction to the square of the ``bare” electron mass (m2 = m20 + δm2) [9, 11, 12, 13],

whereas the renormalized two loop term L(2)iR corresponding to each eigenmode reads

L(2)1R = −5α0m2

0

12π∂L

(1)R

∂m20

− α0

16π3

∫ ∞

0

dττ3

e−m20τ ln

(γm2

0τ) [

s coth(s) +s2

sinh2(s)− 2

]

− α0

32π3

∫ ∞

0dτe−m2

0τ

∫ 1

0dη Q1(s, η)− Q1(s, η)− H1(s, η) , (5.29)

L(2)2R = −5α0m2

0

24π∂L

(1)R

∂m20

− α0

32π3

∫ ∞

0

dττ3

e−m20τ ln

(γm2

0τ) [

3s coth(s) +s2

sinh2(s)− 4− 2s2

3

]

− α0

32π3

∫ ∞

0dτe−m2

0τ

∫ 1

0dη Q2(s, η)− Q2(s, η)− H2(s, η) , (5.30)

L(2)3R = −15α0m2

0

24π∂L

(1)R

∂m20

− α0

32π3

∫ ∞

0

dττ3

e−m20τ ln

(γm2

0τ) [

s coth(s) +3s2

sinh2(s)− 4 +

2s2

3

]

− α0

32π3

∫ ∞

0dτe−m2

0τ

∫ 1

0dη Q3(s, η)− Q3(s, η)− H3(s, η) . (5.31)

Inserting Eqs. (5.25-5.27) into Eq. (5.7), the two loop correction to the Euler-HeisenbergLagrangian is given by

L(2) = − α20

4π2L(0)

∫ ∞

2τ0

dττ

e−m20τ + δm2∂L

(1)R

∂m20

+ L(2)R (5.32)

with L(2)R =

∑3i=1 L

(2)iR .

Considering Eq. (5.1) and Eq. (5.32), we obtain

LEH = L(0)Z−13(2loop) + L

(1)R + δm2∂L

(1)R

∂m20

+ L(2)R (5.33)

with

Z−13(2loop) = Z−1

3(1loop) − limτ0→0

α20

4π2

∫ ∞

2τ0

dττ

e−m20τ . (5.34)

For τ0 → 0 the integral∫∞2τ0

dττ e−m2

0τ = ln(2γm20τ0)

−1 and therefore

Z−13(2loop)

∣∣∣τ0→0

= 1 +α0

3πln

(1

γm20τ0

)− α2

0

4π2ln

(1

2γm20τ0

)

38

Now, we are able to identify

L(1)R (m2) = L

(1)R (m2

0) + δm2 ∂L(1)R

∂m20

∣∣∣∣∣δm2=0

(5.35)

where m is the renormalized electron mass. The renormalized charge and field strengthare introduced by means of the relations e = e0Z1/2

3(2loop) and B = B0Z−1/23(2loop). Under this

condition L(0)R = L(0)Z−1

3(2loop). Note that eB = e0B0. So, the variable s is an invariant underrenormalization. Clearly, m0 must be replaced by m wherever it appears as well as α0 → α.Keeping this in mind,

LEH = L(0)R + L

(1)R +

3∑

i=1

L(2)iR + . . . (5.36)

with

L(2)1R = −5αm4b

96π3

∫ ∞

0

dss2

e−s/b

(s coth(s)− 1− s2

3

)− αm4b2

16π3

∫ ∞

0

dss3

e−s/b ln(γ

sb

)

×[s coth(s) +

s2

sinh2(s)− 2

]− αm4b2

32π3

∫ ∞

0dse−s/b

∫ 1

0dηf1(s, η), (5.37)

L(2)2R = −5αm4b

192π3

∫ ∞

0

dss2

e−s/b

(s coth(s)− 1− s2

3

)− αm4b2

32π3

∫ ∞

0

dss3

e−s/b ln(γ

sb

)

×[3s coth(s) +

s2

sinh2(s)− 4− 2s2

3

]− αm4b2

32π3

∫ ∞

0dse−s/b

∫ 1

0dηf2(s, η), (5.38)

L(2)3R = −15αm4b

192π3

∫ ∞

0

dss2

e−s/b

(s coth(s)− 1− s2

3

)− αm4b2

32π3

∫ ∞

0

dss3

e−s/b ln(γ

sb

)

×[s coth(s) +

3s2

sinh2(s)− 4 +

2s2

3

]− αm4b2

32π3

∫ ∞

0dse−s/b

∫ 1

0dηf3(s, η) (5.39)

where we have replaced the integration variable τ by s, and b = B/Bc. Here we haveused the function fi(s, η) ≡

Qi(s, η)− Qi(s, η)− Hi(s, η)

with Qi(s, η) = Qi(s, η)/(eB)3,

Qi(s, η) = Qi(s, η)/(eB)3 and Hi(s, η) = Hi(s, η)/(eB)3.

5.3 Asymptotic behavior at large magnetic field strength

In the asymptotic region of superstrong magnetic field b À 1, Eq. (5.3) behaves like [11, 13]

L(1)R (b) ' m4b2

24π2

ln

(bγπ

)+

6π2ζ ′(2)

, (5.40)

39

where 6π−2ζ ′(2) = −0.5699610 . . . with ζ(x) the Riemann zeta-function.For the same magnetic field regime the leading term of L

(2)iR is derived in Appendix A.1.

L(2)1R ≈ −αm4b2

16π3N1,

L(2)2R ≈

αm4b2

32π3

[N2 ln

(bγπ

)− 1

3ln2

(bγπ

)+A

],

L(3)3R ≈

αm4b2

32π3

[N3 ln

(bγπ

)+

13

ln2

(bγπ

)+ B

].

(5.41)

These expressions are calculated with accuracy of terms decreasing with b like∼ b−1 ln(b)and faster. Here the numerical constants are N1 = 1.25, N2 = 1

3 − 4ζ′(2)π2 , N3 = 2

3 + 4ζ′(2)π2 ,

A = 4.21 and B = 0.69. Note N2 +N3 = 1.Taking all this into account, the asymptotic behavior of the full two-loop term is

L(2)R =

3∑

i=1

L(2)iR ≈ αm4b2

32π3

[ln

(bγπ

)+ 3.65

], (5.42)

which coincides with the results reported in references [11, 12, 13].

40

Chapter 6

Magnetic properties of the vacuum ina superstrong magnetic field

6.1 Role of the photon polarization modes on the vacuum magne-tization

In presence of an external magnetic field, the zero point vacuum energy Evac is modifiedby the interaction between B and the virtual QED-particles. The latter is determined bythe effective potential coming from the quantum-corrections to the Maxwell Lagrangianwhich is also contained within the finite temperature formalism. According to Eq. (5.36) itis expressed as

Evac = −L(1)R −

3∑

i=1

L(2)iR + . . . (6.1)

Consequently the vacuum acquires a non-trivial magnetization Mvac = −∂Evac/∂B in-duced by the external magnetic field. In what follows we will write

Mvac = M (1)vac + M (2)

vac + . . . (6.2)

in correspondence with the loop-term L(i)R . In this sense, the one loop contribution at very

large magnetic field b À 1 can be computed by means of Eq. (5.40) and gives:

M(1) =∂L

(1)R

∂B≈ m4b

24π2Bc

[2 ln

(bγπ

)+ 1 +

12ζ ′(2)π2

]. (6.3)

The dependence on the external field is shown in Fig. 6.1. The data depicted is obtained inthe field interval 10 ≤ b ≤ 102 · 3π/α. Within this approximation, we find that the vacuum

41

0 20000 40000 60000 80000 100000 120000b

0

5´1014

1´1015

1.5´1015

2´1015

2.5´1015

3´1015

3.5´1015

MH1LHe

rgH

cm3

GLL

Figure 6.1: One loop contribution to the vacuum magnetization density with regard to theexternal field.

reacts paramagnetically and has a non-linear dependence on the external field (see Eq.(6.3)).

The two-loop correction is given by M(2) =∑3

i=1 M(2)i where M(2)

i = ∂L(2)iR /∂B is the

contribution corresponding to a photon propagation mode. Making use of Eqs. (5.41) wefind

M(2)1 ≈ − αm4b

8π3BcN1,

M(2)2 ≈ − αm4b

32π3Bc

[23

ln2

(bγπ

)+

8ζ ′(2)π2

ln(

bγπ

)−N2 − 2A

],

M(2)3 ≈ αm4b

32π3Bc

[23

ln2

(bγπ

)+

(2 +

8ζ ′(2)π2

)ln

(bγπ

)+N3 + 2B

].

(6.4)

According to these results, in a superstrong magnetic field limit, M(2)1 < 0 and M(2)

2 < 0behave diamagnetically whereas M(2)

3 > 0 is purely paramagnetic (see Fig. 6.2). Within therange of magnetic field values for which the photon anomalous magnetic moment is de-fined (1014G . |B| . 1015G) the vacuum magnetization density is M(2)

2 ∼ −109erg/(cm3G).Moreover, while M(2)

1 depends linearly on b, the contributions of the second and thirdpropagation mode depend logarithmically on the external field. Note that the leading be-havior of the complete two-loop contribution is

M(2) ≈ αm4b32π3Bc

[10.8 + 2 ln

(bγπ

)]> 0 (6.5)

which points out a dominance of the third mode.

42

0 20000 40000 60000 80000 100000 120000b

-1´1013

0

1´1013

2´1013

MH2LHe

rgH

cm3

GLL

mode-3

mode-2

mode-1

Figure 6.2: Contribution of the vacuum polarization eigenvalues to the vacuum magneti-zation density with regard to the external field. Here 10 < b < 102 · 3π/α. The dashed linerepresents the complete two-loop contribution to the vacuum.

As it was expected M(1)/M(2) ∼ α−1. This ratio is also manifested between the corre-sponding magnetic susceptibilities (X(i) = ∂M(i)/∂B). Note that

X(1) ≈ m4

24π2B2c

[2 ln

(bγπ

)+ 1.86

]> 0,

X(2) ≈ αm4

32π3B2c

[2 ln

(bγπ

)+ 12.8

]> 0.

(6.6)

For magnetic fields b ∼ 105 corresponding to |B| ∼ 1018G, the magnetic susceptibilityreaches values the order of X(1) ∼ 10−4erg/(cm3G2) which exceeds the values of manylaboratory materials, for example Aluminum (XAl = 2.2 · 10−5erg/(cm3G2)).

6.2 Transverse pressures

Due to the anisotropy generated by B a magnetized vacuum exerts two different pressurecomponents [71, 72]. One of them is positive (P‖ = −Evac) and along B, whereas theremaining is transverse to the external field direction (P⊥ = −Evac −M|B|). For b ∼ 1 thelatter acquires negative values.

At very large magnetic fields (b À 1) the one-loop approximation of P⊥ can be com-puted by making use of Eq. (5.40) and Eq. (6.3). In fact

P(1)⊥ = −m4b2

24π2

[ln

(bγπ

)+ 1 +

6ζ ′(2)π2

]< 0. (6.7)

43

0 20000 40000 60000 80000 100000 120000b

-6´1031

-4´1031

-2´1031

0

2´1031

4´1031

P¦

H2LHd

ync

m2L

mode-3

mode-2

mode-1

Figure 6.3: Contribution of the vacuum polarization eigenvalues to the vacuum transversepressure density with regard to the external field. Here 10 < b < 102 · 3π/α. The dashedline represents the complete two-loop contribution.

Therefore, at asymptotically large values of the external field, the interaction between Band the virtual electron positron pairs generates a negative pressure which would tend toshrink inserted matter in the plane transverse to B.

Again, the two-loop contribution can be written as the sum of the corresponding termsdue to the vacuum polarization modes P(2)

⊥ =∑3

i=1 P(2)⊥i . According to Eqs. (5.41) and Eqs.

(6.4) they read:

P(2)⊥1 ≈

αm4b2

16π3N1 > 0,

P(2)⊥2 ≈

αm4b2

32π3

[13

ln2

(bγπ

)+

(N2 +

8ζ ′(2)π2

)ln

(bγπ

)−N2 −A

]> 0,

P(2)⊥3 ≈ −αm4b2

32π3

[13

ln2

(bγπ

)+

(2−N3 +

8ζ ′(2)π2

)ln

(bγπ

)+N3 + B

]< 0,

(6.8)

with the complete two-loop term given by

P(2)⊥ ≈ −αm4b2

32π3

[4.65 + ln

(bγπ

)]< 0. (6.9)

For b ∼ 105, corresponding to magnetic fields B ∼ 1018G, the transverse pressure gener-ated by the first and second polarization mode is positive and reaches values of the order∼ 1030dyn/cm2 and ∼ 1031dyn/cm2, respectively (see Fig. 6.3). In contrast, the contribu-tion given by the third mode is negative with P(2)

⊥3 ∼ −1031dyn/cm2. In the same context

44

P(2)⊥ ∼ −1031dyn/cm2. The one loop contribution is correspondingly even of the order of

P(1)⊥ ∼ α−1P(2)

⊥ ∼ −1033dyn/cm2.

45

Chapter 7

Summary

In the first part we showed that an infrared photon propagating in a strongly magnetizedvacuum (10 < b ¿ 3π/α), exhibits a non-zero vector anomalous magnetic moment (seeEq. (4.23)) whose intrinsic properties are very similar to those associated with the electronmagnetic moment. This quantity arises due to the interaction between virtual electron-positron quantum pairs with the external magnetic field. Its existence is closely relatedto the gauge invariance. We showed that mγ can be decomposed into two orthogonalcomponents in correspondence with the cylindrical symmetry imposed by the externalfield. These components are opposite in sign, and only the one along B is conserved.We remark that the photon paramagnetism (analyzed in [57]) is only associated with m

‖γ .

Additionally, for a highly energetic photon mγ is a vector opposite to the B−direction,which points out that the photon tends to be quasi-confined in 1 + 1 dimensions.

In the last chapters of this part we discussed the effect of the vacuum polarizationtensor, which modifies the zero-point energy of the vacuum even in the absence of electro-magnetic waves. In the limit of a superstrong magnetic field, the two-loop contribution ofthe magnetization density corresponding to the second and third propagation mode de-pends non-linearly on the external magnetic field and their behavior is diamagnetic andparamagnetic, respectively. On the other hand, the contribution coming from the firstmode is diamagnetic and depends linear on B. We have seen that for very large magneticfield the contribution due to the third mode strongly dominate the analyzed quantities. Inthis magnetic field regime the latter exerts a negative transverse pressures to the externalfield. On the contrary those contributions coming from the first and second virtual modeare positive.

46

Part II

Connection between Hamilton andLagrange formalism in QFT

47

Chapter 8

Introduction

The path integral method within the Lagrange formalism is a fundamental tool to for-mulate Quantum Field Theory. Certainly, the treatment of the same theory within thecanonical operator formalism is somewhat more cumbersome. In fact, additionally to theoperator-ordering problems of gauge theories [79, 80], the formal invariance properties of ageneric theory admit additional transformation laws which are not present in the Lagrangeformulation. The ``momentum fields” in such a case are not related to the fundamentalfields by means of the canonical equations, see, e.g., ref. [3, 81], but constitute separate de-grees of freedom with independent transformation properties and thus symmetries. As aconsequence the number of variables in the path integral increases in comparison to theLagrange formalism. This introduces for most theories considerable additional complica-tions in perturbative as well as non-perturbative calculations [81, 46, 82, 44, 45].

On the other hand, some recent studies indicate that the first order formalism provesto be a successful tool to study the required complete cancellation of the energy diver-gences [41] that emerge in a perturbative treatment of Coulomb gauge Yang-Mills the-ory within the standard Lagrange formalism [42, 45]. Coulomb gauge Yang-Mills theoryhas attracted attention since one possible solution to the confinement problem in QCD isprovided by the ``Gribov-Zwanziger” scenario [40, 41] (cf. also ref. [83]). However, theproblem of renormalizing Coulomb gauge Yang-Mills theory is still unsolved since theseenergy divergences cannot be regularized using any of the standard procedures. Recentlyit was illustrated how these energy divergences cancel at each order in perturbation theory[84]. In order to perform explicit calculations a number of methods have been applied ase.g. introducing a novel method to regularize Feynman integrals in non-covariant gauges[85, 86], employing algebraic renormalizability [41], and trying to recover Coulomb gaugeYang-Mills theory as a limit of an interpolating gauge [87]. On the other hand, there areseveral indications that the canonical or first order formalism is better suited for study-ing Coulomb gauge Yang-Mills theory [46, 82, 44, 45, 41, 87, 88, 89, 47]. Yet, even if such

48

an approach would be successful, due to the dramatically complicated form of the func-tional equations in the first order formalism, an explicit non-perturbative study is far tooinvolved to be computationally feasible. Therefore, an analysis in the second order for-malism would be highly desirable. To this end we provide general connections betweenthe Greens functions in the two formulations that should help to perform the renormaliza-tion in the Lagrange framework according to the insight in the renormalization procedureobtained in the Hamilton framework.

In order to establish general relations between dressed correlation functions in the dif-ferent formulations we exploit the following properties. At vanishing sources associated tothe momentum fields the Effective Action (i.e. the Generating Functional of one-particle-irreducible (1PI) Green’s functions) of the Hamilton formalism reduces to the one of theLagrangean approach. This allows to reduce the set of Dyson-Schwinger equations (DSEs)[48, 49] in the first order formalism to the corresponding set derived from the standard pathintegral representation. An important special case is given if the Hamiltonian is quadraticin the momentum fields. For a corresponding Generating Functional the integration overthe momentum fields can be performed and any m−point function involving this fieldas the average of a polylocal function of the quantum canonical momentum fields can bedetermined. This means that the full correlation functions involving these canonical vari-ables can be found as a functional of those that usually appear in the standard path integralrepresentation. The procedure to find these connections is closely related to the functionalmethod used in the derivation of the DSEs (see e.g. ref. [90]). In this part of our thesiswe give the explicit form of these relations for a general four-dimensional renormalizabletheory. This includes the case that the interaction terms involve the time derivative of thefields. In a final step we resolve the relations between the proper two-point functions inboth frames by considering the inverse of the matrix-valued propagators in the Hamilto-nian approach.

Moreover, we show that similar connections arise for theories where not the kineticbut the interaction part is linearized. Such a bosonization procedure is an important tech-nique used in many parts of physics ranging from hadronic physics to condensed mattersystems. Our results explicitly verify, in agreement with other approaches, that there isno double counting in bosonized theories, but instead a given correlation function in theunderlying theory is exactly given by the sum of all possible contributions involving boththe fundamental and the composite degrees of freedom in the bosonized theory.

This part of the dissertation is organized as follows:

In Chapter 9: the functional equations for DSEs and a Symmetry-Related identity (likeWard-Green-Takahashi, resp. Slavnov-Taylor, identities (STIs) of a gauge theory [91,92, 93, 94, 95]) in the phase space formulation are given. We also present the firstorder DSEs of theories which are quadratic in the momentum fields. In addition,we present a method to determine the correlation functions and the STIs including

49

momentum fields from the respective quantities that usually appear in the standardLagrange representation.

In Chapter 10: we give diagrammatic rules and the general decomposition of the full properfunctions in the Lagrange formalism.

In Chapter 11: we detail the explicit decomposition of the connected and proper two-point functions in the Hamilton framework, respectively.

In Chapter 12: the formalism is applied to the case composite auxiliary fields and Coulombgauge Yang-Mills theory.

Finally we give a summary and outlook to this Part in chapter 13.

50

Chapter 9

Functional equations

9.1 First order Dyson-Schwinger Equations

We start our analysis considering a generic Quantum Field Theory formulated within thefirst order formalism whose Hamiltonian density H (qm(x), pm(x)) depends on the fun-damental fields qm(x) and the conjugate momentum fields pm(x), respectively. Hereafterthey will represent bosonic fields. In the case of gauge invariant theories we will sup-pose that our H includes the additional terms that arise when the constraints associatedto such theories, like e.g. Gauss’s law in Quantum Electrodynamics, are localized. Inthis context each Lagrange multiplier necessary to impose the constraints will be treatedas a fundamental field. Certainly the presence of theses terms leads to main differencesbetween gauge theories and the usual Hamiltonian system. In addition in a fixed gaugeghost fields appear. The usual path integral representation of theses Grassmannian vari-ables is formulated within the first order formalism (see also Sect. 10.2). Thus, in the caseof a gauge theory they will be analysed in an independent way (for details see Sect. 12.2).In this section we set up the problem, and for this these remarks are sufficient, one onlyhas to keep these potential complications in mind. To set up the problem in this section itis sufficient to keep this potential complications in mind.

Let us suppose, in addition, that our fields pm(x) and qm(x) are coupled to a set of clas-sical sources given by Jpm(x) and Jqm(x), respectively. Under such conditions the source de-pendent vacuum-to-vacuum transition amplitude between the asymptotic states |Vac, in〉and |Vac, out〉 looks like

Z[J ] = 〈Vac, out|Vac, in〉J=

∫D[q]D[p] exp

i

~(I [q, p, J ] + iε-terms)

(9.1)

where [q] denotes the collection of all fundamental fields, whereas [p] the corresponding

51

momentum fields. For more details see section 9.2 of the Ref. [3]. Here the argument inthe exponential has the structure

I[q, p, J ] = I0 [q, p] +∫d4xJ(x) · φ(x)

where

φ(x) ≡(pm(x)qm(x)

)and J(x) ≡

(Jpm(x)Jqm(x)

),

include both momentum and fundamental fields and sources. The components of thesources in this notation are distinguished by upper labels, whereas

I0 [q, p] =∫ ∞

−∞dτ

∫d3x [pm(x)qm(x)−H (q(x), p(x))]

. (9.2)

and looks like a classical expression for the Hamiltonian action. Note that the part con-cerning to the ε terms in Eq. (9.1) have the function to produce the necessary iε′s in thedenominators of all propagators such that the correct boundary conditions of the fields atasymptotic time qm(x,±∞) are taken into account (see again Ref. [3]).

The fact that I0[q, p] looks like the action expressed in terms of canonical variables issomewhat misleading since the momentum fields pm(x) are independent variables andtherefore not yet related to the fundamental fields qm(x) or their derivatives. In particular,since the path integral is not saturated by its saddle point(s) they are not constrained toobey the equations of motion of classical Hamiltonian dynamics δI0/δφ = 0 where

δI0δφ

≡

δI0δpm

δI0δqm

=

qm − ∂H∂pm

pm −∇ · ∂H∂(∇qm) + ∂H

∂qm

. (9.3)

The path integrals, however, contain the information about these equations of motion.Indeed, let us consider the operator version of Eq. (9.3). Its vacuum expectation values inthe presence of the external classical sources can be expressed as

δI0δφ

=

∫ D[q]D[p]~iδδφ exp

i~I [q, p, J ]

〈0out|0in〉J − J. (9.4)

Assuming the absence of any boundary terms, the integral of such a functional derivativevanishes. Then by substituting each of the elementary objects present in Eq. (9.3) by thederivative with regard to the respective classical sources, the following functional differ-ential equation arises

δI0δφ

∣∣∣∣φ(x)→~

iδ

δJ(x)

Z[J ] = −J(x)Z[J ]. (9.5)

52

It contains all equations of motions fulfilled by all Green’s functions.Introducing the Generating Functional of connected Green’s functions,WH [J ] = −i ln (Z[J ]),

allows us to rewrite Eq. (9.5) as

δI0δφ

∣∣∣∣φ(x)→ δWH

δJ(x)+~

iδ

δJ(x)

= −J(x), (9.6)

where in the last step we made use of the following identity

F

(δ

δJ

)exp(G(J)) = exp(G(J))F

(δG(J)δJ

+δ

δJ

)(9.7)

for arbitrary functionals F and G. This set of equations constitutes the DSEs for the con-nected Green’s functions.

In the next step we introduce the vacuum expectation values of the momentum andthe fundamental fields in the presence of sources

φ[J ] =1i

δWH

δJ(x)≡

p[J ]

q[J ]

=

1iδWH

δJpm

1iδWH

δJqm

. (9.8)

We assume that it is possible to invert these relations such that the sources are expressed asfunctionals of the vacuum expectation values of Π(x) and Q(x). Hereby the replacementsof variables in Eq. (9.6) become

φ(x) → φ[J ](x) +~i

∫d4x′D[J ](x, x′)

δ

δφ(x′)(9.9)

where

D[J ](x, x′) ≡

∆ppml[J ](x, x′) ∆pq

ml[J ](x, x′)

∆qpml[J ](x, x′) ∆qq

ml[J ](x, x′)

(9.10)

are the source-dependent two-point Green’s function where

∆pq =1i

δWH [J ]δJp(x)δJq(x′)

=δql[J ](x′)δJp(x)

(9.11)

and ∆pp[J ], ∆qq[J ] and ∆qp[J ] are analogously defined.The Effective Action in the first order formalism ΓH [qm, pm] can be defined by the Leg-

endre transform of WH [J ] with respect to the associated ``averaged” fields of the theory

ΓH [φ]def≡ WH [J ]−

∫d4xJ(x) · φ(x). (9.12)

53

Remember that the compact notation implies a twofold Legendre transformation in thetwo independent components q and p. By considering the functional derivative of ΓH [q, p]with respect to pi(x) and qi(x) we obtain as expected

δΓH

δφ(x)= −J(x). (9.13)

Substituting Eq. (9.13) into Eq. (9.6), however, with the replacement Eq. (9.9), we obtainthe desired functional differential equations for ΓH [q, p],

δΓH

δφ(x)=

δI0δφ(x)

∣∣∣∣φ(x)→φ[J ](x)+~

i

Rd4x′D[J ](x,x′) δ

δΦ(x′)

. (9.14)

This relation generates all DSEs for the 1PI Green’s functions of the first order formalism.The method to obtain such a functional relation will be employed several times in thefollowing.

The derivation of the proper propagators is straightforward when keeping in mind thatthey are embedded in a matrix. In fact, taking a derivative with regard φ(x′) in Eq. (9.14)we obtain

G[φ](x′′, x′) ≡

δ2ΓH

δpm(x′′)δpn(x′)δ2ΓH

δpm(x′′)δqn(x′)

δ2ΓH

δ ¯qm(x′′)δpn(x′)δ2ΓH

δ ¯qm(x′′)δqn(x′)

. (9.15)

This expression and Eq. (9.10) are related via the identity∫d4x′′D[J ](x, x′′)G[φ](x′′, x′) = −I, (9.16)

which can be obtained by taking a derivative with respect to the source J in Eq. (9.13).Hereby I = δmnδ

4(x − x′) ⊗ 12×2 denotes the identity in the considered space. Any otherconnected or proper Green’s function can be derived by considering higher order deriva-tives with respect to J and φ in Eq. (9.10) and Eq. (9.15), respectively.

In general, these functions are constrained by the symmetry properties of the initial``action”. In the case of a gauge theory a corresponding derivation leads to the STIs. ForCoulomb gauge Yang-Mills theory it is presented in Sect. 12.2. However in order to illus-trate the potential complications arising in the first order formalism it sufficient to assumethat I0 is invariant under the simultaneous infinitesimal transformations

δφ[φ] ≡ ε ·

Gqm [p, q]

Fpm [p, q]

(9.17)

54

and also assume that they leave the path integration measure invariant. Performing thissubstitution in Eq. (9.1) we can obtain the following ``Symmetry-Related Functional Iden-tity”

0 =∫D[q]D[p]

∫d4xJ(x) · δφ(x) exp

[i

~I [φ, J ]

]. (9.18)

We substitute the fundamental and momentum fields by the respective derivatives withrespect to the classical source. Following a procedure similar to the derivation of the DSEswe arrive at the symmetry-related identity Eq. (9.18) expressed in terms of the full Gener-ating Functional Z[J ]:

0 =∫d4xJ(x) · δφ

[~i

δ

δJ

]Z[J ]. (9.19)

Analogously, the above relation can be rewritten for connected Green’s functions

0 =∫d4xJ(x) · δφ[φ]|

φ→ δWH

δJ+~

iδ

δJ

. (9.20)

As before, Eq. (9.18) can be further transposed in terms of the Effective Action

0 =∫d4x

δΓH

δφ· δφ[φ]

∣∣∣∣φ→φ[J ]+~

i

Rd4yD[J ](x,y) δ

δφ(y)

(9.21)

which tells us that the Effective Action in first order formalism preserves the continuoussymmetries of the initial canonical quantum action.

At this point the DSEs and the symmetry identity Eq. (9.18) of a QFT seem to be morecumbersome than the usual that appear in the standard path integral formulation. How-ever, in the next section we shall show that under certain conditions the above system canbe reduced to the latter one.

9.2 The connection between Lagrange and Hamilton formalism

In what follows we will denote all the variables that a field can depend on by a single latinindex. For instance in case of a gauge field such indices will indicate the space-time pointx, the vectorial index as well as the adjoint gauge group index a. Repeated indices aresummed and integrated over for discrete and continuous variables, respectively. Also allfundamental fields will be denoted by the same letter q. Similarly, p will represent all mo-mentum fields. Let us now consider the path integral representation of a n-point functioninvolving only momentum fields

∆p...pi1...in

=〈0out

∣∣∣T

Πi1 . . . Πin

∣∣∣ 0in〉Ji1−n〈0out|0in〉J =

∫ D[q]D[p]pi1 . . . pin exp[i~I[q, p, J ]

]

i1−n∫ D[q]D[p] exp

[i~I[q, p, J ]

] , (9.22)

55

where T is the time ordering operator and Πi ≡ Πm(x, τ) is the Hermitian momentumfield operators in the Heisenberg picture. This expression can be written via functionalderivatives with respect to Jpi

∆p...pi1...in

=

∫ D[q]D[p]~iδ

δJpi1

. . . ~iδ

δJpin

exp[i~ I[q, p, J ]]

i1−n∫ D[q]D[p] exp

[i~ I[q, p, J ]] . (9.23)

Consider the important class of Hamiltonians that are at most quadratic in the momentumfields

H =12piAij [q]pm + Bi[q]pi + C[q], (9.24)

with a real, symmetric, positive and non-singular matrix A whereas Bi[q] and C[q] are realfunctionals of q. We will in the following only discuss this case which is realized for allrenormalizable as well as most non-renormalizable theories. For this class of Hamiltoniansthe Gaussian integration over p can be performed analytically and as shown in appendixB.1 it yields

∆p...pi1...in

=

∫ D[q]~iδ

δJpi1

. . . ~iδ

δJpin

exp[i~ S[q, J ]

]

i1−n∫ D[q] exp

[i~ S[q, J ]

] . (9.25)

The general result for the new action arising in the exponential is given in Eq. (B.5). Inthe following we will for simplicity discuss the standard case that A is proportional to theunit matrix. The general case where A is a non-trivial operator is realized in the case ofCoulomb gauge QCD and will be discussed section 12.2. In the simplified case the newaction has the structure

S[q, J ] = S0 +12Jpi J

pi + Jpi

δS0

δqi+ Jqi qi (9.26)

where S0 is the standard action

S0[q] =12qiqi − qiBi[q] +

12Bi[q]Bi[q]− C[q]. (9.27)

and we have specialized to the case Aij [q] = δij . Otherwise the inverse A−1[q] wouldappear in the following expressions as a prefactor of Jp and would also make the action S0

non-local.The application of Eq. (9.7) on the integrand present in Eq. (9.25) makes it possible to

write

∆p...pi1...in

=

∫ D[q]Op...pi1...in−1

(Jpin + δS0

δqin

)exp

[i~ S

]

i1−n∫ D[q] exp

[i~ S

] . (9.28)

56

where the operator

Op...pi1...in−1

=n−1∏

l=1

(Jpil +

δS0

δqil+~i

δ

δJpil

)(9.29)

just acts on the function inside the curly brackets. Note that the integrand in the numeratoris a functional dependent on the q’s, Jq’s as well as Jp’s.

Analogous to Eq. (9.6) we can write the partially integrated expression Eq. (9.28) interms of the Generating Functional of connected Green’s functions. Let us specialize to thecase of the one-point Green’s function, i.e., the vacuum expectation values of the momen-tum fields

pi[J ] =δS0

δqi

∣∣∣∣q→ δWH

δJq +~i

δδJq

+ Jpi =δWH

δJpi (x, τ). (9.30)

Note that Eq. (9.13) allows to express Eq. (9.30) as

δΓH

δpi= −pi + δS0

δqi

∣∣∣∣q→q[J ]+~

i∆qq [J ] δ

δq

. (9.31)

Here the dependence on the average momentum fields is made explicit and the action inthe Lagrange formalism appears which depends only on the q-fields. From a functionalpoint of view it is more convenient to express this via Eq. (9.30) as

δΓH

δpi= −

[pi − δS0

δqi

]

φ→φ[J ]+~i∆φφ[J ] δ

δφ

. (9.32)

Now, employing the identity

0 =∫D[q]

δ

δqm

∫D[p] exp

[i

~I[q, p, J ]

](9.33)

we perform the p−integration, so that the above expression reads

0 =∫D[q]

(δS0

δqi+ Jpm

δ2S0

δqiδqm+ Jqi

)exp

[i

~S[q, J ]

], (9.34)

with which it is possible to obtain an analogous form of the field equations

δΓH

δqi=

(δS0

δqi− δΓH

δpm

δ2S0

δqiδqm

)∣∣∣∣φ→φ[J ]+~

i∆φφ[J ] δ

δφ

. (9.35)

Eqs. (9.32) and (9.35) represent alternative forms of the first order DSEs Eq. (9.14).

57

This alternative form of the DSEs allows us to find corresponding equations for the fun-damental proper Green’s functions by taking derivatives with respect to the momentumand/or fundamental fields. This way the proper momentum propagator can be written as

δ2ΓH

δpiδpj= −δij +

δ

δpi

δS0

δqj

∣∣∣∣q→q[J ]+~

i∆qq [J ] δ

δq

, (9.36)

whereas the corresponding mixed Green’s function is

δ2ΓH

δqiδpj=

δ

δqi

δS0

δqj

∣∣∣∣q→q[J ]+~

i∆qq [J ] δ

δq

. (9.37)

In a similar way, the other mixed propagator can be written as

δ2ΓH

δpiδqj=

δ

δpi

δS0

δqj

∣∣∣∣q→q[J ]+~

i∆qq [J ] δ

δq

− δ2ΓH

δpiδpm

δ2S0

δqjδqm

∣∣∣∣φ→φ[J ]+~

i∆φφ[J ] δ

δφ

, (9.38)

whereas the fundamental field propagator is given by

δ2ΓH

δqiδqj=

δ

δqi

δS0

δqj

∣∣∣∣q→q[J ]+~

i∆qq [J ] δ

δq

− δ2ΓH

δqiδpm

δ2S0

δqjδqm

∣∣∣∣φj→φ[J ]+~

i∆φφ[J ] δ

δφ

. (9.39)

At this point we remark that in the derivation of Eqs. (9.38-9.39) we have discardedthose terms that vanish when the sources are set to zero. In addition, note that the prolifer-ated occurrence of Green’s functions involving the p−field arises from functional deriva-tives on ∆qq

ij [J ]. As will become clear below, despite the different appearance the structureof Eqs. (9.38) and (9.39) is related to Eqs. (9.36) and (9.37).

9.3 Pure and mixed momentum correlation functions

In this subsection we derive expressions for first order correlation functions entirely interms of second order correlation functions. To this end, we start from Eq. (9.28). Subse-quently, we again follow a procedure similar to that used in the derivation of Eq. (9.6) thusarriving at a general expression for arbitrary momentum correlation functions,

∆p...pi1...in

= in−1

[Op...pi1...in−1

(Jpin +

δS0

δqin

)∣∣∣∣Jp=0

]

q→ δWδJq +~

iδ

δJq

(9.40)

58

In contrast to the corresponding Eq. (9.30) that leads to the alternative set of DSEs, herethe connected Generating Functional in the standard Lagrange formalism appears whichdoes not depend on Jp anymore. From the above functional equation, any other m-pointGreen’s function with n-external legs associated to the Jp’s is obtained by taking m−nderivatives with regard to Jqm and setting them to zero. In particular, the quantum averageof the momentum fields becomes a functional of the averaged field q,

pi[q] =δS0

δqi

∣∣∣∣qm→qi[Jq ]+~

i∆qq

ij [Jq ] δδqj

. (9.41)

We remark already at this point that the averaged momentum field is generally not givenby the usual definition of a canonical momentum field pcan

i on the level of the EffectiveAction

pi[J = 0] =⟨δS

δq

⟩6= δΓδ ˙q

≡ pcani (9.42)

as shown explicitly in subsection 11.3.In addition, the relation between the pp−correlation functions and those that appear in

the standard formalism can be expressed as

∆ppij = δij + i

(δS0

δqi

δS0

δqj

)∣∣∣∣q→q[Jq ]+~

i∆qq[Jq ] δ

δq

, (9.43)

whereas the corresponding mixed qp−correlation function can be written as

∆qpij [Jq] =

δ

δJqi

(δS0

δqj

)∣∣∣∣qm→ δW

δJq +~i

δδJq

. (9.44)

By using the chain rule δ/δJqm = δqn/δJqmδ/δqn, Eq. (9.44) can be expressed as

∆qpij [Jq] = ∆qq

il

δ

δql

δS0

δqj

∣∣∣∣q→q[Jq ]+~

i∆qq[Jq ] δ

δq

= ∆qq

il

δpjδql

. (9.45)

The other mixed correlator ∆pq is related by the bosonic nature of the fields ∆pqij = ∆qp

ji .Similarly, it is possible to determine the connected three point correlation functions

∆qppijk [Jq] = ∆qq

il

δ

δql∆ppjk,

∆qqpijk [Jq] = ∆qq

im∆qqjl

δ3Γ

δqlδqmδqn∆qpnk +

δ2pkδqmδql

,

∆pppijk [Jq] = iδij pk + iδikpj + iδjkpi + i2

(δS0

δqi

δS0

δqj

δS0

δqk

)∣∣∣∣q→q[Jq ]+~

i∆qq [Jq ] δ

δq

(9.46)

59

where in the last expression we have employed the previous results.Employing Eq. (9.45) and Eq. (9.46) the first and second derivatives of p with respect

to q can be expressed as functionals of the remaining elements. In this way the vacuumexpectation value of p can be expanded. In fact, up to second order in the classical field itreads

pm = − δ2Γδqjδql

∆qplmqj − 1

2!

[δ3Γ

δqiδqjδql∆qplm −

δ2Γδqiδqn

δ2Γδqjδql

∆qqplnm

]qiqj . . . (9.47)

Since ∆qpij can be computed using the procedure shown above, the expression Eq. (9.47)

determines p as a function of dressed correlation functions of the second order formalism.The determination of the proper functions within the canonical formalism as a func-

tional of those that appear in the standard Lagrange framework is rather cumbersome. Inthe more general sense this problem involves connected tensors of rank larger than twoand depends on the possibility to invert the propagator D. Once the individual elementsof this propagator are computed the proper two-point function is completely determinedin terms of the elements of the Lagrange framework. We remark that when the externalsources associated to the p−fields vanish, the inverse of −∆qq is the proper Green’s func-tion that arises in the standard path integral representation.

By direct inversion of the 2× 2 block matrix one obtains

G =

Γppij Γppil ∆pqlmΓqqmj

Γqqil ∆qplmΓppmj Γqqij

H

(9.48)

where

Γppij = − (∆pp + ∆pqΓqq∆qp)−1ij and Γqqij

H = Γqqij + Γqqil ∆qplkΓppkm∆pq

mnΓqqnj . (9.49)

According to Eq. (9.48) there are several equivalent representations of the latter expres-sion. In fact, considering the non-diagonal elements ΓH can be expressed as

ΓqqijH =

(δim + Γqpil ∆pq

lm

)Γqqmj . (9.50)

However, in what follows we will consider the most simple form given by

ΓqqijH = Γqqij + Γqpil

(Γpplm

)−1 Γpqmj , (9.51)

where we have introduced a unity in the form I = (Γpp)−1 Γpp in Eq. (9.49) and further theexpressions for the off-diagonal elements of Eq. (9.48) have been used.

The method to obtain G can be generalized to any proper m−point function. For in-stance, let us suppose that we want to compute the proper three-point function. We denote

60

it asG3 and the correspondingly connected version asD(3). By considering the action of thesymbolic functional derivative δ/δJ on Eq. (9.16) we get the equation D(3)G+ DDG3 = 0.By inversion of D in the second term we obtain the desired form for the proper 3-pointGreen’s function G3 = −GGD(3)G. Proceeding in an analogous way it is possible to ex-press the proper four-point Green’s function G4 and so on. Thereby, the number of vari-ables within the first order formalism can be reduced and from that point of view, theinitially cumbersome problem becomes simpler.

9.4 Connecting the Symmetry-Related identities

Let us now turn to the symmetry identity. To this end we write Eq. (9.18) in the followingform

0 =∫D[q]

Jqmδqm

[~i

δ

δJp, q

]+ Jpmδpm

[~i

δ

δJp, q

]∫D[p] exp

[i

~I[φ, J ]

]. (9.52)

In order to simplify our analysis let us consider theories where the transformations arelinear in the momentum fields. Under this condition and considering the class of Hamil-tonian analyzed so far we get

0 =∫D[q]

Jqmδqm

[Jp +

δS0

δq, q

]+ Jpmδpm

[Jp +

δS0

δq, q

]exp

[i

~S[q, J ]

]. (9.53)

In particular, if Jp = 0 we find that the action S0 is invariant under the symmetry transfor-mation δqm[q] = δqm

[δS0δq , q

], which is a functional of q.

The structure of Eq. (9.53) allows to write the symmetry indetities in a similar form toEq. (9.20) and Eq. (9.21)

0 =∫d4x

δΓH

δpmδpm

[δSδq− δΓH

δp, q

]+δΓH

δqmδqm

[δS0

δq− δΓH

δp, q

]

φ→φ[J ]+~i∆φφ[J ] δ

δφ

.(9.54)

This concludes our formal discussion of the functional symmetry identities. A completederivation, especially for constrained systems as Coulomb gauge Yang-Mills theories, ispresented below in Sect. 12.2.

61

Chapter 10

Decomposition of proper Lagrangecorrelation functions

In this chapter we will show how a general proper correlation function in the Lagrangeformalism can be decomposed into correlation functions in the Hamilton approach.

10.1 Relations between the bare elements

So far we have presented the formalism for a general field theory. In the present and forth-coming sections we will restrict ourselves to the most important class of renormalizablefield theories. In four dimensions the most general renormalizable ``canonical action” fora pure bosonic theory can be expressed as a functional Taylor expansion,

I0[q, p] = Iqp0jipiqj +12Ipp0ijpipj +

12Ipqq0ijkpiqjqk +

12Iqq0ijqiqj

+13!Iqqq0ijkqiqjqk +

14!Iqqqq0ijklqiqjqkql. (10.1)

Here Iqp0ji = ∂τiδij is such thatIqp0jiqj = qi. (10.2)

Clearly, the coefficients Iφ...0i... are field independent. They are given by the functional deriva-tives of I0 evaluated at φ = 0, namely

Iφφ0ij... ≡δ

δφi

δ

δφj

δ

δφk. . . I0

∣∣∣∣φ=0

.

We remark that Ipqq0ijk as well Iqqqq0ijkl are dimensionless tensor couplings whereas Iqq0ij andIqqq0ijk have mass dimension 2 and 1, respectively, and do not involve time derivatives of the

62

fields. Nevertheless, depending on the assumed theory the latter two might depend on∇2 and ∇. We are considering bosonic theories, so that all coefficients are symmetric. Inparticular, the first term in Eq. (9.2) can be written as Ipq0jipjqi where Ipq0ji = −∂τiδij whichleads to the functional relation ∂τjδji = −∂τiδij .

We can identify the coefficients present in the Hamiltonian density Eq. (9.24) as

Aij = −Ipp0ij = δij , Bi = −12Ipqq0ijkqjqk,

C = −12I

qq0ijqiqj − 1

3!Iqqq0ijkqiqjqk − 1

4!Iqqqq0ijklqiqjqkql

(10.3)

Substituting Eqs. (10.3) in Eq. (9.27) and collecting the terms of the same order in q we getthat the action S0 can be written as a polynomial functional,

S0 =12S0ijqiqj +

13!S0ijkqiqjqk +

14!S0ijklqiqjqkql. (10.4)

In this context the following relations between the bare coefficients arise

S0ij ≡ δ2S0

δqiδqj

∣∣∣∣q=0

= Iqp0ilIpq0lj + Iqq0ij

S0ijk ≡ δ3S0

δqiδqjδqk

∣∣∣∣q=0

= Iqqp0iknIpq0nj + qk ↔ qjperm.+ qi ↔ qjperm.+ Iqqq0ijk,

S0ijkl ≡ δ4S0

δqiδqjδqkδql

∣∣∣∣q=0

= Iqqp0iknIpqq0njl + qi ↔ qlperm.+ qk ↔ qlperm.+ Iqqqq0ijkl.

(10.5)

Substituting the above equations in Eq. (10.4) and considering the relation given by Eq.(10.2) allows to find additional relations. In fact, taking the second, third and four func-tional derivatives of S0 and evaluating at q = 0, respectively, we obtain

Sqq0ij ≡δ2S0

δqiδqj

∣∣∣∣q=0

= Iqp0ij = ∂τjδji, S qq0ji ≡δ2S0

δqjδqi

∣∣∣∣q=0

= Ipq0ij = −∂τjδji,

Sqqq0ijk ≡δ3S0

δqiδqjδqk

∣∣∣∣q=0

= Iqqp0ijk,δ3S0

δqiδqjδqk

∣∣∣∣q=0

= Iqqq0ijk

δ4S0

δqiδqjδqkδql

∣∣∣∣q=0

= Iqqqq0ijkl.

(10.6)

In the last two relations we have introduced δ denoting a partial functional differenti-ation acting just on those terms in the action that do not involve the time derivative of thefield.

63

To complete our analysis we point out that from Eq. (B.7) the general form of the quan-tum canonical momentum fields in four dimensional renormalizable theories is given by

pi = Sqq0jiqj +12Sqqq0kjiqjqk. (10.7)

As these polynomial representations allow to identify a priori the bare elements, the ex-pansions given above prove to be very convenient to derive the DSEs for a general bosonictheory in both formulations.

10.2 Diagrammatic decomposition

The relations between the correlation functions in both formulations will be given in thefollowing via explicit diagrammatic expressions. To enable this, we will first introduce ourgraphical representation in terms of the fundamental objects that characterize the theoryin both formulations.

I) As before we consider fields that involve all the irreducible representations in the the-ory. The fundamental fields are represented by solid lines whereas the correspondingmomentum fields are denoted by zigzag lines.

fields momenta

II) Dressed propagators are denoted by thick, whereas bare propagators and externallines by thin lines, respectively. Off-diagonal propagator components are repre-sented by a thick, half solid and half zigzag line.

dressed bare, external

In order to keep the representation of the DSEs in the first order formalism concisewe also include the matrix propagator Eq. (9.10) represented by a double line.

D

III) All proper correlation functions (including proper 2-point functions) in the first andsecond order formalism are denoted by small and large filled blobs,

64

proper, Lagrange proper, Hamilton

whereas bare vertex functions are represented by open blobs.

bare, Lagrange bare, Hamilton

IV) In our analysis it will become useful to introduce the inverse proper momentum 2-point function

Dppij ≡ −(δ2ΓH

δpiδpj

)−1

(10.8)

which will be represented by a thick dotted line. Since it has the form of an alterna-tive momentum propagator it can connect as an internal momentum line to propervertices. Similarly all connected correlators which are 1PI in the fields but merelyconnected via the ”``propagator” Dpp are called p-connected and are denoted by ablob labeled by a P.

P

Dpp p− connected

To express proper functions in the Lagrange formalism in terms of those of the Hamil-ton formalism, we exploit the underlying equivalence between the first and second orderformalism. Due to the equivalence of the Generating Functionals of connected Green’sfunctions at vanishing sources Jp, the Effective Actions in the two formalisms are identicalwhen p is a functional of q that is implicitly given by the stationarity in p

Γ[q] ≡ ΓH [p(q), q] wheneverδΓH

δp= −Jp = 0. (10.9)

Here ΓH still depends explicitly on ˙q and p, where the quantum average of the momentumfields is in general a complicated functional of the fundamental field. Any proper n-pointfunction in the standard formalism can be determined by taking n derivatives with respectto the fields in Eq. (10.9) and evaluated at the vacuum expectation value. In particularapplying the chain rule and Eq. (10.9) the first derivative reads

δΓδqi

=δΓH

δqi+δΓH

δpj

δpjδqi

=δΓH

δqi. (10.10)

In our graphical representation this equation translates to

65

=

.

As usual in the standard formalism, a differentiation with respect to a field is equivalentto attaching an external leg in the graphical representation (cf., e.g., ref. [96, 97]). Next weconsider the second derivative of the action given by

δ2Γδqiδqj

=δpnδqi

δ2ΓH

δpnδqj+δ2ΓH

δqiδqj. (10.11)

The field derivative of the momentum field can alternatively to our previous result Eq.(9.47) also be obtained from a field derivative of the constraint equation in Eq. (10.9)

δ2ΓH

δqiδpj= −δpn

δqi

δ2ΓH

δpnδpj⇒ δpn

δqi= − δ2ΓH

δqiδpj

(δ2ΓH

δpnδpj

)−1

. (10.12)

Inserting this in Eq. (10.11) it is expressed entirely in terms of proper first order Green’sfunctions and takes the symmetric form

δ2Γδqiδqj

=δ2ΓH

δqiδqj− δ2ΓH

δqiδpj

(δ2ΓH

δpnδpj

)−1δ2ΓH

δpnδqj(10.13)

in accordance with Eq. (9.51). The above equation yields the graphical representation ofthe decomposition of the proper two-point Green’s function

= + .

Interestingly, in terms of the first order correlation functions there is in addition to theproper part also a connected contribution due to the fact that the fundamental field mixeswith the corresponding momentum field. This is evident from the off-diagonal elementsEq. (9.44) in the propagator matrix in the first order formalism. Due to the mixing thearising propagator in Eq. (10.12) is not the elementary p-propagator but the propagator fora collective mode described by the inverse of the proper two-point momentum correlationfunction Dpp which is represented by the dotted line and related to the actual momentumpropagator via Eq. (9.49).

The result for the fundamental field derivative of a proper correlation function in theHamilton formalism yields the replacement rule

+→ .

66

In the next step a field derivative can also act on the propagator Dpp. Its derivative isobtained from the derivative of the inverse of an operator as

δDppijδqk

= − δ

δqk

(δ2ΓH

δpiδpj

)−1

=(δ2ΓH

δpiδpm

)−1δ3ΓH

δpmδpnδqk

(δ2ΓH

δpnδpj

)−1

= DppimΓppqmnkDppnj

which yields the graphical replacement rule

→

→

+ .

Applying these two replacement rules in all possible ways on the right hand side of theabove equation for the two-point vertex yields immediately the corresponding decompo-sition of the proper 3-point vertex

δ3Γδqiδqjδqk

=δ2pnδqiδqj

δ2ΓH

δpnδqk+δpnδqi

δpsδqj

δ3ΓH

δpsδpnδqk+δpnδqj

δ3ΓH

δqiδpnδqk

+ qi ↔ qjperm. + +δ3ΓH

δqiδqjδqk, (10.14)

= + ++

=+ + + +

.

This yields directly a symmetric result, whereas the computation without the replacementEq. (10.12) in each step would yield an asymmetric result involving higher derivatives ofp. E.g. the 3-point function involves the second derivative of p which had to be obtainedfrom

δ3ΓH

δqiδqjδpk= − δ2pn

δqiδqj

δ2ΓH

δpnδpk− δpnδqi

δpsδqj

δ3ΓH

δpsδpnδpk

− δpnδqj

δ3ΓH

δqiδpnδpk− qi ↔ qjpermut. (10.15)

To derive the decomposition of higher order Lagrange correlation functions it is usefulto note that both of the above replacement rules involve the external legs in the form of thecomposite expression

67

→

→

→

→

Figure 10.1: The replacement rules that create the general decomposition of a proper cor-relation function in the Lagrange formalism in terms of Hamilton correlation functions.

=n n with ≡ +

. . . ”p− connected”P

P

Figure 10.2: The general result for the decomposition of a proper correlation function inthe Lagrange framework in terms of correlators in the Hamilton formulation.

with ≡ +

denoted by a dashed line. By introducing the composite external leg into the graphicalrepresentation, the decomposition of the 3-point function is given by a single graph withthree of these new external legs. Via the previous rules it is easy to obtain a correspondingreplacement rule for the composite external leg. Thereby the extension of a general n-point function by an additional leg can be obtained from the simplified set of rules in Fig.10.1. Starting from the 3-point function these rules allow to derive the decomposition ofarbitrary proper n-point functions in the Lagrange formalism. In the appendix we showthat the graphs generated by these replacement rules have a very simple structure that canbe summarized by the following general statement:

A proper n-point function in the Lagrange formalism can be decomposed into the sum of allp-connected n-point functions with composite external legs in the Hamilton framework.

This is shown in graphical form in Fig. 10.2.The graphical representation manifestly shows that the propagation, decay and disper-

sion of particles are more complicated when analyzed in the Hamilton framework. At thetree level the arising propagators Dpp are constant and correspondingly the dotted inter-nal lines between the proper vertices represent merely contact terms. Similarly the externalleg corrections involve no additional poles at tree level and the explicit energy dependence

68

arising from the mixed correlator in Eq. (10.12) just cancels the one from the momentumfield derivative of the initial vertex. In the fully dressed action, however, there could ariseadditional poles due to the actual propagation of the momentum fields via higher orderkinetic terms, as well as an explicit energy dependence of the terms that cancels only aftersumming over all contributions.

Via graphical rules, we can generate in a systematical way the connection betweenany proper Green’s function in the standard second-order formalism to the appropriatecorrelation functions in the first-order formalism. We point out already at this point thatthe inverse relations for Green’s functions in the latter context are more complicated andcan involve loops related to connected correlation functions with both mixed and puremomentum fields. The method used in the last section allows us to represent them interms of those appearing in the Lagrange formalism.

10.3 Inclusion of Grassmannian fields

Let us consider a quantum field theory involving Grassmannian fields ci and c†j fulfillingthe anti-commutation rules

ci, c

†j

= δij

c†i , c

†j

= ci, cj = 0. (10.16)

The Grassmannian action for a renormalizable theory in four dimensions has the generalform

I0 = iκijc†i cj + iλijkqic

†jck + iαijc

†icj . (10.17)

The quantum canonical momentum fields associated to the ci are given by pi = δI0δRci

=

iκijc†j , where the suffix R denotes differentiation from the right. It is clear that the usual

path integral representation for Grassmannian field is already of first-order form sincethe momentum fields are treated as independent variables. Purely fermionic correlationfunctions are therefore trivially identical in the two formulations.

Let us now study mixed correlation functions involving bosonic fields represented bythe qi in Eq. (10.17) when the bosonic path integral is written in the canonical form as well.Generally the tensors κij , αij and λijk are real and field independent. The kinetic tensorreads in particular

κij =

δij with c, c† ∈ Fermions

δijσ∂0 with c, c† ∈ Fadeev − Popov Ghosts

where σ ∈ R. Since it is a bosonic variable, p is a functional of fermionic bilinears ci†Γij cjand q. As a consequence the second term in Eq. (10.11) vanishes identically, which confirms

69

the equality of the propagators in both formulations

δ2Γ

δciδc†j

=δ2ΓH

δciδc†j

. (10.18)

Analog to the bosonic case the three-point vertex can be decomposed as

δ3Γ

δqiδcjδc†k

=δpnδqi

δ3ΓH

δpnδcjδc†k

+δ3ΓH

δqiδcjδc†k

. (10.19)

Although the propagators are the same in both formulations the vertices can be differ-ent. The above chain rule again results in the appearance of composite legs for all bosonsin the graphical representation. The decomposition of a general n-point correlation func-tion is then again given by all p-connected graphs where only the bosonic propagatorsDppare involved and only bosonic external legs are composite.

70

Chapter 11

Decomposition of connectedHamilton correlation functions

So far we gave in subsection 9.3 general expressions for correlation functions in the Hamil-ton formalism. Our goal in this section is to evaluate the general decomposition of thetwo-point functions in the Hamilton framework in case of a generic four-dimensionalrenormalizable quantum field theory in terms of Lagrange correlation functions. To dothis, we start by computing the elements of ∆qp

ij .

11.1 The mixed, connected 2-point function

By considering Eq. (9.45) and Eq. (10.7) the mixed connected two-point function in thefirst order formalism can be written as

∆qpij = ∆qq

il

(Sqq0lj −

i

2Sqqq0ukj

δ

δql∆qquk

). (11.1)

where from now on we skip the explicit factor ~. Using partial differentiation over theidentity ∆qq

il Γqqlj = −δij we get

δ

δql∆qquk = ∆qq

umΓqqqmln∆qqnk, (11.2)

so that this element in configuration space looks like

∆qpij = ∆qq

il

(Iqp0lj −

i

2Iqqp0ukj∆

qqumΓqqqmln∆

qqnk

), (11.3)

where Eqs. (10.6) have been used. According to our diagrammatic representation theabove function has the decomposition given in Fig. 11.1.

71

= −

i

2

Figure 11.1: The decomposition of the mixed Hamilton propagator in terms of Lagrangecorrelation functions.

Via the bosonic symmetry of the propagator it is easy to see that

∆pqij =

(Iqp0li −

i

2Iqqp0uki∆

qqumΓqqqmnl∆

qqnk

)∆qqlj . (11.4)

Clearly, Eq. (11.3) and Eq. (11.4) fulfill the condition ∆qpij = ∆pq

ji .Our convention for the Fourier transform of a general two-point function (connected

or proper) obeying translational invariance is

∆ΦΦij ≡ ∆ΦΦ

ij (xi − xj) =∫

dk∆ΦΦij (k)e−ik·(xi−xj), (11.5)

with k · (xi − xj) = k0(x0i − x0j) − k · (xi − xj). As a consequence of this convention andthe equivalence ∆qp

ij = ∆pqji we get the relation

∆qpij (k) = ∆pq

ji (−k); (11.6)

where i and j represent the remaining internal indices. This yields the correspondingequation in momentum space

∆qpij (k) = ∆qq

il (k)(ik0δlj− i

2

∫dωIqqp0ukj(k −ω, ω,−k)

× ∆qqum(ω − k)Γqqqmln(ω − k, k,−ω)∆qq

nk(ω)). (11.7)

In the derivation of the latter equation we have taken into account the Fourier transforma-tion of the proper 3-point function

ΓΦΦΦαβγ =

∫dkα dkβ dkγ (2π)4 δ4(kα + kβ + kγ)ΓΦΦΦ

αβγ (kα, kβ, kγ)e−ikαxα−ikβxβ−ikγxγ

where the δ−function expresses the momentum conservation.

72

= −

i

2

+ −

1

4

−

1

4−

1

2

−

i

2−

i

2

Figure 11.2: The decomposition of the momentum propagator in terms of Lagrange corre-lation functions.

11.2 The momentum propagator

The pure pp−correlator can be computed using Eqs. (9.43) and (10.7). Neglecting thoseterms that will eventually vanish when the sources are set to zero, it reads

∆ppij = δij + Sqq0liSqq0mj∆qq

lm −i

2Sqq0liSqqq0nmj∆

qqlu

δ

δqu∆qqnm −

i

2Sqq0mjSqqq0kli∆

qqku

δ

δqu∆qqlm

− i

2Sqqq0kliSqqq0nmj∆

qqkm∆qq

ln −14Sqqq0kliSqqq0nmj∆

qqku

δ

δqu

[∆qqlx

δ

δqx∆qqnm

]. (11.8)

By iterated application of Eq. (11.2) and considering Eqs. (10.6) this expression yields

∆ppij = δij − i

2Ipqq0ikl∆

qqkm∆qq

lnIqqp0nmj −

14Ipqq0ikl∆

qqku∆

qqlx∆qq

nyΓqqqqyxuz∆

qqzmI

qqp0nmj (11.9)

+ Ipq0il∆qqlmI

qp0mj −

12Ipqq0ikl∆

qqku∆

qqlx∆qq

npΓqqqpuq∆

qqqyΓ

qqqyxz∆

qqzmI

qqp0nmj

− i

2Ipq0il∆

qqlmΓqqqxmy∆

qqxn∆

qqyuI

qqp0nuj −

i

2Ipqq0ikl∆

qqku∆

qqlxΓqqqxuy∆

qqymI

qp0mj

− 14Ipqq0ikl∆

qqku∆

qqlxΓqqqxuy∆

qqym∆qq

nfΓqqqfmz∆

qqzhI

qqp0nhk.

Here we have already suppressed disconnected expressions involving tadpoles that cancelsince Eq. (9.31) gives

δΓH

δpi

∣∣∣∣q,p=0

∼ Iqqp0mni∆qqmn = 0 . (11.10)

The graphical representation of the decomposition of the momentum propagator in theHamilton framework is shown in Fig. 11.2.

Note that the loop graphs in the first line represent precisely the vacuum graphs of then-particle irreducible (nPI) actions in the Lagrange formalism of order one and two withattached external p-legs. In addition to these proper contributions this result involves again

73

connected graphs in the second line. These graphs correspond to the second and thirdline in Eq. (B.16) which can alternatively be expressed by the known mixed correlationfunctions derived in the last subsection as −∆pq

imΓqqml∆qplj and correspondingly represented

as a single graph.After Fourier transformation of Eq. (B.16) we find the representation of ∆pp in momen-

tum space

∆ppij (k) = δij −∆pq

im(k)Γqqml(k)∆qplj (k)− i

2

∫dωIpqq0ikl(k, ω − k,−ω)∆qq

km(ω − k)

× Iqqp0nmj(ω, k − ω,−k)∆qqln(ω)− 1

4

∫dω dµIpqq0ikl(k, µ− k,−µ)∆qq

ku(µ− k)

× ∆qqlx(−µ)Iqqp0nmj(−ω, ω + k,−k)∆qq

ny(−ω)Γqqqqyxuz(ω, µ, k − µ,−ω − k)

× ∆qqzm(−ω − k)− 1

2

∫dω dµIpqq0ikl(k, ω − k,−ω)∆qq

ku(ω − k)∆qqnp(µ+ k)

× Γqqqpuq(−k − µ, k − ω, ω + µ)∆qqlx(−ω)Iqqp0nmj(µ+ k,−µ,−k)∆qq

qy(ω + µ)× Γqqqyxz(−ω − µ, ω, µ)∆qq

zm(µ). (11.11)

The summation over all field components indicated by the subindices in Eq. (11.7) (seealso Eq. (B.16)) leads to multiple possibilities. Yet many of these might not be alloweddue to the constraints imposed by the symmetries of the theory. The decomposition ofhigher order functions involving one or two external momentum fields is simply obtainedby further derivatives of these generating equations. Via the chain rule a derivative w.r.t. tothe source can be transformed into a derivative with respect to the averaged fields yieldingin addition external propagators. Correspondingly the graphical decomposition of suchcorrelation functions can be obtained recursively via the graphical replacement rules inthe Lagrange framework given in [96]. On the other hand, from Eq. (9.40) it is clear thathigher order correlation functions involving n momentum fields involve graphs of looporder n and it is not possible here to give a close result for all n-point functions.

To conclude this section we remark that the multiplication of Γqq on the left hand sideamputates the Green’s function ∆qq.According to Eq. (9.47), this operation allows to derive

δpiδqj

= Iqp0ji −i

2Iqqp0uki∆

qqumΓqqqmjn∆

qqnk. (11.12)

Considering this expression and the diagrammatic rules in the standard path integral rep-resentation we get

δpiδqjδqk

= Iqqp0jki −i

2Iqqp0mui∆

qqmnΓ

qqqnkx∆

qqxyΓ

qqqyjt∆

qqtu (11.13)

+ j ↔ k − i

2Ipqq0imu∆

qqmnΓ

qqqqnjkl∆

qqlu ,

74

which can be represented graphically as

= −

i

2−

i

2−

i

2

We note in passing, that similar to the composite collective propagator and composite legin the last section this ”``vertex” allows to write the graphs in the first line by a single loopgraph. Yet, here this is then surely no explicit decomposition in terms of second ordercorrelation functions anymore.

11.3 The inverse propagators

While in the context of the canonical formulation it is entirely possible to deduce the com-plete set of DSEs directly from Eqs. (9.36-9.38) we shall follow a slightly less obvious pathhere. We proceed to give the diagrammatic and analytic expressions for the proper prop-agators, typical of first order formalism. Subsequently, we should show that such rep-resentations can be encoded into the usual ones, i.e Eqs. (9.36-9.38), which complete theequivalence between both types of derivations.

The result given by Eq. (9.48) allows to analyze the structure of the inverse propagator.By considering the Fourier transformation of the mentioned equation we have that

Γpqij (k) = Γppil (k)∆pqlm(k)Γqqmj(k),

Γppij (k) = − (∆pp(k) + ∆pq(k)Γqq(k)∆qp(k))−1ij ,

ΓHqqij (k) = Γqqij (k)+Γqqil (k)∆qplk (k)Γppkm(k)∆pq

mn(k)Γqqnj(k).

(11.14)

For a theory without a three-point interaction vertex involving the time derivative ofthe fields, one has Γpp = −I and Eq. (9.48) reduces to

G =

−I −ik0I

ik0I Γqq − k20I

. (11.15)

This simplified expression holds for theories like QED and/or self-interacting φ4 theory,where the vacuum expectation value of the momentum field is completely determined interms of q, pi = ˙qi.

Actually, the expressions for Γpp, Γqp and Γpq in Eq. (11.15) encode a more general re-sult because they could be obtained using Eq. (9.36) and Eq. (9.37) without setting the

75

variables of our problem to zero. As a consequence they show that dressed proper func-tions involving more than two external momentum legs are not present in such theories.This means that the mass and the coupling constant receive no contribution coming fromthe higher order corrections besides the usual ones given in the standard framework. Onthe other hand the absence of quantum corrections to Γpp, Γqp and Γpq means that onlywave-function renormalization contributes to these kinetic terms.

As in the present framework the quantum corrections to the propagators and verticesdepend on temporal derivatives. This shows that within the standard formalism there areseveral pieces in the Effective Action that depend on the time derivative of the averagedfield. As a consequence, the canonical momentum fields defined on the level of the Effec-tive Action as pcan

i = δΓ/δ ˙qi differ from those given by the quantum average of p since theinvolved limiting processes do not commute.

11.4 Recovering the first order DSEs

Although the decomposition of proper Hamilton functions given above involves inver-sions that cannot be omitted in terms of second order correlation functions it is possible totransform these equations into a form where such matrix inversions are eliminated. Thisis done by explicitly introducing first order correlation functions on the right hand sideagain and, as we will show in this subsection, leads precisely to the first order DSEs. Inorder to do this we point out that by considering Eq. (11.14), Γpp can be expressed as

Γppij = −[δij − i

2Ipqq0ikl∆

qqkm∆qq

ln

δpjδqnδqm

]−1

. (11.16)

The multiplication of − Γpp−1 from the left hand side allows to write the above equationas

[δis − i

2Ipqq0ikl

(∆qqpkls −∆qq

km∆qqlnΓqqqumn∆

qpus

)]Γppsj = −δij ,

where Eq. (9.46) has been taken into account. The introduction of a I as−∆qqΓqq in the lastterm in the bracket and the use of Eq. (9.46) makes it possible to express the above relationin the following form

Γppij = −δij +i

2Ipqq0ikl

∆qqpklsΓ

ppsj + ∆qqq

klsΓqpsj

, (11.17)

which can be translated to

Γppij = −δij − i

2Ipqq0ikl∆

qφknΓ

φpφnjm∆φq

ml. (11.18)

76

This presents the DSE for the proper momentum 2-point correlation function.It is possible to obtain a similar equation for the pq−propagator. Considering our pre-

vious result, we find

Γpqij =[δil − i

2Ipqq0ikt∆

qqkm∆qq

tn

δ2plδqnδqm

]−1 (Ipq0lj −

i

2Ipqq0luk∆

qqumΓqqqmjn∆

qqnk

). (11.19)

Indeed, the multiplication of− Γpp−1 from the right hand side allows to express Eq. (11.19)as

Γpqij = Ipq0ij −i

2Ipqq0iuk∆

qqumΓqqqmnj∆

qqnk +

i

2Ipqq0iuk∆

qqum∆qq

nk

δplδqmδqn

Γpqlj . (11.20)

This can be rewritten in the following form

Γpqij = Ipq0ij −i

2Ipqq0iuk

∆qqumΓqqqmnj∆

qqnk + ∆qq

um∆qqnkΓ

qqqmnl∆

qplv Γpqvj −∆qqp

uklΓpqlj

. (11.21)

The introduction of a I as −∆qqΓqq in the first and second term inside the brackets allowsto write this as

Γpqij = Ipq0ij +i

2Ipqq0iuk

∆qqqukl

(Γqqlj +Γqqlm∆qp

mnΓpqnj

)+∆qqp

uklΓpqlj

.

By considering Eq. (9.49) we get

Γpqij = Ipq0ij +i

2Ipqq0iuk

∆qqqukl Γqqlj

H + ∆qqpuklΓ

pqlj

(11.22)

which yields the result

Γpqij = Ipq0ij −i

2Ipqq0ikl∆

qφknΓ

φqφnjm∆φq

ml. (11.23)

We remark that the last terms of Eqs. (11.18) and (11.23) include several combinationsof fundamental and momentum fields. The relations given by Eq. (11.18) and Eq. (11.23)are the DSEs of the proper pp- and pq-propagators expressed in terms of the usual elementsof the canonical formulation.

It is straightforward to prove that Eq. (11.18) coincides with the corresponding equationderived using Eq. (9.36) and Eq. (10.7). This is different for Eq. (11.23). In fact, as discussedpreviously, we could in principle calculate these propagators from Eq. (9.38), neverthelessby considering the action Eq. (10.4) and the relations between the bare elements we findthat the structure is more cumbersome than that given in Eq. (11.23). Instead, the latterone is in correspondence with Eq. (9.37) via the symmetry relation Γpqij = Γqpji in case bothq and p are bosonic fields.

77

= −

i

2= −

i

2

,

= −

i

2−

i

2−

1

2−

1

6

Figure 11.3: Diagrammatical representation of the coupled system of DSEs within the firstorder formalism. The double lines represent the matrix propagator so that all possiblegraphs involving the individual propagators arise which are compatible with the symme-tries and the restriction that the pqq-vertex is the only bare vertex involving momentumlines. In this form the formal equivalence to the second order equation is entirely manifest.

The fact that we recover the standard first order propagator DSEs from the decompo-sition of proper Hamilton correlation functions has its origin in the equivalence betweenthe canonical and Lagrange equations of motion at the quantum level. In appendix B.3it is shown that whenever they describe the same dynamical processes, the DSEs derivedfrom them will be equivalent too. However, as we argued in the last subsection, the clas-sical canonical momentum fields defined from the Effective Action and those given by thequantum average of p are not the same. This means that the equations expressed in termof p and q do not correspond to the classical canonical ones.

Based on this statement, the derivation of ΓqqijH is considerably simpler using Eq. (9.39)

than via the procedure performed in the last two cases. Indeed, by considering Eqs. (10.7),(10.4) and the relations between the bare elements, we arrive at

δΓH

δqi=

[Iqp0ilpl + Iqqp0ijkqjpk + Iqq0ijqj +

12Iqqqijk qjqk +

13!Iqqqq0ijklqjqkql

]

q→q+~i∆qφ δ

δφ

.(11.24)

Now, in order to give the explicit form of ΓqqijH , we rewrite Eq. (11.24) as

δΓH

δqi= Iqp0ij pj + Iqq0ij qj + Iqqp0ijkqj pk +

12Iqqq0ijkqj qk +

13!Iqqqq0ijklqj qkql − iIqqp0ijk∆

qpjk[J ]

− i

2Iqqq0ijk∆

qqjk[J ]− 1

6Iqqqq0ijkl∆

qφjm[J ]∆qφ

ku[J ]Γφφφmun[q, p]∆φqnl [Jq]

− i

2Iqqqq0ijklqj∆

qqkl [J ]. (11.25)

Taking the functional derivative with respect to q and setting the vacuum expectation

78

values of the fundamental fields to zero we get

ΓHij = Iqq0ij −i

2Iqqqqijkl ∆qq

kl − iIqqp0jlm∆qφlu Γφqφuin∆φp

nm −i

2Iqqq0jlm∆qφ

lu Γφqφuin∆φqnm

− 12Iqqqqjlmn∆

qφlh Γφφφhfk∆

φφfy∆

qφmuΓ

φqφuiy ∆φq

kn −16Iqqqqjlmn∆

qφlh∆qφ

mfΓφφqφhfik ∆φq

kn. (11.26)

The Fourier transformation of Eqs. (11.18), (11.23) and (11.26) yields finally the corre-sponding DSEs in momentum space

Γppij (k) = −δij − i

2

∫dωIpqq0ikl(k, ω − k,−ω)∆qφ

kn(k − ω) (11.27)

× Γφpφnjm(k − ω,−k, ω)∆φqml(−ω),

Γpqij (k) = −ik0δij − i

2

∫dωIpqq0ikl(k, ω − k,−ω)∆qφ

kn(k − ω) (11.28)

× Γφpφnjm(k − ω,−k,−ω)∆φqml(−ω),

ΓqqijH (k) = Iqq0ij(k)−

i

2

∫dωIqqqq0ijkl(k,−k,−ω, ω)∆qq

kl (ω)− i

∫dωIqqp0ilm(k, ω − k,−ω)

× ∆qφlu (k − ω)Γφqφujn(k − ω,−k, ω)∆φp

nm(−ω)− i

2

∫dωIqqq0ilm(k, ω − k,−ω)

× ∆qφlu (k − ω)Γφqφujm(k − ω,−k, ω)∆φq

nm(−ω)− 12

∫dω dµ∆qφ

lh (ω + µ)

× Iqqqq0ilmn(k,−µ− ω, ω − k, µ)∆φφfy (ω)Γφφφhfk (ω + µ,−ω,−µ)∆qφ

mu(k − ω)

× Γφqφujy (k − ω,−k, ω)∆φqkn(µ)− 1

6

∫dω dµIqqqq0ilmn(k, ω − k − µ,−ω, µ)∆qφ

mf (ω)

× ∆qφlh (k + µ− ω)Γφφqφhfjk (k + µ− ω, ω,−k,−µ)∆φq

kn(µ) (11.29)

We show the graphical representation of the complete set of DSEs in Fig. 11.3 where forconciseness we use the matrix propagator which yields all possible loop graphs involvingphysical vertices in accordance with the symmetries of the action as a consequence of theimplied summation over repeated indices.

79

Chapter 12

Applications

12.1 Theories with auxiliary fields

Before we come to our main application of the general expressions in the context of Cou-lomb gauge QCD, we show in this section that these results also apply to theories involv-ing auxiliary fields which in some sense represent a special case of the previous discussion.Compared to the first order formalism in such theories not the kinetic terms but the inter-action terms are linearized which is in the case of fermionic theories also referred to asbosonization [98]. Yet, the action is here by construction likewise only quadratic in theauxiliary fields and lacks kinetic terms for them. Therefore, our analysis directly appliesand gives general relations between correlation functions in the fundamental theory andthe linearized form involving auxiliary fields.

We will illustrate this in the case of fermionic theories with non-renormalizable, quarticinteractions. Examples for this class of theories are the Nambu−Jona-Lasinio model [99]or the BCS theory of superconductivity [100]. The functional integral is given by

∫Dψ†Dψ exp

(i

~

∫d4x

(Lψ+Jψ(x)ψ(x)+Jψ(x)ψ(x)

))(12.1)

with the general Lagrangian of such a theory with local, quartic interactions

Lψ = ψ(x) (i6∂ −mψ)ψ(x)−∑

i

giψ(ψ(x)Γiψ(x)

)2 (12.2)

where the Γi are Dirac matrices. The linearization of the fermionic interaction can be per-formed by formally introducing a one of the form

I =∏

i

∫DηiDσi exp

(i

∫d4xσi(x)

(ηi(x)−ψ(x)Γiψ(x)

))(12.3)

80

into the fermionic path integral, where this path integral over σi enforces a functionalδ-function that allows to rewrite the non-linear fermionic interaction in terms of ηi. Inte-grating then over the ηi yields the path integral of the corresponding linear sigma model

∫Dψ†DψDσ exp

(i

~

∫d4x

(Lσ + Jψ(x)ψ(x) + Jψ(x)ψ(x) + J iσ(x)σi(x)

))

where we have introduced additional sources for the auxiliary fields σi. After this bosoniza-tion procedure the Lagrangian of the corresponding ”``linear σ-model” reads

Lσ = ψ(x)(i6∂ −mψ + giσΓiσi(x)

)ψ(x) +

m2σ

2σi(x)2 (12.4)

where gψ = g2σ/(2m

2σ). Here, there is, in contrast to the first order formalism, by construc-

tion no mixing between the fundamental and the auxiliary fields.Analog to Eq. (10.9) the Effective Actions of the two theories are again identical at

vanishing sources J iσ = 0Γψ[ψ†, ψ] = Γσ[ψ†, ψ, σi[ψ†, ψ]] (12.5)

where the auxiliary fields are implicitly given by

δΓσδσi

= −J iσ = 0. (12.6)

Via the chain rule of functional differentiation we obtain analogous expressions for Eqs.(10.10)-(10.15) with pi replaced by σi, respectively. Correspondingly, the same graphi-cal rules apply. With the simplification that there can be no mixing due to the differentstatistics of the fields, the external legs are not composite and the arising inverse 2-pointfunction is the ordinary σ-propagator. Denoting as before connected correlation functionsthat are 1PI with respect to the fundamental fields and connected in the auxiliary fields asσ-connected, this leads to the analogous general result:

A proper n-point function in the fundamental theory can be decomposed into the sum of allσ-connected n-point functions in the linearized theory.

In particular the decomposition of the proper 4-fermion vertex in the fundamental theoryEq. (12.2) reads

= + + +

81

where we represent in an analogous way the proper vertices in the fundamental and lin-earized theory by large and small blobs and the ordinary σ-propagator by the dotted line.Our result shows that there is no double-counting in the linearized theory although thesame process is partly described by quark and mesonic dynamics. This redundancy in thedescription can be prevented from the outset by partial re-bosonization [101] in the con-text of the functional renormalization group, whereby the contribution of the fundamentaldegrees of freedeom is entirely absorbed into the bosonized interactions even at the levelof the effective action.

Similarly, since the Lagrangian of the linearized theory is quadratic and lacks kineticterms for the σ-fields by construction, it can be trivially integrated out retaining the sourcesfor the auxiliary fields at this point. This leads to an analog expression to Eq. (9.40) for thecorrelation function of n auxiliary fields 〈σi1(x1) · · ·σin(xn)〉 after the sources are set totheir vacuum expectation value. Due to the absence of 3-point interactions and mixing,the decomposition of the auxiliary σ-propagator is again simplified compared to the resultgiven in Fig. 11.2

= +i −

where the different prefactors of the loop correction arise due to the fermionic nature of thefields. In contrast to the decomposition of the Hamilton propagators there is no differencebetween proper and connected 2-point functions in the case of the auxiliary field due to theabsence of mixing and the proper correlation functions is simply the inverse of the aboveequation.

Note that this expression requires 2-loop integrals involving the proper 4-point vertexin the fundamental theory and is thereby not a convenient way to compute this propaga-tor. In contrast, the introduction of the auxiliary degrees of freedom generally allows todescribe the dynamics far more efficiently. While the tree level propagator is here againstatic and merely given by the mass term in Eq. (12.4), the loop corrections usually inducepoles that make the bosonic auxiliary field a propagating degree of freedom in the fulltheory [103].

12.2 Coulomb Gauge Yang-Mills theory

As detailed in the introduction of this part the main motivation for developing the pre-sented formalism is given by the fact that the first order formalism might be better suitedfor non-perturbative studies of Coulomb gauge Yang-Mills theory. As a first step into thisdirection we will apply the formalism developed so far to give the explicit relations be-tween the two-point correlation functions of the transversal and longitudinal componentsof the conjugate momentum to the ones of the gauge field.

82

The starting point is the ``canonical action” of Coulomb gauge Yang-Mills theory whosestructure has been derived in [46, 41]

I0 =∫

d4x(

pa · Aa − 1

2(pa · pa + Ba · Ba) + pa ·Dabσb (12.7)

− λa∇ ·Aa − ca∇ ·Dabcb).

Here, p is the conjugate momentum of the gauge field Aµ,a ≡ (Aa, σa), c and c are theGrassmann-valued Faddeev-Popov ghost fields introduced by fixing the gauge, and λa isa ``colored” Lagrange multiplier field. Above

Bai = εijk

(∇jAa

k −12gfabcAb

jAck

)

represents the chromomagnetic field, σb the time-component of the gluon field, whereasDac = ∇δac − gfabcAb is the covariant derivative in the adjoint representation, with thestructure constants fabc of the color-group SU(3), and the gauge coupling g. The lasttwo terms constitute the gauge fixing pieces introduced via the Faddeev-Popov proce-dure. However, as was originally pointed out by Gribov [40], the latter is plagued by theexistence of equivalent gauge field configurations that are related by finite gauge transfor-mations. A further condition must be imposed on the configuration space of gauge fieldswhich restricts it to the so called ``Gribov region”:

Ω ≡ A : ∇ ·A = 0 | −∇ ·D ≥ 0 . (12.8)

This simple domain is still not totally free of Gribov copies. Instead one should considerthe fundamental modular region Λ [128]. It turns out though that functional integrals aredominated by configurations on the common boundary of Ω and Λ [46] so that, in practice,it is enough to consider the domain defined by Eq. (12.8).

Now, the term p · A in Eq. (12.7) is equivalent to pq in Eq. (9.2), and the remainingpart can be identified as a classical ``Hamiltonian” density of Coulomb gauge Yang-Millstheory:

HCoul =12

(pa · pa + Ba · Ba)− pa ·Dabσb + λa∇ ·Aa + ca∇ ·Dabcb. (12.9)

Actually HCoul is not the Coulomb gauge Hamiltonian density derived by Christ and Lee[80]. The derivation of the latter through Eq. (12.7) and the corresponding path integralrepresentation requires further steps detailed in Sec. 16.1 (see also Ref. [102] and referencestherein). Although the Christ and Lee Hamiltonian has the desirable property that it in-volves only the physical degrees of freedom, the non-local character of the Color-Coulomb

83

potential prevents an effective analysis of renormalizability in this framework. On the con-trary, the canonical action given in Eq. (12.7) proved useful to study the renormalizationproperties of Coulomb gauge Yang-Mills theory [46, 82, 41, 47].

It is noteworthy that within the context of Eq. (12.7) the ghost sector is fully discon-nected from pa which allows to write the ``canonical action” as a functional Taylor expan-sion involving a pure bosonic piece like Eq. (10.1) and one containing the ghost field viaEq. (10.17). Therefore, we can identify q = (Aa, σa, λa) and employ the general expressionsderived so far.

Except for the inverse, bare ghost two-point function

Iab(cc)0 = −k2δab, (12.10)

the remaining inverse tree level propagators of the theory are presented in Table 12.1. Notethat I0 involves four-vertices that have the following form in momentum space

Iabc(pAσ)0ij = −gfabcδij , I

abc(ccA)0i = igfabckci,

Iabc(AAA)0ijl = igfabc [δij(ka − kb)l + δjl(kb − kc)i + δli(kc − ka)j ] ,

Iabcd(AAAA)0ijlm = −g2

δijδlm

[facefbde − fadefcbe

]+ δilδjm

[fabefcde − fadefbce

]

+ δimδjl[facefdbe − fabef cde

],

with all momenta defined as incoming.The decomposition of the conjugate momentum into transverse and longitudinal parts,

pa = πa−∇ ·Ωa (note that Ωa must not be confused with the Gribov region), makes it con-venient to study the required complete cancellation of the energy divergences [41, 46, 82]that emerge in a perturbative treatment of Coulomb gauge Yang-Mills theory. Certainly,such a decomposition increases the number of fields of the theory and leads to

I0 =∫

d4x(

πa · Aa −∇Ωa · Aa +

12Ωa∇2Ωa −HCoul(p → π) (12.11)

− τa∇ · πa −∇Ωa ·Dabσb),

where τ is a colored Langrange multiplier field which appears via the transversality con-dition of π. In this context, the general structure of the connected two-point functionsis given in Table 12.2. The latter may be found independent of any approximation butbased solely on the principles of BRST invariance, spatial and time-reversal symmetriesand transversality properties of the vector propagators. Each dressing function Λφφ is adimensionless scalar function of k2

0 and k2 except the ghost propagator

∆ab(cc) = δabΛc(k2)

k2 (12.12)

84

I0 Aj pj σ λ

Ai −Tk2 ik0I −k0k −ikpi −ik0I −I −ik 0σ 0 ik 0 0λ ik 0 0 0

Table 12.1: Tree level proper two point functions (without color factors) in momentumspace.

depending only on k2. The tree-level propagators are obtained when

ΛΩΩ = ΛΩλ = 0, ΛAA = ΛAπ = Λππ = Λσσ = ΛσΩ = Λσλ = Λc = 1. (12.13)

For a complete description, the reader is referred to ref. [46].The general results of the preceeding sections can in principle directly applied to Coulomb

gauge QCD as well by decomposing the momentum field into its individual components.However, since the longitudinal momentum field features more complicated bare corre-lation functions that involve additional derivative operators we derive the correspondingexpressions of subsection 9.3 once more taking now the general action (B.5) into account.To this end we first have to relate the bare momentum correlation functions with the cor-responding ones for the individual momentum components. We formally express the de-composition via a the operator X

pi = πi + ∂iΩ = πi + XijΩj and Xij =δpiδΩj

= −∇iδij . (12.14)

Expanded in a local series in terms of the individual momentum components the generalcanonical action takes the form

I0[q, p] = Iqp0jiπiqj +12Ipp0ijπiπj +

12Ipqq0ijkπiqjqk + Iqp0jiXilΩlqj +

12XilIpp0ijXjkΩlΩk

+12Ipqq0ijkXilΩlqjqk +

12Iqq0ijqiqj +

13!Iqqq0ijkqiqjqk +

14!Iqqqq0ijklqiqjqkql (12.15)

with Ipp0ij = −δij . The above expression yields for the transverse tree level correlators

Iπq0ij = Ipq0ij , Iππ0ij = Ipp0ij and Iπqq0ijk =

δ

δπi

δ

δqj

δ

δqkI0

∣∣∣∣π,Ω,q=0

= Ipqq0ijk, (12.16)

as well as analogously for the longitudinal expressions

IΩq0ij = Iqp0jlXli, IΩΩ

0ij = XliIpp0lkXkj = −XliδlkXkj = −∇2jδji (12.17)

85

and

IΩqq0ijk =

δ

δΩi

δ

δqj

δ

δqkI0

∣∣∣∣π,Ω,q=0

= −Ipqq0ijk∇iδii′ . (12.18)

The Gaussian integration over π yields

S[q,Ω, Jπ, JΩ, Jq] = S0 +12Jπi J

πi + Jπi

δS0

δqi+ IqΩ0jiΩiqj (12.19)

+12ΩjI

ΩΩoij Ωi +

12IΩqq0ijkΩiqjqk + JΩ

i Ωi + Jqi qi

where S0 is given by Eq. (62) and J i are the corresponding sources associated to the mo-mentum fields. Similarly, the subsequent Gaussian integration over Ω gives

S[q, Jπ, JΩ, Jq] = S0 +12Jπi J

πi + Jπi

δS0

δqi+ Jqi qi −

12JΩi

(IΩΩ0

)−1

ijJΩi −JΩ

i

(IΩΩ0

)−1

ij

×(IqΩ0ujqu +

12IΩqq0jukquqk

)− 1

2IqΩ0ji

(IΩΩ0

)−1

ilIqΩ0mlqjqm (12.20)

− 12IqΩ0ji

(IΩΩ0

)−1

ilIΩqq0lmuqjqmqu −

18IΩqq0imu

(IΩΩ0

)−1

ijIΩqq0jklqmquqlqk.

By collecting the terms of the same order in q we obtain the bare vertices in the secondorder formalism in terms of those that arise in the first order formalism. Thereby, themaster equation for the momentum propagator Eq. (9.43) becomes in the case of the Ω-field

∆ΩΩij =

(IΩΩ0

)−1

il

[δlm + i

(IqΩ0ulqu +

12IΩqq0lkuqkqu

) (IqΩ0smqs +

12IΩqq0msdqsqd

)] (IΩΩ0

)−1

mj

whereas the corresponding mixed version Eq. (9.45) is given by

∆qΩij = −∆qq

il

δ

δql

(IqΩ0umqu +

12IΩqqmkuqkqu

) (IΩΩ0

)−1

mj. (12.21)

Here the q fields in Eqs. (12.21) and (12.21) again have to be replaced by

q → q[Jq] +~i∆qq[Jq]

δ

δq. (12.22)

The corresponding equations involving the π field are identical to Eqs. (9.43) and (9.45).Performing the same algebraic steps as in sect. 11 yields the corresponding decompositionof the Hamilton propagators in Coulomb gauge QCD. The result is displayed in diagram-matic form in Fig. 12.1, it constitutes the main result of this section. We have also checked

86

= −i , = −i

= −i −

−i −i − −

−−− +−

Figure 12.1: Decomposition of the proper 2-point functions of Coulomb gauge QCD inthe first order formalism in terms of the corresponding correlation functions of the sec-ond order representation. The spatial (A) and temporal (σ) gauge fields are representedby solid respectively dotted lines whereas the corresponding transverse (π) andse longitu-dinal (Ω) momenta by zigzag respectively wavy lines. The equations for the longitudinalmomentum propagator are identical to the those for the transverse component and givenby replacing zigzag by wavy lines.

the structure of these expressions by an explicit projection of the general equations on thecorresponding momentum components in appendix B.4.

Next we will consider the other direction of the connection discussed in Sect. 10, theone which expresses proper Lagrange correlators in terms of Hamiltonian ones. This isinteresting in the context of Coulomb gauge QCD, since in the Hamilton framework acomplete proof of the renormalizability of the theory seems possible due to explicit cancel-lations ensured by powerful Ward identities [41]. Here again it is the particular definitionof the longitudinal momentum that complicates the issue and does not allow to give theseequations explicitly. Nevertheless, one can easily convince oneself that these equations aregiven entirely by tree graphs that introduce no additional divergences. Therefore, oncethe renormalizability of the theory in the first order formalism is established, the corre-sponding connection immediately implies renormalizability of the theory in the Lagrangeframework. Yet the cancellation mechanism of arising divergences might be far from ob-vious in the latter framework and could nevertheless prevent simple truncation schemes.

87

WH Aj πj σ Ω λ τ

Ai tij −ΛAA

(k20−k2)

tij −ik0ΛAπ

(k20−k2)

0 0 −ikik2 0

πi tij ik0ΛAπ

(k20−k2)

tij −k2Λππ

(k20−k2)

0 0 0 −ikik2

σ 0 0 −Λσσk2

ΛσΩk2

−ik0Λσλk2 0

Ω 0 0 ΛσΩk2

ΛΩΩk2

−ik0Λλk2

−1k2

λikj

k2 0 ik0Λσλk2

ik0ΛΩλk2 0 0

τ 0 ikj

k2 0 −1k2 0 0

Table 12.2: General form of propagators in momentum space. The global color factor δab

has been extracted. All unknown functions Λφφ are dimensionless, scalar functions of k20

and k2. Here tij = δij − kikj/k2 is the transverse projector in momentum space.

12.3 Relating the renormalizability in Hamilton and Lagrange Cou-lomb gauge QCD

In the previous subsection we have derived general connections that give Coulomb gaugeGreens functions in the first order formalism in terms of those in the second order formal-ism and vice versa. These general connections will allow to show the following statement:

Coulomb gauge is renormalizable in both formalisms if it is renormalizable in either one of them.

The renormalizability in the second order formalism is trivial if the first order formalismproves renormalizable since there are no loop graphs in the connection, as has alreadybeen noted.

In order to prove the other direction we assume that the Green’s functions of the sec-ond order theory have been properly renormalized. As the following statements hold forboth πa and Ωa we adopt the notation where p stands for both of these fields. We will showthat the explicit expressions for the first order Greens functions given by the connectionsFig. 12.1 and the respective ones for higher order Greens functions are likewise renormal-izable. As a first step we show that there are no energy divergences in these connections.Energy divergences arise from loop graphs whose integrands are energy independent. Inthe case of the propagators given in Fig. 12.1, all loop graphs involve dressed spatial gluonpropagators that according to table II are energy dependent. Without cancellations of thecorresponding propagators these integrals therefore do not feature energy divergences.Actually this holds not only for the propagators but for the corresponding connection of

88

arbitrary correlation functions in the first order formalism. To see this it is useful to con-sider the general structure of the loop graphs arising in these connection equations. Theloop structure of these equations is determined by Eq. (9.40) which describes arbitrarypure momentum correlation functions. Mixed correlation functions are obtained from thisequations by further derivatives with respect to the fields which correspond to attachingadditional external legs to the graphs but do not alter the loop structure. The terms in-volving the momentum sources do not involve any summation and only yield the treelevel correlators on the classical action. In Coulomb gauge Yang-Mills theory, the arisingderivative of the bare action takes the form given in Eq. (10.7), i.e. it involves bare qp-and qqp-vertices where the external momentum leg is attached. The subsequent replace-ment of Eq. (12.22) in Eq. (9.40) produces a series of graphs. This latter replacement termin particular involves a summation (over the index b in Eq. (9.40)) that can lead to loopintegrals.

As discussed in detail in [97] the field derivatives can either act on fields, propaga-tors, or dressed vertices that were created before. Acting on a field simply attaches theassociated propagator to the corresponding leg of the respective bare vertex. Acting on adressed propagator splits it in two, inserts a dressed 3-point vertex and attaches the asso-ciated propagator to it. Finally acting on a dressed vertex attaches an additional leg andconnects the associated propagator to it. Correspondingly, these individual rules effectmerely that the dressed propagators in Eq. (9.40) are connected in all possible ways to thebare vertices or to each other via dressed vertices. All disconnected contributions vanishanalogous to Eq. (11.10). With this graphical picture in mind it is easy to establish a fewgeneral properties of arbitrary connection graphs. Each external momentum leg is eitherattached indirectly via a tree level pq-mixing term and a field propagator to the residualgraph or directly to a bare pσA-vertex. In case the momentum leg is indirectly attached toa loop graph this is done via a dressed field-vertex. Loop graphs arise if a propagator isconnected to the bare vertex it started from or if a chain of connected propagators returnsto its starting point. Therefore, the number of loops in a graph is at most the number of itsexternal momentum legs. Moreover, each loop graph includes at least one bare pσA-vertexand correspondingly each loop graph also contains at least one spatial gluon propagatorwhich according to table II is manifestly energy dependent.

Now let use exclude the possibility of cancellations. Since energy divergences arisefrom the unsuppressed integration over all modes they arise from the UV part of the loopintegral. In the UV limit, however, the dressed proper correlators arising in the loop graphstake due to asymptotic freedom up to possible logarithmic corrections their bare form.Therefore, the perturbative pole of the transverse gluon propagator had to be cancelled bya corresponding factor in a vertex to remove the energy dependence of the propagator andrender the loop energy divergent. Yet, contributions from the UV regime of graphs involv-ing other than primitively divergent vertices vanish, whereas the bare form of the primi-tively divergent vertices, given in Eqs. (12.11) - (12.11), does not involve a corresponding

89

factor that could cancel the bare propagator pole. Correspondingly it is clear that therecannot be cancellations that would remove the energy dependence of the integrand in theUV regime and so the loop integrals do not involve any explicit energy divergences. Allloop integrals are thereby - aside from usual UV divergences that have to be renormalized- finite and well defined.

On dimensional grounds only propagator and 3-point functions involving momentumfields can feature ordinary UV divergences. These primitively divergent correlators are theqp- and pp-propagator in Fig. 12.1 and the qqp-vertices. Note first that they do not featurepower-law divergences. This is trivial for graphs where the external momentum fields areall directly attached to the loop via a bare vertex, since the corresponding bare vertices aremomentum independent. According to the above rules there is also the possibility thatexactly one external momentum field is attached indirectly via a dressed 3-gluon vertex toa loop graph in the connection of one of the primitively divergent correlation functions. Ifthe corresponding external momentum is transverse the 3-gluon vertex is proportional tothe energy and the linear divergent term vanishes in the symmetric integration. Similarlyif the external momentum is longitudinal the corresponding 3-gluon vertex is proportionalto the momentum and the transversality of the gluon propagator cancels this contribution.Correspondingly there is no linear divergence as should be the case for a gauge theory. Theloop integrals appearing in the expressions for the primitively divergent momentum prop-agators can surely feature ordinary logarithmic UV divergences. Moreover, there could inprinciple be overlapping logarithmic divergences in these up to 3-loop graphs. Yet, theimportant point is that there are dedicated additional counterterms in the first order La-grangian that allow to cancel any divergences explicitly by defining renormalized quan-tities. Finally there could in principle be overlapping divergences in subgraphs of higherorder n-point functions, but due to the absence of energy divergences and the explicitrenormalizability of all primitively divergent correlation functions we do not expect thisto be a real issue. Clearly, a rigorous analysis of this problem requests to study the associ-ated Slavnov-Taylor identities (derived in the last subsection). We finally note that since allconnected Greens functions are finite after the renormalization process the correspondingproper Greens functions related via matrix inversion are likewise finite.

12.4 First order formalism and BRS-invariance

Next we consider the functional symmetry identity. The canonical action I0 is BRST-invariant, i.e. invariant under

δAa = 1gD

acccδλ, δσa = −1gD0acccδλ, δca = 1

gλaδλ,

δca = −12 fabccbccδλ, δpa = fabccbδλ [(1− α)pc − αEc] , δλa = 0.

(12.23)

90

Here δλ is a Grassmannian infinitesimal parameter, whereas D0ac = δac∂0 + gfabcσb. Onthe other hand Ea = −∇σa−D0acAc is the chromoelectric field. Note in addition that α issome color-singlet constant which in general could be some function of position.

By considering the above transformation the Slavnov-Taylor identity reads

0 =∫D[φ]

∫d4x

−1

gρaD0abcb +

1gja ·Dabcb − 1

gλaηa − 1

2fabcηacbcc

+ fabccb [(1− α)pc + αEc] · Jp

exp iI0 + iIs , (12.24)

whereIs =

∫d4x

ρaσa + ja ·Aa + caηa + ηaca + pa · Jap

.

Employing a procedure analogous to the one used in Sect. 9.4 we obtain

0 =∫D[φ]

∫d4x

−1

gρaD0abcb +

1gja ·Dabcb − 1

gλaηa − 1

2fabcηacbcc

+ fabccb[(1− α)Jcp −Ec

] · Jp

exp iS0 + iSs , (12.25)

where D[φ] denotes the remaining integration measure of the fields c, c,A and σ. Ex-pressed in the field strength tensor Faµν = ∂µAa

ν − ∂νAaµ + gfabcAb

µAcν one has

S0 =∫

d4x−1

4FaµνFaµν − λa∇ ·Aa − ca∇ ·Dabcb

Ss =∫

d4xρaσa + ja ·Aa + caηa + ηaca − Jap ·Ea + 1

2Jap · Jap

.

(12.26)

In addition we decompose the source Jpa = Jπa − ∇−∇2J

Ωa into transversal and longi-

tudinal components. Employing this decomposition the Slavnov-Taylor identity can bewritten as

0 =∫D[φ]

∫d4x

−1

gρaD0abcb +

1gja ·Dabcb − 1

gλaηa − 1

2fabcηacbcc (12.27)

+ fabccb(1− α)[Jπc · Jπa +

∇−∇2

JΩc

∇−∇2

JΩa

]− fabccb

[EcT · Jπa −Ec

L

∇−∇2

JΩa

]

× exp iS0 + iSs ,

where

EciT = tijEic and EciL =∂i∂j

∇2Ejc. (12.28)

This completes the application of the developed formalism to Coulomb gauge Yang-Millstheory.

91

Chapter 13

Summary

We conclude this part with our main result that given a quantum field theory in the con-text of the first order formalism it is possible to decompose all Green’s functions in termsof those obtained from the second order formalism and vice versa. We have discussed theconnection between the Hamilton and the Lagrange formalism for a general quantum fieldtheory and illustrated the detailed structure of the arising relations in the important caseof a generic four-dimensional renormalizable field theory. Whereas proper Lagrange cor-relation functions are given explicitly to all orders and involve only tree graphs involvingdressed Hamilton correlation functions, the decomposition of Hamilton n-point functionsinvolves loop graphs of loop order n. Although the structure of the latter equations seemsto be somewhat cumbersome, they are still more compact and simple than the usual DSEswithin the Hamilton formalism.

In accordance with the obtained equations we have argued that in theories where thequantum average of the momentum fields is completely determined as p = ˙q the properp−propagators receives quantum corrections only via wave-function renormalization. Ad-ditionally, it has been shown that the canonical momentum fields which can be definedfrom the Effective Action and those given by the quantum average of p are in general dif-ferent.

Clearly, the determination of Green’s functions of Yang-Mills theories in the first orderformalism is considerably more complicated compared to scalar or Abelian gauge theo-ries. The presence of a coupling between the gauge fields and their time derivative as wellas the subtleties of gauge fixing present major complications. In particular the renormaliz-ability still poses a major challenge. The presented general connection of arbitrary Greenfunctions in the two formulations allows to relate the strucure of arising divergences in thedifferent formulations. In particular, due to this relation the following important statementhas been derived: Coulomb gauge is renormalizable in both formalisms if it is renormalizable ineither one of them. This is useful since a proof of the renormalizability of the theory seems

92

more feasible in the first order formalism where energy divergences explicitly cancel.

93

Part III

The boost operators inCoulomb-gauge QCD and the

relativistic form factor

94

Chapter 14

Introduction and overview

Form factors traditionally encode the structure of a composite target as accessible fromelastic reactions. For a scalar target, we can denote the elastic form factor simply as F(Q2),where Q2 = −q2, q = p − p′. It is conventionally normalized as F(0) = Q, the particle’scharge. For a non-relativistic particle with unit charge (Q = 1, q0 ' 0), an expansionaround zero momentum transfer allows for a physical interpretation of the form factor interms of its rest frame charge density ρ(r), given by

F(Q2) = 1− 13!〈r2〉ρQ2 +

15!〈r4〉ρQ4 +O(Q6). (14.1)

Here we used the notation of Ref. [104]. In hadron physics one often quotes also the cur-vature of the form factor [105] via

F(Q2) = 1− 16〈r2〉Q2 + C π

VQ4 +O(Q6). (14.2)

Comparing with Eq. (14.1) we see that C πV = 1

5!〈r4〉.Beyond the charge normalization

∫d3rρ(r) = 〈1〉ρ = 1, the derivative at the origin

provides, in non-relativistic quantum mechanics, the target charge radius

〈r2〉 = −6dFdQ2

∣∣∣∣Q=0

. (14.3)

It is known from nuclear physics, but sometimes ignored in the hadron physics commu-nity, that the non-relativistic interpretation should be modified as effects of boosting thepion wave function begin to appear. Since the pion is such a light target, 1/m2

π correctionsare even larger than the actual 〈r2〉 measured experimentally or computed on a lattice.

The situation is even worse for the curvature, that has been the object of our recentfocuse. Using both chiral perturbation theory and dispersion relations, we have found a

95

Q^2 (GeV^2)0.02 0.04 0.06 0.08 0.1 0.12 0.14

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

Q^2 (GeV^2)0.02 0.04 0.06 0.08 0.1 0.12 0.14

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

Figure 14.1: Linear and parabolic fit to the elastic pion form factor at small momentumtransfer. The linear fit has χ2/dof ' 77/32. The parabolic fit has χ2/dof ' 22/32.

reliable value for the curvature of 4.0 ± 0.5 GeV−4. Many other authors quoted in [106]have also dedicated time to extracting this curvature. It behooves us to examine whetherthis number –and the more common 〈r2〉, have any interpretation in terms of the targetwavefunctions. We show, indeed, that this is far from trivial. In Fig. 14.1 we replot theform factor data from the NA7 experiment at CERN [107], omitting the few points at highermomentum transfer (as they have large error bars that might still be underestimated). Onecan then obtain a direct fit that is sensitive to the ``square radius” and curvature. From theparabolic fit, one can derive F′(0) = −2.017 ± 0.047 GeV−2, or in units of square radius,〈r2〉 = 0.48± 0.01 fm2. The curvature is less well determined, C π

v ' 8± 1 GeV−4. Addinga small systematic error to the data the uncertainty in the curvature could be as high as2 GeV2, still somewhat higher than the result obtained from the Omnes representationquoted above.

Several works in atomic and nuclear physics have pointed out the necessity of 1/mcorrections when talking about the charge radius of a composite target as extracted fromelectron scattering or from the isotope shift [108], for example the so called Darwin-Foldyterms [109]. In general much of the literature features idle discussion about where the var-ious effects are to be adscribed, to the radius, or to its extraction from electron scattering,or to final state rescattering, etc. In the end, a lot of the discussion is linguistic. In nu-clear physics one is not quite yet in possession of a full Hamiltonian for the few-nucleon

96

problem, so the corrections are worked out on a semiempirical basis [110] and in atomicphysics the corrections due to both relativity and interactions are small, and the first orderones can be accessed by means of simple quantum–mechanical reasoning, although someworks exist providing first–principles computations [111].

In hadron physics however, the situation is radically different. Corrections are large,since energies involved match particle masses (the most salient example being the pion)and interactions strong. In addition one is in possession of the Hamiltonian of the fun-damental theory, QCD, and in this article we seek the first brick in the direction of gain-ing understanding of form factors in terms of the canonical quantization of the theory inCoulomb gauge.

Much is known already in light–front [112] and point–form quantization [113, 114],with recent insight from AdS-QCD [115]. However a good discussion of boost operatorsof QCD in a phenomenological context is nowhere to be found in the literature for canoni-cal, Coulomb gauge quantization. In related quark models there has been indeed interest-ing work within the Bakamjian-Thomas construction [116]. Here we attempt to continuestimulating the discussion and bring it a step closer to QCD.

On the other hand the KLOE collaboration [117, 118] has proposed to employ their ex-cellent data set of radiative φ decays to pseudoscalar mesons, φ→ γη, φ→ γη′, to measurethe gluonium content of the η′ as well as the pseudoscalar mixing angle. The theoreticalassumptions behind the analysis are simple considerations about the flavor structure ofthe couplings of gluonium, quarkonium, etc.

While there is no objection to the model analysis of the pseudoscalar mixing angle [119],we believe that a point has been missed by the community in what regards the gluoniumcontent of the η′. This is the fact that the Fock–space expansion for either of the η or η′

|η〉 = c1|qq〉+ c2|gg〉+ c3|qqg〉+ . . . (14.4)

is dependent on the reference frame. When one makes an assumption about the gluonium(gg and larger number of gluons) content of a meson, one is presumably referring to therest-frame of the meson, since the concept of quantum mechanical state requires a quanti-zation surface that is presumably taken to be t = 0 in the rest frame. But, although the η′

meson is produced non–relativistically in φ radiative decays, the analysis hinges also ondecays to the η meson, and this is produced with a velocity v ' 0.55.

As we will show, even if there was no gluonium at all in either of the two mesons, oneshould expect a gluonium content of the η meson boosted to such reference frame of ordersinφPv

2/2 ' 0.1, given that the pseudoscalar mixing angle seems to be close to 38 degrees.This is the well-known effect that the boost operators involve interactions and change theparticle content.

This is not necessarily a fatal flaw of the analysis, but a call of attention that, whileeffective hadron Lagrangians are explicitly Lorentz–invariant, such Fock–space decompo-

97

sitions are tied to a reference frame where canonical QCD quantization is carried out, andthis needs to be specified and consistently handled.

The necessity of specifying a reference frame will become more accute as the naturalnext step in the analysis [120, 121] will be to employ the data basis for radiative J/ψ decaysaccumulated by BES and others, and given the mass of 3097 MeV of the charmoniumground state, the velocity of the produced mesons will now be decidedly relativistic. ForJ/ψ → γη, vη = 0.94, and for J/ψ → γη′, vη′ = 0.83, that are indeed significant.

The rest of this part is organized as follows.

In Chapter 15: we motivate the necessity of constructing the boost operators with a care-ful setup of the form factor for a scalar target. Then in subsection 15.2 we particularizeto the case of the pion, the lightest hadron where relativistic corrections are most sig-nificant. Subsection 15.3 is then dedicated to dispelling some misconception about thecharge radius interpreted in the Breit frame.

In Chapter 16: the Poincare generators in Coulomb gauge QCD are derived. We presenta relatively easy derivation based on modern functional methods in subsections 16.1,16.2 and 16.3.

In Chapter 17: we present the application of these boost operators to φ → ηγ (or similardecays) where it has been until now ignored

and Chapter 18 presents our final remarks.

98

Chapter 15

The relativistic Pion form factor

15.1 Relation between the Pion form factor and the boost ope-rator

The form factor can be expressed, in non–relativistic normalization for the charge density,as

out〈p′|jµ|p〉in =(p + p′)µ

2√

EE′F(q2). (15.1)

The energy-momentum pµ = (mπ,0) corresponds to a pion at rest and time t = −∞ beforethe momentum transfer by the virtual photon, q2, boosts it to a frame with p′.

The charge density is taken at t = 0, that we employ as quantization surface to de-fine the equal-time commutation and anticommutation relations for field operators. Toundertake any computation one needs to propagate the initial state |p〉in to t = 0. Thepropagators that accomplish this operation are usually called Moeller operators in the in-teraction picture, Ω− = U(t = −∞, 0). Likewise the propagation of the final state to thequantization surface is carried out by Ω†+ = U†(t = 0,∞). The product of both operatorswould reconstruct the S matrix, Ω†+Ω− = S, but here an insertion of the current operatoroccurs at time t = 0, see Eq. (15.3) below.

The final–state pion is boosted, and to be able to use its wavefunction in the rest frame(where the charge density is defined) one needs to employ the boost K operator from QCD

in〈p′| =in 〈p|e−iK·ζ (15.2)

that we construct in section 16 below. In terms of these operators the form factor can beexpressed via

in〈p|e−iK·ζΩ†+jµ(0)Ω−|p〉in =

(p + p′)µ

2√

EE′F(q2) (15.3)

99

(the Moller operators introduce interactions both sides of the current insertion). An alter-native form of Ω†+jµ(0)Ω− is for example T

(jµ(0)ei

Rd4xLI

).

The interpretation of the form factor in non-relativistic quantum mechanics (where itcomes about in the Born approximation by expanding the Fourier transform of the poten-tial causing the scattering)

FNonRel(|q|2) = Q− 13!|q|2〈r2〉+

15!|q|4〈r4〉 (15.4)

in terms of charge radii, is only a limiting case that can be recovered from Eq. (15.3) andwe will do it shortly.

Let us now obtain an equivalent q2/m2π expansion of Eq. (15.3) in powers of the mo-

mentum transfer to match Eq. (15.4). The rapidity parameter in Eq. (15.2) depends on thetransferred momentum through

v = tanh ζ ⇒ ζ =12

log(

1 + v

1− v

). (15.5)

From the above equation we can approximate

ζ ' v +v3

3+ . . . O(v5) with v2 ' − q2

m2π

− 34

q4

m4π

. (15.6)

Expanding also E′ in terms of q, and making q ∝ nz (nz = kz/|kz|) infinitesimal so thatone can truncate the Taylor expansion of the boost

exp(iKzζz

)= 1 + iKzζz − 1

2K2zζ

2z −

i

3!K3zζ

3z +

14!

K4zζ

4z + . . . (15.7)

we obtain

F(q2) =2 (EE′)1/2

(p + p′)0〈p|e−iKzζzΩ†+j0(0)Ω−|p〉 (15.8)

'(

1 +q2

2m2π

+q4

32m4π

)〈p|

(1 +

K2z

2m2π

q2 +K2

z

24m4π

q4 +K4

z

4!m4π

q4

)Ω†+ j0Ω−|p〉

(terms odd in Kz are absent for a scalar target because of z → −z reflection symmetry). Wethen have

F(q2) ' F(0) + q2F′(0) +12!

q4F′′(0) + . . . (15.9)

100

where the derivatives are now

F(0) = 〈p|Ω†+j0Ω−|p〉,

F′(0) =1

2m2π

F(0) +1

2m2π

〈p|K2zΩ

†+j0Ω−|p〉

F′′(0) =1

12m4π

〈p|K4zΩ

†+j0Ω−|p〉 − 25

48m4π

F(0) +7

6m2π

F′(0).

(15.10)

15.2 No square radius interpretation for the pion

To reproduce Eq. (15.4) , one first invokes the impulse approximation, by which the quarksfrom the jµ current density directly enter the state’s wavefunction, in effect neglectingthe Moller operators (rescattering). This is known from nuclear physics to be a poor ap-proximation, and many works in hadron physics have also lifted it [122] since it fails, forexample, to reproduce simple instances of Vector Meson Dominance.

In a second step, one takes the non-relativistic limit by sending the mass denominatorsmπ → ∞, and substitutes the Lorentz boost by its Galilean equivalent. Since a Galileanchange of reference frame is generated by U[v] = exp (−imπ r · v), the relevant boost op-erator in quantum mechanics becomes K → KGalilean = −mπ r. Upgrading it to non-relativistic field theory for quarks, and summing over spin, color and flavor, the boostoperator is then

KGalilean =∑

`

∫d3x q`†(x)(−mq`

x)q`(x). (15.11)

(In terms of quadrispinors including antiparticles, there would be an additional Dirac betamatrix (−mqj)β∂/∂ki in the operator).

In this case, one obtains for the form factor a non-relativistic expression in terms ofthe wavefunction of the target. If the target |H〉 is taken as a two-body scalar with non-relativistic wavefunction f(r), and two particles of mass m1, m2 in the center of massframe, with relative position r, and opposite charge (as corresponds for example to thenon-relativistic Hydrogen atom or the neutral π0 in the quark model)

〈r2〉 = e∫

d3xr2|f(r)|2 m1 −m2

m1 + m2. (15.12)

However, returning to Eq. (15.10) for the case of the pion, 1/2m2π ' 26 GeV−2 is not

negligible in any sense against −F′(0) ' 2 GeV−2 taken from the experimental dataquoted above. There need to be large cancellations among the two terms in the secondof Eq. (15.10). Its second non-constant term, that in the non-relativistic limit would corre-spond to the square radius, is therefore of order 20 GeV−2. One needs to conclude that,

101

0 100 200 300 400 500 600mπ(MeV)

0.2

0.25

0.3

0.35

0.4

0.45

0.5

<r2 >

(fm

2 )

0 100 200 300 400 500 600mπ(MeV)

-6

-4

-2

0

2

4

6

<r2 >

mod

(fm

2 )

Figure 15.1: Left panel: accepted [125] value of −6F′(0), the would-be pion square chargeradius, and lattice calculations [126] as a function of the pion mass. Right panel: the twoterms of Eq. (15.10), that are way larger, and a cancelation must occur to give the physicalform factor derivative. It is not clear that one can extract the square charge radius, exceptfor large quark masses where one could think of the lattice data in non–relativistic terms.

in the case of the pion, there is no connection whatsoever between the derivative of thespacelike form factor and the square radius of the rest frame charge distribution.

15.3 The form factor and the Breit frame

In the study of the proton form factor, the Breit frame became very popular [123, 124]. Onereason is that, for a spin 1/2 target, a separation of the charge and magnetization densitiesis possible by employing the Sachs form factors. The point that concerns us here, the pionbeing a scalar target, is that the 1/mπ terms in Eq. (15.10) for F′(0) are absent.

The Breit (or brick–wall) frame is defined as that in which the target bounces off thevirtual photon with opposite incoming and outgoing momentum. As a consequence,EBreit = E′Breit and the factor (p+p′)0

2√

EE′in Eq. (15.3) and following becomes unity. There-

fore in this frame,F(q2) =

⟨+

p2

∣∣∣Ω†+j0(0)Ω−∣∣∣−p

2

⟩(15.13)

102

and therefore

F(0) =⟨−p

2

∣∣∣Ω†+j0Ω−∣∣∣−p

2

⟩,

F′(0) =1

2m2π

⟨−p

2

∣∣∣ K2zΩ

†+j0Ω−

∣∣∣−p2

⟩,

F′′(0) =1

12m4π

⟨−p

2

∣∣∣ K4zΩ

†+j0Ω−

∣∣∣−p2

⟩+

16m2

π

F′(0).

(15.14)

In principle, the large term in F′(0) proportional to F(0) in Eq. (15.10) is absent, althoughthe interpretation of the remaining in terms of a square charge radius is still not possibledue to the interactions (Moller operators) and the necessary change of momentum (andhence frame) of the incoming and outgoing target. Moreover, the curvature in F′′(0) is stillaffected by 1/m2

π corrections. Therefore there is no clear advantage of interpretation of thelattice or experimental data in employing the Breit frame, at least in the case of the pion.

103

Chapter 16

Poincare generators of QCD

Having motivated their phenomenological importance, in this section we derive the gen-erators of the Poincare transformations in Coulomb gauge QCD. The method we followis a modern path-integral approach in the first–order formalism discussed in Part II (withthe path integration extending over phase space (q, p) [81], not only the field coordinatesq),based on the work of Zwanziger [41] and Watson and Reinhardt [46]. First we obtain theresult for Yang-Mills theory in the functional approach. Then we derive the result for theboost generators appropriate for canonical quantization, in agreement with [127]. Finallywe add the quarks to complete the boost operators in canonically quantized QCD.

16.1 The Translation Generators of a pure Yang-Mills theory

Yang-Mills theory is based on the renormalizable Lagrangian density of a spin-1, color–octet field Aa

µ(x)1, and written down in a gauge invariant manner in terms of the Maxwelltensor Faµν = ∂µAa

ν − ∂νAaµ + gfabcAb

µAcν as

SYM =∫

d4xLYM with LYM = −14FaµνF

aµν . (16.1)

The canonical stress-energy tensor conserved by translational invariance of LYM reads

T µνYM =

∂LYM

∂(∂µAaλ)∂νAa

λ − gµνLYM (16.2)

= −Fµaλ∂

νAλa +14gµνFσaλ Fσaλ . (16.3)

1The notation involved in this chapter is the same used in Sec. 12.2.

104

Associated to space-time translational invariance are the conserved Hamiltonian and Mo-mentum

HYM =∫

d3xT 00YM and PYM =

∫d3xT 0i

YM. (16.4)

In the path-integral framework the Green’s functions are generated from the functional

Z =∫DΦexp iSYM (16.5)

Φ representing generically an integral over all fields. Since we are interested in exposingthe explicit transverse gluons, the degrees of freedom of the intuitive Coulomb gauge, wedivide the Faraday-Maxwell tensor in terms of chromoelectric and chromomagnetic fields

Ea = −∂0Aa −∇σa + gfabcAbσc,

Bai = εijk

[∇jAa

k − 12gfabcAb

jAck

].

(16.6)

Coulomb gauge is fixed with the help of the Faddeev-Popov mechanism by first introduc-ing a gauge–fixing term in the action

Z =∫ DΦexp iSYM + iSFP ,

SFP =∫

dx[−ξa∇ ·Aa − ca∇ ·Dabcb

].

(16.7)

In this context we will assume that the gauge field configuration is restricted to the Gribovregion Eq. (12.8). Having this in mind, the canonical momentum associated to the trans-verse Aa fields can be introduced as an auxiliary field thanks to the following identity

expi

∫d4x

12Ea ·Ea

=

∫Dπ exp

i

∫d4x

[−1

2πa · πa − πa ·Ea

]. (16.8)

Two more auxiliary variables Ω, τ will allow dividing the chromoelectric part of theaction into the dynamical transverse vector potential of Coulomb gauge and a constrainedlongitudinal part. One needs the further identity

const =∫DΩδ

(∇ · π +∇2Ω)

=∫D Ω, τ exp

−i

∫d4xτa

(∇ · πa +∇2Ωa)

and a change of variables π → π −∇Ω to arrive to the equivalent form of the Yang-Millsaction

SYM =∫

d4x[−1

2Ba · Ba − τa∇ · πa − 1

2(πa −∇Ωa) · (πa −∇Ωa)

+(πa −∇Ωa) ·(∂0Aa + Dabσb

)]. (16.9)

105

We now perform the integral over the ghost fields in Eq. (16.7) to recover the Jacobianof the change of variables. Likewise we integrate over the Lagrange multipliers ξa and τa

to recover the delta-functions associated to the constraints, that enforce transversality ofthe physical fields (the Coulomb gauge condition ∇ ·A = ∇ · π = 0). Thereafter one canset to zero any terms involving these divergences in the action.

Taking these details into account Z can be expressed as

Z =∫DΦDet

[−∇ ·Dδ4(x− y)]δ (∇ ·A) δ (∇ · π) exp iS (16.10)

with

S =∫

d4x[−1

2Ba · Ba − 1

2πa · πa +

12Ωa∇2Ωa + πa · ∂0Aa

+ σa(∇ ·DabΩb + g%ag

)](16.11)

and %ag = fadeAd · πe the color–charge density carried by the gluons.The functional integral over σa can also be performed to yield the constraint equivalent

to Poisson’s equation in Quantum Electrodynamics in the form of a new delta–function onthe path integral, δ

(−∇ ·DabΩb − g%ag). The Coulomb instantaneous potential of QED,

1/|x − y|, inverse of the Laplacian, is generalized in QCD to the inverse of the Faddeev-Popov operator [

−∇ ·Dab]Mbc = δac, (16.12)

and the constraint can be formally solved for Ω, now taking the place of the scalar potentialσ,

Ωa = gMab%bg. (16.13)

The associated delta–function factorizes in a useful way [129]

δ(−∇ ·DabΩb − g%ag

)= Det

[−∇ ·Dδ4(x− y)]−1

δ(Ωa − gMab%bg

)(16.14)

to cancel the determinant in Eq. (16.10) yielding

Z =∫DADπδ (∇ ·A) δ (∇ · π) exp iS0 (16.15)

where

S0 =∫

d4x[−1

2Ba · Ba − 1

2πa · πa − 1

2g2%bgM

ba(−∇2)Mac%cg + πa · Aa

]. (16.16)

106

Clearly, the term πa · Aa is equivalent to pq in classical mechanics, and the remainingpart can be identified as the classical Hamiltonian of pure Yang-Mills theory

HYM =∫

HYM(x, t)d3x

HYM = 12πa(x, t) · πa(x, t) + 1

2Ba(x, t) · Ba(x, t) + 12g2%bgM

ba(−∇2)Mac%cg.(16.17)

We now take our expressions to the canonical functional quantization formalism. Atthis point π ceases to be an integration variable and must be considered as the standardclassical canonical momentum πa = ∂LYM/∂(A)a.After canonical qunatization, the trans-verse field πa becomes the momentum conjugate Πa to the transversal field Aa. This ishowever not trivial, but the problem has already been solved in the past.

Indeed, comparing this Hamiltonian with that of Christ and Lee [80], or in recent pa-pers [130, 88, 131, 132], one notices the absence of the Faddeev-Popov determinant in ourexpression. In order to introduce J [A] = Det(−∇ · D) we first note that the path integralrepresentation in Eq. (16.15) involves a Cartesian integration measure over gauge field.Let us consider, then, the corresponding quantized version of Eq. (16.17)

ˆHYM = 12

ˆΠa†(x) · ˆΠa(x) + 12 B

a(x) · B

a(x) + 1

2g2fbdeAd(x) · ˆΠe†(x)Mba(−∇2)Mac ˆ%cg(16.18)

in a Hilbert space where the scalar product of the complex-valued wavefunctionals Ψ[A]is taken as ``Cartesian”:

〈Ψ1|Ψ2〉 =∫DAΨ∗

1[A]Ψ2[A] . (16.19)

(Note that in Eq. (16.18) we have fixed t = 0).We now, introduce

Ψi[A] ≡ J −1/2[A]Ψi[A], (16.20)

which is the wavefunctional one would find if, as Christ and Lee did, one had first quan-tized in σ = 0 gauge and then transformed to Coulomb gauge, since in this variable changethe scalar product picks up the determinant Eq. (16.19)

〈Ψ1|Ψ2〉 =∫DAJ [A]Ψ∗

1[A]Ψ2[A] (16.21)

that we would have instead absorbed in the wavefunctional. However, by doing thechange of wave functional Eq. (16.20) one should at the same time preserve the expec-

tation value of observables 〈Ψ1| ˆO|Ψ2〉 = 〈Ψ1|O|Ψ2〉 with

O = J −1/2 ˆOJ 1/2. (16.22)

107

This has to be true, in particular for the Hamiltonian density. As a consequence

H = J −1/2 ˆH J 1/2,

H = 12J −1/2 ˆΠa†(x) ˆΠa(x)J 1/2 + 1

2 Ba(x) · B

a(x) + 1

2g2J −1/2 ˆ%bgMba(−∇2)Mac ˆ%cgJ 1/2.

(16.23)

Replacing the operator ˆΠa(x) by the transformed one

ˆΠa(x) = J 1/2Πa(x)J −1/2 (16.24)

we find that the quantum Hamiltonian density features the Faddeev-Popov determinant

HYM =12J −1Πa(x)J Πa(x) +

12

Ba(x) · B

a(x) +

12g2J −1%bgJ Mba(−∇2)Mac%cg. (16.25)

In this context the dynamical operators in Eq. (16.25) are Aa(x) and Πa(x) = −iδ/δAa(x),that do satisfy the equal time commutation relation

[Aai (x), Πb

j(y)]

= iδab(δij − ∇i∇j

∇2

)δ3(x− y). (16.26)

We want to remark that Πa(x) is not a Hermitian operator for the ``Curvilinear” scalarproduct Eq. (16.21). Indeed, Πa†(x) = J −1Πa(x)J 6= Πa(x) (see details in appendix C.1.)

We turn our attention to PYM defined in Eq. (16.4). Taking into account Eq. (16.3) wecan express

PYM =∫

d3xE`a(x, t)∇A`a(x.t) (16.27)

The structure of the above expression does not differ from those obtained for the case of apure U(1)−gauge theory. New here is the sum running over the color indices.

We then split the chromoelectric field into transverse and longitudinal part Ea ≡ Eatr +∇Ωa. From a classical point of view Eatr = −πa. Under these conditions

PYM = −∫

d3xπ`a(x, t)∇A`a(x, t) + ∂`Ωa(x, t)A`a(x, t)

. (16.28)

Next, we integrate by part the second term of Eq. (16.28) and use the identity ∇ ·Aa = 0.As a consequence, it vanishes identically and only the first term of Eq. (16.28) remains.The quantized version in the flat gauge field configuration takes the form

ˆPYM = −∫

d3x ˆΠ`a(x, t)∇A`a(x, t) (16.29)

108

Finally, after applying the transformation PYM = J −1/2 ˆPYMJ 1/2 and by considering Eq.(16.24) we obtain

PYM = −∫

d3xΠ`a(x, t)∇A`a(x, t) (16.30)

(in agreement with [127] and generalizing the momentum vector of electrodynamics).

16.2 The Lorentz Generators of a pure Yang-Mills theory

In this subsection we obtain the generators associated to the homogeneous Lorentz group.We again invoke Noether’s current for the field Aλ(x), (with special transformation δAλ` =1/2ωαβ (Jαβ)

λσ Aσ`)

J µαβYM (x) =

∂LYM

∂(∂µAνa)

∂λAνa(J αβ)λσx

σ − (J αβ)νςAςa

− (J αβ)µνx

νLYM

where (Jαβ)`¯ = δ`αgβ ¯− δ`βgα¯ is the vectorial representation of the Lie algebra generatorof SO(3, 1). Recalling Eq. (16.3) we can write

J µαβYM = Lµαβ + Sµαβ,

LµαβYM = xαT µβ

YM − xβT µαYM,

Sµαβ = FµβaAαa − FµαaAβa.

(16.31)

Here Lλµν is the orbital part whereas Sλµν corresponds to the intrinsic spin. Note thatJ µαβ

YM include both rotations and boosts belonging to the Lorentz group. We will start byextracting the first. This is obtained from Eq. (16.31) by fixing µ = 0 and taking the spatialpart of the remaining tensor. As a consequence the structure of the angular momentumdoes not differ from an Abelian group except by a sum running over the color indices

JYM = LYM + SYM,

LYM =∫

d3xx×

[E`a(x, t)∇A`a(x, t)

],

SYM = −∫

d3x Ea(x, t)×Aa(x, t) ,

(16.32)

where we have used Fi0a = Eai. We again decompose Ea → −πa + ∇Ωa. Terms arisingfrom the longitudinal part, ∇Ωa are canceled each other. In the end,

JYM = −∫

d3xx×

[Π`a(x, t)∇A`a(x, t)

]− Π

a(x, t)× Aa(x, t)

.

109

Next we address the boost generator in pure Yang-Mills theory, fixing µ = α = 0 in Eq.(16.31). This procedure leads to

J 00iYM = x0T 0i

YM − xiT 00YM + Fi0aσa. (16.33)

As above, Fi0a = Eai and T 00YM must be understood as the Hamiltonian density in Eq.

(16.17). Quantizing again at t = 0, the corresponding charge is

KYM = KYM + KYM,

KYM = −∫

d3xx

(12Ba ·Ba +

12πa · πa +

12g2%bgM

ba(−∇2)Mac%cg

),

KYM =∫

d3xEa(x)σa(x).

(16.34)

Returning to Eq. (16.1), the classical Euler-Lagrange equation of motion associated toσ, Gauss’s law, is a constraint imposed on the quantization (above in Eq. (16.14) we usedit in the form of the classical Poisson equation)

∇ · Ea − gfabcAb · Ec = Dab ·Eb = 0. (16.35)

Substitution of Ea = −A−Dabσb in the first term of Eq. (16.35) returns

−∇ ·Dabσb = g%ag (16.36)

where we have used the trasversality condition (∇ ·Aa = 0) and identified %ag = fabcAb ·πc.Once more we decompose the electric field into transverse and longitudinal parts

Ea = −πa + ∇Ωa (16.37)

in the middle term of Eq. (16.35) to derive the equation

Dab ·∇Ωb = g%ag (16.38)

(at this point σ and Ω are interchangeable, but we keep the distinction for another instant).Taking into account Eq. (16.37) the spin piece of the boost, Eq. (16.34), becomes KYM =K⊥ + K‖ with

K⊥ = −g∫

d3xπaMab%ag, and K‖ = g∫

d3x(∇Ωa)σa . (16.39)

We express K⊥ in a more symmetric form

K⊥ = −g2

∫d3xπaMab%bg −

g2

∫d3x%bgM

baπa. (16.40)

110

As for the longitudinal part, Eq. (16.39), substitution of solutions of Eq.(16.38) and Eq.(16.36) give

K‖ = −g2

∫d3x∇(%bgM

ba)Mac%cg . (16.41)

After integration by parts K‖ = g2∫

d3x%bgMba∇(Mac%cg). Therefore K‖ is a total derivative

and vanishes identically.Wrapping up, the combination of Eq. (16.34) and Eq. (16.40) in KYM allows us to

express it as

KYM = −∫

d3xx

(12Ba ·Ba +

12πa · πa +

g2

2%bgM

ba(−∇2)Mac%cg

)(16.42)

+g2πaMab%bg +

g2%bgM

baπa.

To match the expression of Besting and Schutte [127] it suffices to add to Eq. (16.42) avanishing term proportional to K‖: 0 = Ng2

∫d3x∇(%bgM

ba)Mac%cg, and choosingN = 1/2,so that the corresponding boost operator for a flat scalar product can be written as

ˆKYM = −∫

d3xx

(12Ba · Ba +

12

ˆΠa† · ˆΠa

)− g2

2ˆ%b†g Mba(∂`x∂`)Mac ˆ%cg

+g

2ˆΠa†Mab ˆ%bg +

g2ˆ%b†

g Mba ˆΠa. (16.43)

Making the change KYM = J −1/2 ˆKYMJ 1/2 and minding the correct ordering of thevarious, non–commuting operators we obtain

KYM = −∫

d3x

12J −1Π

aJ xΠa+

12BaxBa − 1

2g2J −1%b

gJ Mba(∂kx∂k)Mac%cg

+12gJ −1Π

aJ Mab%bg +

12gJ −1%b

gJ MbaΠa. (16.44)

Eq. (16.44) coincides with the expression derived in [127]. To derive the above expression

we have replaced the operator ˆΠa(x) by the transformed one (see Eq. (16.24)).

16.3 Inclusion of the Quark sector

To complete the discussion on QCD we include the matter sector interacting with thegauge field. This is achieved by the matter Lagragian

LMatter = ˆq`(iγµ∂µ −m)q` + gˆq`γµλa

2Aaµq` (16.45)

111

with ˆq = q†γ0 and λa/2 representing the SU(3) Gell-Mann matrices.Its contribution to the stress-energy tensor is

T µνMatter =

∂LMatter

∂(∂µq`)∂ν q` − gµνLMatter (16.46)

= iˆq`γµ∂ν q` − gµν ˆq`(iγλ∂λ −m)q` − ggµν ˆq`γµλa

2Aaµq`

The conserved charges associated to space-time translational invariance are then

HMatter =∫

d3xT 00Matter =

∫d3x

HDirac + HInt

, (16.47)

PMatter =∫

d3xT 0iMatter =

∫d3xq`†(x) (−i∇) q`(x)

where HDirac is the free Dirac Hamiltonian density:

HDirac(x) = q`†(x)(−iα ·∇ + βm)q`(x), (16.48)

whereas

HInt(x) = gq`†(x)λa

2α · Aa(x)q`(x)− gq`†(x)

λa

2q`(x)σa(x)

arises due to the interaction between the gauge field and Quarks. The last term of the aboveequation introduces the quark color charge density %aq = q`† λ

a

2 q` that must be summedto the right hand side (source) of Poisson’s equation in Eq. (16.13) and following. As aconsequence Ωa → Ωa = Mab%b with %a = %ag + %aq . The second term of Eq. (16.49) adds upto the second term of Eq. (16.25) and both together have the same functional form as thelatter but with %ag replaced %a = %ag + %aq .

The complete QCD Hamiltonian and momentum densities are the sum of both

HQCD = HYM + HMatter, (16.49)

PQCD = PYM + PMatter (16.50)

where HYM is the corresponding operational version of Eq. (16.17) with HYM given by Eq.(16.25) In turn, Eq. (16.50) combines Eq. (16.30) and Eq. (16.48).

We now turn to the angular momentum and boost operators. The conserved currentdue to quarks is obtained from the tensor

J µαβMatter(x) = xαT µβ

Matter − xβT µαMatter +

i

4∂LMatter

∂(∂µq`)σαβ q` − i

4ˆq`σαβ

∂LMatter

∂(∂µˆq`). (16.51)

112

Here ∂LMatter/∂(∂µq`) = iˆqγµ whereas ∂LMatter/∂(∂µˆq`) = −iγµq therefore

J µαβMatter(x) = L

µαβMatter + S

µαβMatter,

LµαβMatter ≡ xαT µβ

Matter − xβT µαMatter,

SµαβMatter = −1

4ˆq`

γµ, σαβ

q`.

(16.52)

By fixing µ = 0 and taking the remaining indices as spatial, we obtain the quark angularmomentum, LiMatter = 1/2εijk

∫d3x(xjT 0k

Matter − 12 q`†σjkq`), where T 0k

Matter is the momen-tum density in Eq. (16.48). Therefore

JMatter = LMatter + SMatter,

LMatter =∫

d3x

q`†(x) [x× (−i∇)] q`(x),

SMatter =∫

d3xq†`(x)(

12Σ

)q`(x)

(16.53)

with Σi = 12εijkσjk.

Following a procedure similar to that used in deriving Eq. (16.33) one obtains the boostdensity easily. The generator is obtained integrating over the (t = 0) surface Ki

Matter =∫d3x

−xiT 00Matter − 1

2 q`†σ0iq`. Therefore

KMatter = KMatter + KMatter,

KMatter = −∫

d3xx

q`†(x) (−iα ·∇ + βm) q`(x) +g2q`†(x)λaα · Aa(x)q`(x)

,

KMatter =∫

d3xq`†(x)(i2α

)q`(x)

(16.54)

with αi = iσ0i. For complete QCD including quarks, one needs to substitute again %ag by%a = %ag + %aq in Eq. (16.33), the boost generator for pure Yang-Mills theory. Finally onefinds

JQCD = JYM + JMatter, (16.55)

KQCD = KYM + KMatter (16.56)

(keeping in mind that %ag → %a = %ag + %aq).

113

We also record the (possibly Bogoliubov rotated) normal mode expansions of the dy-namical fields

Aa(x) =∫

dk√2ωk

[aa(k) + aa†(−k)

]eik·x,

Πa(x) = −i

∫dk

√ωk

2

[aa(k)− aa†(−k)

]eik·x,

q`(x) =∑

λ

∫dkeik·x

(B`

kλUkλ + D`†−kλV−kλ

),

q`†(x) =∑

λ

∫dke−ik·x

(B`†

kλU†kλ + D`

−kλV†−kλ

)

(16.57)

where ωk = |k|. In momentum space Eq. (16.26) reduces to

[aai (k), ab†j (k′)

]= (2π)3δab

(δij − kikj

k2

)δ3(k− k′). (16.58)

Transversality translates into k · aa(k) = k · aa†(k). The operators aa(k) and aa†(k) includethe gluon polarization vectors aa ≡ ∑

r=1,2 εr(k)aar(k) and the usual creation aa†r (k) andannihilation aar(k) operators (the normalization is as usual εirεis = δrs, εirε

jr = δij −kikj/k2).

Similarly, quark creation and annihilation operators satisfy the anticommutation rela-tions

Bkλ,B†k′λ′

=

Dkλ,D

†k′λ′

= (2π)3δ3(k− k′)δλλ′ (16.59)

(others zero).The renormalization of the boost generators also deserves a comment. It is clear that

radiative corrections to the classical, conformal theory are infinite and the quantum the-ory is only well defined after a scale is chosen. Regularization and renormalization infixed-gauge Hamiltonian dynamics is arduous and is being pursued independently withattention to the Slavnov-Taylor identities [133]. One should like to find a Lorentz-invariantregulator that would leave the Poincare algebra intact (which could be established by look-ing into Schwinger’s condition). Whatever strategy one adopts to regularize the Hamilto-nian, for example a lattice regularization, one can similarly apply it to the boost operator.A large part of the same, coming from xiH, is automatically regulated and requires noadditional counterterms other than those in H, as the additional power of xi lowers themass-dimension. This leaves the spin terms like K, and in principle, one should expectmass counterterms of equal or lower mass dimension than those spin terms to appear. Theissue is postponed to future work.

114

Chapter 17

φ-radiative decays

17.1 The boosted wavefunctions of decay products

We consider again the radiative decay φ → γη as an example, but it is obvious that thediscussion is general. The boosted meson with velocity v is given (in terms of the rapidityζ defined in Eq. (15.5)) by

|ηv〉 = eiK·ζ |η0〉 (17.1)

' |η0〉+ iK · ζ|η0〉 − (K · ζ)2

2|η0〉 . . .

We want to establish that a gluonium term, not taken into account in past analysis, arisesjust because of the change of reference frame, even if |η0〉 contained only purely qq config-urations. Therefore we need to see what pieces of the QCD boost operator connect the qq

and gg Fock subspaces.At linear order in v, the two spaces are indeed disconnected, since the interacting part

of K that could effect the change is proportional to the color charge densities in quarks

and gluons respectively, %q%g ∝ 1/2q†λaqfabcAb · Πc

, and upon closing the quark line,color arithmetic sets the contribution to zero since Tr(λ) = 0. One needs then to resourceto the second order term, K

2, where the square of the instantaneous Coulomb potential

contributes, as does the square of the gauge coupling to transverse gluons.We focuss on the latter

Kz = −g2

∫d3xzq`†(x)λaα · A

a(x)q`(x) (17.2)

to show the phenomenon, not because the contribution of the α · A Hamiltonian is largerthan the %% piece, but just because of simplicity: this contribution is the one with the small-est number of loops that turns out not to vanish, which we here demonstrate. An added

115

bonus is that, although formally in Coulomb gauge, the piece we calculate will be presentin other gauges since it involves only the transverse gluons. The calculational methodsare analogous to those of the estimate of the glueball width in Coulomb-gauge models ofQCD [134].

With v ' 0.55 for our particular case, ζ2

2 ' 0.15. For the rest–frame state is sufficient totake the color singlet, flavor singlet (through mixing) component of the η meson

|η0〉 = sin (φP)∫

d3k Y00(Ωk)fη0(|k|) (17.3)

×∑µ1µ2

⟨12µ1

12− µ2

∣∣∣∣ 00⟩

(−1)1/2+µ2 B†kµ1

D†−kµ2

|0〉

with color wavefunction δi1i2/(Nc)1/2, and flavor wavefunction understood to be the sin-glet projection.

Substituting the normal–mode expansion Eq. (16.57) for the fields q and Aa

in Eq. (17.2),one obtains, in terms of BCS spinors U, V, a gluonium component of the η moving withvelocity v,

|ηgv〉 = g2 ζ

2

2sin (φP)

∫dq

∫dp

∫d3k

∫d3k′ (17.4)

× ∂pz

(δ(3)(k + p− q)

)∂pz

(δ(3)(k′ + q− p)

)

×∑

λµ1µ2

1√4π

12√ωkωk′

⟨12µ1

12− µ2

∣∣∣∣ 00⟩

(−1)1/2+µ2

× fη0(|q|)V†−qλαj λ

a

2Upµ1V

†−pµ2

αj′ λb

2V−qλa

†ajk a†bj

′k′ |0〉.

The color algebra is easy since Tr(λaλb) = 2δab. The spin work however is a little moretedious. Let us adopt a definition of the spinors in terms of a BCS angle, defined as sk ≡sinϕ(|k|) = m(|k|)

E(|k|) , with E(|k|) = [k2 + m(|k|)2]1/2 and ck ≡ cosϕ(|k|) = |k|E(|k|) . The spinors

read then

Ukλ =1

21/2

[(1 + sk)1/2χλ

(1− sk)1/2σ · nkχλ

](17.5)

V−kλ =1

21/2

[ −(1− sk)1/2σ · nkχλ(1 + sk)1/2χλ

](17.6)

in terms of (bidimensional) Pauli matrices and Pauli spinors satisfying the closure relation∑λ χλχ

†λ = 12×2.

116

The spin combination needed yields

4∑

λ

V†−qλαjUpµ1V

†−pµ2

αj′V−qλ =

∑

λ

χ†λ

(1− sp)cqσ · nqσjσ · npχµ1χ

†µ2

σ · npσj′

+ (1− sq)cpσ · nqσjσ · npχµ1χ

†µ2σj

′σ · nq

− (1 + sq)cpσjχµ1χ†µ2

σ · npσj′

− (1 + sp)cqσjχµ1χ†µ2σj

′σ · nq

χλ. (17.7)

Then we take into account that the Clebsch-Gordan coefficient in Eq. (17.4) is⟨

12µ1

12− µ2

∣∣∣∣ 00⟩

(−1)1/2+µ2 = − 1√2δµ1µ2 (17.8)

which forces a trace over the Pauli matrices. Employing cyclicity of the trace and Tr(σiσjσk) =2iεijk, and returning to the running mass m(|p|) and energy E(|p|), we have

|ηgv〉 =

v2

2(4παs) sin (φP)

∫dq dp

iεijj′

(32πNc)1/2

fη0(|q|)qim(|p|)− pim(|q|)

E(|q|)E(|p|) (17.9)

× ∂qz

(1√

2ωq−pa†ajq−p

)(−∂qz)

(1√

2ωp−qa†ajp−q

)|0〉.

where αs = g2/(4π) is the strong fine structure constant.

17.2 The gluonium content

In calculating matrix elements of the boost operator one encounters unusual features. Thepresence of xi turns, in the usual momentum representation, in a derivative respect toki, that affects in the end the δ-functions from the field (anti) commutators. While morethan one such momentum-conservation functions appear, the following construction canbe found often:

∫dkF(k)δ(k)∂2

kδ(k) that we at present ignore how to handle. If we ignorethe term with second derivatives acting on the Dirac delta functions, we can complete therest of the calculation by the usual per-parts integration method that systematically trans-fers derivatives from the δ-functions to the other functions being integrated. To proceed

117

we write the overlap as

〈ηgv |ηg

v〉 = (4παs)2(ζ2

2

)2

sin2 (φP)∫

dq dp dq′ dp′1

16πNc

εijlεi′j′l′

2EpEqEp′Eq′

× f∗η0(|q|)fη0(|q′|)[m(|p|)qi −m(|q|)pi]

[m(|p′|)q′i′ −m(|q′|)p′i′

]

×∫

d3k1d3k2d3k′1d3k′2

(2ω1ω2ω′1ω′2)

1/2∂kz1δ

(3)(k1 + q− p)∂kz2δ(3)(k2 + p− q)

× ∂k′z1δ(3)(k′1 + q′ − p′)∂k′z2

δ(3)(k′2 + p′ − q′)(2π)6δ(3)(k1 − k′1)

× tjj′(k1)δ(3)(k2 − k′2)tll

′(k2) + δ(3)(k1 − k′2)tjl

′(k1)δ(3)(k2 − k′1)

× tlj′(k2)

. (17.10)

There are several things to remark concerning this expression. First, note that tij(k) ≡(δij−kikj/k2) is the transverse projector in momentum space which ensures transversality.Since the suite of Kronecker deltas will force k1 = −k2, and due to the spin contractionwith antisymmetric εijl symbols, the terms quartic in kn will not contribute.

It is also convenient to define a shorthand notation regrouping the meson wavefunctionand the quark propagator pieces,

G(p,q) ≡ fη0(|q|)m(|p|)q−m(|q|)p

EpEq. (17.11)

Finally, as explained above,

∂kz1δ(3)(k1 + q− p)∂kz2δ

(3)(k2 + q− p)× [. . . ] (17.12)→ δ(3)(k1 + q− p)δ(3)(k1 + q− p)∂pz∂qz [. . . ]

ignoring the effect of the derivatives over other delta functions, that would normally leadto the δ∇2δ term. Due to invariance under rotations, one can also substitute ∂2/∂2

kzby the

Laplacian 1/3∇2.After isolating the δ’s, one encounters two factors of the type

19∇2

k1∇2

k2

1

2ωk12ωk2

tjl′(k1)tj

′l(k2)

)

with ∆ ≡ (q-p)/|q-p|. The above term can be handled by means of

∇2k

1

2ωktij(k)

=

tij(k)ω2

k

((ω′k)2

4ωk− ω

′′k

2− ω′k|k|

)− 2ω|k|2

(δij − 3

kikj

k2

). (17.13)

118

Since there are two insertions of the boost operator, this is a three-loop calculation. Theintegration variables can be chosen as q, p and q’. The four, p′ becomes a linear combina-tion. There are two terms with a different value of p′ in each, that we will denote with asubindex p′1 = p− q + q′and p′2 = −p + q + q′.

We can then write down our partial calculation for the overlap as

〈ηgv |ηg

v〉〈0|0〉 = (4παs)2

(ζ2

2

)2

sin2 (φP)14π

136Nc

∫dq dp dq′

S (q,p,q′)−S ′(q,p,q′)

(17.14)

with spin work S (q,p,q′) that can be obtained by

εijlεi′j′l′

2(δjj

′ − 3∆j∆j′)(δll′ − 3∆l∆l′)GiGi′ = −2G ·G′ + 3(∆ ·G)(∆ ·G′) (17.15)

and similar relations, to yield (ω ≡ ωq−p)

S (q,p,q′) =

[1ω2

((ω′)2

4ω− ω

′′

2− ω′

|q− p|

)]2

[G(p,q)·∆q−p][G(p′1,q

′)·∆q−p

]

− 2ω|q− p|2

[1ω2

((ω′)2

4ω− ω

′′

2− ω′

|q− p|

)]−G(p,q)·G(p′1,q

′)

+ 2 [G(p,q)·∆q−p)][G(p′1,q

′)·∆q−p)]

+4

ω2|q− p|4 −2G(p,q)·

G(p′1,q′) + 3 [G(p,q)·∆q−p]

[G(p′1,q

′)·∆q−p

](17.16)

and with S ′(q,p,q′) obtainable replacing G(p′1,q′) by the second solution to the momen-

tum δs G(p′2,q′). With all these factors, one can proceed to a numerical evaluation of

Eq. (17.14). Given the now explicit invariance under rotations of the integrand, the nine-dimensional integral can be reduced by choosing the z axis along p, and the x axis so thatp, q are in the xz plane. The remaining integration variables are then the three moduli|p|, |q|, |q′|, and the three integration angles θq, Ωq′ . This six-dimensional integral can becarried out with the help of VEGAS, the standard adaptive Montecarlo algorithm [135].The running quark masses m(q) and gluon (dispersive) energies ω(|k|) appearing satisfytheir respective gap equations. We take them to be solutions of the truncated versions in[136] and [137], as later work, especially in the gluon sector, has improved the analysis butnot changed the numerical results qualitatively.

Exploration of the six-dimensional numerical integral =, defined by

〈ηgv |ηg

v〉 = (4παs)2(ζ2

2

)2

sin2 (φP)= (17.17)

119

and taking into account conceivable uncertainties in the quarkonium η wavefunction,quark mass gap function, and gluon dispersive function, allows us to make an order ofmagnitude estimate (note =, after extraction of the volume factor (2π)3δ(3)(0) is a pure, di-mensionless number) log= = 2.6+0.5

−2.0. The prefactor shows the explicit dependence in thepseudoscalar mixing angle (through which the singlet component of the η arises), whichwe take at 39 degrees, the rapidity ζ ' 0.62 of the boosted η meson, and αs at some lowscale. This last value is not known with certainty, but since it is estimated to be 0.4 in τdecays, we obtain a lower bound to the prefactor of = which is 0.4.

Putting all together, we obtain

log〈ηgv |ηg

v〉 = 1.8+0.5−2.0 (17.18)

which means that 〈ηgv |ηg

v〉 is larger than 1 by somewhat more than one sigma. This is ofcourse not possible, as the sum of all coefficients of the wavefunction in Eq.(14.4)

∑i c

2i = 1,

but one has to take into account that this is only a partial calculation including only one ofthe relevant terms of the boost operator, and that model uncertainties creep in the choiceof wave and gap functions.

What one should conclude is that there is no reason why 〈ηgv |ηg

v〉 should be orders ofmagnitude smaller than one. By the size of the controled factors one could guess it to beof order 0.1. That is, we believe we can argue with certain confidence that a gluoniumcomponent of the η meson is induced by boosting it to a moving reference frame, as in φradiative decays, even if this component was not present in the rest-frame wavefunction.

120

Chapter 18

Summary

In this part of the dissertation the boost generators of QCD quantized in Coulomb gaugein the path integral formalism has been derived. These confirm the result of Besting andSchutte [127]for pure Yang-Mills theory, to which we add the fermion fields to completethe boost operator of canonically quantized QCD in Coulomb gauge.

This should call the attention of the community to the dynamical nature of these oper-ators. We have focused on two aspects of current interest for the interpretation of hadrondata in terms of their quark and gluon constituents. The first is the lack of a charge-distribution interpretation of the pion form factor. The operator matrix elements that oneobtains from such experiments are a much more complicated generalization. The secondis the fact that Fock-space expansions of hadron wavefunctions are tied to the rest frame ofthe hadron, and they change with its velocity. This is of current interest for the theoreticalunderstanding of pseudoscalar mesons produced in φ radiative decays at Frascatti.

Many more applications come to mind where theoretical interpretation is complicatedby the dynamical nature of the boost operators. However there is no known practical wayof handling them to obtain even model results. We look forward to any progress in thisdirection by lattice methods or other means. 1

1Of course, Light Front quantization or explicitly covariant approaches such as the Dyson-Schwinger equa-tions are free of this problem by construction but they have other issues of their own that need to be overcome.Coulomb gauge QCD remains an attractive avenue of thought for non-perturbative problems since the phys-ical Hilbert space makes transparent the application of the variational principle.

121

Conclusions and outlook

In this thesis it has been shown that a low energy photon acquires an anomalous magneticmoment mγ induced by a strong magnetic field |B| ∼ 1013 − 1016G. We have pointed outthat this entity is a signal of the Lorentz symmetry breaking due to the presence of an exter-nal background field. With this in mind we conclude that purely electromagnetic effects ofthe photon-spin emerge through the interaction between the virtual electron-positron pairand the external magnetic field. We have pointed out that mγ does not contribute to themagnetization but modifies the angular momentum of a free photon. The latter might berelevant in the future, since it may help to guide experiment which would assert the non-linear character of QED in a strong magnetic field. With the results concerning mγ , we areable to assert that, in the presence of an external background field there should exist analternative conserved quantity to classify a photon state which manifests the symmetriesof an anisotropic vacuum. We are now concentrating our effort to systematically get thelatter states including the mγ−effects.

Partially connected to the photon anomalous magnetic moment, we have obtained themagnetic response of the virtual photon propagation modes in a highly magnetized vac-uum. This was possible by considering the individual contribution of each vacuum polar-ization eigenmodes in the Euler-Heisenberg Lagrangian. Our results display that not allvirtual-photons react magnetically in the same form. Indeed, we obtained that, whilst onemode features a dominant paramagnetic behavior, the remaining two modes have an asso-ciated negative magnetization. We want to point out that, although the decomposition ofthe two-loop term in the Euler-Heisenberg Lagrangian was considered in a the magneticbackground it remains valid also for the electric case. This fact should allow to study therole of the vacuum polarization modes in electron-positron production in a superstrongelectric field. A detailed analysis of this issue will be presented in a forthcoming work.

Another progress achieved in this thesis concernes the connections between the Green’sfunctions in the first and second order formalisms. Functional methods have been used toderive these remarkable functional connections as well as their symmetry-related identi-ties. Additionally, a formal implementation of the diagrammatic rules within the Hamil-tonian formalism has been developed. As a demonstration of the general nature of thepresented formalism, it has been shown that the results obtained in this work can also be

122

applied to the case of theories involving auxiliary fields from a linearization of the inter-action part of the action. This yields general relations between correlation functions in thefundamental theory and the linearized form involving auxiliary fields. A major differenceis that in this case the fundamental and the auxiliary fields do not mix which simplifies theconnection considerably.

In addition, we have been able to express the transversal and longitudinal momentumpropagators arising in first order Coulomb gauge Yang-Mills Theory in terms of Green’sfunction of the Lagrange framework. Indeed, our results show that there are only threeindependent two-point dressing functions: the (transverse) spatial gluon propagator, thetemporal gluon propagator and the ghost propagator. In particular, due to this relationthe following important statement has been derived: Coulomb gauge is renormalizable inboth formalisms if it is renormalizable in either one of them. This is useful since a proof ofthe renormalizability of the theory seems more feasible in the first order formalism whereenergy divergences explicitly cancel.

The outlook for future research in the context of Coulomb gauge Yang-Mills theoryis promising since the obtained connections should help to understand the more intricatecancellation mechanism and thereby how to explicitly renormalize the theory in the secondorder formalism, where actual analytic or numerical calculations are considerably simpler.The most direct continuation of this work is to consider a potential connection between the``Gribov-Zwanziger” scenario of confinement and Green’s functions involved in the the-ory. This is typically investigated with the help of an infrared analysis. Unfortunately, thelatter analysis has not been successful within a range of truncation schemes of Coulombgauge QCD in the Hamilton formalism. To profit from the experience gained in Landaugauge the relation between the functional equations of the Hamilton and Lagrange formal-ism may provide the decisive clue. An analysis of these aspects will be studied in future.

Finally, in the last part of our work, the boost operator of Coulomb gauge QCD wasderived making use of the some results of the path-integral method in the first order for-malism. Its application on the pion states allowed us to remark that the meaning of pionsquare charge radii is completely blurred out as consequence of field interactions. In addi-tion, we have shown that the expansion coefficients of the η meson in Fock space dependon the reference frame. This fact asserts that the boost and the particle number operatorsdo not commute. However, the dynamical nature of the boost operators still awaits furtherclarification.

123

Appendix A

A.1 Asymptotic behavior of L(2)iR at large magnetic field strength

The aim of this appendix is to find the asymptotic behavior of L(2)iR for b À 1. For this

purpose, it is convenient to write

L(2)iR = L

(2)1iR + L

(2)2iR (A.1)

with the first term being given by

L(2)11R = −5αm2

12π∂L

(1)R

∂m2− αm2b2

16π3Σ1 − αm2

2πln

(bγπ

)∂L

(1)R

∂m2,

L(2)12R = −5αm2

24π∂L

(1)R

∂m2− αm4b2

32π3Σ2 − α

2πln

(bγπ

) L

(1)R +

m2

2∂L

(1)R

∂m2

,

L(2)13R = −15αm2

24π∂L

(1)R

∂m2− αm4b2

32π3Σ3 +

α

2πln

(bγπ

) L

(1)R − 3m2

2∂L

(1)R

∂m2

,

(A.2)

with

Σ1 =∫ ∞

0

dss3

e−s/b ln( sπ

)(s coth(s) +

s2

sinh2(s)− 2

),

Σ2 =∫ ∞

0

dss3

e−s/b ln( sπ

)(3s coth(s) +

s2

sinh2(s)− 4− 2s2

3

),

Σ3 =∫ ∞

0

dss3

e−s/b ln( sπ

)(s coth(s) + 3

s2

sinh2(s)− 4 +

2s2

3

).

(A.3)

124

The second term in Eq. (A.1) is defined as:

L(2)2i = −αm4b2

32π3Gi(b) with Gi(b) =

∫ 1

0dη

∫ ∞

0dse−s/bfi(s, η) (A.4)

with fi(s, η) ≡Qi(s, η)− Qi(s, η)− Hi(s, η)

.

A.1.1 Leading behavior of L(2)1iR in an asymptotically large magnetic field

In order to determine the leading asymptotic-magnetic field term of L(2)1iR , we substitute

Eq. (5.40) into Eqs. (A.2), which gives:

L(2)11R ' αm4b2

16π3

[13

ln(

bγπ

)+

518− Σ1

],

L(2)12R ' αm4b2

32π3

[(13− 4ζ ′(2)

π2

)ln

(bγπ

)− 2

3ln2

(bγπ

)+

518− Σ2

],

L(3)12R ' αm4b2

32π3

[(1 +

4ζ ′(2)π2

)ln

(bγπ

)+

23

ln2

(bγπ

)+

56

+ Σ3

].

(A.5)

Note that Σ2 and Σ3 can be expressed as

Σ2 = Σ1 + 2Σ,

Σ3 = Σ1 − 2Σ− 2bdΣdb

,

(A.6)

with

Σ =∫ ∞

0

dss3

ln( sπ

)e−s/b

[s coth(s)− 1− s2

3

]. (A.7)

To derive the second expression in Eq. (A.6) we have used the identity

s coth(s) +s2

sinh2(s)− 2 = −s3

dds

[1s2

(s coth(s)− 1− s2

3

)]

and an integration by parts.Note that Σ1 converges even without the exponential factor which approaches to 1 for

b →∞. By using MATHEMATICA code we find Σ1 ' 0.19.

125

Σ does not involve singularities in the integrands at s = 0, but would diverge at s →∞if one sets the limiting value exp(−s/b) = 1. For that reason we divide the integrationdomain into two parts:

Σ = Σ(L) + Σ(H) (A.8)

with Σ1 ' 0.19

Σ(L) =∫ T

0

dss3

ln( sπ

)e−s/b

[s coth(s)− 1− s2

3

],

Σ(H) =∫ ∞

T

dss3

ln( sπ

)e−s/b

[s coth(s)− 1− s2

3

],

(A.9)

and T an arbitrary positive number.Now, we can omit the exponential in ΣL since the resulting integral converges anyway:

Σ(L) '∫ T

0

dss3

ln( sπ

)[s coth(s)− 1− s2

3

]. (A.10)

Substantial simplification is achieved by splitting the integrand of ΣH into its parts andneglecting the exponential factor exp (−s/b) whenever is possible

Σ(H) '∫ ∞

T

dss2

ln( sπ

)coth(s)−

∫ ∞

T

dss3

ln( sπ

)− 1

3

∫ ∞

T

dss

ln( sπ

)exp(−s/b). (A.11)

For s →∞ the leading term of coth(s) ' 1. Having this in mind, we compute the integrals:∫ ∞

T

dss2

ln( sπ

)cosh(s)|s→∞ =

1 + ln(

Tπ

)

T,

∫ ∞

T

dss3

ln( sπ

)=

1− 2 ln(

Tπ

)

4T2,

(A.12)

and∫ ∞

T

dss

ln( sπ

)e−s/b ' π2

12+

12

ln2

(bπγ

)− 1

2ln2

(Tπγ

)

− ln(

Tγ

)ln (γ)− 1

2ln2

(π

γ

). (A.13)

In the latter we have neglected terms decreasing as ∼ T/b. Numerical calculation usingMATHEMATICA code gives T ' π by minimizing Σ (dΣ/dT = 0). Therefore,

Σ ' −16

ln2

(bπγ

)+ 0.28 (A.14)

126

and according to Eq. (A.6)

Σ1 = 0.19, Σ2 ' −13

ln2

(bπγ

)+ 0.75,

Σ3 ' 13

ln2

(bπγ

)− 0.04.

(A.15)

The substitution of the latter into Eqs. (A.5) yields

L(2)11R ' αm4b2

16π3

[13

ln(

bγπ

)+ 0.09

],

L(2)12R ' αm4b2

32π3

[N2 ln

(bγπ

)− 1

3ln2

(bγπ

)− 0.47

],

L(3)12R ' αm4b2

32π3

[(1 +

4ζ ′(2)π2

)ln

(bγπ

)+

13

ln2

(bγπ

)+ 0.87

].

A.1.2 Leading behavior of L(2)2iR in an asymptotically large magnetic field

The asymptotic behavior of L(2)2iR is obtained from the integral Gi. We start our analysis by

dividing the integration domain into two regions:

Gi =∫ T

0ds

∫ 1

0dη . . .+

∫ ∞

Tds

∫ 1

0dη . . . (A.16)

with T > 0. We denote the corresponding integrals by G(L)i and G(H)

i , respectively. In thelatter we replace the upper integration limit over η−variable by a parameter η0 = 1−T/s >0. In order to find the behavior of G(L)

i we write

G(L)i =

1b

∫ T

0ds

∫ 1

0dηfi (s, η) exp(−s/b) (A.17)

'∫ T

0ds

∫ 1

0dηfi (s, η) (A.18)

where we have set exp(s/b) ' 1 since fi converges within the corresponding integrationdomain. For T → 0 the behavior of the remaining integrand is

f1 (s, η) ' 1180

(η2 − 13)s, f2 (s, η) ' 1270

(η2 − 3)s, f3 (s, η) ' − 1540

(η2 + 27)s. (A.19)

127

The integrals over s and η are trivial to perform and give:

G(L)1 ≈ − 19

540T2, G(L)

2 ≈ − 2405

T2, G(L)3 ≈ − 41

1620T2. (A.20)

Let us consider now, the contributions coming from the second integration domainwhich concerns to G(H)

i . In order to this we consider the asymptotic expression of fi(s, η)for s → ∞. First of all, the asymptotic expansion of Eqs. (2.40) in powers of exp(−s) andexp(sη) produces an expansion of Eq. (2.39) in a sum of contributions coming from thethresholds, the singular behavior in the threshold points originating from the divergenciesof the s−integration in Eq. (2.39) near s = ∞ as it was developed in [21]. The leading termsin the expansion of Eqs. (2.40) at s →∞ are

(σ1(s, η)sinh s

)∣∣∣∣s→∞

' 1 + η

4e−(1+η)s +

1− η

4e−(1−η)s

(σ2(s, η)sinh s

)∣∣∣∣s→∞

' 1− η2

4,

(σ3(s, η)sinh s

)∣∣∣∣s→∞

' 2 exp (−2s) .

(A.21)

These expressions are in correspondence with the lowest threshold (n = 0, n′ = 1 orviceversa for i = 1, n = n′ = 1 for i = 3, and n = n′ = 0 for i = 2). The fact thatM(∞, η) ' 1/2 and N(∞, η) À M(∞, η) allow us to write

V(s, η) ' − 2N2(s, η)

+2 ln [2N(s, η)]

N2(s, η)' 2 ln [2N]

N2

W(s, η) ' 4N(s, η)

− 2 ln [2N(s, η)]N2(s, η)

' 4N(s, η)

(A.22)

with V(s, η)+W(s, η) ' 4N−1(s, η). Considering only the terms which decrease most slowlyas a function of s, we find

Q1(s, η)∣∣∣s→∞

' 4e−(1−η)s

(1 + η)s+

4e−(1+η)s

(1− η)s,

Q2(s, η)∣∣∣s→∞

' Q1(s, η)|s→∞ , Q3(s, η)∣∣∣s→∞

' 8 exp(−2s)(1− η2)s

.

(A.23)

128

Additionally, the most significant terms arising from Qi + Hi for s →∞ are

[Q1(s, η) + H1(s, η)

]∣∣∣s→∞

' − 23s

+16e−2s

(1− η2)s,

[Q2(s, η) + H2(s, η)

]∣∣∣s→∞

' 8e−2s

(1− η2)s,

[Q3(s, η) + H3(s, η)

]∣∣∣s→∞

' − 13s

+24e−2s

(1− η2)s

. (A.24)

The behavior of fi = exp(−b/s)(Qi − Qi − Hi) for s →∞ is find out by considering Eqs.(A.23-A.24). In fact

f1|s→∞ ' 4e−(1−η)s

(1 + η)s+

4e−(1+η)s

(1− η)s− 16e−2s

(1− η2)s+

23s,

f2|s→∞ ' 4e−(1−η)s

(1 + η)s+

4e−(1+η)s

(1− η)s− 8e−2s

(1− η2)s,

f3|s→∞ ' −16 exp(−2s)(1− η2)s

+13s.

(A.25)

Except for the last term in f1|s→∞ and f3|s→∞ , we may use the exp(−s/b) ' 1 so that

G(H)1 '

∫ ∞

Tds

∫ η0

−η0dη

4e−(1+η)s

(1− η)s− 8e−2s

(1− η2)s+

e−s/b

3s

,

G(H)2 ' 4

∫ ∞

Tds

∫ η0

−η0dη

e−(1+η)s

(1− η)s− e−2s

(1− η2)s

,

G(H)3 '

∫ ∞

Tds

∫ η0

0dη

− 16e−2s

(1− η2)s+

e−s/b

3s

.

(A.26)

129

Performing the integration over η, G(H)i are given by

G(H)1 ' 4

∫ ∞

Tds

e−2s

s[Ei(2s− T)− Ei(T)]− 8

∫ ∞

Tds ln

(2sT− 1

)e−2s

s

+23Ei

(−T

b

)− 2T

3

∫ ∞

Tds

e−s/b

s2, (A.27)

G(H)2 ' 4

∫ ∞

Tds

e−2s

s[Ei(2s− T)− Ei(T)]− 4

∫ ∞

T

dss

ln(

2sT− 1

)e−2s, (A.28)

G(H)3 ' −8

∫ ∞

T

dss

ln(

2sT− 1

)e−2s +

13Ei

(−T

b

)− T

3

∫ ∞

Tds

e−s/b

s2(A.29)

where Ei(−T/b) is the exponential-integral function whose asymptotic expansion for verylarge magnetic field b →∞ is

∫ ∞

T

dss

exp(− s

b

)'= ln

(bγπ

)− ln

(Tπ

). (A.30)

This expression is calculated with accuracy of terms that decrease with b. To be consis-tent the last integral of Eq. (A.27) and Eq. (A.29) can be neglected as well, since the termsdecrease as fast as ∼ b−1 ln b and ∼ b−1. The remaining integrals present in Eq. (A.27) andEq. (A.29) depend on the parameter T. Taking all this into account, the leading asymptoticbehavior of Gi for b →∞ reads

G1 ≈ 23

ln(

bγπ

)+ C1, G2 ≈ C2, G3 ≈ 1

3ln

(bγπ

)+ C3. (A.31)

where we have used Eq. (A.30). Here, the numerical constants C1,3 are determined byimposing the condition dC1,3/dT = 0 with

C1 = 4∫ ∞

Tds

e−2s

s[Ei(2s− T)− Ei(T)]− 19T2

540− 8

∫ ∞

Tds ln

(2sT− 1

)e−2s

s

− 23

ln(

Tπ

), (A.32)

C3 = −8∫ ∞

T

dss

ln(

2sT− 1

)e−2s − 1

3ln

(Tπ

)− 41T2

1620. (A.33)

This yieldsC1 ≈ 2.67, and C3 ≈ 0.18. (A.34)

Note that there is not value fulfilling the condition dC2/dT = 0. However, in order tocompute it, we first set d

dT

∑3i=1 Gi = 0. This condition leads to T ' 0.46 and

G =3∑

i=1

Gi ≈ ln(

bγπ

)− 1.82. (A.35)

130

Obviously, G2 = G−G1 −G3. Thus, by taking into account Eqs. (A.31) and Eq. (A.34)we obtain C2 ≈ −4.68. Such that

G1 ≈ 23

ln(

bγπ

)+ 2.67, G2 ≈ −4.62,

G3 ≈ 13

ln(

bγπ

)+ 0.18.

(A.36)

Substitution of Eqs. (A.36) in Eq. (A.4) allows to obtain

L(2)21R ≈ −αm4b2

16π3

[13

ln(

bγπ

)+ 1.34

], (A.37)

L(2)22R ≈ −αm4b2

32π3C2 with C2 = −4.68, (A.38)

L(2)23R ≈ −αm4b2

32π3

[13

ln(

bγπ

)+ 0.18

]. (A.39)

Finally, the behavior of L(2)iR presented in Eq. (5.41) is derived by inserting the expression

bellow Eqs. (A.15) and Eqs. (A.37-A.39) into Eq. (A.1).

131

Appendix B

B.1 p−integration of Z[J ].

In this appendix we perform the integration over the momentum fields in the vacuum-vacuum transition amplitude in presence of the classical sources. In order to do this, weassume a general quadratic Hamiltonian density given by Eq.(9.24). In order to evalu-ate the Gaussian functional integral over the momentum fields we first consider a finitedimensional generating functional

Z[J ] = NJ

∫ ∞

−∞

N−1∏

j=1

d4qjN∏

j=0

d4pj exp[i

~(I0 − C + J iqq

i)]. (B.1)

with the momentum field dependent part of the action

I0 = −12piAijpj +

(qi − Bi + J ip

)pi. (B.2)

Completing the squares in the above expression we obtain

I0 =12πi (−Aij)πj +

12

[qi − Bi + J ip

]A−1ij

[qj − Bj + J jp

],

where πi ≡ pi−A−1ij

(qj − Bj + J jp

). By considering the change of the integration variables

from pi to πi we obtain

Z[J ] = NJ

∫ N−1∏

i=1

d4qj exp[i

~S[q, J ]

](B.3)

where

NJ = NJ

∫ N−1∏

i=0

d4πj exp[− i

2πiAijπj

](B.4)

132

andS[q, J ]=S0[q] +

12J ipA−1

ij [q]J jp + J iqqi + J ipA−1

ij [q](qj−Bj [q]) .

Here, the action S0 can be obtained by considering the limit of S[q, J ] when J → 0.Explicitly it reads

S0[q] =12qiA−1

ij [q]qj − qiA−1ij [q]Bj [q]−

∫d4xV[q], (B.5)

whereV[q] = −1

2Bi[q]A−1

ij [q]Bj [q] + C[q]. (B.6)

The canonical momentum fields defined as

pcann ≡ δS0

δqi= A−1

ij [q](qj − Bj [q]) (B.7)

allow us to identify and substitute the last term in S[q, J ] by its respective definition bywhich we obtain the desired form of Eq. (9.26).

B.2 Proof of the general form of the decomposition of proper La-grange correlation functions

In this appendix we prove the statement made in subsection 10.2 that the replacementrules precisely generate the p-connected correlators. As usual for a statement over theintegers this is done by induction. The statement is trivially fulfilled in the case n = 3given explicitly before. Now let us assume it is fulfilled for all integers ≤ n and showthat this implies its validity for n + 1. In the p-connected diagrams with n legs we singleout the proper vertex we started the iteration with. This vertex can be connected to otherconnected clusters so that all arising graphs have the general form

n P = n−m∑n

m=0∑⌊m/2⌋

s=0∑D(mi) [

P

P

]s

ms

m1

Here we suppress all external legs and only give their number next to the correspondingvertex. The sum over s counts the connected clusters attached to the considered propervertex which in addition has n − m external legs and the implicitly given sum labeledD(mi) runs over all ways to distribute the remaining m external legs to the s indistin-guishable connected vertices (taking into account that a vertex has at least 3 legs).

133

Next we consider the attachment of an additional external leg via the rules given in Fig.10.1, applied in all possible ways. The above representation then goes over to a n+1-pointfunction of the following form

n−m

P

P

P

1

[

[

mi ] )n P →∑n

m=2∑⌊m/2⌋

s=1∑D(mi)

∑si=1 (

ms

m1

+

+∑n

m=0∑⌊m/2⌋

s=0∑D(mi) ( +

∑n−mi=1[

P

P

n−m + 1 ]

ms

m1

]n−m

P

P

PR mi + 1

ms

m1

[

P

P

2 n−m ] )

m1

ms

Here the connected cluster with n + 1 legs in the first class of graphs is labeled by theadditional index R (for reduced) since one of the legs is connected to the p-propagatorinstead to a composite external leg. The derivative of this propagator is explicitly presentvia the second class of graphs. When the new proper vertex is absorbed in the connectedcluster, according to the induction assumption (mi + 1 < n) these two terms togetherprecisely yield the full p-connected vertex with m+ 1 external legs and without the indexR. Since each connected cluster had before already at least two external legs, together withthe new one there are now at least three. Terms where besides the new external leg there isonly one other external leg at a connected cluster are explicitly given by the fourth class ofgraphs involving a new ``connected cluster” that consists only of the proper 3-point vertex.Finally, the general expression at order n+ 1 contains also graphs where the new externalleg is attached to the proper vertex itself, given by the third class of graphs. Altogether, thesum of the different classes of graphs precisely yields all necessary graphs at order n + 1,which completes the proof.

B.3 First order Dyson-Schwinger equations

In this appendix we derive the DSEs in the first order formalism directly. In order to dothis we compute Eq. (9.14) with the canonical action given in Eq. (10.1)

δΓH

δpi=δI0δpi

∣∣∣∣φ→φ+~

i∆φφ δ

δφ

=[−pi + Iqp0liql +

12Iqqp0lkiqlqk

]

φ→φ+~i∆φφ δ

δφ

. (B.8)

We can identify the sum of the last two terms inside the bracket as the quantum canonicalmomentum fields, which allows to write the above relation as Eq. (9.31).

134

The substitution of Eq. (B.8) in Eq. (9.35), allows to write the latter as

δΓH

δqi=

[δS0

δqi+

(pm− δS0

δqm

)δ2S0

δqiδqm

]

φ→φ+~i∆φφ δ

δφ

(B.9)

where

δS0

δqi= Sqq0ijqj +

12Sqqq0ijkqjqk +

13!Sqqqq0ijklqjqkql + Iqp0jlI

pq0liqj + Iqq0ijqj + Iqqp0ijmI

pq0mkqjqk

+12Iqqp0jkmI

pq0miqjqk +

12Iqqq0ijkqjqk +

13!Iqqqq0ijklqjqkql +

12Iqqp0ijmI

ppqq0mklqjqkql (B.10)

and(pm − δS0

δqm

)δ2S0

δqiδqm= Iqp0impm + Iqqp0ijmpmqj − Iqp0jlI

pq0liqj − Iqqp0ijmI

pq0mkqjqk

− 12Iqqp0jkmI

pq0miqjqk −

12Iqqp0ijmI

ppq0mklqjqkql. (B.11)

In the last two equations we have used the elementary decomposition of the bare elementsand Eq. (10.7). Plugging Eqs. (B.10) and (B.11) into Eq. (B.9) we arrive at the DSE Eq.(11.24).

B.4 Explicit projection on the individual momentum components

Defining the transverse projector tij = δij − ∂i∂j/∇2 one has πai = tijpaj and Ωa = ∇−∇2 · pa.

As no time derivative appears one can obtain ∆Aπij , ∆

ππij , ∆

σΩij , and ∆ΩΩ

ij by projecting p inthe two-point functions ∆Ap

ij , ∆σpij and ∆pp

ij .For instance to obtain ∆Aπ

ij we decode the result given in Eq. (11.3). Let us first identifythe fundamental field q with index i with Aai . We then expand the remaining sums overthe fields involved in the q’s considering all possible combinations which generate the barevertex I(pσA)

0ijk . The bosonic symmetry of the latter allows to write

∆Apij = ∆AA

il

(IAp0lj − iIσAp0ukj∆

σσumΓσAAmln ∆AA

nk

). (B.12)

Here no internal propagator like ∆Aλij appears since no proper vertex functions with λ−derivative

exist. The relation between this function and ∆Aπij is given by

∆Aπij = tjr∆Ap

ir = trj∆AAil

(IAp0lr − iIσAp0ukr∆

σσumΓσAAmln ∆AA

nk

). (B.13)

135

Obviously the case corresponding to ∆σΩij is obtained by replacing in Eq. (B.12) A → σ

where appropriate and considering the relation

∆σΩij =

∂j

∇2∆σpij =

∂j

∇2∆σσil

(Iσp0lj − iIσAp0ukj∆

σσumΓσσAmln∆AA

nk

). (B.14)

Note that the sum over r in Eq. (B.13) and over j in Eq. (B.14) must be understood overthe discrete spatial indices only.

Clearly, by demanding a similar procedure we obtain that

∆ππij = tii

′tjj

′∆ppi′j′ and ∆ΩΩ

ij =∂i′

∇2

∂j′

∇2∆ppi′j′ . (B.15)

with

∆ppij = δij −∆pq

imΓqqml∆qplj − iIpAσ0ikl ∆

AAkm∆σσ

ln IσAp0nmj − IpAσ0ikl ∆

AAku ∆σσ

lx ∆σσnyΓ

σσAAyxuz ∆AA

zmIσAp0nmj

− IpAσ0ikl ∆AAku ∆σσ

lx ∆σσnpΓ

σAqpuq ∆qq

qyΓqσAyxz ∆AA

zmIσAp0nmj − IpAσ0ikl ∆

AAku ∆σσ

lx ∆AAnp

× ΓAAqpuq ∆qqqyΓ

qσσyxz∆

σσzmI

σAp0nmj . (B.16)

Note that the sum over all field components qi in the internal progators leads to multiplepossibilities.

The Fourier transformation of these expressions read therefore

∆ab(Aπ)ij (k) = ∆ar(AA)

il (k)(ik0tlj(k)δrb − itjv(k)

∫dωIcdb(σAp)0uv (k − ω, ω,−k)

× ∆de(AA)um (ω − k)Γerh(AAσ)

ml (ω − k, k,−ω)∆hc(σσ)(ω)), (B.17)

∆ab(σΩ)(k) = ∆al(σσ)(k)(−δlb − kj

k2

∫dωIcdb(σAp)0uj (k − ω, ω,−k)∆de(AA)

um (ω − k)

× Γelh(Aσσ)m (ω − k, k,−ω)∆hc(σσ)(ω)

), (B.18)

136

∆ab(ππ)ij (k) = tij(k)δab −∆ae(πA)

im (k)Γec(AA)ml (k)∆cb(Aπ)

lj (k)− tii′(k)tjj′(k)

×i

∫dωIacd(pAσ)

0i′k (k, ω − k,−ω)∆cf(AA)km (ω − k)Iefb(σAp)0mj′ (ω, k − ω,−k)

× ∆de(σσ)(ω) +∫

dωdµIacd(pAσ)0i′k (k, µ− k,−µ)∆cn(AA)

ku (µ− k)∆df(σσ)(−µ)

× ∆ey(σσ)(−ω)Γyfnl(σσAA)uz (ω, µ, k − µ,−ω − k)∆lf(AA)

zm (−ω − k)

× Iefb(σAp)0mj′ (−ω, ω + k,−k) +

∫dωdµIacd(pAσ)

0i′k (k, ω − k,−ω)∆cl(AA)ku (ω − k)

× Γvlh(σAq)up (−k − µ, k − ω, ω + µ)∆dx(σσ)(−ω)∆ev(σσ)(µ+ k)∆hs(qq)py (ω + µ)

× Γsxt(qσA)yz (−ω − µ, ω, µ)∆tf(AA)

zm (µ)Iefb(σAp)0mj′ (µ+ k,−µ,−k)

+∫

dωdµIacd(pAσ)0i′k (k, ω − k,−ω)∆cl(AA)

ku (ω − k)∆ev(AA)zm (µ+ k)∆dx(σσ)(−ω)

× Γvlh(AAq)zup (−k − µ, k − ω, ω + µ)∆hs(qq)py (ω + µ)Γsxt(qσσ)

y (−ω − µ, ω, µ)

× ∆tf(σσ)(µ)Iefb(σAp)0mj′ (µ+ k,−µ,−k), (B.19)

∆ab(ΩΩ)(k) =1k2δab −∆ae(Ωσ)

im (k)Γec(σσ)ml (k)∆cb(σΩ)

lj (k)− kikj

k4

i

∫dω∆de(σσ)(ω)

× Iacd(pAσ)0ik (k, ω − k,−ω)∆cf(AA)

km (ω − k)Iefb(σAp)0mj (ω, k − ω,−k)

+∫

dωdµIacd(pAσ)0ik (k, µ− k,−µ)∆cn(AA)

ku (µ− k)∆df(σσ)(−µ)

× ∆ey(σσ)(−ω)Γyfnl(σσAA)uz (ω, µ, k − µ,−ω − k)∆lf(AA)

zm (−ω − k)

× Iefb(σAp)0mj (−ω, ω + k,−k) +

∫dωdµIacd(pAσ)

0ik (k, ω − k,−ω)

× ∆cl(AA)ku (ω − k)Γvlh(σAq)up (−k − µ, k − ω, ω + µ)∆dx(σσ)(−ω)

× ∆ev(σσ)(µ+ k)∆hs(qq)py (ω + µ)Γsxt(qσA)

yz (−ω − µ, ω, µ)∆tf(AA)zm (µ)

× Iefb(σAp)0mj (µ+ k,−µ,−k) +

∫dωdµIacd(pAσ)

0ik (k, ω − k,−ω)

× ∆cl(AA)ku (ω − k)Γvlh(AAq)zup (−k − µ, k − ω, ω + µ)∆ev(AA)

zm (µ+ k)

× ∆dx(σσ)(−ω)∆hs(qq)py (ω + µ)Γsxt(qσσ)

y (−ω − µ, ω, µ)∆tf(σσ)(µ)

× Iefb(σAp)0mj (µ+ k,−µ,−k) (B.20)

The bare Greens functions involving the full momentum field can now be expressed by thecorresponding expression for the individual components via the explicit expressions givenin section 12.2 which results in the equations given in diagrammatic form in Fig. 12.1.

137

Appendix C

C.1 The Fadeev-Popov Determinant and the Hermiticity of Πa(x).

Let us first recheck for the reader not familiar with functional quantization how Hermitic-ity of the canonical momentum comes about, in total analogy with elementary quantummechanics,

〈Ψ1|Πa(x)|Ψ2〉 =∫DAJΨ∗

1[A]1i

δ

δAa(x)Ψ2[A] . (C.1)

Again integrating by parts

〈Ψ1|Πa(x)|Ψ2〉 = −∫DA

[1i

δ

δAa(x)JΨ∗

1[A]

]Ψ2[A]

=∫DA

[Πa(x)JΨ1[A]

]∗Ψ2[A] . (C.2)

The introduction of a unit factor 1 = JJ −1 in front of the bracket of the above expressionleads to

〈Ψ1|Πa(x)|Ψ2〉 =∫DAJ

[J −1Πa(x)JΨ1[A]

]∗Ψ2[A] .

(C.3)

It is to notice that

〈Ψ1|Πa†(x)|Ψ2〉 = 〈Πa(x)Ψ1|Ψ2〉 (C.4)

=∫DAJ

[Πa†(x)Ψ1[A]

]∗Ψ2[A]

and we compare Eq. (C.3) with Eq. (C.4) to find that with the curvilinear scalar product,the canonical momentum is not Hermitian, Πa†(x) = J −1Πa(x)J .

138

Bibliography

[1] P. W. Higgs. ``Broken symmetries and mass generation of gauge bosons.” Phys. Rev. Lett.13, 508 (1964).

[2] E. P. Wigner. ``On Unitary Representations Of The Inhomogeneous Lorentz Group.” AnnalsMath. 40, 149 (1939). [Nucl. Phys. Proc. Suppl. 6 (1989) 9].

[3] S. Weinberg. ``The Quantum theory of fields. Vol. 1: Foundations.” Cambridge, UK: Univ.Pr., (1995), 609 p.

[4] S. Weinberg. ``Feynman Rules for Any Spin.” Phys. Rev. 133, B1318 (1963).

[5] S. Weinberg. ``Feynman Rules for Any Spin. II. Massless Particles.” Phys. Rev. 134, B882(1964).

[6] W. K. Tung, `` Relativistic Wave Equations and Field Theory for Arbitrary Spin.” Phys. Rev.156, 1385 (1967).

[7] Roman U. Sexl and Helmuth K. Urbantke, ``Relativity, Group, Particles.” Springer-Verlag Wien New York, (2001), 500 p

[8] E. Leader, ``Spin in Particle Physics” Cambridge, UK: Univ. Pr., (2001), 521 p

[9] J. Schwinger. ``On gauge invariance and vacuum polarization.” Phys. Rev. 82, 664, (1951)

[10] W. Heisenberg and H. Euler. ``Consequences of Dirac’s Theory of positron.” Z. Phys. 98,714 (1936).

[11] V. I. Ritus. ``Lagrangian of Intense Electromagnetic Field and Quantum Electrodynamics atShort Distances.” Sov. Phys. JETP 42, 774 (1976).

[12] V. I. Ritus. ``Effective Lagrange Function of Intense Electromagnetic Field in QED.” Pro-ceedings of Workshop on Frontier Tests of Quantum Electrodynamics and Physics ofthe Vacuum, Sandansky, Bulgaria.(1998). hep-th/9812124

139

[13] W. Dittrich and M. Reuter. ``Effective Lagrangians in Quantum Electrodynamics.”Springer, (1985).

[14] D. Fliegner, M. Reuter M. G. Schmidt, and C. Schubert. ``Two-loop Euler-HeisenbergLagrangian in dimensional regularization.” Theor. Math. Phys. 113, 1442 (1997).

[15] B. Koers and M. G. Schmidt. ``The effective two-loop Euler-Heisenberg action for scalar andspinor QED in a general constant background field.” Eur. Phys. J. C6, 175 (1999).

[16] E. F. Ferrer and V. de la Incera. ``Dynamically Induced Zeeman Effect in Massless QED.”Phys. Rev. Lett. 102, 050402 (2009). [arXiv:0807.4744 [hep-ph]].

[17] E. F. Ferrer and V. de la Incera. ``Dynamically Generated Anomalous Magnetic Momentin Massless QED.” Nucl. Phys. B 824, 217 (2010). [arXiv:0905.1733 [hep-ph]].

[18] S. Villalba Chavez. ``Photon Magnetic Moment and Vacuum Magnetization in an Asymp-totically Large Magnetic Field.” arXiv:0910.5149 [hep-th].

[19] S. L. Adler, J. N. Bahcall C. G. Callan and M. N. Rosenbluth. ``Photon Splitting in aStrong Magnetic Field.” Phys. Rev. Lett. 25, 1061 (1970).

[20] S. L. Adler, ``Photon splitting and photon dispersion in a strong magnetic field.” AnnalsPhys. 67, 599 (1971).

[21] A. E. Shabad. ``Cyclotronic Resonance in the Vacuum Polarization.” Letter al Nuovo Ci-mento 2, 457 (1972).

[22] A. E. Shabad. ``Photon dispersion in a strong magnetic field.” Ann. Phys. 90, 166 (1975).

[23] A. E. Shabad and V. V. Usov, ``Propagation of γ-radiation in strong magnetic fields ofpulsars.” Astrophys. Space Sci., 102, 327, 1984.

[24] A. E. Shabad and V. V. Usov. ``Photon Dispersion in a Strong Magnetic Field with Positro-nium Formation: theory.” Astrophys. Space Sci. 128, 377 (1986).

[25] A. E. Shabad. ``Interaction of electromagnetic radiation with supercritical magnetic field.”Sov. Phy. JETP 98, 186 (2004).

[26] F. Bastianelli and C. Schubert. ``One Loop Photon-Graviton Mixing in an ElectromagneticField: Part 1.” JHEP 02, 069 (2005). [arXiv:gr-qc/0412095].

[27] F. Bastianelli, U. Nucamendi, C. Schubert and V. M. Villanueva. ``One LoopPhoton-Graviton Mixing in an Electromagnetic Field: Part 2.” JHEP 11, 099 (2007).[arXiv:0710.5572 [gr-qc]].

140

[28] F. Bastianelli, U. Nucamendi, C. Schubert and V. M. Villanueva. J. Phys. A:Math. Theor. ``Photon-graviton mixing in an electromagnetic field.” 41, 164048 (2008).[arXiv:0809.0652 [hep-th]].

[29] V. P. Gusynin, V. A. Miransky and I. A. Shovkovy. ``Dynamical chiral symmetry breakingby a magnetic field in QED.” Phys. Rev. D 52, 4747 (1995).

[30] V. P. Gusynin, V. A. Miransky and I. A. Shovkovy. ``Dimensional reduction and catalysisof dynamical symmetry breaking by a magnetic field.” Nucl. Phys. B 462, 249 (1996).

[31] V. P. Gusynin, V. A. Miransky and I. A. Shovkovy. ``Large N dynamics in QED in amagnetic field.” Phys. Rev. D 67, 107703 (2003).

[32] C. N. Leung and S. Y. Wang. ``Gauge independent approach to chiral symmetry breakingin a strong magnetic field.” Nucl. Phys. B 747, 266 (2006).

[33] A. E. Shabad and V. V. Usov. ``Electric field of a point-like charge in a strong magnetic fieldand ground state of a hydrogen-like atom.” Phys, Rev. D 77, 025001 (2008).

[34] A. E. Shabad and V. V. Usov. ``Modified Coulomb Law in a Strongly Magnetized Vacuum.”Phys, Rev. Lett. 98, 180403 (2007).

[35] A. E. Shabad and V. V. Usov, ``String-Like Electrostatic Interaction from QED with InfiniteMagnetic Field.” Talk given at 13th Lomonosov Conference on Elementary ParticlePhysics, Moscow, Russia, 23-29 Aug 2007. arXiv:0801.0115 [hep-th].

[36] N. Sadooghi and A. Sodeiri Jalili. ``New look at the modified Coulomb potential in a strongmagnetic field.” Phys, Rev. D. 76, 065013 (2007). [arXiv:0705.4384 [hep-th]].

[37] T. Muta. `` Foundations of Quantum Chromodynamics: An Introduction to PerturbativeMethods in Theories.” World Scientific Publishing Co. Pte. Ltd., (1998), 418 p.

[38] M. E. Peskin, and D. V. Schroeder. ``An introduction to quantum field theory.” PerseusBooks Publishing, L.L.C., (1995).

[39] C. S.Fischer. J. Phys. G: Nucl. Part. Phys. ``Infrared Properties of QCD from Dyson-Schwinger equations.” 32, R253 (2006). [arXiv:0605173 [hep-th]].

[40] V. N. Gribov. ``Quantization of non-Abelian gauge theories.”. Nucl. Phys. B , 139, 1, 1978.

[41] D. Zwanziger. ``Renormalization in the Coulomb gauge and order parameter for confinementin QCD.” Nucl. Phys. B, 518, 237, 1998.

[42] R. N. Mohapatra. ``Feynman Rules For The Yang-Mills Field: A Canonical QuantizationApproach. I.” Phys. Rev. D, 4, 378, 1971.

141

[43] R. N. Mohapatra. ``Feynman Rules For The Yang-Mills Field: A Canonical QuantizationApproach. II.” Phys. Rev. D, 4, 1007, 1971.

[44] A. Andrasi. ``The gluon propagator in the Coulomb gauge.” Eur. Phys. J. C , 37, 307, 2004.[arXiv:hep-th/0311118].

[45] A. Andrasi and J. C. Taylor. ``Cancellation of energy-divergences in Coulomb gauge QCD.”Eur. Phys. J. C , 41, 377, 2005. [arXiv:hep-th/0503099].

[46] P. Watson and H. Reinhardt. ``Propagator Dyson-Schwinger equations of Coulomb gaugeYang-Mills theory within the first order formalism.” Phys. Rev. D, 75, 045021, 2007.[arXiv:hep-th/0612114].

[47] P. Watson and H. Reinhardt. ``Two-Point Functions of Coulomb Gauge Yang-Mills The-ory.” Phys. Rev. D, 77, 025030, 2008. [arXiv[hep-th]:/0709.3963].

[48] F. J. Dyson. ``The S matrix in quantum electrodynamics.” Phys. Rev., 75, 1736, 1949.

[49] J. S. Schwinger. ``On the Green’s functions of quantized fields. 1 and 2.” Proc. Nat. Acad.Sci., 37, 452 and 455, 1951.

[50] R. M. Manchester, G. B. Hobbs A. Teoh and M. Hobbs. ``The Australia Telescope NationalFacility Pulsar Catalogue.” Astron. J. 129, 1993 (2005).

[51] C. Kouveliotou et al. ``An X-ray pulsar with a superstrong magnetic field in the soft gamma-ray.” Nature 393, 235 (1998).

[52] J. S. Bloom, S. R. Kulkarni, F. A. Harrison, T. Prince, E. S. Phinney and D. A. Frail. ``Ex-pected characteristics of the subclass of Supernova Gamma-ray Bursts (S-GRBs).” Astron. J.506, L105 (1998).

[53] F. Sauter. ``Uber das Verhalten eines Elektrons im homogenen elektrischen Feld nach derrelativistischen Theorie Diracs.” Z. Phys. 69, 742 (1931).

[54] F. Hebenstreit, R. Alkofer and H. Gies. ``Pair Production Beyond the Schwinger Formulain Time-Dependent Electric Fields.” Phys. Rev. D 78, 061701 (2008). [arXiv:0807.2785[hep-ph]].

[55] F. Hebenstreit, R. Alkofer, G. V. Dunne and H. Gies. ``Momentum signatures forSchwinger pair production in short laser pulses with sub-cycle structure.” Phys. Rev. Lett.102, 150404 (2009). [arXiv:0901.2631 [hep-ph]].

[56] I. A. Batalin and A. E. Shabad. ``Green’s function of a photon in a constant homogeneouselectromagnetic field of general form.” Sov. Phys. JETP 33, 483 (1971).

142

[57] H. P. Rojas and E. R. Querts. ``Is the Photon Paramagnetic?” Phys. Rev. D 79, 093002(2009).

[58] Selym Villalba Chavez and H. P. Rojas. ``On the Photon Anomalous Magnetic moment.”[arXiv:hep-th/0604059]; S. Villalba-Chavez and H. Perez-Rojas. ``Has the photon ananomalous magnetic moment?” [arXiv:hep-th/0609008].

[59] A. V. Kuznetsov, N. V. Mikheev and M. V. Osipov. ``Electron mass operator in a strongmagnetic field.” Mod. Phys. Lett. A, 17, 231, (2002).

[60] H. Belich, L. P. Colatto, T. Costa Soares J. A. Helayel Neto and M. T. D. Orlando. ``Mag-netic Moment Generation from non-minimal couplings in a scenario with Lorentz-SymmetryViolation.” Eur. Phys. J. C 62, 425 (2009). [arXiv:0806.1253 [hep-th]].

[61] D. Uzdensky, C. Forest, H. Ji R. Townsend and M. Yamada. ``Magnetic Fields in StellarAstrophysics.” arXiv:0902.3596 [astro-ph.SR].

[62] M. Chaichian, S. S. Masood, C. Montonen A. Perez Martinez and H. Perez Ro-jas. ``Quantum-magnetic and gravitational collapse.” Phys. Rev. Lett. 84, 5261 (2000).[arXiv:hep-ph/9911218].

[63] A. Perez Martinez, H. Perez Rojas and H. J. Mosquera Cuesta. ``Magnetic collapse of aneutron gas: Can magnetars indeed be formed.” Eur. Phys. J. C 29, 111 (2003). [arXiv:astro-ph/0303213].

[64] H. Perez Rojas, A. Perez Martinez and H. J. Mosquera Cuesta. ``Possible Origin OfMagnetic Fields In Very Dense Stars.” Chin. Phys. Lett. 21, 2117 (2004).

[65] H. Perez Rojas, A. Perez Martinez and H. J. Mosquera Cuesta. ``Bose-Einstein Conden-sation Of Charged Particles And Self-Magnetization.” Int. J. Mod. Phys. D 13, 1207 (2004).

[66] A. A. Sokolov, I. M. Ternov, V. Ch. Zhukovski and A. V. Borisov. ``ElectrodinamicaCuantica.” Moscu: Mir, (1989).

[67] V. I. Denisov, I. P. Denisova and S. I. Svertilov. ``The nonlinear-electrodynamic bendingof the x-ray and gamma-ray in the magnetic field of pulsars and magnetars.” Dokl. Akad.Nauk Ser. Fiz. 380, 435 (2001). [arXiv:astro-ph/0110705].

[68] V. I. Denisov and S. I. Svertilov. ``Nonlinear Electromagnetic And Gravitational ActionsOf Neutron Star Fields On Electromagnetic Wave Propagation.” Phys. Rev. D 71, 063002(2005).

[69] J. Schwinger. ``Gauge Invariance And Mass. 2.” Phys. Rev. 128, 2425 (1962).

[70] H. P. Rojas. ``Electroweak plasma in a magnetic field.” Act. Phys. Pol. B 17, 861 (1986).

143

[71] H. Perez Rojas and E. Rodrıguez Querts. ``Vacuum pressure and energy in strong mag-netic field.” Proc. Int. Workshop on Strong Magnetic Field and Neutron Stars, 189,(2003).

[72] H. Perez Rojas and E. Rodrıguez Querts. ``Negative pressures in QED vacuum in anexternal magnetic field.” Int. J. Mod. Phys A21, 3761, 2006.

[73] H. Perez Rojas and E. Rodrıguez Querts. ``Magnetic fields in quantum degenerate systemsand in vacuum.” Int. J. Mod. Phys D16, 165,2007.

[74] H. Gies. ``QED effective action at finite temperature: Two loop dominance.” Phys. Rev. D61, 085021, (2000).

[75] H. Bacry, P. Combe and J. L. Richard. ``Group-Theoretical Analysis of Elemetary Particlesin and External Electromagnetic Field. I. - The relativistic Particle in a Constant and UniformField.” Nuovo Cim. A67, 267 (1970).

[76] N. Sadooghi and A. J. Salim. ``Axial anomaly of QED in strong magnetic field and non-commutative anomaly.” Phys. Rev. D 74, 085032 (2006).

[77] H. Perez Rojas and A. E. Shabad. ``Polarization of relativistic electron and positron gas ina strong magnetic field. Propagation of electromagnetic wave.” Ann. Phys., 121, 432, (1979).

[78] A. E. Shabad and V. V. Usov. “Convexity of effective Lagrangian in nonlinear electrody-namics as derived from causality.” [arXiv:0911.0640 [hep-th]].

[79] J. S. Schwinger. ``NonAbelian gauge fields. Relativistic invariance.” Phys. Rev., 127, 324,1962.

[80] N. H. Christ and T. D. Lee. ``Operator Ordering And Feynman Rules In Gauge Theories.”Phys. Rev. D, 22, 939, 1980.

[81] Selym Villalba Chavez. ``On the connection between Hamilton and Lagrange formalism inQuantum Field Theory.” arXiv:0807.2146 [hep-th].

[82] P. Watson and H. Reinhardt. ``Perturbation Theory of Coulomb Gauge Yang-Mills The-ory Within the First Order Formalism.” Phys. Rev. D, 76, 125016, 2007. [arXiv:0709.0140[hep-th]].

[83] R. Alkofer and J. Greensite. ``Quark Confinement: The Hard Problem of Hadron Physics.”J. Phys. G, 34, S3, 2007. [arXiv:hep-ph/0610365].

[84] A. Niegawa, M. Inui and H. Kohyama. ``Cancellation of energy divergences and renor-malizability in Coulomb gauge QCD within the Lagrangian formalism.” Phys. Rev. D, 74,105016, 2006. [arXiv:hep-th/0607207].

144

[85] G. Leibbrandt and J. Williams. ``Split Dimensional Regularization for the CoulombGauge.” Nucl. Phys. B, 475, 469, 1996. [arXiv:hep-th/9807024]. [arXiv:hep-th/9601046].

[86] G. Heinrich and G. Leibbrandt. ``Split dimensional regularization for the Coulomb gaugeat two loops.” Nucl. Phys. B, 575, 359, 2000. [arXiv:hep-th/9807024]. [arXiv:hep-th/9911211].

[87] L. Baulieu and D. Zwanziger. ``Renormalizable non-covariant gauges and Coulomb gaugelimit.” Nucl. Phys. B, 548, 527, 1999. [arXiv:hep-th/9807024].

[88] A. P. Szczepaniak and E. S. Swanson. ``Coulomb gauge QCD, confinement, and the con-stituent representation.” Phys. Rev. D, 65, 025012, 2002. [arXiv:hep-ph/0107078].

[89] A. Andrasi and J. C. Taylor. ``Renormalization of Hamiltonian QCD.” [arXiv:0704.1420[hep-th]].

[90] R. Alkofer and L. von Smekal. ``The infrared behavior of QCD Green’s functions: Confine-ment, dynamical symmetry breaking, and hadrons as relativistic bound states.” Phys. Rept.,353, 281, 2001. [arXiv: hep-ph/0007355].

[91] J. C. Ward. ``An Identity in Quantum Electrodynamics.” Phys. Rev., 78, 182, 1950.

[92] H. S. Green. ``A Pre-renormalized quantum electrodynamics.” Proc. Phys. Soc. A ., 66, 873,1953.

[93] Y. Takahashi. ``On the generalized Ward identity.” Nuovo Ciment., 6, 371, 1957.

[94] J. C. Taylor. ``Ward Identities And Charge Renormalization Of The Yang-Mills Field.” Nucl.Phys. B, 33, 436, 1971.

[95] A. A. Slavnov. ``Ward Identities In Gauge Theories.” Theor. Math. Phys., 10, 99, 1972.

[96] R. Alkofer, M. Q. Huber and K. Schwenzer. ``On infrared singularities in Landau gaugeYang-Mills theory.” arXiv:0801.2762 [hep-th].

[97] R. Alkofer, M. Q. Huber and K. Schwenzer. ``Algorithmic derivation of Dyson-SchwingerEquations.” Comput. Phys. Commun., 180, 965, 2009. [arXiv:0808.2939 [hep-th]].

[98] J. Hubbard. ``Calculation of partition functions.” Phys. Rev. Lett., 3, 77, 1959.

[99] Y. Nambu and G. Jona-Lasinio. ``Dynamical model of elementary particles based on ananalogy with superconductivity. I.” Phys. Rev., 122, 345, 1961.

[100] J. Bardeen, L. N. Cooper and J. R. Schrieffer. ``Theory Of Superconductivity.” Phys.Rev., 108, 1175, 1957.

145

[101] H. Gies and C. Wetterich. ``Renormalization flow of bound states.” Phys. Rev. D, 65,065001, 2002.

[102] M. G. Rocha, F. J. Llanes-Estrada, D. Schuette and S. Villalba-Chavez. ``Boost operatorsin Coulomb-gauge QCD: the pion form factor and Fock expansions in φ radiative decays.”arXiv:0910.1448 [hep-ph].

[103] J. Meyer, K. Schwenzer, H. J. Pirner and A. Deandrea. ``Renormalization group flow inlarge N(c).” Phys. Lett. B, 526, 79, 2002. [arXiv:hep-ph/0110279].

[104] M. E. Rose. ``The Charge Distribution in Nuclei and the Scattering of High Energy Elec-trons.” Phys. Rev., 73, 279, 1948.

[105] J. Gasser and U. G. Meißner. ``Chiral expansion of pion form-factors beyond one loop.”Nucl. Phys. B, 357, 90, 1991.

[106] F. K. Guo, C. Hanhart, F. J. Llanes-Estrada and U. G. Meißner. ``Quark mass dependenceof the pion vector form factor.” Phys. Lett. B, 678, 90, 2009. [arXiv:0812.3270 [hep-ph]]

[107] S. R. Amendolia et al [NA7 Collaboration]. ``A Measurement Of The Space - Like PionElectromagnetic Form-Factor.” Nucl. Phys. B 277, 168, (1986).

[108] A. Hubert. ``Hydrogen-Deuterium S-1- S-2 Isotope Shift and the Structure of theDeuteron.” Phys. Rev. Lett. 80, 468, (1998).

[109] J. L. Friar, J. Martorell and D. W. L. Sprung. ``Nuclear Sizes and the Isotope Shift.” Phys.Rev. A, 56, 4579, 1997. [arXiv:nucl-th/9707016]

[110] J. L. Friar. ``Corrections To The Impulse Approximation. 1. Dispersion And Recoil EffectsIn Elastic Electron Scattering. 2. Meson Exchange Currents And Relativistic Effects.” Ann.Phys. 81, 332-363, (1973).

[111] S. J. Brodsky and J. R. Primack. ``The Electromagnetic Interactions Of Composite Sys-tems.” Ann. Phys. 52, 315, (1969).

[112] H. M. Choi and C. R. Ji. ``Conformal Symmetry and Pion Form Factor: Space- and Time-like Region.” Phys. Rev. D 77, 113004, (2008). [arXiv:0803.2604 [hep-ph]].

[113] E. P. Biernat, W. Schweiger, K. Fuchsberger and W. H. Klink. ``Electromagnetic mesonform factor from a relativistic coupled-channel approach.” [arXiv:0902.2348 [nucl-th]].

[114] W. Plessas, S. boffi, L. Y. Glozman, W. Klink, M. Radici and R. F. Wagenbrunn. ``Nu-cleon properties in a semirelativistic chiral quark model.” Nucl. Phys. A 699, 312, (2002).

146

[115] S. J. Brodsky and G. F. Teramond. ``Light-Front Dynamics and AdS/QCD Correspon-dence: The Pion Form Factor in the Space- and Time-Like Regions.” Phys. Rev. D 77,056007, (2008). [arXiv:0707.3859 [hep-ph]].

[116] A. Szczepaniak, C. R. Ji and S. R. Cotanch. ``Nucleon structure in a relativistic quarkmodel.” Phys. Rev. C 52, 2738, (1995).

[117] B. Di Micco [KLOE Collaboration]. ``Eta, eta-prime mixing angle and eta-prime gluoniumcontent extraction from the KLOE R(phi) measurement.” Eur. Phys. J. A, 38, 129, (2008).

[118] B. Di Micco [KLOE Collaboration]. ``Measurement Of The Pseudoscalar Mixing AngleAnd The Eta-Prime Gluonium Content With The Kloe Detector.” Acta Phys. Polon. Supp.2, 63, (2009).

[119] A. Bramon, R. Escribano and M. D. Scadron. ``Radiative V P gamma transitions and etaeta’ mixing.” Phys. Lett. B 503, 271, (2001). [arXiv:hep-ph/0012049].

[120] R. Escribano and J. Nadal. ``On the gluon content of the eta and eta’ mesons.” JHEP 0705,006, (2007). [arXiv:hep-ph/0703187].

[121] R. Escribano. ``Eta’ gluonic content and J/psi→VP decays.” Nucl. Phys. Proc. Suppl.181-182, 226, (2008). [arXiv:0807.4205 [hep-ph]].

[122] P. Maris and P. C. Tandy. ``The pi, K+, and K0 electromagnetic form factors.” Phys. Rev.C 62, 055204, (2000). [arXiv:nucl-th/0005015].

[123] R. G. Sachs. ``High-Energy Behavior of Nucleon Electromagnetic Form Factors.” Phys.Rev. 126, 2256, (1962).

[124] J. J. Kelly. ``Nucleon charge and magnetization densities from Sachs form factors.” Phys.Rev. C 66, 065203, (2002). [arXiv:hep-ph/0204239].

[125] C. Amsler et al. [Particle Data Group]. ``Review of particle physics.” Phys. Lett. B 667,1, (2008).

[126] P. A. Boyle et al. ``The pion’s electromagnetic form factor at small momentum transfer infull lattice QCD.” JHEP 0807, 112, (2008). [arXiv:0804.3971 [hep-lat]];

S. Aoki et al. [TWQCD collaboration]. ``Pion form factors from two-flavor lattice QCDwith exact chiral symmetry.” [arXiv:0905.2465 [hep-lat]];

R. Frezzotti, V. Lubicz and S. Simula. ``Electromagnetic form factor of the pion fromtwisted-mass lattice QCD at Nf=2.” [arXiv:0812.4042 [hep-lat]].

[127] P. Besting and D. Schutte. ``Relativistic Invariance of Coulomb Gauge Yang-Mills The-ory.” Phys. Rev. D 42, 594, (1990).

147

[128] D. Zwanziger. ``Non-perturbative Faddeev-Popov formula and infrared limit of QCD.”Phys. Rev. D 69, 016002, (2004). [arXiv:hep-ph/0303028].

[129] H. Reinhardt and P. Watson. ``Resolving temporal Gribov copies in Coulomb gauge Yang-Mills theory.” Phys. Rev. D 79, 045013, (2009). [arXiv:0808.2436 [hep-th]].

[130] C. Feuchter and H. Reinhardt. ``Variational solution of the Yang-Mills Schroedinger equa-tion in Coulomb gauge.” Phys. Rev. D 70, 105021, (2004). [arXiv:hep-th/0408236].

[131] A. P. Szczepaniak. ``Confinement and gluon propagator in Coulomb gauge QCD.” Phys.Rev. D 69, 074031, (2004). [arXiv:hep-ph/0306030].

[132] D. R. Campagnari, H. Reinhardt and A. Weber. ``Perturbation theory in the Hamilto-nian approach to Yang-Mills theory in Coulomb gauge.” Phys. Rev. D 80, 025005, (2009).[arXiv:0904.3490 [hep-th]].

[133] P. Watson and H. Reinhardt. ``Slavnov-Taylor identities in Coulomb gauge Yang-Millstheory.” [arXiv:0812.1989 [hep-th]].

[134] P. Biscudo, S. R. Cotanch, F. J. Llanes-Estrada, and D. G. Robertson. ``The newf0(1810) in omega Phi at BES: A good glueball candidate.” Eur. Phys. J. C 52, 363, (2007).[arXiv:hep-ph/0602172].

[135] G. P. Lepage, CLNS-80/447, 1980. ``Vegas: An Adaptive Multidimensional IntegrationProgram.”

[136] F. J. Llanes-Estrada, and S. R. Cotanch. ``Relativistic many-body Hamiltonian approachto mesons.” Nucl. Phys. A 697, 303, (2002). [arXiv:hep-ph/0101078].

[137] A. Szczepaniak, E. S. Swanson, C. R. Ji, and S. R. Cotanch. ``Glueball Spectroscopy in aRelativistic Many-Body Approach to Hadron Structure.” Phys. Rev. Lett. 76, 2011, (1996).[arXiv:hep-ph/9511422].

148