PP-13-02

April, 2008 PROGRESS IN PHYSICS Volume 2

SPECIAL REPORT

New Approach to Quantum Electrodynamics

Sze Kui Ng

Department of Mathematics, Hong Kong Baptist University, Hong Kongand School of Engineering and Information Technology, Deakin University, Australia

E-mail: [email protected]

It is shown that a photon with a specific frequency can be identified with the Dirac mag-netic monopole. When a Dirac-Wilson line forms a Dirac-Wilson loop, it is a photon.This loop model of photon is exactly solvable. From the winding numbers of this loop-form of photon, we derive the quantization properties of energy and electric charge. Anew QED theory is presented that is free of ultraviolet divergences. The Dirac-Wilsonline is as the quantum photon propagator of the new QED theory from which we canderive known QED effects such as the anomalous magnetic moment and the Lamb shift.The one-loop computation of these effects is simpler and is more accurate than that inthe conventional QED theory. Furthermore, from the new QED theory, we have deriveda new QED effect. A new formulation of the Bethe-Salpeter (BS) equation solves thedifficulties of the BS equation and gives a modified ground state of the positronium. Bythe mentioned new QED effect and by the new formulation of the BS equation, a termin the orthopositronium decay rate that is missing in the conventional QED is found,resolving the orthopositronium lifetime puzzle completely. It is also shown that thegraviton can be constructed from the photon, yielding a theory of quantum gravity thatunifies gravitation and electromagnetism.

1 Introduction

It is well known that the quantum era of physics began withthe quantization of energy of electromagnetic field, fromwhich Planck derived the radiation formula. Einstein thenintroduced the light-quantum to explain the photoelectric ef-fects. This light-quantum was regarded as a particle calledphoton [13]. Quantum mechanics was then developed, ush-ering in the modern quantum physics era. Subsequently, thequantization of the electromagnetic field and the theory ofQuantum Electrodynamics (QED) were established.

In this development of quantum theory of physics, thephoton plays a special role. While it is the beginning of quan-tum physics, it is not easy to understand as is the quantummechanics of other particles described by the Schrodingerequation. In fact, Einstein was careful in regarding thelight-quantum as a particle, and the acceptance of the light-quantum as a particle called photon did not come about untilmuch later [1]. The quantum field theory of electromagneticfield was developed for the photon. However, such difficul-ties of the quantum field theory as the ultraviolet divergencesare well known. Because of the difficulty of understandingthe photon, Einstein once asked: What is the photon? [1].

On the other hand, based on the symmetry of the electricand magnetic field described by the Maxwell equation andon the complex wave function of quantum mechanics, Diracderived the concept of the magnetic monopole, which is hy-pothetically considered as a particle with magnetic charge, inanalogy to the electron with electric charge. An importantfeature of this magnetic monopole is that it gives the quanti-

zation of electric charge. Thus it is interesting and importantto find such particles. However, in spite of much effort, nosuch particles have been found [4, 5].

In this paper we shall establish a mathematical model ofphoton to show that the magnetic monopole can be identifiedas a photon. Before giving the detailed model, let us discusssome thoughts for this identification in the following.

First, if the photon and the magnetic monopole are dif-ferent types of elementary quantum particles in the electro-magnetic field, it is odd that one type can be derived from theother. A natural resolution of this oddity is the identificationof the magnetic monopole as a photon.

The quantum field theory of the free Maxwell equationis the basic quantum theory of photon [6]. This free fieldtheory is a linear theory and the models of the quantum parti-cles obtained from this theory are linear. However, a stableparticle should be a soliton, which is of the nonlinear na-ture. Secondly, the quantum particles of the quantum the-ory of Maxwell equation are collective quantum effects in thesame way the phonons which are elementary excitations ina statistical model. These phonons are usually considered asquasi-particles and are not regarded as real particles. Regard-ing the Maxwell equation as a statistical wave equation ofelectromagnetic field, we have that the quantum particles inthe quantum theory of Maxwell equation are analogous to thephonons. Thus they should be regarded as quasi-photons andhave properties of photons but not a complete description ofphotons.

In this paper, a nonlinear model of photon is established.In the model, we show that the Dirac magnetic monopole

Sze Kui Ng. New Approach to Quantum Electrodynamics 15

Volume 2 PROGRESS IN PHYSICS April, 2008

can be identified with the photon with some frequencies. Weprovide a U(1) gauge theory of Quantum Electrodynamics(QED), from which we derive photon as a quantum Dirac-Wilson loop W (z; z) of this model. This nonlinear loopmodel of the photon is exactly solvable and thus may be re-garded as a quantum soliton. From the winding numbers ofthis loop model of the photon, we derive the quantizationproperty of energy in Plancks formula of radiation and thequantization property of charge. We show that the quanti-zation property of charge is derived from the quantizationproperty of energy (in Plancks formula of radiation), whenthe magnetic monopole is identified with photon with certainfrequencies. This explains why we cannot physically find amagnetic monopole. It is simply a photon with a specific fre-quency.

From this nonlinear model of the photon, we also con-struct a model of the electron which has a mass mechanismfor generating mass of the electron. This mechanism of gen-erating mass supersedes the conventional mechanism of gen-erating mass (through the Higgs particles) and makes hypoth-esizing the existence of the Higgs particles unnecessary. Thisexplains why we cannot physically find such Higgs particles.

The new quantum gauge theory is similar to the conven-tional QED theory except that the former is not based on thefour dimensional space-time (t;x) but is based on the propertime s in the theory of relativity. Only in a later stage in thenew quantum gauge theory, the space-time variable (t;x) isderived from the proper time s through the Lorentz metricds

2

= dt

2

dx2 to obtain space-time statistics and explainthe observable QED effects.

The derived space variable x is a random variable in thisquantum gauge theory. Recall that the conventional quan-tum mechanics is based on the space-time. Since the spacevariable x is actually a random variable as shown in the newquantum gauge theory, the conventional quantum mechanicsneeds probabilistic interpretation and thus has a most myste-rious measurement problem, on which Albert Einstein onceremarked: God does not play dice with the universe. Incontrast, the new quantum gauge theory does not involve thementioned measurement problem because it is not based onthe space-time and is deterministic. Thus this quantumgauge theory resolves the mysterious measurement problemof quantum mechanics.

Using the space-time statistics, we employ Feynman dia-grams and Feynman rules to compute the basic QED effectssuch as the vertex correction, the photon self-energy and theelectron self-energy. In this computation of the Feynman in-tegrals, the dimensional regularization method in the conven-tional QED theory is also used. Nevertheless, while the con-ventional QED theory uses it to reduce the dimension 4 ofspace-time to a (fractional) number n to avoid the ultravioletdivergences in the Feynman integrals, the new QED theoryuses it to increase the dimension 1 of the proper time to anumber n less than 4, which is the dimension of the space-

time, to derive the space-time statistics. In the new QED the-ory, there are no ultraviolet divergences, and the dimensionalregularization method is not used for regularization.

After this increase of dimension, the renormalizationmethod is used to derive the well-known QED effects. Unlikethe conventional QED theory, the renormalization method isused in the new QED theory to compute the space-time statis-tics, but not to remove the ultraviolet divergences, since theultraviolet divergences do not occur in the new QED theory.From these QED effects, we compute the anomalous mag-netic moment and the Lamb shift [6]. The computation doesnot involve numerical approximation as does that of the con-ventional QED and is simpler and more accurate.

For getting these QED effects, the quantum photon prop-agator W (z; z0), which is like a line segment connectingtwo electrons, is used to derive the electrodynamic interac-tion. (When the quantum photon propagatorW (z; z0) forms aclosed circle with z= z0, it then becomes a photonW (z; z).)From this quantum photon propagator, a photon propagator isderived that is similar to the Feynman photon propagator inthe conventional QED theory.

The photon-loopW (z; z) leads to the renormalized elec-tric charge e and the massm of electron. In the conventionalQED theory, the bare charge e0

is of less importance than therenormalized charge e, in the sense that it is unobservable. Incontrast, in this new theory of QED, the bare charge e0

andthe renormalized charge e are of equal importance. While therenormalized charge e leads to the physical results of QED,the bare charge e0

leads to the universal gravitation constantG. It is shown that e=ne

e

0

, where ne

is a very large wind-ing number and thus e0

is a very small number. It is furthershown that the gravitational constant G=2e20

which is thusan extremely small number. This agrees with the fact that theexperimental gravitational constant G is a very small num-ber. The relationships, e=ne

e

0

and G=2e20

, are a part ofa theory unifying gravitation and electromagnetism. In thisunified theory, the graviton propagator and the graviton areconstructed from the quantum photon propagator. This con-struction leads to a theory of quantum gravity. In short, a newtheory of quantum gravity is developed from the new QEDtheory in this paper, and unification of gravitation and elec-tromagnetism is achieved.

In this paper, we also derive a new QED effect from theseagull vertex of the new QED theory. The conventionalBethe-Salpeter (BS) equation is reformulated to resolve itsdifficulties (such as the existence of abnormal solutions [732]) and to give a modified ground state wave function ofthe positronium. By the new QED effect and the reformu-lated BS equation, another new QED effect, a term in the or-thopositronium decay rate that is missing in the conventionalQED is discovered. Including the discovered term, the com-puted orthopositronium decay rate now agrees with the ex-perimental rate, resolving the orthopositronium lifetime puz-zle completely [3352]. We note that the recent resolution of

16 Sze Kui Ng. New Approach to Quantum Electrodynamics


this orthopositronium lifetime puzzle resolves the puzzle onlypartially due to a special statistical nature of this new term inthe orthopositronium decay rate.

This paper is organized as follows. In Section 2 we give abrief description of a new QED theory. With this theory, weintroduce the classical Dirac-Wilson loop in Section 3. Weshow that the quantum version of this loop is a nonlinear ex-actly solvable model and thus can be regarded as a soliton.We identify this quantum Dirac-Wilson loop as a photon withthe U(1) group as the gauge group. To investigate the prop-erties of this Dirac-Wilson loop, we derive a chiral symmetryfrom the gauge symmetry of this quantum model. From thischiral symmetry, we derive, in Section 4, a conformal fieldtheory, which includes an affine Kac-Moody algebra and aquantum Knizhnik-Zamolodchikov (KZ) equation. A mainpoint of our model on the quantum KZ equation is that wecan derive two KZ equations which are dual to each other.This duality is the main point for the Dirac-Wilson loop tobe exactly solvable and to have a winding property which ex-plains properties of photon. This quantum KZ equation canbe regarded as a quantum Yang-Mills equation.

In Sections 5 to 8, we solve the Dirac-Wilson loop in aform with a winding property, starting with the KZ equations.From the winding property of the Dirac-Wilson loop, we de-rive, in Section 9 and Section 10, the quantization of energyand the quantization of electric charge which are properties ofphoton and magnetic monopole. We then show that the quan-tization property of charge is derived from the quantizationproperty of energy of Plancks formula of radiation, when weidentify photon with the magnetic monopole for some fre-quencies. From this nonlinear model of photon, we also de-rive a model of the electron in Section 11. In this model ofelectron, we provide a mass mechanism for generating massto electron. In Section 12, we show that the photon with a spe-cific frequency can carry electric charge and magnetic charge,since an electron is formed from a photon with a specific fre-quency for giving the electric charge and magnetic charge. InSection 13, we derive the statistics of photons and electronsfrom the loop models of photons and electrons.

In Sections 14 to 22, we derive a new theory of QED,wherein we perform the computation of the known basic QEDeffects such as the photon self-energy, the electron self-energyand the vertex correction. In particular, we provide simplerand more accurate computation of the anomalous magneticmoment and the Lamb shift. Then in Section 23, we com-pute a new QED effect. Then from Section 24 to Section25, we reformulate the Bethe-Salpeter (BS) equation. Withthis new version of the BS equation and the new QED effect,a modified ground state wave function of the positronium isderived. Then by this modified ground state of the positron-ium, we derive in Section 26 another new QED effect, a termmissing in the theoretic orthopositronium decay rate of theconventional QED theory, and show that this new theoreticalorthopositronium decay rate agrees with the experimental de-

cay rate, completely resolving the orthopositronium life timepuzzle [3352].

In Section 27, the graviton is derived from the photon.This leads to a new theory of quantum gravity and a new uni-fication of gravitation and electromagnetism. Then in Section28, we show that the quantized energies of gravitons can beidentified as dark energy. Then in a way similar to the con-struction of electrons by photons, we use gravitons to con-struct particles which can be regarded as dark matter. Weshow that the force among gravitons can be repulsive. Thisgives the diffusion phenomenon of dark energy and the accel-erating expansion of the universe [5357].

2 New gauge model of QED

Let us construct a quantum gauge model, as follows. In prob-ability theory we have the Wiener measure which is a mea-sure on the space C[t0

; t

1

] of continuous functions [58]. Thismeasure is a well defined mathematical theory for the Brow-nian motion and it may be symbolically written in the follow-ing form:d = e

L

0

dx ; (1)

where L0

:=

1

2

R

t

1

t

0

dx

dt

2

dt is the energy integral of theBrownian particle and dx= 1N

Q

t

dx(t) is symbolically aproduct of Lebesgue measures dx(t) and N is a normalizedconstant.

Once the Wiener measure is defined we may then defineother measures on C[t0

; t

1

], as follows [58]. Let a potentialterm 12

R

t

1

t

0

V dt be added to L0

. Then we have a measure 1

on C[t0

; t

1

] defined by:

d

1

= e

1

2

R

t

1

t

0

V dt

d : (2)

Under some condition on V we have that 1

is well de-fined on C[t0

; t

1

]. Let us call (2) as the Feynman-Kac for-mula [58].

Let us then follow this formula to construct a quantummodel of electrodynamics, as follows. Then similar to theformula (2) we construct a quantum model of electrodynam-ics from the following energy integral:

R

s

1

s

0

Dds :=

R

s

1

s

0

h

1

2

@A

1

@x

2

@A

2

@x

1

@A

1

@x

2

@A

2

@x

1

+

+

dZ

ds

+ ie

0

(

P

2

j=1

A

j

dx

j

ds

)Z

dZ

ds

ie

0

(

P

2

j=1

A

j

dx

j

ds

)Z

i

ds ;

(3)

where the complex variable Z =Z(z(s)) and the real vari-ables A1

=A

1

(z(s)) and A2

=A

2

(z(s)) are continuousfunctions in a form that they are in terms of a (continuouslydifferentiable) curve z(s)=C(s)= (x1(s); x2(s)); s0

6 s 6s

1

; z(s

0

)= z(s

1

) in the complex plane where s is a parameterrepresenting the proper time in relativity. (We shall also write



z(s) in the complex variable form C(s)= z(s)=x1(s)++ ix

2

(s); s

0

6 s 6 s1

.) The complex variable Z =Z(z(s))represents a field of matter, such as the electron (Z denotesits complex conjugate), and the real variables A1

=A

1

(z(s))

and A2

=A

2

(z(s)) represent a connection (or the gauge fieldof the photon) and e0

denotes the (bare) electric charge.The integral (3) has the following gauge symmetry:

Z

0

(z(s)) := Z(z(s)) e

ie

0

a(z(s))

A

0

j

(z(s)) := A

j

(z(s)) +

@a

@x

j

; j = 1; 2

(4)

where a= a(z) is a continuously differentiable real-valuedfunction of z.

We remark that this QED theory is similar to the conven-tional Yang-Mills gauge theories. A feature of (3) is that itis not formulated with the four-dimensional space-time but isformulated with the one dimensional proper time. This onedimensional nature let this QED theory avoid the usual ul-traviolet divergence difficulty of quantum fields. As most ofthe theories in physics are formulated with the space-time letus give reasons of this formulation. We know that with theconcept of space-time we have a convenient way to under-stand physical phenomena and to formulate theories such asthe Newton equation, the Schrodinger equation, e.t.c. to de-scribe these physical phenomena. However we also know thatthere are fundamental difficulties related to space-time suchas the ultraviolet divergence difficulty of quantum field the-ory. To resolve these difficulties let us reexamine the conceptof space-time. We propose that the space-time is a statisticalconcept which is not as basic as the proper time in relativity.Because a statistical theory is usually a convenient but incom-plete description of a more basic theory this means that somedifficulties may appear if we formulate a physical theory withthe space-time. This also means that a way to formulate a ba-sic theory of physics is to formulate it not with the space-timebut with the proper time only as the parameter for evolution.This is a reason that we use (3) to formulate a QED theory. Inthis formulation we regard the proper time as an independentparameter for evolution. From (3) we may obtain the con-ventional results in terms of space-time by introducing thespace-time as a statistical method.

Let us explain in more detail how the space-time comesout as a statistics. For statistical purpose when many electrons(or many photons) present we introduce space-time (t;x) asa statistical method to write ds2 in the form

ds

2

= dt

2

dx2: (5)

We notice that for a given ds there may have many dtand dxwhich correspond to many electrons (or photons) suchthat (5) holds. In this way the space-time is introduced as astatistics. By (5) we shall derive statistical formulas for manyelectrons (or photons) from formulas obtained from (3). Inthis way we obtain the Dirac equation as a statistical equa-tion for electrons and the Maxwell equation as a statistical

equation for photons. In this way we may regard the con-ventional QED theory as a statistical theory extended fromthe proper-time formulation of this QED theory (From theproper-time formulation of this QED theory we also have atheory of space-time statistics which give the results of theconventional QED theory). This statistical interpretation ofthe conventional QED theory is thus an explanation of themystery that the conventional QED theory is successful inthe computation of quantum effects of electromagnetic inter-action while it has the difficulty of ultraviolet divergence.

We notice that the relation (5) is the famous Lorentz met-ric. (We may generalize it to other metric in General Rela-tivity.) Here our understanding of the Lorentz metric is thatit is a statistical formula where the proper time s is morefundamental than the space-time (t;x) in the sense that wefirst have the proper time and the space-time is introducedvia the Lorentz metric only for the purpose of statistics. Thisreverses the order of appearance of the proper time and thespace-time in the history of relativity in which we first havethe concept of space-time and then we have the concept ofproper time which is introduced via the Lorentz metric. Oncewe understand that the space-time is a statistical concept from(3) we can give a solution to the quantum measurement prob-lem in the debate about quantum mechanics between Bohrand Einstein. In this debate Bohr insisted that with the prob-ability interpretation quantum mechanics is very successful.On the other hand Einstein insisted that quantum mechan-ics is incomplete because of probability interpretation. Herewe resolve this debate by constructing the above QED the-ory which is a quantum theory as the quantum mechanics andunlike quantum mechanics which needs probability interpre-tation we have that this QED theory is deterministic since itis not formulated with the space-time.

Similar to the usual Yang-Mills gauge theory we can gen-eralize this gauge theory with U(1) gauge symmetry to non-abelian gauge theories. As an illustration let us considerSU(2)U(1) gauge symmetry where SU(2)U(1) denotesthe direct product of the groups SU(2) and U(1).

Similar to (3) we consider the following energy integral:

L :=

R

s

1

s

0

1

2

Tr (D

1

A

2

D

2

A

1

)

(D

1

A

2

D

2

A

1

)+

+ (D

0

Z

)(D

0

Z)

ds ;

(6)

where Z =(z1

; z

2

)

T is a two dimensional complex vector;A

j

=

P

3

k=0

A

k

j

t

k

(j=1; 2) where Akj

denotes a componentof a gauge field Ak; tk= i T k denotes a generator ofSU(2) U(1) where T k denotes a self-adjoint generator ofSU(2) U(1) (here for simplicity we choose a conventionthat the complex i is absorbed by tk and tk is absorbed byA

j

; and the notation Aj

is with a little confusion with the no-tation Aj

in the above formulation of (3) where Aj

; j=1; 2

are real valued); andDl

=

@

@x

l

e

0

(

P

2

j=1

A

j

dx

j

ds

) for l=1; 2;

andD0

=

d

ds

e

0

(

P

2

j=1

A

j

dx

j

ds

) where e0

is the bare electric



charge for general interactions including the strong and weakinteractions.

From (6) we can develop a nonabelian gauge theory assimilar to that for the above abelian gauge theory. We havethat (6) is invariant under the following gauge transformation:

Z

0

(z(s)) := U(a(z(s)))Z(z(s))

A

0

j

(z(s)) := U(a(z(s)))A

j

(z(s))U

1

(a(z(s)))+

+ U(a(z(s)))

@U

1

@x

j

(a(z(s))); j = 1; 2

(7)

where U(a(z(s)))= ea(z(s)); a(z(s))=P

k

e

0

a

k

(z(s))t

k

for some functions ak. We shall mainly consider the casethat a is a function of the form a(z(s))=P

k

Re!k(z(s))tk

where !k are analytic functions of z. (We let the function!(z(s)) :=

P

k

!

k

(z(s))t

k and we write a(z)=Re!(z).)The above gauge theory is based on the Banach spaceX of continuous functions Z(z(s)), Aj

(z(s)), j=1; 2; s0

6s 6 s1

on the one dimensional interval [s0

; s

1

].Since L is positive and the theory is one dimensional (and

thus is simpler than the usual two dimensional Yang-Millsgauge theory) we have that this gauge theory is similar to theWiener measure except that this gauge theory has a gaugesymmetry. This gauge symmetry gives a degenerate degreeof freedom. In the physics literature the usual way to treatthe degenerate degree of freedom of gauge symmetry is to in-troduce a gauge fixing condition to eliminate the degeneratedegree of freedom where each gauge fixing will give equiv-alent physical results [59]. There are various gauge fixingconditions such as the Lorentz gauge condition, the Feynmangauge condition, etc. We shall later in the Section on the Kac-Moody algebra adopt a gauge fixing condition for the abovegauge theory. This gauge fixing condition will also be used toderive the quantum KZ equation in dual form which will beregarded as a quantum Yang-Mill equation since its role willbe similar to the classical Yang-Mill equation derived fromthe classical Yang-Mill gauge theory.

3 Classical Dirac-Wilson loop

Similar to the Wilson loop in quantum field theory [60] fromour quantum theory we introduce an analogue of Wilson loop,as follows. (We shall also call a Wilson loop as a Dirac-Wilson loop.)Definition A classical Wilson loopWR

(C) is defined by:

W

R

(C) :=W (z

0

; z

1

) := Pe

e

0

R

C

A

j

dx

j

; (8)

where R denotes a representation of SU(2); C()= z() isa fixed closed curve where the quantum gauge theories arebased on it as specific in the above Section. As usual thenotation P in the definition ofWR

(C) denotes a path-orderedproduct [6062].

Let us give some remarks on the above definition of Wil-son loop, as follows.

1. We use the notationW (z0

; z

1

) to mean the Wilson loopW

R

(C) which is based on the whole closed curve z(). Herefor convenience we use only the end points z0

and z1

of thecurve z() to denote this Wilson loop (We keep in mind thatthe definition of W (z0

; z

1

) depends on the whole curve z()connecting z0

and z1

).Then we extend the definition ofWR

(C) to the case thatz() is not a closed curve with z0

, z1

. When z() is not aclosed curve we shall callW (z0

; z

1

) as a Wilson line.2. In constructing the Wilson loop we need to choose a

representation R of the SU(2) group. We shall see that be-cause a Wilson line W (z0

; z

1

) is with two variables z0

andz

1

a natural representation of a Wilson line or a Wilson loopis the tensor product of the usual two dimensional represen-tation of the SU(2) for constructing the Wilson loop.

A basic property of a Wilson line W (z0

; z

1

) is that for agiven continuous path Aj

; j=1; 2 on [s0

; s

1

] the Wilson lineW (z

0

; z

1

) exists on this path and has the following transitionproperty:W (z

0

; z

1

) =W (z

0

; z)W (z; z

1

) (9)

where W (z0

; z

1

) denotes the Wilson line of a curve z()which is with z0

as the starting point and z1

as the endingpoint and z is a point on z() between z0

and z1

.This property can be proved as follows. We have thatW (z

0

; z

1

) is a limit (whenever exists) of ordered product ofe

A

j

4x

j

and thus can be written in the following form:

W (z

0

; z

1

) = I +

R

s

00

s

0

e

0

A

j

(z(s))

dx

j

(s)

ds

ds+

+

R

s

00

s

0

e

0

A

j

(z(s

2

))

dx

j

(s

2

)

ds

h

R

s

2

s

0

e

0

A

j

(z(s

3

))

dx

j

(s

3

)

ds

ds

3

i

ds

2

+

(10)

where z(s0)= z0

and z(s00)= z1

. Then since Ai

are contin-uous on [s0; s00] and xi(z()) are continuously differentiableon [s0; s00] we have that the series in (10) is absolutely con-vergent. Thus the Wilson line W (z0

; z

1

) exists. Then sinceW (z

0

; z

1

) is the limit of ordered product we can writeW (z

0

; z

1

) in the form W (z0

; z)W (z; z

1

) by dividing z()into two parts at z. This proves the basic property of Wil-son line. Remark (classical and quantum versions of Wilson loop)From the above property we have that the Wilson lineW (z

0

; z

1

) exists in the classical pathwise sense where Ai

areas classical paths on [s0

; s

1

]. This pathwise version of theWilson lineW (z0

; z

1

); from the Feynman path integral pointof view; is as a partial description of the quantum version ofthe Wilson line W (z0

; z

1

) which is as an operator when Ai

are as operators. We shall in the next Section derive and de-fine a quantum generator J of W (z0

; z

1

) from the quantumgauge theory. Then by using this generator J we shall com-pute the quantum version of the Wilson lineW (z0

; z

1

).We shall denote both the classical version and quantum

version of Wilson line by the same notationW (z0

; z

1

) when



there is no confusion. By following the usual approach of deriving a chiral sym-

metry from a gauge transformation of a gauge field we havethe following chiral symmetry which is derived by applyingan analytic gauge transformation with an analytic function !for the transformation:

W (z

0

; z

1

) 7! W

0

(z

0

; z

1

) =

= U(!(z

1

))W (z

0

; z

1

)U

1

(!(z

0

)) ;

(11)

whereW 0(z0

; z

1

) is a Wilson line with gauge field:

A

0

=

@U(z)

@x

U

1

(z)+U(z)A

U

1

(z) : (12)

This chiral symmetry is analogous to the chiral symmetryof the usual guage theory where U denotes an element of thegauge group [61]. Let us derive (11) as follows. Let U(z) :=:= U(!(z(s))) and U(z + dz) U(z) + @U(z)@x

dx

wheredz = (dx

1

; dx

2

). Following [61] we have

U(z + dz)(1 + e

0

dx

A

)U

1

(z) =

= U(z + dz)U

1

(z) + e

0

dx

U(z+dz)A

U

1

(s)

1+

@U(z)

@x

U

1

(z)dx

+ e

0

dx

U(z+dz)A

U

1

(s)

1 +

@U(z)

@x

U

1

(z)dx

+ e

0

dx

U(z)A

U

1

(z)

=: 1 +

@U(z)

@x

U

1

(z)dx

+ e

0

dx

U(z)A

U

1

(z)

=: 1 + e

0

dx

A

0

:

(13)

From (13) we have that (11) holds.As analogous to the WZW model in conformal field the-

ory [65, 66] from the above symmetry we have the followingformulas for the variations !

W and !

0

W with respect tothis symmetry (see [65] p.621):

!

W (z; z

0

) =W (z; z

0

)!(z) (14)and

!

0

W (z; z

0

) = !

0

(z

0

)W (z; z

0

) ; (15)

where z and z0 are independent variables and !0(z0)=!(z)when z0= z. In (14) the variation is with respect to the z vari-able while in (15) the variation is with respect to the z0 vari-able. This two-side-variations when z , z0 can be derived asfollows. For the left variation we may let ! be analytic in aneighborhood of z and extended as a continuously differen-tiable function to a neighborhood of z0 such that !(z0)= 0 inthis neighborhood of z0. Then from (11) we have that (14)holds. Similarly we may let !0 be analytic in a neighborhoodof z0 and extended as a continuously differentiable function toa neighborhood of z such that !0(z)= 0 in this neighborhoodof z. Then we have that (15) holds.

4 Gauge fixing and affine Kac-Moody algebra

This Section has two related purposes. One purpose is tofind a gauge fixing condition for eliminating the degeneratedegree of freedom from the gauge invariance of the abovequantum gauge theory in Section 2. Then another purpose isto find an equation for defining a generator J of the Wilsonline W (z; z0). This defining equation of J can then be usedas a gauge fixing condition. Thus with this defining equationof J the construction of the quantum gauge theory in Section2 is then completed.

We shall derive a quantum loop algebra (or the affine Kac-Moody algebra) structure from the Wilson line W (z; z0) forthe generator J of W (z; z0). To this end let us first con-sider the classical case. Since W (z; z0) is constructed fromSU(2) we have that the mapping z ! W (z; z0) (We con-sider W (z; z0) as a function of z with z0 being fixed) has aloop group structure [63, 64]. For a loop group we have thefollowing generators:

J

a

n

= t

a

z

n

n = 0 ;1;2; : : : (16)

These generators satisfy the following algebra:

[J

a

m

; J

b

n

] = if

abc

J

c

m+n

: (17)

This is the so called loop algebra [63, 64]. Let us thenintroduce the following generating function J :

J(w) =

X

a

J

a

(w) =

X

a

j

a

(w) t

a

; (18)

where we define

J

a

(w) = j

a

(w)t

a

:=

1

X

n=1

J

a

n

(z)(w z)

n1

: (19)

From J we have

J

a

n

=

1

2i

I

z

dw (w z)

n

J

a

(w) ; (20)

whereH

z

denotes a closed contour integral with center z. Thisformula can be interpreted as that J is the generator of theloop group and that Jan

is the directional generator in the di-rection !a(w)= (w z)n. We may generalize (20) to thefollowing directional generator:

1

2i

I

z

dw !(w)J(w) ; (21)

where the analytic function !(w)=P

a

!

a

(w)t

a is regardedas a direction and we define

!(w)J(w) :=

X

a

!

a

(w)J

a

: (22)

Then since W (z; z0) 2 SU(2), from the variational for-mula (21) for the loop algebra of the loop group of SU(2) we



have that the variation of W (z; z0) in the direction !(w) isgiven by

W (z; z

0

)

1

2i

I

z

dw !(w)J(w) : (23)

Now let us consider the quantum case which is based onthe quantum gauge theory in Section 2. For this quantum casewe shall define a quantum generator J which is analogous tothe J in (18). We shall choose the equations (34) and (35) asthe equations for defining the quantum generator J . Let usfirst give a formal derivation of the equation (34), as follows.Let us consider the following formal functional integration:

hW (z; z

0

)A(z)i :=

:=

R

dA

1

dA

2

dZ

dZe

L

W (z; z

0

)A(z);

(24)

where A(z) denotes a field from the quantum gauge theory.(We first let z0 be fixed as a parameter.)

Let us do a calculus of variation on this integral to derivea variational equation by applying a gauge transformation on(24) as follows. (We remark that such variational equationsare usually called the Ward identity in the physics literature.)

Let (A1

; A

2

; Z) be regarded as a coordinate system of theintegral (24). Under a gauge transformation (regarded as achange of coordinate) with gauge function a(z(s)) this co-ordinate is changed to another coordinate (A01

; A

0

2

; Z

0

). Assimilar to the usual change of variable for integration we havethat the integral (24) is unchanged under a change of variableand we have the following equality:R

dA

0

1

dA

0

2

dZ

0

dZ

0

e

L

0

W

0

(z; z

0

)A

0

(z) =

=

R

dA

1

dA

2

dZ

dZe

L

W (z; z

0

)A(z) ;

(25)

whereW 0(z; z0) denotes the Wilson line based on A01

and A02

and similarlyA0(z) denotes the field obtained fromA(z)with(A

1

; A

2

; Z) replaced by (A01

; A

0

2

; Z

0

).Then it can be shown that the differential is unchanged

under a gauge transformation [59]:

dA

0

1

dA

0

2

dZ

0

dZ

0

= dA

1

dA

2

dZ

dZ : (26)

Also by the gauge invariance property the factor eL isunchanged under a gauge transformation. Thus from (25) wehave

0 = hW

0

(z; z

0

)A

0

(z)i hW (z; z

0

)A(z)i ; (27)

where the correlation notation hi denotes the integral withrespect to the differential

e

L

dA

1

dA

2

dZ

dZ (28)

We can now carry out the calculus of variation. From thegauge transformation we have the formula:

W

0

(z; z

0

) = U(a(z))W (z; z

0

)U

1

(a(z

0

)) ; (29)

where a(z)=Re!(z). This gauge transformation gives avariation ofW (z; z0)with the gauge function a(z) as the vari-ational direction a in the variational formulas (21) and (23).Thus analogous to the variational formula (23) we have thatthe variation of W (z; z0) under this gauge transformation isgiven by

W (z; z

0

)

1

2i

I

z

dw a(w)J(w) ; (30)

where the generator J for this variation is to be specific. ThisJ will be a quantum generator which generalizes the classicalgenerator J in (23).

Thus under a gauge transformation with gauge functiona(z) from (27) we have the following variational equation:

0 =

D

W (z; z

0

)

h

a

A(z)+

+

1

2i

I

z

dwa(w)J(w)A(z)

iE

;

(31)

where a

A(z) denotes the variation of the field A(z) in thedirection a(z). From this equation an ansatz of J is that Jsatisfies the following equation:

W (z; z

0

)

h

a

A(z) +

1

2i

I

z

dwa(w)J(w)A(z)

i

= 0 : (32)

From this equation we have the following variationalequation:

a

A(z) =

1

2i

I

z

dwa(w)J(w)A(z) : (33)

This completes the formal calculus of variation. Now(with the above derivation as a guide) we choose the follow-ing equation (34) as one of the equation for defining the gen-erator J :

!

A(z) =

1

2i

I

z

dw !(w)J(w)A(z) ; (34)

where we generalize the direction a(z)=Re!(z) to the ana-lytic direction !(z). (This generalization has the effect of ex-tending the real measure of the pure gauge part of the gaugetheory to include the complex Feynman path integral since itgives the transformation ds ! ids for the integral of theWilson lineW (z; z0).)

Let us now choose one more equation for determine thegenerator J in (34). This choice will be as a gauge fixingcondition. As analogous to the WZW model in conformalfield theory [6567] let us consider a J given by

J(z) := k

0

W

1

(z; z

0

) @

z

W (z; z

0

) ; (35)

where we define @z

= @

x

1

+ i@

x

2 and we set z0= z after thedifferentiation with respect to z; k0

> 0 is a constant whichis fixed when the J is determined to be of the form (35) andthe minus sign is chosen by convention. In the WZW model[65, 67] the J of the form (35) is the generator of the chiral



symmetry of the WZW model. We can write the J in (35) inthe following form:

J(w) =

X

a

J

a

(w) =

X

a

j

a

(w) t

a

: (36)

We see that the generators ta of SU(2) appear in this formof J and this form is analogous to the classical J in (18). Thisshows that this J is a possible candidate for the generator Jin (34).

Since W (z; z0) is constructed by gauge field we need tohave a gauge fixing for the computations related toW (z; z0).Then since the J in (34) and (35) is constructed byW (z; z0)we have that in defining this J as the generator J ofW (z; z0)we have chosen a condition for the gauge fixing. In this paperwe shall always choose this defining equations (34) and (35)for J as the gauge fixing condition.

In summary we introduce the following definition.Definition The generator J of the quantum Wilson lineW (z; z

0

) whose classical version is defined by (8), is an op-erator defined by the two conditions (34) and (35). Remark We remark that the condition (35) first defines Jclassically. Then the condition (34) raises this classical J tothe quantum generator J .

Now we want to show that this generator J in (34) and(35) can be uniquely solved. (This means that the gauge fix-ing condition has already fixed the gauge that the degeneratedegree of freedom of gauge invariance has been eliminated sothat we can carry out computation.)

Let us now solve J . From (11) and (35) the variation !

J

of the generator J in (35) is given by [65, p. 622] and [67]:

!

J = [J; !] k

0

@

z

! : (37)

From (34) and (37) we have that J satisfies the followingrelation of current algebra [6567]:

J

a

(w)J

b

(z) =

k

0

ab

(w z)

2

+

X

c

if

abc

J

c

(z)

(w z)

; (38)

where as a convention the regular term of Ja(w)Jb(z) isomitted. Then by following [6567] from (38) and (36) wecan show that the Jan

in (18) for the corresponding Laurentseries of the quantum generator J satisfy the following Kac-Moody algebra:

[J

a

m

; J

b

n

] = if

abc

J

c

m+n

+ k

0

m

ab

m+n;0

; (39)

where k0

is usually called the central extension or the level ofthe Kac-Moody algebra.Remark Let us also consider the other side of the chiralsymmetry. Similar to the J in (35) we define a generatorJ

0 by:J

0

(z

0

) = k

0

@

z

0

W (z; z

0

)W

1

(z; z

0

) ; (40)

where after differentiation with respect to z0 we set z= z0.

Let us then consider the following formal correlation:

hA(z

0

)W (z; z

0

)i :=

:=

Z

dA

1

dA

2

dZ

dZA(z

0

)W (z; z

0

) e

L

;

(41)

where z is fixed. By an approach similar to the above deriva-tion of (34) we have the following variational equation:

!

0

A(z

0

) =

1

2i

I

z

0

dwA(z

0

)J

0

(w)!

0

(w) ; (42)

where as a gauge fixing we choose the J 0 in (42) be the J 0 in(40). Then similar to (37) we also have

!

0

J

0

= [J

0

; !

0

] k

0

@

z

0

!

0

: (43)

Then from (42) and (43) we can derive the current algebraand the Kac-Moody algebra for J 0 which are of the same formof (38) and (39). From this we have J 0= J .

Now with the above current algebra J and the formula(34) we can follow the usual approach in conformal fieldtheory to derive a quantum Knizhnik-Zamolodchikov (KZ)equation for the product of primary fields in a conformal fieldtheory [6567]. We derive the KZ equation for the productof n Wilson lines W (z; z0). Here an important point is thatfrom the two sides of W (z; z0) we can derive two quantumKZ equations which are dual to each other. These two quan-tum KZ equations are different from the usual KZ equationin that they are equations for the quantum operatorsW (z; z0)while the usual KZ equation is for the correlations of quan-tum operators. With this difference we can follow the usualapproach in conformal field theory to derive the followingquantum Knizhnik-Zamolodchikov equation [65, 66, 68]:

@

z

i

W (z

1

; z

0

1

) W (z

n

; z

0

n

) =

=

e

2

0

k

0

+g

0

P

n

j,i

P

a

t

a

i

t

a

j

z

i

z

j

W (z

1

; z

0

1

) W (z

n

; z

0

n

) ;

(44)

for i=1; : : : ; n where g0

denotes the dual Coxeter numberof a group multiplying with e20

and we have g0

=2e

2

0

forthe group SU(2) (When the gauge group is U(1) we haveg

0

=0). We remark that in (44) we have defined tai

:= t

a and:

t

a

i

t

a

j

W (z

1

; z

0

1

) W (z

n

; z

0

n

) :=W (z

1

; z

0

1

)

[t

a

W (z

i

; z

0

i

)] [t

a

W (z

j

; z

0

j

)] W (z

n

; z

0

n

) :

(45)

It is interesting and important that we also have the fol-lowing quantum Knizhnik-Zamolodchikov equation with re-spect to the z0i

variables which is dual to (44):

@

z

0

i

W (z

1

; z

0

1

) W (z

n

; z

0

n

) =

=

e

2

0

k

0

+g

0

P

n

j,iW (z1; z0

1

) W (z

n

; z

0

n

)

P

a

t

a

i

t

a

j

z

0

j

z

0

i

(46)



for i=1; : : : ; n where we have defined:

W (z

1

; z

0

1

) W (z

n

; z

0

n

)t

a

i

t

a

j

:=W (z

1

; z

0

1

)

[W (z

i

; z

0

i

)t

a

] [W (z

j

; z

0

j

)t

a

] W (z

n

; z

0

n

) :

(47)

Remark From the quantum gauge theory we derive theabove quantum KZ equation in dual form by calculus of vari-ation. This quantum KZ equation in dual form may be con-sidered as a quantum Euler-Lagrange equation or as a quan-tum Yang-Mills equation since it is analogous to the classi-cal Yang-Mills equation which is derived from the classicalYang-Mills gauge theory by calculus of variation.

5 Solving quantum KZ equation in dual form

Let us consider the following product of two quantum Wilsonlines:G(z

1

; z

2

; z

3

; z

4

) :=W (z

1

; z

2

)W (z

3

; z

4

) ; (48)

where the quantum Wilson lines W (z1

; z

2

) and W (z3

; z

4

)

represent two pieces of curves starting at z1

and z3

and endingat z2

and z4

respectively.We have that this product G(z1

; z

2

; z

3

; z

4

) satisfies theKZ equation for the variables z1

, z3

and satisfies the dualKZ equation for the variables z2

and z4

. Then by solvingthe two-variables-KZ equation in (44) we have that a form ofG(z

1

; z

2

; z

3

; z

4

) is given by [6971]:

e

^

t log[(z

1

z

3

)]

C

1

; (49)

where ^t := e2

0

k

0

+g

0

P

a

t

a

t

a andC1

denotes a constant matrixwhich is independent of the variable z1

z

3

.We see that G(z1

; z

2

; z

3

; z

4

) is a multi-valued analyticfunction where the determination of the sign depended onthe choice of the branch.

Similarly by solving the dual two-variable-KZ equationin (46) we have that G is of the form

C

2

e

^

t log[(z

4

z

2

)]

; (50)

where C2

denotes a constant matrix which is independent ofthe variable z4

z

2

.From (49), (50) and letting:

C

1

= Ae

^

t log[(z

4

z

2

)]

; C

2

= e

^

t log[(z

1

z

3

)]

A ; (51)

where A is a constant matrix we have thatG(z1

; z

2

; z

3

; z

4

) isgiven by:

G(z

1

; z

2

; z

3

; z

4

) = e

^

t log[(z

1

z

3

)]

Ae

^

t log[(z

4

z

2

)]

; (52)

where at the singular case that z1

= z

3

we definelog[(z

1

z

3

)] = 0. Similarly for z2

= z

4

.Let us find a form of the initial operator A. We notice

that there are two operators

(z

1

z

3

) := e

^

t log[(z

1

z

3

)]

and

(z

4

z

2

)= e

^

t log[(z

4

z

2

)] acting on the two sides of

A respectively where the two independent variables z1

; z

3

of

are mixed from the two quantum Wilson linesW (z1

; z

2

)

andW (z3

; z

4

) respectively and the the two independent vari-ables z2

; z

4

of

are mixed from the two quantum Wilsonlines W (z1

; z

2

) and W (z3

; z

4

) respectively. From this wedetermine the form of A as follows.

Let D denote a representation of SU(2). Let D(g) rep-resent an element g of SU(2) and let D(g) D(g) denotethe tensor product representation of SU(2). Then in the KZequation we define

[t

a

t

a

][D(g

1

)D(g

1

)] [D(g

2

)D(g

2

)] :=

:= [t

a

D(g

1

)D(g

1

)] [t

a

D(g

2

)D(g

2

)]

(53)

and

[D(g

1

)D(g

1

)] [D(g

2

)D(g

2

)][t

a

t

a

] :=

:= [D(g

1

)D(g

1

)t

a

] [D(g

2

)D(g

2

)t

a

] :

(54)

Then we let U(a) denote the universal enveloping alge-bra where a denotes an algebra which is formed by the Liealgebra su(2) and the identity matrix.

Now let the initial operator A be of the form A1

A

2

A

3

A

4

with Ai

; i=1; : : : ; 4 taking values in U(a). In thiscase we have that in (52) the operator

(z

1

z

3

) acts on Afrom the left via the following formula:

t

a

t

a

A = [t

a

A

1

]A

2

[t

a

A

3

] A

4

: (55)

Similarly the operator

(z

4

z

2

) in (52) acts onA fromthe right via the following formula:

At

a

t

a

= A

1

[A

2

t

a

] A

3

[A

4

t

a

] : (56)

We may generalize the above tensor product of two quan-tum Wilson lines as follows. Let us consider a tensor productof n quantum Wilson lines: W (z1

; z

0

1

) W (z

n

; z

0

n

) wherethe variables zi

, z0i

are all independent. By solving the twoKZ equations we have that this tensor product is given by:

W (z

1

; z

0

1

) W (z

n

; z

0

n

) =

=

Y

ij

(z

i

z

j

)A

Y

ij

(z

0

i

z

0

j

) ;

(57)

whereQ

ij

denotes a product of

(z

i

z

j

) or

(z

0

i

z

0

j

)

for i; j = 1; : : : ; n where i , j. In (57) the initial opera-tor A is represented as a tensor product of operators Aiji

0

j

0 ,i; j; i

0

; j

0

=1; : : : ; n where each Aiji

0

j

0 is of the form of theinitial operator A in the above tensor product of two-Wilson-lines case and is acted by

(z

i

z

j

) or

(z

0

i

z

0

j

) on itstwo sides respectively.

6 Computation of quantum Wilson lines

Let us consider the following product of two quantum Wilsonlines:G(z

1

; z

2

; z

3

; z

4

) :=W (z

1

; z

2

)W (z

3

; z

4

) ; (58)



where the quantum Wilson lines W (z1

; z

2

) and W (z3

; z

4

)

represent two pieces of curves starting at z1

and z3

and endingat z2

and z4

respectively. As shown in the above Section wehave that G(z1

; z

2

; z

3

; z

4

) is given by the following formula:

G(z

1

; z

2

; z

3

; z

4

) = e

^

t log[(z

1

z

3

)]

Ae

^

t log[(z

4

z

2

)]

; (59)

where the product is a 4-tensor.Let us set z2

= z

3

. Then the 4-tensorW (z1

; z

2

)W (z

3

; z

4

)

is reduced to the 2-tensorW (z1

; z

2

)W (z

2

; z

4

). By using (59)the 2-tensorW (z1

; z

2

)W (z

2

; z

4

) is given by:

W (z

1

; z

2

)W (z

2

; z

4

) =

= e

^

t log[(z

1

z

2

)]

A

14

e

^

t log[(z

4

z

2

)]

;

(60)

where A14

=A

1

A

4

is a 2-tensor reduced from the 4-tensorA=A

1

A

2

A

3

A

4

in (59). In this reduction the ^t operatorof = e^t log[(z1z2)] acting on the left side of A1

and A3

in A is reduced to acting on the left side of A1

and A4

in A14

.Similarly the ^t operator of = e^t log[(z4z2)] acting on theright side of A2

and A4

in A is reduced to acting on the rightside of A1

and A4

in A14

.Then since ^t is a 2-tensor operator we have that ^t is as

a matrix acting on the two sides of the 2-tensor A14

whichis also as a matrix with the same dimension as ^t. Thus and are as matrices of the same dimension as the matrixA

14

acting on A14

by the usual matrix operation. Then since^

t is a Casimir operator for the 2-tensor group representationof SU(2) we have that and commute with A14

since and are exponentials of ^t. (We remark that and arein general not commute with the 4-tensor initial operator A.)Thus we have

e

^

t log[(z

1

z

2

)]

A

14

e

^

t log[(z

4

z

2

)]

=

= e

^

t log[(z

1

z

2

)]

e

^

t log[(z

4

z

2

)]

A

14

:

(61)

We let W (z1

; z

2

)W (z

2

; z

4

) be as a representation of thequantum Wilson lineW (z1

; z

4

):

W (z

1

; z

4

) :=W (z

1

; w

1

)W (w

1

; z

4

) =

= e

^

t log[(z

1

w

1

)]

e

^

t log[(z

4

w

1

)]

A

14

:

(62)

This representation of the quantum Wilson lineW (z1

; z

4

)

means that the line (or path) with end points z1

and z4

isspecific that it passes the intermediate point w1

= z

2

. Thisrepresentation shows the quantum nature that the path is notspecific at other intermediate points except the intermediatepoint w1

= z

2

. This unspecification of the path is of the samequantum nature of the Feynman path description of quantummechanics.

Then let us consider another representation of the quan-tum Wilson line W (z1

; z

4

). We consider the three-productW (z

1

; w

1

)W (w

1

; w

2

)W (w

2

; z

4

) which is obtained from the

three-tensor W (z1

; w

1

)W (u

1

; w

2

)W (u

2

; z

4

) by two reduc-tions where zj

, wj

, uj

, j=1; 2 are independent variables.For this representation we have:

W (z

1

; w

1

)W (w

1

; w

2

)W (w

2

; z

4

)= e

^

t log[(z

1

w

1

)]

e

^

t log[(z

1

w

2

)]

e

^

t log[(z

4

w

1

)]

e

^

t log[(z

4

w

2

)]

A

14

:

(63)

This representation of the quantum Wilson lineW (z1

; z

4

)

means that the line (or path) with end points z1

and z4

is spe-cific that it passes the intermediate points w1

and w2

. Thisrepresentation shows the quantum nature that the path is notspecific at other intermediate points except the intermediatepoints w1

and w2

. This unspecification of the path is ofthe same quantum nature of the Feynman path description ofquantum mechanics.

Similarly we may represent W (z1

; z

4

) by path with endpoints z1

and z4

and is specific only to pass at finitely manyintermediate points. Then we let the quantum Wilson lineW (z

1

; z

4

) as an equivalent class of all these representations.Thus we may write:

W (z

1

; z

4

) =W (z

1

; w

1

)W (w

1

; z

4

) =

=W (z

1

; w

1

)W (w

1

; w

2

)W (w

2

; z

4

) =

(64)

Remark Since A14

is a 2-tensor we have that a naturalgroup representation for the Wilson line W (z1

; z

4

) is the 2-tensor group representation of the group SU(2).

7 Representing braiding of curves by quantum Wilsonlines

Consider again the G(z1

; z

2

; z

3

; z

4

) in (58). We have thatG(z

1

; z

2

; z

3

; z

4

) is a multi-valued analytic function where thedetermination of the sign depended on the choice of thebranch.

Let the two pieces of curves represented byW (z1

; z

2

) andW (z

3

; z

4

) be crossing at w. In this case we write W (zi

; z

j

)

asW (zi

; z

j

) = W (z

i

; w)W (w; z

j

) where i = 1; 3, j = 2; 4.Thus we have

W (z

1

; z

2

)W (z

3

; z

4

) =

=W (z

1

; w)W (w; z

2

)W (z

3

; w)W (w; z

4

) :

(65)

If we interchange z1

and z3

, then from (65) we have thefollowing ordering:

W (z

3

; w)W (w; z

2

)W (z

1

; w)W (w; z

4

) : (66)

Now let us choose a branch. Suppose these two curvesare cut from a knot and that following the orientation of aknot the curve represented by W (z1

; z

2

) is before the curverepresented byW (z3

; z

4

). Then we fix a branch such that theproduct in (59) is with two positive signs:

W (z

1

; z

2

)W (z

3

; z

4

) = e

^

t log(z

1

z

3

)

Ae

^

t log(z

4

z

2

)

: (67)



Then if we interchange z1

and z3

we have

W (z

3

; w)W (w; z

2

)W (z

1

; w)W (w; z

4

) =

= e

^

t log[(z

1

z

3

)]

Ae

^

t log(z

4

z

2

)

:

(68)

From (67) and (68) as a choice of branch we have

W (z

3

; w)W (w; z

2

)W (z

1

; w)W (w; z

4

) =

= RW (z

1

; w)W (w; z

2

)W (z

3

; w)W (w; z

4

) ;

(69)

where R = ei^t is the monodromy of the KZ equation. In(69) z1

and z3

denote two points on a closed curve such thatalong the direction of the curve the point z1

is before the pointz

3

and in this case we choose a branch such that the angle ofz

3

z

1

minus the angle of z1

z

3

is equal to .

Remark We may use other representations of the productW (z

1

; z

2

)W (z

3

; z

4

). For example we may use the followingrepresentation:

W (z

1

; w)W (w; z

2

)W (z

3

; w)W (w; z

4

) =

= e

^

t log(z

1

z

3

)

e

2

^

t log(z

1

w)

e

2

^

t log(z

3

w)

Ae

^

t log(z

4

z

2

)

e

2

^

t log(z

4

w)

e

2

^

t log(z

2

w)

:

(70)

Then the interchange of z1

and z3

changes only z1

z

3

to z3

z

1

. Thus the formula (69) holds. Similarly all otherrepresentations of W (z1

; z

2

)W (z

3

; z

4

) will give the sameresult.

Now from (69) we can take a convention that the order-ing (66) represents that the curve represented by W (z1

; z

2

)

is up-crossing the curve represented byW (z3

; z

4

) while (65)represents zero crossing of these two curves.

Similarly from the dual KZ equation as a choice of branchwhich is consistent with the above formula we have

W (z

1

; w)W (w; z

4

)W (z

3

; w)W (w; z

2

) =

=W (z

1

; w)W (w; z

2

)W (z

3

; w)W (w; z

4

)R

1

;

(71)

where z2

is before z4

. We take a convention that the order-ing in (71) represents that the curve represented byW (z1

; z

2

)

is under-crossing the curve represented by W (z3

; z

4

). Herealong the orientation of a closed curve the piece of curverepresented by W (z1

; z

2

) is before the piece of curve rep-resented byW (z3

; z

4

). In this case since the angle of z3

z

1


z

3

is equal to we have that the an-gle of z4

z

2


z

4

is also equal to and this gives the R1 in this formula (71).

From (69) and (71) we have

W (z

3

; z

4

)W (z

1

; z

2

) = RW (z

1

; z

2

)W (z

3

; z

4

)R

1

; (72)

where z1

and z2

denote the end points of a curve which isbefore a curve with end points z3

and z4

. From (72) wesee that the algebraic structure of these quantum Wilson linesW (z; z

0

) is analogous to the quasi-triangular quantum group[66, 69].

8 Computation of quantum Dirac-Wilson loop

Consider again the quantum Wilson lineW (z1

; z

4

) given byW (z

1

; z

4

)=W (z

1

; z

2

)W (z

2

; z

4

). Let us set z1

= z

4

. In thiscase the quantum Wilson line forms a closed loop. Now in(61) with z1

= z

4

we have that the quantities e^t log(z1z2)

and e^t log(z1z2) which come from the two-side KZ equa-tions cancel each other and from the multi-valued property ofthe log function we have:

W (z

1

; z

1

) = R

N

A

14

; N = 0;1;2; : : : (73)

where R= ei^t is the monodromy of the KZ equation [69].

Remark It is clear that if we use other representation of thequantum Wilson loop W (z1

; z

1

) (such as the representationW (z

1

; z

1

)=W (z

1

; w

1

)W (w

1

; w

2

)W (w

2

; z

1

)) then we willget the same result as (73).

Remark For simplicity we shall drop the subscript of A14

in (73) and simply write A14

=A.

9 Winding number of Dirac-Wilson loop as quanti-zation

We have the equation (73) where the integerN is as a windingnumber. Then when the gauge group is U(1) we have

W (z

1

; z

1

) = R

N

U(1)

A ; (74)

where RU(1)

denotes the monodromy of the KZ equation forU(1). We have

R

N

U(1)

= e

iN

e

2

0

k

0

+g

0

; N = 0;1;2; : : : (75)

where the constant e0

denotes the bare electric charge (andg

0

=0 for U(1) group). The winding number N is as thequantization property of photon. We show in the follow-ing Section that the Dirac-Wilson loop W (z1

; z

1

) with theabelian U(1) group is a model of the photon.

10 Magnetic monopole is a photon with a specific fre-quency

We see that the Dirac-Wilson loop is an exactly solvable non-linear observable. Thus we may regard it as a quantum solitonof the above gauge theory. In particular for the abelian gaugetheory with U(1) as gauge group we regard the Dirac-Wilsonloop as a quantum soliton of the electromagnetic field. Wenow want to show that this soliton has all the properties ofphoton and thus we may identify it with the photon.

First we see that from (75) it has discrete energy levels oflight-quantum given by

h := N

e

2

0

k

0

; N = 0; 1; 2; 3; : : : (76)



where h is the Plancks constant; denotes a frequency andthe constant k0

> 0 is determined from this formula. Thisformula is from the monodromy RU(1)

for the abelian gaugetheory. We see that the Plancks constant h comes out fromthis winding property of the Dirac-Wilson loop. Then sincethis Dirac-Wilson loop is a loop we have that it has the polar-ization property of light by the right hand rule along the loopand this polarization can also be regarded as the spin of pho-ton. Now since this loop is a quantum soliton which behavesas a particle we have that this loop is a basic particle of theabove abelian gauge theory where the abelian gauge propertyis considered as the fundamental property of electromagneticfield. This shows that the Dirac-Wilson loop has properties ofphoton. We shall later show that from this loop model of pho-ton we can describe the absorption and emission of photon byan electron. This property of absorption and emission is con-sidered as a basic principle of the light-quantum hypothesisof Einstein [1]. From these properties of the Dirac-Wilsonloop we may identify it with the photon.

On the other hand from Diracs analysis of the magneticmonopole we have that the property of magnetic monopolecomes from a closed line integral of vector potential of theelectromagnetic field which is similar to the Dirac-Wilsonloop [4]. Now from this Dirac-Wilson loop we can definethe magnetic charge q and the minimal magnetic charge qmin

which are given by:

eq := enq

min

:= n

e

e

0

n

n

m

e

0

k

0

; n = 0; 1; 2; 3; : : : (77)

where e :=ne

e

0

is as the observed electric charge for somepositive integer ne

; and qmin

:=

n

m

e

0

k

0

for some positive in-teger nm

and we write N =nne

n

m

; n = 0; 1; 2; 3; : : : (byabsorbing the constant k0

to e20

we may let k0

=1).This shows that the Dirac-Wilson loop gives the property

of magnetic monopole for some frequencies. Since this loopis a quantum soliton which behaves as a particle we have thatthis Dirac-Wilson loop may be identified with the magneticmonopole for some frequencies. Thus we have that photonmay be identified with the magnetic monopole for some fre-quencies. With this identification we have the following in-teresting conclusion: Both the energy quantization of elec-tromagnetic field and the charge quantization property comefrom the same property of photon. Indeed we have:

nh

1

:= n

n

e

n

m

e

2

0

k

0

= eq ; n = 0; 1; 2; 3; : : : (78)

where 1

denotes a frequency. This formula shows that theenergy quantization gives the charge quantization and thusthese two quantizations are from the same property of thephoton when photon is modelled by the Dirac-Wilson loopand identified with the magnetic monopole for some frequen-cies. We notice that between two energy levels neqmin

and(n+ 1)eq

min

there are other energy levels which may be re-garded as the excited states of a particle with charge ne.

11 Nonlinear loop model of electron

In this Section let us also give a loop model to the electron.This loop model of electron is based on the above loop modelof the photon. From the loop model of photon we also con-struct an observable which gives mass to the electron and isthus a mass mechanism for the electron.

LetW (z; z) denote a Dirac-Wilson loop which representsa photon. Let Z denotes the complex variable for electron in(3). We then consider the following observable:

W (z; z)Z : (79)

Since W (z; z) is solvable we have that this observableis also solvable where in solving W (z; z) the variable Z isfixed. We let this observable be identified with the electron.Then we consider the following observable:

Z

W (z; z)Z : (80)

This observable is with a scalar factor ZZ where Z de-notes the complex conjugate ofZ and we regard it as the massmechanism of the electron (79). For this observable we modelthe energy levels with specific frequencies ofW (z; z) as themass levels of electron and the massm of electron is the low-est energy level h1

with specific frequency 1

of W (z; z)and is given by:mc

2

= h

1

; (81)

where c denotes the constant of the speed of light and thefrequency 1

is given by (78). From this model of the massmechanism of electron we have that electron is with massmwhile photon is with zero mass because there does not havesuch a mass mechanism ZW (z; z)Z for the photon. Fromthis definition of mass we have the following formula relat-ing the observed electric charge e of electron, the magneticcharge qmin

of magnetic monopole and the mass m of elec-tron:mc

2

= eq

min

= h

1

: (82)

By using the nonlinear model W (z; z)Z to represent anelectron we can then describe the absorption and emission ofa photon by an electron where photon is as a parcel of energydescribed by the loop W (z; z), as follows. Let W (z; z)Zrepresents an electron and let W1

(z

1

; z

1

) represents a pho-ton. Then the observableW1

(z

1

; z

1

)W (z; z)Z represents anelectron having absorbed the photon W1

(z

1

; z

1

). This prop-erty of absorption and emission is as a basic principle of thehypothesis of light-quantum stated by Einstein [1]. Let usquote the following paragraph from [1]:

. . . First, the light-quantum was conceived of as a par-cel of energy as far as the properties of pure radiation(no coupling to matter) are concerned. Second, Ein-stein made the assumption he call it the heuristicprinciple that also in its coupling to matter (that is,in emission and absorption), light is created or anni-hilated in similar discrete parcels of energy. That, I



believe, was Einsteins one revolutionary contributionto physics. It upset all existing ideas about the interac-tion between light and matter. . .

12 Photon with specific frequency carries electric andmagnetic charges

In this loop model of photon we have that the observed elec-tric charge e :=ne

e

0

and the magnetic charge qmin

are car-ried by the photon with some specific frequencies. Let ushere describe the physical effects from this property of pho-ton that photon with some specific frequency carries the elec-tric and magnetic charge. From the nonlinear model of elec-tron we have that an electronW (z; z)Z also carries the elec-tric charge when a photon W (z; z) carrying the electric andmagnetic charge is absorbed to form the electron W (z; z)Z.This means that the electric charge of an electron is from theelectric charge carried by a photon. Then an interaction (asthe electric force) is formed between two electrons (with theelectric charges).

On the other hand since photon carries the constant e20

ofthe bare electric charge e0

we have that between two photonsthere is an interaction which is similar to the electric forcebetween two electrons (with the electric charges). This in-teraction however may not be of the same magnitude as theelectric force with the magnitude e2 since the photons maynot carry the frequency for giving the electric and magneticcharge. Then for stability such interaction between two pho-tons tends to give repulsive effect to give the diffusion phe-nomenon among photons.

Similarly an electronW (z; z)Z also carries the magneticcharge when a photonW (z; z) carrying the electric and mag-netic charge is absorbed to form the electronW (z; z)Z. Thismeans that the magnetic charge of an electron is from themagnetic charge carried by a photon. Then a closed-loop in-teraction (as the magnetic force) may be formed between twoelectrons (with the magnetic charges).

On the other hand since photon carries the constant e20

ofthe bare electric charge e0

we have that between two photonsthere is an interaction which is similar to the magnetic forcebetween two electrons (with the magnetic charges). This in-teraction however may not be of the same magnitude as themagnetic force with the magnetic charge qmin

since the pho-tons may not carry the frequency for giving the electric andmagnetic charge. Then for stability such interaction betweentwo photons tends to give repulsive effect to give the diffusionphenomenon among photons.

13 Statistics of photons and electrons

The nonlinear modelW (z; z)Z of an electron gives a relationbetween photon and electron where the photon is modelledbyW (z; z) which is with a specific frequency forW (z; z)Z

to be an electron, as described in the above Sections. Wewant to show that from this nonlinear model we may also de-rive the required statistics of photons and electrons that pho-tons obey the Bose-Einstein statistics and electrons obey theFermi-Dirac statistics. We have thatW (z; z) is as an operatoracting on Z. LetW1

(z; z) be a photon. Then we have that thenonlinear model W1

(z; z)W (z; z)Z represents that the pho-tonW1

(z; z) is absorbed by the electronW (z; z)Z to form anelectron W1

(z; z)W (z; z)Z. Let W2

(z; z) be another pho-ton. The we have that the modelW1

(z; z)W

2

(z; z)W (z; z)Z

again represents an electron where we have:

W

1

(z; z)W

2

(z; z)W (z; z)Z =

=W

2

(z; z)W

1

(z; z)W (z; z)Z :

(83)

More generally the modelQ

N

n=1

W

n

(z; z)W (z; z)Z rep-resents that the photons Wn

(z; z); n=1; 2; : : : ; N areabsorbed by the electron W (z; z)Z. This model shows thatidentical (but different) photons can appear identically and itshows that photons obey the Bose-Einstein statistics. Fromthe polarization of the Dirac-Wilson loop W (z; z) we mayassign spin 1 to a photon represented byW (z; z).

Let us then consider statistics of electrons. The observ-able ZW (z; z)Z gives mass to the electron W (z; z)Z andthus this observable is as a scalar and thus is assigned withspin 0. As the observableW (z; z)Z is betweenW (z; z) andZ

W (z; z)Z which are with spin 1 and 0 respectively we thusassign spin 12

to the observable W (z; z)Z and thus electronrepresented by this observableW (z; z)Z is with spin 12

.Then letZ1

andZ2

be two independent complex variablesfor two electrons and let W1

(z; z)Z

1

and W2

(z; z)Z

2

repre-sent two electrons. Let W3

(z; z) represents a photon. Thenthe model W3

(z; z)(W

1

(z; z)Z

1

+W

2

(z; z)Z

2

) means thattwo electrons are in the same state that the operatorW3

(z; z)

is acted on the two electrons. However this model means thata photon W (z; z) is absorbed by two distinct electrons andthis is impossible. Thus the modelsW3

(z; z)W

1

(z; z)Z

1

andW

3

(z; z)W

2

(z; z)Z

2

cannot both exist and this means thatelectrons obey Fermi-Dirac statistics.

Thus this nonlinear loop model of photon and electrongives the required statistics of photons and electrons.

14 Photon propagator and quantum photon propagator

Let us then investigate the quantum Wilson line W (z0

; z)

with U(1) group where z0

is fixed for the photon field. Wewant to show that this quantum Wilson line W (z0

; z) maybe regarded as the quantum photon propagator for a photonpropagating from z0

to z.As we have shown in the above Section on computation

of quantum Wilson line; to compute W (z0

; z) we need towrite W (z0

; z) in the form of two (connected) Wilson lines:W (z

0

; z)=W (z

0

; z

1

)W (z

1

; z) for some z1

point. Then we



have:

W (z

0

; z

1

)W (z

1

; z) =

= e

^

t log[(z

1

z

0

)]

Ae

^

t log[(zz

1

)]

;

(84)

where ^t= e2

0

k

0

for the U(1) group (k0

is a constant and we

may for simplicity let k0

=1) where the term e^t log[(zz0)]

is obtained by solving the first form of the dual form of the KZequation and the term e^t log[(z0z)] is obtained by solvingthe second form of the dual form of the KZ equation.

Then we may writeW (z0

; z) in the following form:

W (z

0

; z) =W (z

0

; z

1

)W (z

1

; z) = e

^

t log

(z

1

z

0

)

(zz

1

)

A : (85)

Let us fix z1

with z such that:

jz

1

z

0

j

jz z

1

j

=

r

1

n

2

e

(86)

for some positive integer ne

such that r1

6 n2e

; and we letz be a point on a path of connecting z0

and z1

and then aclosed loop is formed with z as the starting and ending point.(This loop can just be the photon-loop of the electron in thiselectromagnetic interaction by this photon propagator (85).)Then (85) has a factor e20

log

r

1

n

2

e

which is the fundamentalsolution of the two dimensional Laplace equation and is anal-ogous to the fundamental solution e2

r

(where e := e0

n

e

de-notes the observed (renormalized) electric charge and r de-notes the three dimensional distance) of the three dimensionalLaplace equation for the Coulombs law. Thus the opera-tor W (z0

; z)=W (z

0

; z

1

)W (z

1

; z) in (85) can be regardedas the quantum photon propagator propagating from z0

to z.We remark that when there are many photons we may in-

troduce the space variable x as a statistical variable via theLorentz metric ds2= dt2 dx2 to obtain the Coulombs lawe

2

r

from the fundamental solution e20

log

r

1

n

2

e

as a statisticallaw for electricity (We shall give such a space-time statisticslater).

The quantum photon propagator (85) gives a repulsive ef-fect since it is analogous to the Coulombs law e2

r

. On theother hand we can reverse the sign of ^t such that this photonoperator can also give an attractive effect:

W (z

0

; z) =W (z

0

; z

1

)W (z

1

; z) = e

^

t log

(zz

1

)

(z

1

z

0

)

A ; (87)

where we fix z1

with z0

such that:

jz z

1

j

jz

1

z

0

j

=

r

1

n

2

e

(88)

for some positive integer ne

such that r1

> n2e

; and we againlet z be a point on a path of connecting z0

and z1

and then aclosed loop is formed with z as the starting and ending point.(This loop again can just be the photon-loop of the electronin this electromagnetic interaction by this photon propagator

(85).) Then (87) has a factor e20

log

r

1

n

2

e

which is the funda-mental solution of the two dimensional Laplace equation andis analogous to the attractive fundamental solution e2

r

of thethree dimensional Laplace equation for the Coulombs law.

Thus the quantum photon propagator in (85), and in (87),can give repulsive or attractive effect between two points z0

and z for all z in the complex plane. These repulsive or at-tractive effects of the quantum photon propagator correspondto two charges of the same sign and of different sign respec-tively.

On the other hand when z= z0

the quantum Wilson lineW (z

0

; z

0

) in (85) which is the quantum photon propagatorbecomes a quantum Wilson loop W (z0

; z

0

) which is identi-fied as a photon, as shown in the above Sections.

Let us then derive a form of photon propagator from thequantum photon propagator W (z0

; z). Let us choose a pathconnecting z0

and z= z(s). We consider the following path:

z(s) = z

1

+ a

0

(s

1

s)e

i

1

(s

1

s)

+

+ (s s

1

)e

i

1

(s

1

s)

;

(89)

where 1

> 0 is a parameter and z(s0

)= z

0

for some propertime s0

; and a0

is some complex constant; and is a stepfunction given by (s)= 0 for s< 0, (s)= 1 for s > 0. Thenon this path we have:

W (z

0

; z) =

=W (z

0

; z

1

)W (z

1

; z) = e

^

t log

(zz

1

)

(z

1

z

0

)

A =

= e

^

t log

a

0

[(ss

1

)e

i

1

(s

1

s)

+(s

1

s)e

i

1

(s

1

s)

]

(z

1

z

0

)

A =

= e

^

t log b[(ss

1

)e

i

1

(s

1

s)

+(s

1

s)e

i

1

(s

1

s)

]

A =

= b

0

(s s

1

)e

i

^

t

1

(s

1

s)

+ (s

1

s)e

i

^

t

1

(s

1

s)

A

(90)

for some complex constants b and b0

. From this chosen ofthe path (89) we have that the quantum photon propagator isproportional to the following expression:

1

2

1

(s s

1

)e

i

1

(ss

1

)

+ (s

1

s)e

i

1

(ss

1

)

(91)

where we define 1

=

^

t

1

= e

2

0

1

> 0. We see that this isthe usual propagator of a particle x(s) of harmonic oscillatorwith mass-energy parameter 1

> 0 where x(s) satisfies thefollowing harmonic oscillator equation:

d

2

x

ds

2

=

2

1

x(s) : (92)

We regard (91) as the propagator of a photon with mass-energy parameter 1

. Fourier transforming (91) we have thefollowing form of photon propagator:

i

k

2

E

1

; (93)



where we use the notation kE

(instead of the notation k) todenote the proper energy of photon. We shall show in thenext Section that from this photon propagator by space-timestatistics we can get a propagator with the kE

replaced bythe energy-momemtum four-vector k which is similar to theFeynman propagator (with a mass-energy parameter 1

> 0).We thus see that the quantum photon propagator W (z0

; z)

gives a classical form of photon propagator in the conven-tional QED theory.

The

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