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QUANTUM FIELD THEORYa cyclist tourlecture notes - Spring 2005

Predrag Cvitanovic

Incomplete and unassuming notes for the Georgia Tech graduate quantumfield theory course (PHYS-7147 - Spring 2005), version 3.3, Feb 14 2005.

What reviewers say. N. Bohr - The most important work since thatSchrodinger killed the cat. R.P. Feynman - Great doorstop!

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Contents

1 Path integral formulation of Quantum Mechanics 3

1.1 Quantum mechanics: a brief review . . . . . . . . . . . . . 41.2 Matrix-valued functions . . . . . . . . . . . . . . . . . . . . 61.3 Short time propagation . . . . . . . . . . . . . . . . . . . . 81.4 Path integral . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Free propagation . . . . . . . . . . . . . . . . . . . . . . . . 10

2 WKB quantization 11

2.1 WKB ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Method of stationary phase . . . . . . . . . . . . . . . . . . 132.3 WKB quantization. . . . . . . . . . . . . . . . . . . . . . . 142.4 Beyond the quadratic saddle point . . . . . . . . . . . . . . 16resume 18 - references 18 - exercises 20

3 Lattice field theory 21

3.1 Saddle-point expansions are asymptotic . . . . . . . . . . . 223.2 Path integrals . . . . . . . . . . . . . . . . . . . . . . . . . 253.3 Field theory - setting up the notation. . . . . . . . . . . . . 263.4 Saddle-point expansions . . . . . . . . . . . . . . . . . . . . 27

3.5 Free field theory . . . . . . . . . . . . . . . . . . . . . . . . 303.6 Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . 303.7 Propagator in the space representation . . . . . . . . . . . . 323.8 Periodic lattices . . . . . . . . . . . . . . . . . . . . . . . . 363.9 Continuum field theory . . . . . . . . . . . . . . . . . . . . 403.10 Summary and outlook . . . . . . . . . . . . . . . . . . . . . 40exercises 42

A Group theory 43

A.1 Invariant matrices and reducibility . . . . . . . . . . . . . . 43

Index 47

1

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Chapter 1

Path integral formulation of

Quantum Mechanics

1.1 Quantum mechanics: a brief review . . . . . . . 4

1.2 Matrix-valued functions . . . . . . . . . . . . . . 6

1.3 Short time propagation . . . . . . . . . . . . . . . 8

1.4 Path integral . . . . . . . . . . . . . . . . . . . . . 8

1.5 Free propagation . . . . . . . . . . . . . . . . . . . 10

We introduce Feynman path integral and construct semiclassical approxi-mations to quantum propagators and Greens functions.

Have: the Schrodinger equation, that is the (infinitesimal time) evolu-tion law for any quantum wavefunction:

i

t(t) = H(t) . (1.1)

Want: (t) at any finite time, given the initial wave function (0).As the Schrodinger equation (1.1) is a linear equation, the solution can

be written down immediately:

(t) = ei

Ht (0) , t 0 .Fine, but what does this mean? We can be a little more explicit; usingthe configuration representation (q, t) = q|(t) and the configurationrepresentation completness relation

1 = dqD

|q

q

|(1.2)

we have

(q, t) = q|(t) =

dq q|e i Ht |qq|(0) , t 0 . (1.3)

In sect. 1.1 we will solve the problem and give the explicit formula (1.9)for the propagator. However, this solution is useless - it requires knowingall quantum eigenfunctions, that is it is a solution which we can implementprovided that we have already solved the quantum problem. In sect. 1.4 we

shall derive Feynmans path integral formula for K(q, q, t) = q|e i Ht |q.

3

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4CHAPTER 1. PATH INTEGRAL FORMULATION OF QUANTUM MECHANIC

1.1 Quantum mechanics: a brief review

We start with a review of standard quantum mechanical concepts prerequi-site to the derivation of the semiclassical trace formula: Schrodinger equa-tion, propagator, Greens function, density of states.

In coordinate representation the time evolution of a quantum mechanicalwave function is governed by the Schrodinger equation (1.1)

i

t(q, t) = H(q,

i

q)(q, t), (1.4)

where the Hamilton operator H(q, iq) is obtained from the classicalHamiltonian by substitution p iq. Most of the Hamiltonians weshall consider here are of form

H(q, p) = T(p) + V(q) , T(p) =p2

2m, (1.5)

appropriate to a particle in a D-dimensional potential V(q). If, as is oftenthe case, a Hamiltonian has mixed terms of pq, consult any book on field

theory. We are interested in finding stationary solutions(q, t) = eiEnt/n(q) = q|eiHt/|n ,

of the time independent Schrodinger equation

H(q) = E(q) , (1.6)

where En, |n are the eigenenergies, respectively eigenfunctions of the sys-tem. For bound systems the spectrum is discrete and the eigenfunctionsform an orthonormal

dqD n(q)m(q) =

dqD n|qq|m = nm (1.7)

and completen

n(q)n(q

) = (q q) ,

n

|nn| = 1 (1.8)

set of Hilbert space functions. For simplicity we will assume that thesystem is bound, although most of the results will be applicable to opensystems, where one has complex resonances instead of real energies, andthe spectrum has continuous components.

A given wave function can be expanded in the energy eigenbasis

(q, t) = n cneiEnt/n(q) ,

where the expansion coefficient cn is given by the projection of the initialwave function onto the nth eigenstate

cn =

dqD n(q)(q, 0) = n|(0).

The evolution of the wave function is then given by

(q, t) =

n

n(q)eiEnt/

dq

Dn(q)(q, 0).

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1.1. QUANTUM MECHANICS: A BRIEF REVIEW 5

Figure 1.1: Propagation from q to q in timet = t+ t receives contributions from all pathsfrom q to q for time t followed by propagationfrom q to q in time t.

We can write this as

(q, t) =

dq

DK(q, q, t)(q, 0),

K(q, q, t) =

nn(q) e

iEnt/n(q)

= q|e i Ht |q = n

q|neiEnt/n|q , (1.9)

where the kernel K(q, q, t) is called the quantum evolution operator, orthe propagator. Applied twice, first for time t1 and then for time t2, itpropagates the initial wave function from q to q, and then from q to q

K(q, q, t1 + t2) =

dq K(q, q, t2)K(q, q, t1) (1.10)

forward in time, hence the name propagator, see figure 1.1. In non-relativistic quantum mechanics the range of q is infinite, meaning that thewave can propagate at any speed; in relativistic quantum mechanics this is

rectified by restricting the forward propagation to the forward light cone.Since the propagator is a linear combination of the eigenfunctions of the

Schrodinger equation, the propagator itself also satisfies the Schrodingerequation

i

tK(q, q, t) = H(q,

i

q)K(q, q, t) . (1.11)

The propagator is a wave function defined for t 0 which starts out att = 0 as a delta function concentrated on q

limt0+

K(q, q, t) = (q q) . (1.12)

This follows from the completeness relation (1.8).

The time scales of atomic, nuclear and subnuclear processes are tooshort for direct observation of time evolution of a quantum state. For thisreason, in most physical applications one is interested in the long time be-havior of a quantum system.

In the t limit the sharp, well defined quantity is the energy E (orfrequency), extracted from the quantum propagator via its Laplace/Fouriertransform, the energy dependent Greens function

G(q, q, E+i) =1

i

0

dt ei

Et

tK(q, q, t) =

n

n(q)n(q)E En + i .(1.13)

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6CHAPTER 1. PATH INTEGRAL FORMULATION OF QUANTUM MECHANIC

Here is a small positive number, ensuring that the propagation is forwardin time (and the very existence of the integral).

This completes our lightning review of quantum mechanics.

Feynman arrived to his formulation of quantum mechanics by thinkingof figure 1.1 as a multi-slit experiment, with an infinitesimal slit placedat everyq point. The Feynman path integral follows from two observations:

1. For short time the propagator can be expressed in terms of classicalfunctions (Dirac).

2. The group property (1.10) enables us to represent finite time evolutionas a product of many short time evolution steps (Feynman).

1.2 Matrix-valued functions

How are we to think of the quantum operator

H = T + V , T = p2/2m , V = V(q) , (1.14)

corresponding to the classical Hamiltonian (1.5)?

Whenever you are confused about an operator, think matrix. Ex-pressed in terms of basis functions, the propagator is an infinite-dimensionalmatrix; if we happen to know the eigenbasis of the Hamiltonian, (1.9) isthe propagator diagonalized. Of course, if we knew the eigenbasis the prob-lem would have been solved already. In real life we have to guess that somecomplete basis set is good starting point for solving the problem, and gofrom there. In practice we truncate such matrix representations to finite-dimensional basis set, so it pays to recapitulate a few relevant facts about

matrix algebra.The derivative of a (finite-dimensional) matrix is a matrix with elements

A(x) =dA(x)

dx, Aij(x) =

d

dxAij(x) . (1.15)

Derivatives of products of matrices are evaluated by the chain rule

d

dx(AB) =

dA

dxB + A

dB

dx. (1.16)

A matrix and its derivative matrix in general do not commute

ddx A2 = dAdx A + A dAdx . (1.17)

The derivative of the inverse of a matrix follows from ddx (AA1) = 0:

d

dxA1 = 1

A

dA

dx

1

A. (1.18)

As a single matrix commutes with itself, any function of a single variablethat can be expressed in terms of additions and multiplications generalizesto a matrix-valued function by replacing the variable by the matrix.

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1.2. MATRIX-VALUED FUNCTIONS 7

In particular, the exponential of a constant matrix can be defined eitherby its series expansion, or as a limit of an infinite product:

eA =

k=0

1

k!Ak , A0 = 1 (1.19)

= limN

1 +

1

NA

N

(1.20)

The first equation follows from the second one by the binomial theorem, sothese indeed are equivalent definitions. For finite N the two expressionsdiffer by orderO(N2). That the terms of order O(N2) or smaller do notmatter is easy to establish for A x, the scalar case. This follows fromthe bound

1 +x

N

N 0, but forg < 0, the potential is unbounded for large , and the integrand explodes.Hence the partition function in not analytic at the g = 0 point.

Is the whole enterprise hopeless? As we shall now show, even though di-

vergent, the perturbation series is an asymptotic expansion, and an asymp-totic expansion can be extremely good [3.9]. Consider the residual errorafter inclusion of the first n perturbative corrections:

Rn =

Z(g) n

m=0

gmZm

=

d

2e

2/2

eg4/4 n

m=0

1

m!

g

4

m4m

d

2e

2/2 1

(n + 1)! g4

4 n+1

= gn+1 |Zn+1| . (3.7)

The inequality follows from the convexity of exponentials, a generalization 3.5page 42

of the inequalityex 1+x. The error decreases as long as gn |Zn| decreases.From (3.6) the minimum is reached at4g nmin 1, with the minimum error

gnZn|min

4g

e1/4g. (3.8)

As illustrated by the figure 3.1, a perturbative expansion can be, for allpractical purposes, very accurate. In QED such argument had led Dyson tosuggest that the QED perturbation expansions are good to nmin 1/ 137

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24 CHAPTER 3. LATTICE FIELD THEORY

Figure 3.1: Plot of the saddle-point estimate of Zn vs. the exact result (3.5) forg = 0.1, g = 0.02, g = 0.01.

terms. Due to the complicated relativistic, spinorial and gauge invariancestructure of perturbative QED, there is not a shred of evidence that this isso. The very best calculations that humanity has been able to perform sofar stop at n 5.

Commentary

Remark 3.1 Asymptotic series.

The Taylor expansion in g fails, as g is precisely on the border ofanalyticity. The situation can sometimes be rescued by a Borelre-summation.

If you really care, an asymptotic series can be improved by re-sumations beyond all orders, a technically daunting task (seeM. Berrys papers on such topics as re-summation of the Weylseries for quantum billiards).

Pairs of nearby and coalescing saddles should be treated by uni-form approximations, where the Airy integrals

Z0[J] =1

2i

C

dx ex3/3!+Jx

play the role the Gaussian integrals play for isolated saddles [3.4].In case at hand, the phase transition c = 0 c = 0 is aquartic inflection of this type, and in the Fourier representationof the partition function one expects instead of |det M| 12 explicitdependence on the momentum k

1

4 . Whether anyone has triedto develop a theory of the critical regime in this way I do notknow.

If there are symmetries that relate terms in perturbation expan-sions, a perturbative series might be convergent. For example,individual Feynman diagrams in QED are not gauge invariant,

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3.2. PATH INTEGRALS 25

Figure 3.2: In the classical 0 limit thepath integral (1.33) is dominated by the minimaof the integrands exponent. The location qc

of a minimum is determined by the extremumcondition S[q

c] + J = 0 .

only their sums are, and QED n expansions might still turnout to be convergent series [3.10].

Expansions in which the field is replaced by N copies of theoriginal field are called 1/N expansions. The perturbative coef-ficients in such expansions are convergent term by term in 1/N.

3.2 Path integrals

The path integral (1.33) is an ordinary multi-dimensional integral. In theclassical 0, the action is large (high price of straying from the beatenpath) almost everywhere, except for some localized regions of the q-space.Highly idealized, the action looks something like the sketch in figure 3.2 (inorder to be able to draw this on a piece of paper, we have supressed a largenumber of q coordinates).

Such integral is dominated by the minima of the action. The minimumvalue S[q] configurations qc are determined by the zero-slope, saddle-pointcondition

d

d

S[qc] + J = 0 . (3.9)

The term saddle refers to the general technique of evaluating such inte-grals for complexq; in the statistical mechanics applications qc are locationsof the minima ofS[q], not the saddles. If there is a number of minima, onlythe one (or the nc minima related by a discrete symmetry) with the lowestvalue of S[qc]qc J dominates the path integral in the low temperaturelimit. The zeroth order, classical approximation to the partition sum (1.33)is given by the extremal configuration alone

Z[J] = eW[J]

c

eWc[J] = eWc[J]+lnnc

Wc[J] = S[qc] + qc

J . (3.10)

In thesaddlepoint approximation the corrections due to the fluctuationsin the qc neighborhood are obtained by shifting the origin of integration to

q qc + q ,the position of the c-th minimum of S[q] q J, and expanding S[q] in aTaylor series around qc.

For our purposes it will be convenient to separate out the quadratic partS0[q], and collect all terms higher than bilinear in q into an interaction

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term SI[q]

S0[q] =

q

M1

,q ,

SI[q] = ( ),,qqq + . (3.11)Rewrite the partition sum (1.33) as

eW[J]

= eWc[J] [dq]e 12 qTM1q+SI[q] .

As the expectation value of any analytic function

g(q) =

gn1n2...qn11 q

n22 /n1!n2!

can be recast in terms of derivatives with respect to J[dq]g(q)e

12

qTM1q = g( ddJ

)

[dq]e

12

qTM1q+qJ

J=0

,

we can move SI[q] outside of the integration, and evaluate the Gaussianintegral in the usual way

3.3page 42

eW[J]

= eWc[J]

eSI[

d

dJ] [dq]e 12 qTM1q+qJ

J=0

= |det M| 12 eWc[J]eSI[ ddJ] e 12 JTMJ

J=0. (3.12)

M is invertible only if the minima in figure 3.2 are isolated, and M1

has no zero eigenvalues. The marginal case would require going beyondthe Gaussian saddlepoints studied here, typically to the Airy-function typestationary points [3.4]. In the classical statistical mechanics S[q] is a real-valued function, the extremum ofS[q] at the saddlepointqc is the minimum,all eigenvalues ofM are strictly positive, and we can drop the absolute valuebrackets | | in (3.12).

2.4

page 20

Expanding the exponentials and evaluating the d

dJ

derivatives in (3.12)yields the fluctuation corrections as a power series in 1/ = T.

The first correction due to the fluctuations in the qc neighborhood isobtained by approximating the bottom of the potential in figure 3.2 by aparabola, that is, keeping only the quadratic terms in the Taylor expansion(3.11).

3.3 Field theory - setting up the notation

The partition sum for a lattice field theory defined by a Hamiltonian H[]is

Z[J] = [d]e(H[]h)[d] =

d12

d22

,

where = 1/T is the inverse temperature, and h is an external probe thatwe can twiddle at will site-by-site. For a theory of the Landau type theHamiltonian is

HL[] = r2

+c

2 + u

Nd=1

4 . (3.13)

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3.4. SADDLE-POINT EXPANSIONS 27

Unless stated otherwise, we shall assume the repeated index summation con-vention throughout. We find it convenient to bury now some factors of

2 into the definition of Z[J] so they do not plague us later on whenwe start evaluating Gaussian integrals. Rescaling (const) changes[d] (const)N[d], a constant prefactor in Z[J] which has no effect onaverages. Hence we can get rid of one of the Landau parameters r, u, andc by rescaling. The accepted normalization convention is to set the gradient

term to 12()2 by h c1/2h, c1/2, and theHL in (3.13) is replacedby

H[] = 12

+m202

+g04!

4

m20 =r

c, g0 = 4!

u

c2. (3.14)

Dragging factors of around is also a nuisance, so we absorb them bydefining the action and the sources as

S[] = H[] , J = h .

The actions we learn to handle here are of form

S[] = 12

(M1) + SI[] ,

SI[] =1

3!123 1 2 3 +

1

4!1234 1 2 3 4 + . (3.15)

Why we chose such awkward notation M1 for the matrix of coefficients ofthe term will become clear in due course (or you can take a peak at(3.25) now). Our task is to compute the partition function Z[J], the freeenergy W[J], and the full n-point correlation functions

Z[J] = eW[J] = [d]eS[]+J (3.16)= Z[0]

1 + n=1

12n

G12nJ1 J2 . . . J n

n!

,G12n = 1 2 . . . n =

1

Z[0]

d

dJ1. . .

d

dJnZ[J]

J=0

. (3.17)

The bare mass m0 and the bare coupling g0 in (3.14) parameterize therelative strengths of quadratic, quartic fields at a lattice point vs. contribu-tion from spatial variation among neighboring sites. They are called bareas the 2- and 4-point couplings measured in experiments are dressed byfluctuation contributions.

3.4 Saddle-point expansions

The path integral (3.16) is an ordinary multi-dimensional integral. In the limit, or the T 0 low temperature limit, the action is large (highprice of straying from the beaten path) almost everywhere, except for somelocalized regions of the-space. Highly idealized, the action looks somethinglike the sketch in figure 3.3 (in order to be able to draw this on a piece ofpaper, we have suppressed a large number of coordinates).


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Figure 3.3: For the low temperature T = 1/ the path integral (3.16) is domi-nated by the minima of the integrands exponent. The location c of a minimum isdetermined by the extremum condition S[c] + J = 0 .

Such integral is dominated by the minima of the action. The minimumvalue S[] configurations c are determined by the zero-slope, saddle-pointcondition

d

d S[c] + J = 0 . (3.18)

The term saddle refers to the general technique of evaluating such inte-grals for complex; in the statistical mechanics applicationsc are locationsof the minima of S[], not the saddles. If there is a number of minima,only the one (or the nc minima related by a discrete symmetry) with thelowest value of S[c]c J dominates the path integral in the low tem-perature limit. The zeroth order, mean field approximation to the partitionsum (3.16) is given by the extremal configuration alone

Z[J] = eW[J]

c

eWc[J] = eWc[J]+lnnc

Wc[J] = S[c] + c J . (3.19)

In thesaddle-point approximation the corrections due to the fluctuationsin the c neighborhood are obtained by shifting the origin of integration to

c + ,

the position of the c-th minimum of S[] J, and expanding S[] in aTaylor series around c. For our purposes it will be convenient to separateout the quadratic part S0[], and collect all terms higher than bilinear in into an interaction term SI[]

S0[] =

r

2c+ 12

u

c2(c)

2

+

2

,

,

SI[] = uc2

Nd=1

4 . (3.20)


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3.4. SADDLE-POINT EXPANSIONS 29

Spatially nonuniform c are conceivable. The mean field theory assump-tion is that the translational invariance of the lattice is not broken, and cis independent of the lattice point, c c. In the4 theory consideredhere, it follows from (3.18) that c = 0 for r > 0, and c =

|r|/4ufor r < 0 . There are at most nc = 2 distinct

c configuration with thesame S[c], and in the thermodynamic limit we can neglect the mean fieldentropy ln nc in (3.19) when computing free energy density per site [3.3],

f[J] = limN

W[J]/Nd . (3.21)

We collect the matrix of bilinear coefficients in

(M1) = m20 c , m20 = m20 + 12u(c)2 (3.22)in order to be able to rewrite the partition sum (3.16) as

eW[J] = eWc[J]

[d]e12

TM1+SI[] .

As the expectation value of any analytic function

g() = gn1n2...n11

n22 /n1!n2!

can be recast in terms of derivatives with respect to J[d]g()e

12

TM1 = g( ddJ

)

[d]e

12

TM1+J

J=0

,

we can move SI[] outside of the integration, and evaluate the Gaussianintegral in the usual way

3.3page 42

eW[J] = eWc[J]eSI[ddJ

]

[d]e

12

TM1+J

J=0

= |det M| 12 eWc[J]eSI[ ddJ] e 12 JTMJ

J=0. (3.23)

M is invertible only if the minima in figure 3.3 are isolated, and M1

has no zero eigenvalues. The marginal case would require going beyondthe Gaussian saddle-points studied here, typically to the Airy-function typestationary points [3.4]. In the classical statistical mechanics S[] is a real-valued function, the extremum of S[] at the saddle-point c is the mini-mum, all eigenvalues ofM are strictly positive, and we can drop the absolutevalue brackets | | in (3.23).

As we shall show in sect. 3.6, expanding the exponentials and evaluatingthe ddJ derivatives in (3.23) yields the fluctuation corrections as a powerseries in 1/ = T.

The first correction due to the fluctuations in the c neighborhood isobtained by approximating the bottom of the potential in figure 3.3 by a

parabola, that is, keeping only the quadratic terms in the Taylor expansion(3.20). For a single minimum the free energy is in this approximation

W[J]1-loop = Wc[J] +1

2tr ln M , (3.24)

where we have used the matrix identity ln det M = tr ln M, valid for anyfinite-dimensional matrix. This result suffices to establish the Ginzburg cri-terion (explained in many excellent textbooks) which determines when theeffect of fluctuations is comparable or larger than the mean-field contribu-tion alone.

3.4page 42

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3.5 Free field theory

There are field theory courses in which months pass while free non-interactingfields are beaten to pulp. This text is an exception, but even so we get ourfirst glimpse of the theory by starting with no interactions, SI[] = 0. Thefree-field partition function (which sometimes ekes living under the nameGaussian model) is

Z0[J] = eW0[J] =

[d]e

12

TM1+J = |det M| 12 e 12 JTMJ

W0[J] =1

2JT M J + 1

2tr ln M . (3.25)

The full n-point correlation functions (3.17) vanish for n odd, and for neven they are given by products of distinct combinations of 2-point correla-tions

G = (M)

G1234 = (M)12 (M)34 + (M)13 (M)24 + (M)14 (M)23

G12n = (M)12 (M)n1n + (M)13 (M)n1n + (3.26)Keeping track of all these dummy indices (and especially when they turninto a zoo of of continuous coordinates and discrete indices) is a pain, andit is much easier to visualize this diagrammatically. Defining the propagatoras a line connecting 2 lattice sites, and the probe J as a source/sink fromwhich a single line can originate

(M)12 = 1 2 , J = , (3.27)

we expand the free-field theory partition function (3.25) as a Taylor seriesin JT M1 J

Z0[J]Z0[0]

= 1 + . (3.28)

In the diagrammatic notation the non-vanishing n-point correlations (3.26)are drawn as

(11 terms) . (3.29)

The total number of distinct terms contributing to the noninteracting full 3.1page 42

n-point correlation is 1 3 5 (n 1) = (n 1)!!, the number of ways thatn source terms J can be paired into n/2 pairs M.

3.6 Feynman diagrams

For field theories defined at more than a single point the perturbative correc-tions can be visualized by means of Feynman diagrams. It is not clear that


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3.6. FEYNMAN DIAGRAMS 31

this is the intelligent way to proceed [3.5], as both the number of Feynmandiagrams and the difficulty of their evaluation explodes combinatorially, butas 99% of physicists stop at a 1-loop correction, for the purpose at handthis is a perfectly sensible way to proceed.

3.6.1 Hungry pac-men munching on fattened Js

The saddle-point expansion is most conveniently evaluated in terms of Feyn-man diagrams, which we now introduce. Expand both exponentials in (3.23)

eSI[ddJ

] e12

JTMJ =

1 +1

4!+

1

2

1

(4!)2+

1 + (3.30)

Here we have indicated ddJ as a pac-man [3.6] that eats J, leaving a deltafunction in its wake

d

dJjJ = j

. (3.31)

For example, the rightmost pac-man in the

(ddJ

)4 interaction termquartic in derivative has four ways of munching a J from the free-fieldtheory 12

12J

T M J2 term, the next pac-man has three Js to bite intoin two distinct ways, and so forth:

1

4!

1

23=

1

3!

1

23=

1

3!

1

23 + 2 =

1

23=

1

8. (3.32)

In the hum-drum field theory textbooks this process of tying together verticesby propagators is called the Wick expansion. Professionals have smarter

3.7page 42

ways of generating Feynman diagrams [3.2], but this will do for the problemat hand.

It is easy enough to prove this to all orders [3.7], but to this order youcan simply check by expanding the exponential (3.16) that the free energyW[J] perturbative corrections are the connected, diagrams with J = 0

3.6page 42

W[0] = S[c]+ 12

tr ln M+ 18

+ 116

+ 148

.(3.33)

According to its definition, every propagator line M connecting two verticescarries a factor of T = 1/, and every vertex a factor of 1/T. In the4

theory the diagram with n vertices contributes to the order Tn of the pertur-bation theory. In quantum theory, the corresponding expansion parameteris.

To proceed, we have to make sense of M, and learn how to evaluatediagrammatic perturbative corrections.

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3.7 Propagator in the space representation

In order to describe the spatial variation in (3.14) we need to define alatticederivative.

3.7.1 Lattice derivatives

Consider a smooth function (x) evaluated on d-dimensional lattice

= (x) , x = a = lattice point, Zd , (3.34)where a is the lattice spacing and there are Nd points in all. A vector specifies a lattice configuration. Assume the lattice is hyper-cubic, and letn {n1, n2, , nd} be the unit lattice cell vectors pointing along the dpositive directions, |n| = 1 . The lattice partial derivative is then

() =(x + an) (x)

a=

+n a

.

Anything else with the correct a 0 limit would do, but this is the simplestchoice. We can rewrite the derivative as a linear operator, by introducingthe hopping operator (or shift, or step) in the direction

(h)j = +n,j . (3.35)

As h will play a central role in what follows, it pays to understand what itdoes, so we write it out for the 1-dimensional case in its full [NN] matrixglory:

h =

0 10 1

0 1. . .

0 11 0

. (3.36)

We will assume throughout that the lattice is periodic in each n direction;this is the easiest boundary condition to work with if we are interested inlarge lattices where surface effects are negligible. We will (perhaps) brieflydiscuss other boundary conditions in sect. ??.

Applied on the lattice configuration = (1, 2, , N), the hoppingoperator shifts the lattice by one site, h = (2, 3, , N, 1). Its trans-pose shifts the entries the other way, so the transpose is also the inverse

h1 = hT . (3.37)

The lattice derivative can now be written as a multiplication by a matrix:

=1

a (h 1)j j .In the 1-dimensional case the [NN] matrix representation of the lattice

derivative is:

=1

a

1 11 1

1 1. . .

11 1

. (3.38)


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3.7. PROPAGATOR IN THE SPACE REPRESENTATION 33

To belabor the obvious: On a finite lattice of N points a derivative is sim-ply a finite [NN] matrix. Continuum field theory is a world in whichthe lattice is so fine that it looks smooth to us. Whenever someone callssomething an operator you say it is just a matrix, no big deal, I knowwhat a matrix is. OK?

3.7.2 Lattice Laplacian

In order to get rid of some of the lattice indices in (3.15) it is convenient toemploy vector notation for the terms bilinear in , and keep the rest lumpedinto interaction,

S[] = m20

2T

2a2[(h 1) ]T (h 1) + SI[] . (3.39)

In the Landau case (3.14) the quartic termSI[] is local site-by-site, 1234 =4! u 12 23 34 , so this general quartic coupling is a little bit of anoverkill, but by the time we get to the Fourier-transformed theory, it willmake sense as a momentum conserving vertex (3.67).

In the continuum integration by parts moves around; on a lattice thisamounts to a matrix transposition

[(h 1) ]T [(h 1) ] = T (h1 1) (h 1) .If you are wandering where the integration by parts minus sign is, it isthere in discrete case at well. It comes from the identity T = h1. Thecombination = h12

= 1a2

d=1

(h1 1) (h 1) = 2

a2

d=1

1 1

2(h1 + h)

(3.40)

is the lattice Laplacian. We shall show in sect. 3.9 that this Laplacian has

the correct continuum limit. It is the simplest spatial derivative allowed forx x symmetric actions. In the 1-dimensional case the [NN] matrixrepresentation of the lattice Laplacian is:

=1

a2

2 1 11 2 1

1 2 11

. . .

11 1 2

. (3.41)

The lattice Laplacian measures the second variation of a field across

three neighboring sites. You can easily check that it does what the secondderivative is supposed to do by applying it to a parabola restricted to thelattice, = (), where () is defined by the value of the continuumfunction (x) = x2 at the lattice point .

3.7.3 Inverting the Laplacian

Evaluation of perturbative corrections in (3.23) requires that we come togrips with the free or bare propagator M . While the inverse propagatorM1 is a simple one-step difference operator (3.22), its inverse is a messier


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object. A way to compute is to start expanding M as a power series in theLaplacian

M =1

m20 1 =

1

m20

k=0

c

m20

km . (3.42)

As is a finite matrix, the expansion is convergent for sufficiently largem20 . To get a feeling for what is involved in evaluating such series, evaluate2 in the 1-dimensional case:

2 =1

a4

6 4 1 1 44 6 4

1 4 6 4 14 . . .

44 1 1 4 6

. (3.43)

What 3, 4, contributions look like is now clear; as we include higherand higher powers of the Laplacian, the propagator matrix fills up; whilethe inverse propagator is differential operator connecting only the nearest

neighbors, the propagator is integral operator, connecting every lattice siteto any other lattice site [3.8]. 3.8page 42

This matrix can be evaluated as is, on the lattice, and sometime it isevaluated this way, but in case at hand a wonderful simplification followsfrom the observation that the lattice action is translationally invariant. Wewill show how this works in sect. 3.8.

3.7.4 Why is propagator called propagator?

In statistical mechanics, M is the (bare) 2-point correlation. In quantumfield theory, it is called a propagator. Why?

Until now the collective indices have stood for all particle labels; space-

time location, spin, field type and so on. Statistical mechanics is formulatedin a Euclidean world in which there is no time, just space. What do we meanby propagation in such a space?

Our formulation is inevitably phenomenological: we have no idea whatthe structure of our space on distances much shorter than interatomicmight be. The very space-time might be discrete rather than continuous, orit might have geometry different from the one we observe at the accessibledistance scales. The formalism we use should reflect this ignorance. We dealwith this problem by coarse graining the space into small cells and requiringthat our theory be insensitive to distances comparable to or smaller thanthe cell sizes.

Our next problem is that we have no idea why there are particles, andwhy or how they propagate. The most we can say is that there is some prob-ability that a particle hops from one space-time cell to another space-timecell. At the beginning of the century, the discovery of Brownian motionshowed that matter was not continuous but was made up of atoms. In par-ticle physics we have no indication of having reached the distance scalesin which any new space-time structure is being sensed: hence for us thishopping probability has no direct physical significance. It is simply a phe-nomenological parameter: in the continuum limit it will be replaced by themass of the particle.

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3.7. PROPAGATOR IN THE SPACE REPRESENTATION 35

skip the rest of this section, unfinished

We assume for the time being that the state of a particle is specified byits space-time position, and that it has no further labels (such as spin orcolor): a= (x1,x x d ). What is it like to be free? A free particle existsonly in itself and for itself; it neither sees nor feels the others; it is, in thischilly sense, free. But if it is not at once paralyzed by the vast possibilitiesopened to it, it soon becomes perplexed by

with all possible paths connecting the two cells:

A.. = sIhLNij(L) (3.44)

Nij (L) is the number of all paths of length L connecting lattice sites i andj. Define a stepping matrix

(S)ij = 6i+n,j . (3.45)

If a particle is introduced into the i-th cell by a source

Jk = ik

the stepping matrix moves it into a neighboring cell:

(SJ)k = 6i + n,ki + n

The operator

d(h S)ij =

it[(S)ij + (S)jil = 1

h = (h,h, , h), h) (3.46)generates all paths of length 1 with probability h:

(h S)J = hi thcell(The examples are drawn in two dimensions). The paths of length 2 aregenerated by

2(has) 21 = h2

I and so on. Note that the k-th component of the vector(h S)LJ countsthe number of paths of length L connecting the i-th and the k-th spacetimecells. The total probability that the particle stops in the k-th cell is given by

s ZNS?J.I ki a(5.4)

The value of the field at a space-time point it measures the probabilityof observing the particle introduced into the system by the source J. TheEuclidean free scalar particle propagator (5.1) is given by

A s (5.5)ij = OAKor, in the continuum limit do by

3.8page 42dik (x y)A(X, Y)dkeI +m2(2)dk2 (3.47)

Interpreting as a field is consistent with the previous definition of a freefield, equations (2.22) and (2.25).


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3.8 Periodic lattices

Our task now is to transform M into a form suitable to evaluation of Feyn-man diagrams. The theory we will develop in this section is applicable onlyto translationally invariant saddle-point configurations.

Consider the effect of a h translation on the action (3.20)

S[h] = 12

T hTM1h g04!

Nd=1

(h)4 .

As M1 is constructed from h and its inverse, M1 and h commute, andthe bilinear term is h invariant. In the quartic term h permutes cyclicallythe terms in the sum, so the total action is translationally invariant

S[h] = S[] = 12

T M1 g04!

Nd=1

4 .

If a function (in this case, the action S[]) defined on a vector space (inthis case, the configuration ) commutes with a linear operator h, then theeigenvalues of h can be used to decompose the vector space into invariantsubspaces. For a hyper-cubic lattice the translations in different directionscommute, hh = hh, so it is sufficient to understand the spectrum ofthe 1-dimensional shift operator (3.36). To develop a feeling for how thisreduction to invariant subspaces works in practice, let us continue in thehumble spirit of sect. 3.1, by expanding the scope of our deliberations to alattice consisting of 2 points.

3.8.1 A 2-point lattice diagonalized

The action of the shift operator h (3.36) on a 2-point lattice = (1, 2)

is to permute the two lattice sites

h =

0 11 0

.

As exchange repeated twice brings us back to the original configuration,h2 = 1, and the characteristic polynomial of h is

(h + 1)(h 1) = 0 ,with eigenvalues = 1, 2 = 1. Construct now the symmetrization, anti-symmetrization projection operators

P1 =

h

21

1 2 =h

(

1)

1 (1) =1

2 (1 + h) =

1

2 1 11 1 (3.48)P2 =

h 11 1 =

1

2(1 h) = 1

2

1 1

1 1

. (3.49)

Noting that P1 + P2 = 1, we can project the lattice configuration onto thetwo eigenvectors of h:

= 1 = P1 + P2 ,12

=

(1 + 2)2

12

11

+

(1 2)2

12

1

1

. (3.50)


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3.8. PERIODIC LATTICES 37

As P1P2 = 0, the symmetric and the antisymmetric configurations trans-form separately under any linear transformation constructed fromh and itspowers.

In this way the characteristic equation h2 = 1 enables us to reduce the2-dimenional lattice configuration to two 1-dimensional ones, on which thevalue of the shift operatorh is = 1, 1, and the eigenvectors are 1

2(1, 1),

1

2(1,

1). We have inserted

2 factors only for convenience, in order that

the eigenvectors be normalized unit vectors.

3.8.2 Discrete Fourier transforms

Now let us generalize this reduction to a 1-dimensional periodic lattice withN sites.

Each application of h translates the lattice one step; in N steps thelattice is back in the original configuration

hN = 1 ,

so the eigenvalues of h are the N distinct N-th roots of unity

hN 1 =N1k=0

(h k1) = 0 , = ei 2N . (3.51)

As the eigenvalues are all distinct and N in number, the space is decom-posed into N one-dimensional subspaces. The general theory (expounded inappendixA.1) associates with thek-th eigenvalue ofh a projection operatorthat projects a configuration onto k-th eigenvector of h,

Pk = j=k

h

j 1

k j . (3.52)

A factor (h j 1) kills the j-th eigenvector j component of an arbitraryvector in expansion = + jj + . The above product kills everythingbut the eigendirection k, and the factor

j=k(k j ) ensures that Pk is

normalized as a projection operator. The set of the projection operators iscomplete

k

Pk = 1 (3.53)

and orthonormal

PkPj = kj Pk (no sum on k) . (3.54)

Constructing explicit eigenvectors is usually not a the best way to fritterones youth away, as choice of basis is largely arbitrary, and all of thecontent of the theory is in projection operators [3.11]. However, in case athand the eigenvectors are so simple that we can forget the general theory,and construct the solutions of the eigenvalue condition

h k = kk (3.55)

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by hand:

1N

0 10 1

0 1. . .

0 1

1 0

1k

2k

3k...

(N

1)k

= k1N

1k

2k

3k...

(N

1)k

The 1/

N factor is chosen in order that k be normalized unit vectors

k k =1

N

N1k=0

1 = 1 , (no sum on k)

k =1N

1, k, 2k, , (N1)k

. (3.56)

The eigenvectors are orthonormal

k j = kj , (3.57)

as the explicit evaluation of k j yields the Kronecker delta function fora periodic lattice

kj =1

N

N1=0

ei2N

(kj) . (3.58)

The sum is over the N unit vectors pointing at a uniform distribution ofpoints on the complex unit circle; they cancel each other unless k = j (modN), in which case each term in the sum equals 1.

The projection operators can be expressed in terms of the eigenvectors(3.55), (3.56) as

(Pk) = (k)(k) =

1

Nei

2N()k , (no sum on k) . (3.59)

The completeness (3.53) follows from (3.58), and the orthonormality (3.54)from (3.57).

k, the projection of the configuration on the k-th subspace is givenby

(Pk ) = k (k) , (no sum on k)

k = k =

1N

N1

=0ei

2N

k (3.60)

We recognize k as the discrete Fourier transform of . Hopefully redis-covering it this way helps you a little toward understanding why Fouriertransforms are full of eixp factors (they are eigenvalues of the generator oftranslations) and when are they the natural set of basis functions (only ifthe theory is translationally invariant).

Now insert the identity

Pk = 1 wherever profitable:

M = 1M1 =kk

PkMPk =kk

k(k M k)k .


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3.8. PERIODIC LATTICES 39

The matrix

Mkk = (k M k) (3.61)

is the Fourier space representation of M. No need to stop here - the termsin the action (3.15) that couple 3, 4, fields also have the Fourier spacerepresentations

12n 1 2 n = k1k2kn k1 k2 kn ,k1k2kn = 12n(k1 )1 (k2 )2 (kn)n

=1

Nn/2

1n

12n ei 2

N(k11++knn) .(3.62)

According to (3.57) the matrix Uk = (k) =1N

ei2N

k is a unitary ma-

trix, and the Fourier transform is a linear, unitary transformation U U =Pk = 1 with Jacobian det U = 1. The form of the path integral (3.16)

does not change under k transformation, and from the formal pointof view, it does not matter whether we compute in the Fourier space or in

the configuration space that we started out with. For example, the trace ofM is the trace in either representation

tr M =

M =kk

(PkMPk)

=kk

(k)(k M k)(k) =

kk

kkMkk = tr M .(3.63)

From this it follows that tr Mn = tr Mn, and from the tr ln = ln tr relationthat det M = det M. In fact, any scalar combination of s, Js andcouplings, such as the partition function Z[J], has exactly the same form

in the configuration and the Fourier space.OK, a dizzying quantity of indices. But whats the pay-back?

3.8.3 Lattice Laplacian diagonalized

Now use the eigenvalue equation (3.55) to convert h matrices into scalars.

IfM commutes withh, then(k Mk) = Mkkk , and the matrix M actsas a multiplication by the scalar Mk on thek-th subspace. For example, forthe 1-dimensional version of the lattice Laplacian (3.40) the projection onthe k-th subspace is

(k k) =2

a2 12 (k + k) 1 (k k)=

2

a2

cos

2

Nk

1

kk (3.64)

In the k-th subspace the bare propagator (3.42) is simply a number, and,in contrast to the mess generated by (3.42), there is nothing to invertingM1:

(kMk) = (G0)kkk = 1

kk

m20 2a2d

=1

cos

2N k

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40 References

where k = (k1, k2, , k) is a d-dimensional vector in the Nd-dimensionaldual lattice.

Going back to the partition function (3.23) and sticking in the factorsof 1 into the bilinear part of the interaction, we replace the spatial J by itsFourier transformJk, and the spatial propagator (M) by the diagonalizedFourier transformed (G0)k

JT M J = k,k

(JT k)(k M k)(k J) = k

Jk(G0)kJk .(3.66)

Whats the price? The interaction term SI[] (which in (3.23) was local inthe configuration space) now has a more challenging k dependence in theFourier transform version (3.62). For example, the locality of the quarticterm leads to the 4-vertex momentum conservation in the Fourier space

SI[] =1

4!1234 1 2 3 4 = u

Nd=1

()4

= u1

N3d/2

N

{ki} 0,k1+k2+k3+k4 k1 k2 k3 k4 . (3.67)3.9 Continuum field theory

3.10 Summary and outlook

We have formulated the standard field-theoretic perturbation theory. Fromhere, any generalization and future direction is wide open.

Acknowledgement. I am indebted to Benny Lautrup both for myfirst introduction to a lattice field theory, and for the interpretation of the

Fourier transform as the spectrum of the shift operator.

References

[3.1] P.M. Chaikin and T.C. Lubensky, Principles of condensed matter physics(Cambridge University Press, Cambridge 1995).

[3.2] P. Cvitanovic, Field theory, notes prepared by E. Gyldenkerne (Nordita,Copenhagen, January 1983);ChaosBook.org/FieldTheory.

[3.3] For classically chaotic field theories we will not be so lucky - there the num-ber of contributing saddles grows exponentially with the lattice size.

[3.4] N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals(Dover, New York, 1986).

[3.5] P. Cvitanovic, Chaotic field theory: a sketch, Physica D, (2000), to appear.

[3.6] www.classicgaming.com/pac-man/home.html

[3.7] Read section 2.F of ref. [3.2].

[3.8] Read chapter 5 of ref. [3.2], pp. 61-64 up to eq. (5.6), and do the exercise5.A.1. (reproduced as the exercise 3.8 here).

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References 41

[3.9] R.B. Dingle, Asymptotic Expansions: their Derivation and Interpretation(Academic Press, London, 1973).

[3.10] P. Cvitanovic, Asymptotic estimates and gauge invariance, Nucl. Phys.B127, 176 (1977).

[3.11] P. Cvitanovic, Group theory,ChaosBook.org/GroupTheory, a monograph in preparation.

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42 References

ExercisesExercise 3.1 Free-field theory combinatorics. Check that there indeed are

no combinatorial prefactors in the expansion (3.29).Exercise 3.2 Quality of asymptotic series. Use the saddle-point method toevaluate Zn

Zn =(1)nn!4n

d

2e

2/2+4n ln

Find the smallest error for a fixed g; plot both your error and the the exact result(3.5) for g = 0.1, g = 0.02, g = 0.01. The prettiest plot makes it into these notes as

figure 3.1!Exercise 3.3 Complex Gaussian integrals. Read sect. 3.B, do exercise 3.B.1

of ref. [3.2].Exercise 3.4 Prove lndet = tr ln. (link here the lndet = tr ln problem sets,

already done).Exercise 3.5 Convexity of exponentials. Prove the inequality (3.7). Matthias

Eschrig suggest that a more general proof be offered, applicable to any monotone

descreasing sequence with alternating signs.Exercise 3.6 Wick expansion for 4 theories. Derive (3.33), check the

combinatorial signs.Exercise 3.7 Wick expansions. Read sect. 3.C and do exercise 3.C.2 of

ref. [3.2].Exercise 3.8 Propagators in the configuration space. Define the finite

difference operator by f(x + f(x -a)f(x) 2 a where a is the lattice spacing. Show that1 d h (So + Sri 2d+a3P if ji ,where 2 = 91111 is the finite difference Laplacian. Show that the Euclidean

scalar lattice propagator (??) is given by-1 ha 2 ij sThe mass in the continuum propagator (5.6) is related to the hopping parameter

by s(5.9)

If the particle does not like hopping (h O), the mass is infinite and there isno propagation. If the particle does not like stopping (s -0), the mass is zero and

the particle zips all over the space. Diagonalize 32 by Fourier transforming and derive

(5.6).

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Appendix A

Group theory

A.1 Invariant matrices and reducibility . . . . . . . . 43

A.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . 43

A.1.2 Projection operators . . . . . . . . . . . . . . . . 44

A.1.3 Decomposition of representations . . . . . . . . . 45

A.1 Invariant matrices and reducibility

The basic idea is simple; any hermitian (or, for real matrices, any symmet-ric) matrix can be brought to diagonal form. If this matrix is an invariantmatrix, it decomposes the space into direct sums of lower-dimensional sub-spaces. This topic is developed at great length in ref. [A.1]. For the humbleapplication at hand, even this appendix is an overkill, but it might giveyou a feeling for how the Fourier analysis fits into the general theory of

invariance groups and their representations.

A.1.1 Definitions

LetV be thedefining d-dimensional complex vector space, V = {x | x V}the conjugate space, andG a group acting linearly onV. The action ofg Gon a vector x V is given by a [dd] matrix representation G

xa = Gbaxb a, b = 1, 2, . . . , d . (A.1)

The repeated indices are always summed over unless explicitly statedotherwise.

We distinguish the components of defining space vectors, resp. conjugatevectors, by lower, resp. upper indices

x = (x1, x2, . . . , xn) , x Vx = (x1, x2, . . . , xn) , x V . (A.2)

The two spaces are related by complex conjugation, which is here indicatingby raising the vector index:

xa = (xa) .

43
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44 APPENDIX A. GROUP THEORY

The action of g G on a vector q V is given by the conjugate represen-tation G

xa = xb(G)ab , (G)ab (Gba) (A.3)

A.1.2 Projection operators

A matrix is hermitian if its elements satisfy

(M)ab = Mab . (A.4)

For M a hermitian matrix, there exists a diagonalizing unitary matrix C:

CM C =

1 00 1

0 0

0

2 0 . . . 00 2...

. . ....

0 . . . 2

...

0 03 . . ....

. . .

, i = j .(A.5)

Here i are the r distinct roots of the minimal characteristic polynomialr

i=1

(M i1) = 0 . (A.6)

In the matrix (M 21) the eigenvalues corresponding to 2 are replacedby zeroes:

C(M 21)C =

1 21 2

1 20

. . .0

3 23

so the product over all factors (M 11)(M 31) . . . with exception ofthe (M 21) factor has non-zero entries only in the subspace associatedwith 1:

C

j=1(M j 1)C =

j=1(1 j)

11

1 00

. . .

0

.

In this way we can associate with each distinct root i a projection operatorPi

Pi =j=i

M j 1i j , (A.7)

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A.1. INVARIANT MATRICES AND REDUCIBILITY 45

which is identity on the ith subspace, and zero elsewhere. For example, theprojection operator onto the 1 subspace is

P1 = C

11

10

0 0. . .

0

C . (A.8)

It follows from the characteristic equation (A.6) that i is the eigenvalueof M on Pi subspace:

M Pi = iPi , (no sum on i) , (A.9)

and that any matrix polynomial f(M) takes scalar value f(i) on the Pisubspace

f(M)Pi = f(i)Pi . (A.10)

The matrices Pi are orthonormal PiPj = ijPj , (no sum on j) , (A.11)

and satisfy a completeness relation

ri=1

Pi = 1 . (A.12)

As tr (CPiC+) = tr Pi, the dimension of the i-th subspace is given by

di = tr Pi . (A.13)

A.1.3 Decomposition of representations

A matrix M is invariant if it commutes with all group transformations[G, M] = 0. Projection operators (A.7) constructed fromM are polynomialsin M, so they also commute with all g G:

[G, Pi] = 0 , (A.14)

(remember that Pi are [d d] matrices). Hence a [d d] matrix represen-tation can be written as a direct sum of [di

di] matrix representations

G = 1G1 =

i,j

PiGPj =

i

PiGPi =

i

Gi . (A.15)

In the diagonalized representation (A.8), the matrix G has a block di-agonal form:

CGC =

G1 0 00 G2 00 0

. . .

, G = i

CiGiCi . (A.16)

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46 References

Representation Gi acts only on the di dimensional subspace Vi consistingof vectors Piq, q V. In this way an invariant [d d] hermitian matrixM with r distinct eigenvalues induces a decomposition of a d-dimensionalvector space V into a direct sum of di-dimensional vector subspaces Vi

VM

V1 V2 . . . Vr . (A.17)

On the one-dimensional spaces the group acts trivially, G = 1.If a projection operator projects onto a zero-dimensional subspace, it

must vanish identically

di = 0 = Pi = 0 . (A.18)This follows from (A.8); di is the number of 1s on the diagonal on theright-hand side. The general form of Pi is

Pi =r

k=1

ckMk (A.19)

where Mk

are the invariant matrices used in construction of the projec-tor operators, and ck are numerical coefficients. Vanishing of Pi thereforeimplies linear dependence among invariant matrices Mk.

References

[A.1] P. Cvitanovic, Group Theory, ChaosBook.org/GroupTheory, a monographin preparation

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Index

action, 12Airy

equation, 17function, 17

Airy function, 17, 20, 26at a bifurcation, 20

Airy integral, 17

bifurcationAiry function approximation,

20Bohr-Sommerfeld quantization, 17

chain rulematrix, 6

connection formulas, 17

degree of freedom, 13delta function

Dirac, 5

eigen

function, 4eigenvaluezero, 17, 26

evolution operatorquantum, 5

extremal point, 14

Fresnel integral, 14, 20

Gaussianintegral, D-dimensional, 22

Greens function

energy domain, 5trace, 6

Hamiltonsprincipal function, 9

Hamiltonian 4

Lagrangian, 8Laplace

method, 21transform, 5

mechanicsquantum, 4

propagator, 5

quantization

WKB, 11, 14quantum

evolution, 5propagator, 5

quantum mechanics, 4

Schrodingerpicture, 7

Schrodinger equation, 4stationary

phase approximation, 13, 20Sterling formula, 20

Trotter product formula, 7

Wentzel-Kramers-Brillouin, 11WKB

connection formulas, 17quantization, 11, 14

zero eigenvalue, 17, 26