Finite temperature field theory principles and applications

Topics include functional integral representation of the partition function, diagrammatic expansions, lin- ear response theory, screening and plasma oscillations, spontaneous symmetry br

This book develops the basic formalism and theoretical techniques for ing relativistic quantum field theory at high temperature and density Specific physical theories treated include QED, QCD, electroweak theory, and effective nuclear field theories of hadronic and nuclear matter Topics include functional integral representation of the partition function, diagrammatic expansions, lin- ear response theory, screening and plasma oscillations, spontaneous symmetry breaking, the Goldstone theorem, resummation and hard thermal loops, lattice gauge theory, phase transitions, nucleation theory, quark–gluon plasma, and color superconductivity Applications to astrophysics and cosmology include white dwarf and neutron stars, neutrino emissivity, baryon number violation in the early universe, and cosmological phase transitions Applications to relativistic nucleus–nucleus collisions are also included.

study-J OSEPH I K APUSTA is Professor of Physics at the School of Physics and omy, University of Minnesota, Minneapolis He received his Ph.D from the Uni- versity of California, Berkeley, in 1978 and has been a faculty member at the University of Minnesota since 1982 He has authored over 150 articles in refereed journals and conference proceedings Since 1997 he has been an associate editor

Astron-for Physical Review C He is a Fellow of the American Physical Society and of

the American Association for the Advancement of Science The first edition of

Finite-Temperature Field Theory was published by Cambridge University Press

in 1989; a paperback edition followed in 1994.

C HARLES G ALE is James McGill Professor at the Department of Physics, McGill University, Montreal He received his Ph.D from McGill University in 1986 and joined the faculty there in 1989 He has authored over 100 articles in refereed journals and conference proceedings Since 2005 he has been the Chair of the Department of Physics at McGill University He is a Fellow of the American Physical Society.

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1

Finite-Temperature Field Theory

Principles and Applications

c a m b r i d g e u n i v e r s i t y p r e s s Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ ao Paulo

Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York

www.cambridge.org Information on this title: www.cambridge.org/9780521820820

C

J I Kapusta and C Gale 2006 This publication is in copyright Subject to statutory exception

and to the provisions of relevant collective licensing agreements,

no reproduction of any part may take place without the written permission of Cambridge University Press.

First published 1989 First paperback edition 1994 Second edition 2006 Printed in the United Kingdom at the University Press, Cambridge

A catalog record for this publication is available from the British Library

ISBN-13 978-0-521-82082-0 hardback ISBN-10 0-521-82082-0 hardback

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

3.4 First-order corrections to Π and ln Z 41

v

3.7 Remarks on real time perturbation theory 51

6.5 Exact formula for screening length in QED 97

6.9 Kubo formulae for viscosities and conductivities 107

7.1 Charged scalar field with negative mass-squared 117

8.3 Perturbative evaluation of partition function 146

8.5 Gluon propagator and linear response 152

9.1 Isolating the hard thermal loop contribution 1799.2 Hard thermal loops and Ward identities 1859.3 Hard thermal loops and effective perturbation theory 187

11.5 Summary 236

12.2 Self-energy from experimental data 248

13.3 Nonrelativistic thermal nucleation 296

Contents ix

16.5 Electroweak phase transition and baryogenesis 402

A1.2 Microcanonical and canonical ensembles 418

What happens when ordinary matter is so greatly compressed that theelectrons form a relativistic degenerate gas, as in a white dwarf star? Whathappens when the matter is compressed even further so that atomic nucleioverlap to form superdense nuclear matter, as in a neutron star? Whathappens when nuclear matter is heated to such great temperatures thatthe nucleons and pions melt into quarks and gluons, as in high-energynuclear collisions? What happened in the spontaneous symmetry break-ing of the unified theory of the weak and electromagnetic interactionsduring the big bang? Questions like these have fascinated us for a longtime The purpose of this book is to develop the fundamental principlesand mathematical techniques that enable the formulation of answers tothese mind-boggling questions The study of matter under extreme con-ditions has blossomed into a field of intense interdisciplinary activity andglobal extent The analysis of the collective behavior of interacting rela-tivistic systems spans a rich palette of physical phenomena One of theultimate goals of the whole program is to map out the phase diagram ofthe standard model and its extensions

This text assumes that the reader has completed graduate level courses

in thermal and statistical physics and in relativistic quantum field theory.Our aims are to convey a coherent picture of the field and to prepare thereader to read and understand the original and current literature Thebook is not, however, a compendium of all known results; this would havemade it prohibitively long We start from the basic principles of quantumfield theory, thermodynamics, and statistical mechanics This develop-ment is most elegantly accomplished by means of Feynman’s functionalintegral formalism Having a functional integral expression for the parti-tion function allows a straightforward derivation of diagrammatic rules forinteracting field theories It also provides a framework for defining gaugetheories on finite lattices, which then enables integration by Monte Carlo

xi

techniques The formal aspects are illustrated with applications drawnfrom fields of research that are close to the authors’ own experience Eachchapter carries its own exercises, reference list, and select bibliography.

The book is based on Finite-Temperature Field Theory, written by one

of us (JK) and published in 1989 Although the fundamental principleshave not changed, there have been many important developments sincethen, necessitating a new book

We would like to acknowledge the assistance of Frithjof Karsch andSteven Gottlieb in transmitting some of their results of lattice computa-tions, presented in Chapter 10, and Andrew Steiner for performing thenumerical calculations used to prepare many of the figures in Chapter

11 We are grateful to a number of friends, colleagues, and students fortheir helpful comments and suggestions and for their careful reading of themanuscript, especially Peter Arnold, Eric Braaten, Paul Ellis, Philippe deForcrand, Bengt Friman, Edmond Iancu, Sangyong Jeon, Keijo Kajantie,Frithjof Karsch, Mikko Laine, Stefan Leupold, Guy Moore, Ulrich Mosel,Robert Pisarski, Brian Serot, Andrew Steiner, and Laurence Yaffe

1 Review of quantum statistical mechanics

Thermodynamics is used to describe the bulk properties of matter in ornear equilibrium Many scientists, notably Boyle, Carnot, Clausius, Gay-Lussac, Gibbs, Joule, Kelvin, and Rumford, contributed to the develop-ment of the field over three centuries Quantities such as mass, pressure,energy, and so on are readily defined and measured Classical statisticalmechanics attempts to understand thermodynamics by the application ofclassical mechanics to the microscopic particles making up the system.Great progress in this field was made by physicists like Boltzmann andMaxwell Temperature, entropy, particle number, and chemical potentialare thus understandable in terms of the microscopic nature of matter.Classical mechanics is inadequate in many circumstances however, andultimately must be replaced by quantum mechanics In fact, the ultravio-let catastrophe encountered by the application of classical mechanics andelectromagnetism to blackbody radiation was one of the problems thatled to the development of quantum theory The development of quan-tum statistical mechanics was achieved by a number of twentieth centuryphysicists, most notably Planck, Einstein, Fermi, and Bose The purpose

of this chapter is to give a mini-review of the basic concepts of quantumstatistical mechanics as applied to noninteracting systems of particles.This will set the stage for the functional integral representation of thepartition function, which is a cornerstone of modern relativistic quantumfield theory and the quantum statistical mechanics of interacting particlesand fields

1.1 Ensembles

One normally encounters three types of ensemble in equilibrium statistical

mechanics The microcanonical ensemble is used to describe an isolated system that has a fixed energy E, a fixed particle number N , and a fixed

1

volume V The canonical ensemble is used to describe a system in contact with a heat reservoir at temperature T The system can freely exchange

energy with the reservoir, but the particle number and volume are fixed

In the grand canonical ensemble the system can exchange particles as well

as energy with a reservoir In this ensemble the temperature, volume, and

chemical potential μ are fixed quantities The standard thermodynamic

relations are summarized in appendix section A1.1

In the canonical and grand canonical ensembles, T −1 = β may be

thought of as a Lagrange multiplier that determines the mean energy

of the system Similarly, μ may be thought of as a Lagrange multiplier

that determines the mean number of particles in the system In a tivistic quantum system, where particles can be created and destroyed,

rela-it is most straightforward to compute observables in the grand canonicalensemble For that reason we use the grand canonical ensemble through-out this book There is no loss of generality in doing so because onemay pass over to either of the other ensembles by performing an inverse

Laplace transform on the variable μ and/or the variable β See appendix

section A1.2

Consider a system described by a Hamiltonian H and a set of

con-served number operators ˆN i (A hat or caret is used to denote an tor for emphasis or whenever there is the possibility of an ambiguity.) InQED, for example, the number of electrons minus the number of positrons

opera-is a conserved quantity, not the number of electrons or positrons

sepa-rately, because of reactions like e+e − → e+e+e − e − These number

oper-ators must be Hermitian and must commute with H as well as with each

other They must also be extensive (scale with the volume of the system)

in order that the usual macroscopic thermodynamic limit can be taken.The statistical density matrix ˆρ is the fundamental object in equilibrium

is used to compute the ensemble average of any desired observable, resented by the operator ˆA, via

rep-A =  ˆ A = Tr ˆAˆ ρ

where Tr denotes the trace operation

The grand canonical partition function

Z = Z(V, T, μ1, μ2, ) = Tr ˆ ρ (1.3)

1.2 One bosonic degree of freedom 3

is the single most important function in thermodynamics From it all thethermodynamic properties may be determined For example, the pressure,particle number, entropy, and energy are, in the infinite-volume limit,given by

1.2 One bosonic degree of freedom

As a simple example consider a time-independent single-particle quantummechanical mode that may be occupied by bosons Each boson in that

mode has the same energy ω There may be 0, 1, 2, or any number of

bosons occupying that mode There are no interactions between the ticles This system may be thought of as a set of noninteracting quantizedsimple harmonic oscillators It will serve as a prototype of the relativisticquantum field theory systems to be introduced in later chapters We areinterested in computing the mean particle number, energy, and entropy.Since the system has no volume there is no physical pressure

par-Denote the state of the system by |n, which means that there are n

bosons in the system The state |0 is called the vacuum The properties

of these states are

One may think of the bras n| and kets |n as row and column vectors,

respectively, in an infinite-dimensional vector space These vectors form acomplete set The operation in (1.5) is an inner product and the number

1 in (1.6) stands for the infinite-dimensional unit matrix

It is convenient to introduce creation and annihilation operators, a † and a, respectively The creation operator creates one boson and puts it

in the mode under consideration Its action on a number eigenstate is

Similarly, the annihilation operator annihilates or removes one boson,

unless n = 0, in which case it annihilates the vacuum,

Apart from an irrelevant phase, the coefficients appearing in (1.7) and

(1.8) follow from the requirements that a † and a be Hermitian conjugates and that a † a be the number operator ˆ N That is,

Next we need a Hamiltonian Up to an additive constant, it must be

ω times the number operator Starting with a wave equation in

nonrela-tivistic or relanonrela-tivistic quantum mechanics the additive constant emergesnaturally One finds that

N + 12

(1.13)The additive term 12ω is the zero-point energy Usually this term can

be ignored Exceptions arise when the vacuum changes owing to a ground field, such as the gravitational field or an electric field, as in theCasimir effect We shall drop this term in the rest of the chapter and leave

back-it as an exercise to repeat the following analysis wback-ith the inclusion of thezero-point energy

The states|n are simultaneous eigenstates of energy and particle

num-ber We can assign a chemical potential to the particles This is possiblebecause there are no interactions to change the particle number Thepartition function is easily computed:

1.3 One fermionic degree of freedom 5

and the mean energy E is ωN Note that N ranges continuously from zero

to infinity as μ ranges from −∞ to ω Values of the chemical potential, in

this system, are restricted to be less than ω on account of the positivity

of the particle number or, equivalently, the Hermiticity of the numberoperator

There are two interesting limits One is the classical limit, where the

occupancy is small, N  1 This occurs when T  ω − μ In this limit

the exponential in (1.15) is large and so

The other is the quantum limit, where the occupancy is large, N  1.

This occurs when T  ω − μ.

1.3 One fermionic degree of freedom

Now consider the same problem as in the previous section but for fermionsinstead of bosons This is a prototype for a Fermi gas, and later on willhelp us to formulate the functional integral expression for the partitionfunction involving fermions These could be electrons and positrons inQED, neutrons and protons in nuclei and nuclear matter, or quarks inQCD

The Pauli exclusion principle forbids the occupation of a single-particlemode by more than one fermion Thus there are only two states of thesystem, |0 and |1 The action of the fermion creation and annihilation

operators on these states is as follows:

Up to an arbitrary phase factor, the coefficients in (1.17) are chosen so

that α and α † are Hermitian conjugates and α † α is the number operator

ˆ

N :

ˆ

It follows that the creation and annihilation operators satisfy the commutation relation

N −1 2



(1.21)This form follows from the Dirac equation Notice that the zero-pointenergy is equal in magnitude but opposite in sign to the bosonic zero-point energy In this chapter we drop this term for fermions, as we havefor bosons

The partition function is computed as in (1.14) except that the sum

terminates at n = 1 on account of the Pauli exclusion principle:

Z = Tr e −β(H−μ ˆ N ) = Tr e−β(ω−μ) ˆ N

=1

and the mean energy E is ωN Note that N ranges continuously from zero

to unity as μ ranges from −∞ to ∞ Unlike bosons, for fermions there is

no restriction on the chemical potential

As with bosons, there are two interesting limits One is the classical

limit, where the occupancy is small, N  1 This occurs when T  ω − μ:

which is the same limit as for bosons The other is the quantum limit

When T → 0 one obtains N → 0 if ω > μ and N → 1 if ω < μ.

1.4 Noninteracting gases

Now let us put particles, either bosons or fermions, into a box with sides of

length L We neglect their mutual interactions, although in principle they

must interact in order to come to thermal equilibrium One can imagineincluding interactions, waiting until the particles come to equilibrium,and then slowly turning off the interactions Such a noninteracting gas

is often a good description of the atmosphere around us, electrons in ametal or white dwarf star, blackbody photons in a heated cavity or in

1.4 Noninteracting gases 7the cosmic microwave background radiation, phonons in low-temperaturematerials, neutrons in a neutron star, and many other situations.

In the macroscopic limit the boundary condition imposed on the surface

of the box is unimportant For definiteness we impose the condition thatthe wave function vanishes at the surface of the box (Also frequently usedare periodic boundary conditions.) The vanishing of the wave function onthe surface means that an integral number of half-wavelengths must fit in

the distance L:

λ x = 2L/j x λ y = 2L/j y λ z = 2L/j z (1.25)

where j x , j y , j z are all positive integers The magnitude of the x

com-ponent of the momentum is |p x | = 2π/λ x = πj x /L, and similarly for the

y and z components Amazingly, quantum mechanics tells us that these

relations hold for both nonrelativistic and relativistic motion, for bothbosons and fermions

The full Hamiltonian is the sum of the Hamiltonians for each mode onaccount of the assumption that the particles do not interact We use a

shorthand notation in which j represents the triplet of numbers (j x , j y , j z)that uniquely specifies each mode Thus the Hamiltonian and numberoperator are

j

Hj

(1.26)ˆ

In the macroscopic limit, L → ∞, it is permissible to replace the sum

from j x = 1 to∞ with an integral from j x = 1 to∞ (The correction to

this approximation is proportional to the surface area L2 and the relative

contribution is therefore of order 1/L.) We can then use dj x = Ld |p x |/π

where the upper sign (+) refers to fermions and the lower sign (−) refers

to bosons From (1.4) and (1.31) we obtain the pressure, particle number,and energy:

These formulæ for N and E have the simple interpretation of

phase-space integrals over the mean particle number and energy of each mode,respectively

The dispersion relation ω = ω(p) determines the energy for a given momentum For relativistic particles ω =

p2+ m2, where m is the mass The nonrelativistic limit is ω = m + p2/2m For phonons the dispersion

relation is ω = csp, where cs is the speed of sound in the medium.There are a number of interesting and physically relevant limits Con-

sider the dispersion relation ω =

p2+ m2 The classical limit sponds to low occupancy of the modes and is the same for bosons (1.16)and fermions (1.24) The momentum integral for the pressure can be per-formed and written as

e(μ −m)/T classical nonrelativistic limit (1.34)

Knowing the pressure as a function of temperature and chemical potential

we can obtain all other thermodynamic functions by differentiation or byusing thermodynamic identities

The zero-temperature limit for fermions requires that μ > m,

other-wise the vacuum state is approached In this limit all states up to the

6π2

In the nonrelativistic limit,

5 F

3

2P nonrelativistic limitElectrons and nucleons have spin 1/2 and these expressions need to bemultiplied by 2 to take account of that! The low-temperature limit forbosons will be discussed in the next chapter

Massless bosons with zero chemical potential have pressure

tion A + B → C + D can occur then not only must the reverse

reac-tion, C + D → A + B, occur but it must happen at the same rate.

Detailed balance implies relationships between the chemical potentials It

is shown in standard textbooks that, for the reactions just mentioned, the

chemical potentials obey μ A + μ B = μ C + μ D For a long-lived resonance

that decays according to X → A + B, the formation process A + B → X

must happen at the same rate The chemical potentials are related by

μ X = μ A + μ B Generally any reactions that are allowed by the tion laws can and will occur These conservation laws restrict the number

conserva-of linearly independent chemical potentials Consider, for example, a tem whose only relevant conservation laws are for baryon number andelectric charge There are only two independent chemical potentials, one

sys-for baryon number (μ B ) and one for electric charge (μ Q) Any particle

in the system has a chemical potential which is a linear combination ofthese:

μ i = b i μ B + q i μ Q (1.38)

Here b i is the baryon number and q i the electric charge of the particle

of type i These chemical potentials are all measured with respect to the total particle energy including mass (The chemical potential μNRi , as cus-tomarily defined in nonrelativistic many-body theory, is related to ours by

μNRi = μ i − m i.) Bosons that carry no conserved quantum number, such

as photons and π0 mesons, have zero chemical potential Antiparticleshave a chemical potential opposite in sign to particles

The electrically charged mesons π+ and π − have electric charges of+1 and−1 and therefore equal and opposite chemical potentials, μ Qand

−μ Q , respectively The total conserved charge is the number of π+mesons

minus the number of π − mesons:

1.1 Prove that the state |n given in (1.12) is normalized to unity.

1.2 Referring to (1.17), let|0 and |1 be represented by the basis vectors

in a two-dimensional vector space Find an explicit 2× 2 matrix

representation of the abstract operators α and α †in this vector space.1.3 Calculate the partition function for noninteracting bosons, includingthe zero-point energy From it calculate the mean energy, particlenumber, and entropy Repeat the calculation for fermions

Bibliography 111.4 Calculate the average energy per particle of a noninteracting gas ofmassless bosons with no chemical potential Repeat the calculationfor massless fermions.

1.5 Derive an expression like (1.39) or (1.40) for the entropy Repeat thecalculation for fermions

Bibliography

Thermal and statistical physics

Reif, F (1965) Fundamentals of Statistical and Thermal Physics

(McGraw-Hill, New York).

Landau, L D., and Lifshitz, E M (1959) Statistical Physics (Pergamon Press,

Oxford).

Many-body theory

Abrikosov, A A., Gorkov, L P., and Dzyaloshinskii, I E (1963) Methods of

Quantum Field Theory in Statistical Physics (Prentice-Hall, Englewood

Cliffs).

Fetter, A L and Walecka, J D (1971) Quantum Theory of Many-Particle

Systems (McGraw-Hill, New York).

Negele, J W and Orland, H (1988) Quantum Many-Particle Systems

(Addison-Wesley, Redwood City).

Numerical evaluation of thermodynamic integrals

Johns, S D., Ellis, P J., and Lattimer, J M., Astrophys J 473, 1020 (1996).

2 Functional integral representation of the

partition function

The customary approach to nonrelativistic many-body theory is to ceed with the method of second quantization begun in the first chap-ter There is another approach, the method of functional integrals, which

pro-we shall follow here Of course, what can be done in one formalism canalways be done in another Nevertheless, functional integrals seem to bethe method of choice for most elementary particle theorists these days,and they seem to lend themselves more readily to nonperturbative phe-nomena such as tunneling, instantons, lattice gauge theory, etc For gaugetheories they are practically indispensable However, there is a certainamount of formalism that must be developed before we can start to dis-cuss physical applications In this chapter, we shall derive the functionalintegral representation of the partition function for interacting relativisticnon-gauge field theories As a check on the formalism, as well as to obtainsome feeling for how functional integrals work, we shall then rederive somewell-known results on relativistic ideal gases for bosons and fermions

2.1 Transition amplitude for bosons

Let ˆφ(x, 0) be a Schr¨ odinger-picture field operator at time t = 0 and let

ˆ

π(x, 0) be its conjugate momentum operator The eigenstates of the field

operator are labeled|φ and satisfy

ˆ

where φ(x) is the eigenvalue, as indicated, a function of x We also have

the usual completeness and orthogonality conditions,



12

2.1 Transition amplitude for bosons 13

In a natural generalization one goes from a denumerably finite number

of degrees of freedom N in quantum mechanics to a continuously

infi-nite number of degrees of freedom in quantum field theory:N

i=1 p i x i →



d3x π(x)φ(x).

For the dynamics one requires a Hamiltonian, which is now a functional

of the field and of its conjugate momentum:

H =



Now suppose that a system is in a state |φ a  at a time t = 0 After a

time tf it evolves into e−iHtf|φ a , assuming that the Hamiltonian has no

explicit time dependence The transition amplitude for going from a state

|φ a  to a state |φ b  after a time tf is thusφ b |e −iHtf|φ a .

For statistical mechanical purposes we will be interested in cases where

the system returns to its original state after the time tf To obtain apractical definition of the transition amplitude we use the following pre-

scription: we divide the time interval (0, tf ) into N equal steps of duration

Δt = tf/N Then, at each time interval we insert a complete set of states,

alternating between (2.2) and (2.5):

φ a |e −iHtf|φ a  = lim

N →∞

 N i=1

H i =



d3x H (π i (x), φ i(x)) (2.14)Putting it all together we get

φ a |e −iHtf|φ a = lim

N →∞

 N i=1

where φ N +1 = φ a = φ1 The advantage of alternating between π and φ for

the insertion of a complete set of states is that the Hamiltonian in (2.13)and (2.15) is evaluated at a single point in time

2.2 Partition function for bosons 15Taking the continuum limit of (2.15), we finally arrive at the importantresult

The symbols [dπ] and [dφ] denote functional integration as defined in

(2.15) The integration over π(x, t) is unrestricted, but the integration over φ(x, t) is such that the field starts at φ a (x) at t = 0 and ends at

φ a (x) at t = tf Note that all references to operators have gone

2.2 Partition function for bosons

can express Z as an integral over fields and their conjugate momenta by

making use of (2.16) In order to make that connection, we switch to

an imaginary time variable τ = it The trace operator in (2.17) simply means that we must integrate over all φ a Finally, if the system admits aconserved charge then we must make the replacement

[dφ]

× exp

 β0

strained in such a way that φ(x, 0) = φ(x, β) This follows from the trace

operation, setting φ a (x) = φ(x, 0) = φ(x, β) There is no restriction over

the π integration The expression for the partition function (2.19) can

readily be generalized to an arbitrary number of fields and conservedcharges

2.3 Neutral scalar field

The most general renormalizable Lagrangian for a neutral scalar field φ

We shall evaluate the partition function by returning to the discretizedversion:

periodic

The momentum integrals can be evaluated immediately since they are

simply products of Gaussian integrals We divide position space into M3small cubes with V = L3, L = aM , a → 0, M → ∞, M being an integer.

For convenience and to make sure that Z remains explicitly dimensionless

at each stage of the calculation, we write π j = A j /(a3Δτ ) 1/2and integrate

A j from −∞ to ∞ We get

 ∞

−∞

dA j 2π exp

2.3 Neutral scalar field 17for each cube Thus far we have

M,N→∞ (2π)

−M3N/2

 N i=1

Z = N 

periodic

[dφ] exp

 β0

by any constant will not change the thermodynamics

Next, we turn to the case of noninteracting fields by letting U(φ) = 0.

Interactions will be discussed in a later chapter We define

∞ n= −∞ p

ei(p·x+ω n τ ) φ n(p) (2.30)

where ω n = 2πnT , owing to the constraint of periodicity that φ(x, β) =

φ(x, 0) for all x The normalization in (2.30) is chosen such that each

Fourier amplitude is dimensionless Substituting (2.30) into (2.29) andrecalling that the field is real, we find that

S = −1

2β2

n p

(ω2n + ω2)φ n (p)φ ∗ n(p) (2.31)

with ω =

p2+ m2 The integrand depends only on the magnitude of

the field, A n (p) = |φ n (p) | Integrating out the phases, we get

where N  is a constant Here D equals β2(−∂2/∂τ2 − ∇2 + m2) in (x, τ )

space and β2(ω2n + ω2) in (p, ω n ) space, and (φ, Dφ) is the inner product

on the function space The expression (2.34) follows from the formula for

Riemann integrals with a constant matrix D:

 ∞

−∞ dx1 · · · dx ne−x i D ij x j = π n/2 (det D) −1/2 (2.35)One may also derive (2.33) using (2.34)

ln

(2πn)2+ β2ω2

=

 β2ω2 1

dθ2

θ2+ (2πn)2 + ln

1 + (2πn)2

(2.37)and

∞ n=−∞

ln Z = −

p

 βω1

dθ

1

2.4 Bose–Einstein condensation 19Carrying out the integral and dropping terms that are independent oftemperature and volume, we finally get

This expression is identical to the bosonic version of (1.31) with μ = 0,

except that (2.40) includes the zero-point energy Both

An interesting system is obtained by considering a theory with a charged

scalar field Φ The field Φ is then complex and describes bosons of

pos-itive and negative charge, i.e., they are each other’s antiparticle TheLagrangian density in this case is

L = ∂ μΦ∗ ∂ μΦ− m2Φ∗Φ− λ(Φ ∗Φ)2 (2.43)This expression has an obvious U(1) symmetry:

where α is a real constant This is a global symmetry since the multiplying

phase factor is independent of spacetime location

By Noether’s theorem, there is a conserved current associated witheach continuous symmetry of the Lagrangian We can find this current

by letting the phase factor α depend on the spacetime coordinate for a

moment In this case the U(1) transformation is

L → L  = ∂ μ(Φ∗eiα(x) )∂ μ(Φe−iα(x))− m2Φ∗Φ− λ(Φ ∗Φ)2

Since ∂ L  /∂α = 0, it follows that the “current” ∂ L  /∂(∂ μ α) = Φ ∗ Φ∂ μ α +

iΦ ∗ ∂ μΦ− iΦ∂ μΦ∗ is conserved We recover our original theory by letting

α(x) = constant The conserved current density is then

j μ = i(Φ ∗ ∂ μΦ− Φ∂ μΦ∗ (2.47)

with ∂ μ j μ= 0 The conservation law may be verified independently using

the equation of motion for Φ The full current and density are J μ=



d3x j μ (x) and Q =

d3x j0(x).

It is convenient to decompose Φ into real and imaginary parts using

the real fields φ1 and φ2, Φ = (φ1+ iφ2)/ √

2 In terms of the conjugate

momenta π1 = ∂φ1 /∂t, π2= ∂φ2 /∂t, the Hamiltonian density and charge

Q =



d3x(φ2π1− φ1π2) (2.49)The partition function is

Z =



[dπ1][dπ2]

periodic

[dφ1][dφ2]× exp

 β0

where we have used a chemical potential associated with the conserved

charge Q Integrating out the conjugate momenta, we get

L(φ1, φ2, ∂ μ φ1, ∂ μ φ2; μ = 0) + μj0(φ1, φ2, i∂φ1/∂τ, i∂φ2/∂τ )

by an amount μ2Φ∗ Φ, owing to the momentum dependence of j0

The expression (2.51) cannot be evaluated in closed form unless λ = 0.

In this case, the functional integral becomes Gaussian and can then beworked out analogously to that for the free scalar field

n p

ei(p ·x+ω n τ ) φ 2;n(p)

Here ζ and θ are independent of (x, τ ) and determine the full infrared

behavior of the field; that is, φ1;0(p = 0) = φ2;0(p = 0) = 0 This allows

for the possibility of condensation of the bosons into the zero-momentumstate Condensation means that in the infinite-volume limit a finite frac-

tion of the particles resides in the n = 0, p = 0 state.

Setting λ = 0 and substituting (2.52) into (2.51) after an integration



Carrying out the integrations,

ln Z = βV (μ2− m2)ζ2+ ln(det D) −1/2 (2.54)The second term can be handled as follows:

Putting all this together,

The last two terms in (2.56) are precisely of the form (2.36) All we

need to do is recall (2.40) and make the substitutions ω → ω − μ and

ω → ω + μ, respectively, for the two terms in (2.56) We obtain

expression but θ does not, as expected from the U(1) symmetry of the Lagrangian In this context, since the parameter ζ is not determined a

priori, it should be treated as a variational parameter that is related to

the charge carried by the condensed particles At fixed β and μ, ln Z is

an extremum with respect to variations of such a free parameter Thus

∂ ln Z

∂ζ = 2βV (μ

2− m2)ζ = 0 (2.58)

which implies that ζ = 0 unless |μ| = m, in which case ζ is undetermined

by this variational condition When|μ| < m we simply recover the results

obtained in Chapter 1, namely (1.31)

To determine ζ when |μ| = m, note that the charge density ρ = Q/V

(The case μ = −m is handled analogously.) Here the separate

contribu-tions from the condensate (the zero-momentum mode) and the thermal

excitations are manifest If the density ρ is kept fixed and the ture is lowered, μ will decrease until the point μ = m is reached If the temperature is lowered even further then ρ ∗ (β, μ = m) will be less than

In the limit m → 0, we have |μ| → 0 and Tc→ ∞ When m = 0, all the

charge resides in the condensate, at all temperatures, and none is carried

by the thermal excitations

There is a second-order phase transition at Tc This can be shown

rig-orously by a careful examination of the behavior of the chemical potential

μ(ρ, T ) as a function of T near Tc with ρ fixed This analysis is left as

an exercise A more intuitive way to see this involves the general Landau

theory of phase transitions [1] The order parameter ζ drops continuously

to zero as Tc is approached from below and remains zero above Tc ically, the reason for a phase transition is the following At T = 0, all the

Phys-conserved charge can reside in the zero-momentum mode on account ofthe bosonic character of the particles (This would not be the case forfermions.) As the temperature is raised, some of the charge is excited out

of the condensate Eventually, the temperature becomes great enough tocompletely melt, or thermally disorder, the condensate There is no rea-

son for ζ to drop to zero discontinuously; hence the transition is second

order

2.5 Fermions

We now turn our attention to (Dirac) fermions In relativistic quantummechanics, we know that electrons or muons are described by a four-

component spinor ψ The components are identified as ψ α , with α

run-ning from 1 to 4 The motion of a free electron is characterized by a

respec-complex functions in relativistic quantum mechanics but become

opera-tors in a field theory As usual, p · x = p μ x μ = Et − p · x Equation (2.64)

The Dirac matrices γ μ, which are defined by the anticommutators

{γ μ , γ ν } = 2g μν, are in the standard convention

Each of these is a 4× 4 matrix: “1” denotes the unit 2 × 2 matrix and σ

denotes the triplet of Pauli matrices In (2.66), ¯ψ = ψ † γ0 and μ ∂ μ=

γ μ ∂/∂x μ Written out explicitly,

The Lagrangian has a global U(1) symmetry, so that ψ → ψe −iα and

ψ † → ψ †eiα Following Noether’s theorem, there is a conserved currentassociated with this symmetry To find it, we proceed in the same way

as we did for the charged scalar field theory We allow α to depend on

x, treating it as an independent field Under the above phase

transforma-tion,L → L + ¯ ψ [

∂ μ (∂ L/∂[∂ μ α(x)]) − ∂L/∂α(x) = 0, we find the conservation law

∂ μ j μ= 0

For relativistic quantum mechanics in the absence of interactions this is

a trivial result because of (2.65)

In the field theory we treat ψ as a basic field The momentum conjugate

to this field is, from (2.68),

of the field in imaginary time τ and with the nature of the “classical” (in

the path-integral formulation) fields ψ(x, τ ) and ψ † (x, τ ) over which we

integrate

The canonical commutation relations for bosons are

ˆ

ψ † α (x, t), ˆ ψ † β (y, t)



= 0

These commutation relations are the only ones allowed by the tal spin-statistics theorem in relativistic quantum field theory In the limit

fundamen- → 0 the field operators are replaced by their eigenvalues For the case of bosons, those eigenvalues are actually c-number functions, as illustrated

in (2.1) We have expressed the partition function as a functional

inte-gral over these c-number functions, or “classical fields” For the case of

fermions, the  → 0 limit is rather peculiar since the eigenvalues

replac-ing the field operators anticommute with each other! This is of courseconnected with the Pauli exclusion principle and with the famous spin-statistics theorem Note that (2.74) instructs us to integrate over these

“classical” but anticommuting functions The mathematics necessary tohandle this situation was studied by Grassmann There are Grassmannvariables, Grassmann algebra, and Grassmann calculus

For a single Grassmann variable η, there is only one anticommutator

to define the algebra,

The first of these says that the integral is invariant under the shift η →

η + a, and the second is just a convenient normalization.

In a more general setting, we may have a set of Grassmann variables

η i , i = 1, 2, N , and a paired set η † i The algebra is defined by

{η i , η j } = {η i , η j † } = {η i † , η † j } = 0 (2.79)The most general function of these variables may be written as

2.5 Fermions 27bibliography at the end of this chapter refers the interested reader tomore detailed treatments.

For our purposes, the only integral we need is



dη1† dη1· · · dη N † dη Neη † Dη = det D (2.82)

where D is an N × N matrix This formula is simple to prove if N = 1 or

2 The general case is left as an exercise for the reader

As with bosons, it is most convenient to work in (p, ω n) space instead

of (x, τ ) space In imaginary time we can write

ψ α (x, τ ) = √1

i(p·x+ω n τ ) ψ˜α;n(p) (2.83)

where both n and p run over negative and positive values For an arbitrary

function defined over the interval 0≤ τ ≤ β, the discrete frequency ω n can take on the values nπT For bosons we argued that we must take

ω n = 2πnT in order that φ(x, τ ) be periodic, which followed from the trace

operation in the partition function This can be verified by examining theproperties of the thermal Green’s function for bosons defined by

GB(x, y; τ1, τ2) = Z −1Tr

ˆ

ρT τˆ

where θ is the step-function Using the fact that T τ commutes with ˆρ =

e−βK , where K ≡ H − μ ˆ Q, and the cyclic property of the trace we find

For fermions, however, instead of (2.85) one has (in direct analogy withthe real time Green’s functions)

GF(x, y; τ, 0) = −GF(x, y; τ, β) (2.88)This implies that

that the system returns to its original state after a “time” β Since the sign

of ψ is just an overall phase and hence is not observable, the right-hand

side of (2.89) describes the same physical state as the left-hand side.Now we are ready to evaluate (2.74) Inserting (2.83) and using (2.82)

−μ Q...

Fetter, A L and Walecka, J D (1971) Quantum Theory of Many-Particle

Systems (McGraw-Hill, New York).

Negele, J W and Orland, H (1988)... phase and hence is not observable, the right-hand

side of (2.89) describes the same physical state as the left-hand side.Now we are ready to evaluate (2.74) Inserting (2.83) and using