An Introduction To Thermal Field Theory
An Introduction toThermal Field TheoryYuhao YangSeptember 23, 2011Supervised by Dr. Tim EvansSubmitted in partial fulfilment of the requirements for the degree ofMaster of Science in Quantum Fields and Fundamental ForcesDepartment of PhysicsImperial College London
AbstractThis thesis aims to give an introductory review of thermal field theories. We review the imaginary time formalism to generalize field theoriesat finite temperature. We study the scalar φ4 theory and gauge theoriesin a thermal background. We study the propagators and self energiesat one-loop approximation and understand how particles acquire thermalmasses and the consequences. We show in a hot plasma, there are collective excitation modes absent in the zero temperature case, and a pointcharge is screened by the thermal effects. However, a naive perturbation theory would break down for soft external momenta owing to theso-called hard thermal loop corrections. Diagrams of higher orders couldcontribute at the leading order, and hence it is necessary to develop aneffective theory. We introduce the basic ideas of the resummation schemeand make several remarks on its applications.
AcknowledgementsI would sincerely thank Dr. Tim Evans for his supervision on my thesis.I learned a lot from discussions with him. Dr. Evans is also my personaltutor of the MSc program. He has been giving me a lot of encouragementsduring the past few months, which helped me greatly.I would also thank the lecturers of the courses that I have been takingthis year, for their guiding me into the exciting and rich fields of theoreticalphysics that I am always curious of.At last, I would thank my parents for giving me the spiritual and financial supports under all circumstances, without whom I could not imaginewhat I would achieve.
Contents1 Introduction: Why Thermal Field Theory?12 Review of Quantum Statistical Mechanics23 Scalar Field Theory at Finite Temperature63.13.2Free scalar fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . .63.1.1Partition function . . . . . . . . . . . . . . . . . . . . . . . . .63.1.2Scalar propagators . . . . . . . . . . . . . . . . . . . . . . . .9Interactions and perturbative theory . . . . . . . . . . . . . . . . . .143.2.1Feynman rules . . . . . . . . . . . . . . . . . . . . . . . . . . .143.2.2Progapators and self-energies . . . . . . . . . . . . . . . . . .163.2.3Thermal mass and phase transitions . . . . . . . . . . . . . . .183.2.4Partition function . . . . . . . . . . . . . . . . . . . . . . . . .204 Gauge Theories at Finite Temperature4.122Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .224.1.1Partition function . . . . . . . . . . . . . . . . . . . . . . . . .224.1.2Electron propagators . . . . . . . . . . . . . . . . . . . . . . .26Quantum electrodynamics . . . . . . . . . . . . . . . . . . . . . . . .284.2.1QED Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . .284.2.2Partition function for photons . . . . . . . . . . . . . . . . . .29Photon propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . .334.3.1Vacuum photon propagator . . . . . . . . . . . . . . . . . . .334.3.2Longitudinal and transverse projections . . . . . . . . . . . . .354.3.3Full photon propagator . . . . . . . . . . . . . . . . . . . . . .36Photon self energy . . . . . . . . . . . . . . . . . . . . . . . . . . . .374.4.1Photon self energy . . . . . . . . . . . . . . . . . . . . . . . .374.4.2Computation of F. . . . . . . . . . . . . . . . . . . . . . . .384.4.3Computation of G . . . . . . . . . . . . . . . . . . . . . . . .40Photon Collective Modes . . . . . . . . . . . . . . . . . . . . . . . . .424.5.1Longitudinal modes . . . . . . . . . . . . . . . . . . . . . . . .434.5.2Transverse modes . . . . . . . . . . . . . . . . . . . . . . . . .444.6Debye screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .464.7Electron self energy . . . . . . . . . . . . . . . . . . . . . . . . . . . .494.8Quantum chromodynamics . . . . . . . . . . . . . . . . . . . . . . . .514.24.34.44.5i
5 Hard Thermal Loops and Resummation Rules5.15.254The breakdown of a naive perturbative theory . . . . . . . . . . . . .545.1.1Scalar theories . . . . . . . . . . . . . . . . . . . . . . . . . . .545.1.2Gauge theories . . . . . . . . . . . . . . . . . . . . . . . . . .56The effective perturbative theory . . . . . . . . . . . . . . . . . . . .57A Frequency Sums60ii
1Introduction: Why Thermal Field Theory?The conventional quantum field theory is formalized at zero temperature. The theoretical predictions under this framework, for example the cross sections of particlecollisions in an accelerator, are extremely good to match experimental data. However,our real world is certainly of non zero temperature, it is natural to ask to what extentthe contributions from the temperature start to be relevant? What new phenomenaabsent at zero temperature could arise in a thermal background?It may occur to us the heavy-ion collision experiments at LHC and at RHIC. Anew state of matter, which is called the quark gluon plasma [1], is predicted to becreated in the collisions. The consistence between experimental data and theoreticalpredictions of the Standard Model has to be verified. A useful theoretical frameworkto study this hot quark gluon plasma would be the thermal field theory.We may also think of the early stages of the universe, or astrophysical objectssuch as white dwarfs and neutron stars, where the temperature is sufficiently high.Thermal field theory would be responsible for our understanding of the phenomenaas phase transitions and cosmological inflation in the early universe, the evolution ofa neutron star. On the other hand, cosmology and astrophysics are good test fieldsfor theoretical studies to verify practical calculations.These things that we can think of necessitate formal studies of field theories atfinite temperature. This dissertation is devoted to giving an introductory review ofthermal field theories. Most of the topics covered in this article are covered in someexcellent textbooks [2] [3] [4].The organization of this article is as follows. We give a brief review of quantumstatistical mechanics in Section 2. We then develop the imaginary time formalism tostudy scalar field and gauge theories at finite temperature in Section 3 and 4, respectively. We will see in Section 5 the breakdown of a naive perturbatiion expansionand the necessity to formulate an effective theory to resum the contributions fromthe so-called hard thermal loops at all orders.1
2Review of Quantum Statistical MechanicsIn statistical mechanics the concept of thermal ensembles is of great importance. Thecanonical ensemble will be particularly useful to describe a system in equilibrium ofour interest in this introductory review. The canonical ensemble describes a systemin contact with a heat reservoir at a fixed temperature T . Energy can be exchangedbetween the system and the reservoir, but the particle number N and volume V arefixed. We may also use the grand canonical ensemble, where the system exchangeboth energy and particles with the reservoir at temperature T , while the chemicalpotential µ of particles and volume V are fixed. We can think of the canonicalensemble as a special case of the grand canonical ensemble in which the particleshave vanishing chemical potentials.In order to formalize quantum field theory at non-zero temperature, for simplicity,we use the canonical ensemble by assuming that the chemical potentials are zero. Wewill define and calculate quantities such as the density operator, the partition functionin terms of the canonical ensemble.Consider now a dynamical system characterized by a Hamiltonian H. The equilibrium state of the system of volume V is described by the canonical density operatorρ exp( βH)(2.1)The partition function, a very important quantity playing the central role in ourstudies of finite temperature field theory, is defined asZ Trρ(2.2)All thermodynamical quantities can be generated from the partition function. Forexample, ln Z V ln ZN T µ ln ZS T TPressureP TParticle numberEntropy(2.3)(2.4)(2.5)Recall in the zero temperature quantum field theory, the expectation value of agiven operator A ishAi0 Xhn A ni(2.6)nwhere ni are a complete set of orthonormal states. However, in a heat bath, theoperator expectation should be calculated as the ensemble average with a Boltzmann2
weight factorhAiβ 11Xhn A nie βH Tr e βH AZ nZ(2.7)or with the use of the density matrix, we writehAiβ TrAρTrρ(2.8)We can also think of a system characterized by a Hamiltonian H and a set ofconserved charges Q with particles of non-zero chemical potential. In this case weshall switch to use the grand canonical ensemble and redefine the density operatorρ exp [ β(H µN )](2.9)The definitions of the other quantities follow similarly as the zero chemical potentialcase, where we use the canonical ensemble. The use of grand canonical ensembleenables us to extend our studies on cases with non-trivial chemical potentials. Thecanonical ensemble can be thought of as a special case of the canonical ensemble withvanishing chemical potentials, but there are subtitles to care about.We are now ready to derive a fundamental relation in finite temperature theory.Consider the two-point correlation function 1 βHTr eφ(x, t)φ(y, 0)Z 1 Tr φ(x, t)e βH eβH φ(y, 0)e βHZ 1 Tr φ(x, t)e βH ei( iβH) φ(y, 0)e i( iβH)Z 1 Tr φ(x, t)e βH φ(y, iβ)Z 1 Tr e βH φ(y, iβ)φ(x, t)Z hφ(y, iβ)φ(x, t)iβhφ(x, t)φ(y, 0)iβ (2.10)where we used the cyclic permutation property of a trace of operator products. Wesurprisingly see that imaginary temperature plays the role as a time variable. If wedefine the imaginary time variablet iττ it(2.11)then the relation above can be rewritten ashφ(x, τ )φ(y, 0)iβ hφ(y, β)φ(x, τ )iβ3(2.12)
This is called the Kubo-Martin-Schwinger relation, or the KMS relation in short. Itfollows immediately from this relation thatφ(x, 0) φ(x, β)(2.13)where sign corresponds to whether the fields commute or anti-commute with eachother, or in other words, whether the fields are bosonic or fermionic. The KMSrelation shows that the fields are periodic or anti-periodic in imaginary time with β.It is convenient to cope with the fields in the frequency-momentum space. Owingto the periodicity constraint on the fields, the Fourier expansionXφ(x, τ ) φ(x, ωn )eiωn τ(2.14)nis no longer a continuous Fourier integral but a Fourier series instead. In order tosatisfy the KMS relation (2.13), we can only take the discrete frequencies2πnβ2π(n 1)ωn βωn for bosonic fields(2.15)for fermionic fields(2.16)n are integers , · · · , 2, 1, 0, 1, 2, · · · , . These frequencies are called the Matsubara frequencies, named after the Matsubara who first formally constructed a thermal field theory in the imaginary time formalism [5].We also develop a path integral form for the partition function. The advantage ofa path integral representation lies in the convenience within this framework to dealwith gauge theories than using operator formalism, especially for non-Abelian gaugetheories such as QCD. By noting thate βH e iR iβ0Hdt e Rβ0Hdτ(2.17)We may think of exp( βH) as an evolution operator in imaginary time with τ it.Recall the standard formalism of path integrals which can be found in many quantumfield theory textbooks [6], we have000U (q , t ; q, t) hq e iH(t0 t)Z qi " ZDq(t ) exp i#t00000dt L(t)(2.18)tWe write down the expression for partition function ZZZ βHZ Dφhφ e φi Dφ exp β dτ L(τ )(2.19)0All paths φ(x, τ ) satisfying the boundary condition (2.13) shall be evaluated in thepath integral.4
The subtleties of dealing with fermions by introducing anti-commuting Grassmannvariables will be discussed more carefully in Section 3.1.As a brief ending remark of this section, we see that it is very useful to think ofthe temperature as the imaginary time, but the origin of the correspondence betweenthese two very distinct arguments is an interesting question. This might merely be acoincidence that the evolution operator e iHt in quantum mechanics is related to theBoltzmann distribution factor eβH in statistical physics by an analytical continuation,but there might be some deeper reasons that we do not well understand.5
3Scalar Field Theory at Finite Temperature3.1Free scalar fields3.1.1Partition functionWe begin with the simplest possible model of a free real scalar field11L µ φ µ φ m2 φ222(3.1)Our choice of convention for the metric is gµν diag( 1, 1, 1, 1). The conjugatemomentum to the field operator isπ(x) L φ̇(x) φ̇(3.2)As in the zero temperature case, the field operator and its conjugate momentumcan be Fourier expanded in terms of a set of creation and annihilation operatorsZ d3 k1 ik·x† ikx φ(x) ae ae(3.3)kk(2π)3 2EkrZ d3 kEk ik·x† ikxπ(x) iae ae(3.4)kk(2π)32The equal time commutation relation is imposed as[φ(t, x), π(t, y)] iδ (3) (x y)δab(3.5)[ap , a†q ] (2π)3 δ (3) (p q)δrs(3.6)or equivalentlyWe want to compute the partition function to obtain thermodynamicals for freescalar fields. Plug the Lagrangian (3.1) into (2.19), we have Z iβ Z Z111 2 2322dt d x ( 0 φ) ( φ) m φZ Dφ exp i2220 Z β Z Z111 2 2322 Dφ exp dτ d x ( τ φ) ( φ) m φ2220(3.7)(3.8)To work under the frequency-momentum space, we Fourier expand the fields. Recall that the KMS relation has imposed a periodicity constraint, the Fourier integralwe had in the conventional field theories should be replaced by a Fourier seriesrβ X X i(ωn τ p·x)φ(x) eφ(ωn , p)(3.9)V n p6
where the allowed Matsubara frequencies are discreteωn 2πnβ(3.10)Substitute the Fourier expansion (3.9) into the partition function, we obtain)( ZZZXX ω 2 p2 m2β βnDφ expdτ d3 xφn,p ei(ωn τ p·x)φm,k ei(ωm τ k·x)V 02n,pm,k(3.11)Using the representations for the δ-functionsZ βdτ ei(ωm ωn )τ βδ(m n)0Zd3 xei(k p)x V δ (3) (p k)(3.12)(3.13)We carry out the integration over dτ , d3 x and then the summation over m, k withthe Kronecker δ-functions, we get(Zβ2 X 2Z Dφ exp (ω p2 m2 )φ n, p φn,p2 n,p n)(Zβ2 X 2(ωn p2 m2 )φ n,p φn,p Dφ exp 2 n,p ZYβ2 222 dφn exp (ωn p m )φn,p φn,p2n,pY 1/2β 2 (ωn2 p2 m2 ) N·)(3.14)n,pSome unimportant integration constant N is independent of temperature and therefore can be dropped.We are interested in the logarithm of the partition function from which we calculate all the physical measurables. We have 1Xln β 2 (ωn2 ωp2 )2 n,p 1X ln (2πn)2 β 2 ωp22 n,p)(Z 2 2β ωp 1Xdx2 ln (2πn)2 1222 n,p(2πn) x1ln Z (3.15)The last step can be checked by completing the integral. The reason for doing this isthat we can rewrite the frequency sum as a contour integral. The discrete Matsubara7
frequencies correspond to a collection of poles produced by a well chosen hyperboliccotangent function on the imaginary axis on the complex plane. The details of thistechnique is given in the appendix. Using the result (A.7) and dropping terms havingno temperature dependence, we obtain Z 2 21 X β ωp 2 12ln Z dx1 x2 p 12xe 1 X Z βωp 11 dx x2 e 11p X 1 βωp βωp ln 1 e2pIn the continuum limit, we haveZln Z VPpd3 p(2π)3R V(3.16)d3 p/(2π)3 , so this can be rewritten as 1 βωp βωp ln 1 e2(3.17)The first term is nothing but the familiar zero-point energy, which is divergent sinceit sums over an infinity number of zero-point modes. But we can simply neglect itbecause the effects of its contribution cannot be measured experimentally.We have derived the explicit expression for the partition function, now we caneasily calculate thermodynamicals from it, for instance, the pressure of the scalarparticles.d3 pln(1 e βωp )(3.18)(2π)3pTake the high energy limit, where p m, ωp p2 m2 p , we findZ 1d3 p βωp 2βωp 3βωpP e e e ···β(2π)3Z 1dp 2 X e nβωpp β2π 2 n 1 nZ dp 2 X e nβp1p β2π 2 n 1 n1TP ln Z VβZ T4 X 1 2π n 1 n4 π2 4T90(3.19)The T 4 behavior can be expected from a dimensional analysis by noting that T isthe only characteristic parameter in the free scalar field theory. The pressure is halfof the familiar result of black body radiation. This could be understood from our8
knowledge of statistical mechanics. The mean energy of a system is proportional tothe independent degrees of freedom. The real scalar particles are spin zero particlesand thus have only one degree of freedom, while the photons have two transversepropagating modes.3.1.2Scalar propagatorsLet us study the propagators carefully. We define the two-point correlatorsD (x, y) hφ(x)φ(y)iβ(3.20)D (x, y) hφ(y)φ(x)iβ D (y, x)(3.21)We insert complete sets of eigenstates of the Hamiltonian ni’s and express1 X βHhen φ(x) mihm φ(y) niZ n,m1 X βEn ehn eipx φ(0)e ipx mihm eipy φ(0)e ipy niZ n,m1 X βEn i(pn pm )(x y)2 eehn φ(0) miZ n,mD (x, y) (3.22)which shows that the correlator is a function of (x y). Its Fourier transform isZ1 X βEn2 ehn φ(0) mi (2π)4 δ (4) (k pm pn ) (3.23)D (k) d4 xeikx D (x) Z n,mThe domain of validity of D is determined by requiring the convergence of the sum.For simplicity, we look at only the time argument1 X βEn i(En Em )tD (t) eehn φ(0) miZ n,m2(3.24)from which we read off D (t) is defined on the strip β Im(t) 0 for a complexvariable t.Similarly, we get1 X βEn i(pm pn )(x y)eehn φ(0) miZ n,m1 X βEn i(Em En )t2D (t) eehn φ(0) miZ n,mD (x, y) 2(3.25)(3.26)and find that D (t) is defined within 0 Im(t) β.There is a relation between D (t) and D (t)D (t iβ) D (t)9(3.27)
which can be checked by comparing (3.24) and (3.26). This relation is just the KMSrelation dressed in a different form.In the Fourier frequency space, we haveZ0 0D (k ) dteik t D (t)ZZ0 0ik0 t D (k ) dte D (t) dteik t D (t iβ)(3.28)(3.29)Comparing these two, we obtain a relation between them0D (k 0 ) e βk D k 0(3.30)We also define the spectral density 0 ρ(k 0 ) D (k 0 ) D (k 0 ) eβk 1 D (k 0 )(3.31)with which we may rewrite the correlators asρ(k 0 ) [1 n(k 0 )]ρ(k 0 )1 e βk0ρ(k 0 )D (k 0 ) βk0 n(k 0 )ρ(k 0 )e 1D (k 0 ) (3.32)(3.33)where n(E) is the Bose distribution factorn(E) 1 1eβE(3.34)We find for the spectral densityZ 0 0ρ(k ) dteik t D (t) D (t) Z X 02 dteik te βEn hn φ(0) mi ei(En Em )t e i(En Em )t Xen,m βEnhn φ(0) mi2 δ(k 0 Em En ) δ(k 0 Em En )(3.35)n,mfrom which we see that the spectral density is odd in k 0ρ( k 0 ) ρ(k 0 )(3.36)Going back to the definition of D , we findD (t) D (t) h[φ(t), φ(0)]iβ10(3.37)
Differentiate with respect to t, we get from the LHSZ dddk 0 ik0 t D (t) D (t) eD (k 0 ) D (k 0 )dtdt2πZdk 0 0 ik0 t 0 ik eρ(k )2π(3.38)and from the RHSdh[φ(t), φ(0)]iβ h[φ(0), π(t)]iβdt(3.39)Recall that the equal time commutation relation is imposed as[φ(t), π(t)] i(3.40)Therefore, differentiating both sides of (3.37) and taking the t 0 limit, we obtainZdk 0 0 0k ρ(k ) 1(3.41)2πshowing that the spectral density is bounded for large k 0 . As k 0 , we shallexpect ρ(k 0 ) 1/(k 0 )2 0 at least.For the free scalar fields we are dealing with, we can explicitly calculate thespectral density. We have1ρF (k ) Z0Zdteik0tXeβEn hn φ(t)φ(0) φ(0)φ(t) nin 2π δ(k 0 Ek ) δ(k 0 Ek )2Ek 2π (k 0 )δ (k 0 )2 Ek2(3.42)It is sometimes convenient to define the propagators in terms of imaginary timevariables, so that we can work under the Euclidean space. This is usually referred toas the Matsubara propagator in literature. We define (τ ) 1Xhn eβH φ(τ )φ(0) niZ n(3.43)whose Fourier transform isβZdτ eiωn τ (τ ) (iωn ) (3.44)0The Matsubara frequencies ωn 2πn/β can be read off from the condition (τ ) 1 X iωn τe (iωn ) (τ β)β n11(3.45)
The relation between imaginary and real time propagators can be found as (τ ) D (t iτ )Zdk 0 k0 τ e[1 n(k 0 )]ρ(k 0 )2π(3.46)(3.47)So we can computeβ
An Introduction to Thermal Field Theory Yuhao Yang September 23, 2011 Supervised by Dr. Tim Evans Submitted in partial ful lment of the requirements for the degree of Master of Science in Quantum Fields and Fundamental Forces Department of Physics Imperial College London. Abstract This thesis aims to give an introductory review of thermal eld theo- ries. We review the imaginary time formalism .
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