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If instead of fulfilling the homogeneous KleinGordon equation the field cp would satisfy (0 + m2 )cp(x) = j(x), Eq. (388) would be replaced by (Ox' + m2 )Tcp(x')cpt(x) = GF(x'  x) = Tj(x')cpt(x)  ib 4 (x'  x) (389) It follows that the vacuum expectation value
i<OI Tcp(x')cpt(x) 10) (390) is one of the Green functions of the KleinGordon operator. A little calculation shows that it is indeed the Feynman scalar propagator encountered in Chap. 1: GF (x  y)   d k 1 (21lf k2 _ m2
+ ie e ik'(xy) (391) We conclude that the quantization of a free, relativistic scalar field has produced a satisfactory description of spinless noninteracting particles, obeying the BoseEinstein statistics, with or without charge. In the absence of interactions the number of field quanta is conserved. 315 Thermodynamic Eqnilibrinm
In a rest frame with total threemomentum zero it may happen that the correct description of the above quantized system, instead of being the vacuum state, is a thermodynamic equilibrium at temperature T = l/kfJ (k is Boltzmann's constant) and chemical potential Jl. This is well defined as long as the total number N of quanta is conserved. We return to a fixed volume quantization with discrete momenta: n" n2, n3 integers
whereupon integrals are replaced by sums
We rescale the annihilation and creation operators by defining
a(k) J2Wk V Ak
in such a way that
with Kronecker symbols instead of delta functions' We have therefore
dk wkat(k)a(k) = LkWkAtAk
dk at(k)a(k) = LkAtAk
To simplify we have assumed the system to be neutral. The grand partition function !!t is given by a trace in Fock space QUANTIZATIONFREE FIELDS
Each mode contributes a factorized term, and since Tr eAAtA
(1  eA)l, In the large volume limit the thermodynamical potential is
From thermodynamics we know that if p is the pressure n p=
{Jp V. Therefore
(392) particle density
d k (2n)3In (1  eP(a>k # ) while the mean values of the energy density
E/V (without zero point oscillations) and
Fi/Vare
(393) the familiar expressions corresponding to Bose's statistics. In the case m = 0, and when the number of particles is not conserved (that is, J1 = 0), we have a situation analogous with blackbody radiation except for polarization. The energy density becomes Integrating by parts we obtain
 3 (2n)3 k In (1  e  PI k I) (394) a classical relationship, familiar from blackbody radiation, stating that the average energy momentum tensor 0#v which has only diagonal elements 0 00 = E/V, 0ij = p(jij (i,j = 1,2,3) is traceless: 0##=y3p=0
Denoting by parentheses thermodynamic averages (we assume now J1 arbitrary) (A) = L,._ex=p,{_~{JE,.,} <,IXIcA,IIX_> L. exp {  {JE.} let us study the propagation at finite temperature. Set f(w) = Then 1 ePa> _ 0 but keep the mass
(AtA k ) = (jk,kf(Wk) (AkAt ) = (jk.d1 (AkAd In the infinite volume limit
+ f(wdJ = =
(jk.k f(  Wk) (AtAt ) QUANTUM FIELD THEORY
and the timeordered product takes the form (395) This expression exhibits the propagator at finite temperature as the sum of the zero temperature Feynman propagator plus a temperaturedependent solution of the homogeneous KleinGordon equation which vanishes like e Pm when fJ > ro(T > 0). This propagator is interesting to study dynamical disturbances away from equilibrium. The structure of the partition function suggests that we may encounter, in applications of perturbation theory to statistical systems, a different, "Euclidean," propagator. Starting from the basic dynamical field variables at fixed time, we define an evolution for imaginary times Xo =
ixo
where Xo is real, and at first restricted to the interval [0, fJ] according to
<p(x) = eiHxo <p(0, x) e iHxo = eHXo <p(0, x) e Hxo
Operator ordering with respect to Xo will be denoted by the symbol f and will satisfy
It can be continued as a periodic function of Xo  yo of period
fJ. Indeed, assume at first
Yo restricted
to the interval [0, fJ]. Then from the cyclic character of the trace
[f<p(x)<p(yl]xo~o = (Tr ePH)l Tr [e PH <p(y)<p(O, xl] (Tr ePH)l Tr [e PH ePH <p(0, x) e PH <p(yl] = [f<p(x)<p(yl]xo~P When this continuation is performed the D(xo  Yo) function is to be understood as applied to periodic functions (functions on a circle) with 2nD (IX) = Ln e'na. We can then write (396) In the last expression use has been made of minkowskian notations with X O = ixo pure imaginary as well as kO. If we denote by the symbol J(P) d4k/(2n)4 the combined discrete sum and integral

