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b(t' - t)[<p(x' ), <pt(x)] ib(t' - t)d(O, x' - x)
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<p(X'), <p t(X)] = ib(t' - t)
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(Ox'
+ m2)T<p(x')<pt(x) =
(3-88)
QUANTUM FIELD THEORY
If instead of fulfilling the homogeneous Klein-Gordon equation the field cp would satisfy (0 + m2 )cp(x) = j(x), Eq. (3-88) would be replaced by
(Ox'
+ m2 )Tcp(x')cpt(x) =
GF(x' - x) =
Tj(x')cpt(x) - ib 4 (x' - x)
(3-89)
It follows that the vacuum expectation value
i<OI Tcp(x')cpt(x) 10)
(3-90)
is one of the Green functions of the Klein-Gordon 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'(x-y)
(3-91)
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.
3-1-5 Thermodynamic Eqnilibrinm
In a rest frame with total three-momentum 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
QUANTIZATION-FREE 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
(3-92) particle density
d k (2n)3In (1 -
e-P(a>k -# )
while the mean values of the energy density
E/V (without zero point oscillations) and
Fi/Vare
(3-93)
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)
(3-94)
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##=-y-3p=0
Denoting by parentheses thermodynamic averages (we assume now J1 arbitrary) (A) = L-,._ex-=-p-,{_-~{JE-,.,----} <--,IXIc-A-,-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,k-f(Wk)
(AkAt ) = (jk.d1 (AkAd In the infinite volume limit
+ f(wdJ = =
(jk.k f( - Wk)
(AtAt )
QUANTUM FIELD THEORY
and the time-ordered product takes the form (3-95) This expression exhibits the propagator at finite temperature as the sum of the zero temperature Feynman propagator plus a temperature-dependent solution of the homogeneous Klein-Gordon 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 e-PH)-l Tr [e- PH <p(y)<p(O, xl]
(Tr e-PH)-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
(3-96)
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
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