The effective potential at finite temperature
33
For massless fermions (with only one helicity state of the particle), the calculation
(using Weyl spinors) gives half of this answer.
Pulling all of this together, the free energy density of an ideal ultra relativistic
gas (T » m) is given by
7 ) 7r 2 T4
:1"= - ( NB +-NF - -
(2.19)
8
90
where NB and N F are, respectively, the numbers ofbosonic and fermionic degrees
of freedom. (NB = 1 for a real scalar field, NB = 2 for a real gauge field, N F = 4
for a Dirac particle where there are two helicity states for the particle and two for
the antiparticle and NF = 2 for a Weyl field.) The pressure, entropy density and
energy density follow from (2.5), (2.6) and (2.4).
7) 7r 2 T4
P =
(2.20)
( NB + gNF ---g{)
7) 27r 2 T3
(
s= NB+-NF - -
(2.21)
8
45
and
7) 7r 2 T4
(2.22)
p= ( NB+gNF 30'
2.3 The effective potential at finite temperature
In quantum field theory at zero temperature, the expectation value rpc of a scalar
field rp (also referred to as the classical field) is determined by minimizing
the effective potential V (rpcl. The effective potential contains a tree-level
potential term, which can be read off from the Hamiltonian density, and quantum
corrections from various loop orders. The one-loop quantum correction is
calculated by shifting the fields rp by their expectation values rpc and isolating the
terms .cquad(rpc, jJ) in the Lagrangian density which are quadratic in the shifted
fields jJ. If we write
V(rpcl = Vo(rpc) + Vl(rpc)
(2.23)
where Vo is the tree-level contribution and VI is the one-loop quantum correction
then, for a single scalar field,
exp ( -if d4XVl(rpC») = f vjJexp(if d 4 X.cQUad (rpc,jJ») (2.24)
where, as in section 2.2, f VjJ, denotes a path integral. (The derivation and
evaluation of (2.24) can be found elsewhere [I ].)