32
Phase transitions in the early universe
and N (fJ) is a temperature-dependent normalization. The integral f V4> is a path
integral. Such integrals may be thought of as the generalization of an integration
f~oo dYI f~ dY2 ... f~oo dYn over the finite number of components of an ncomponent column vector y to an integration over the continuous infinity of
components of a function 4>(r, x). Evaluation of the path integral gives for the
contribution of a real scalar field to the free energy
-fJF = InZ
= - f d 3 x f (~~3 (~/p2+m2+ln[l-exp(_p/p2+ m2)]).
(2.14)
When the mass of the scalar field is negligible compared with the temperature (an
ideal ultra relativistic gas of scalar particles), the free energy density simplifies to
]f2T4
: F = - -
when T» m.
(2.15)
90
For gauge vector bosons, there are some subtleties because, for a typical
choice of gauge, the Lagrangian involves all four degrees of freedom of the gauge
field AIL(X) and also involves the Fadeev-Popov ghost fields which occur in the
construction of a consistent renormalizable theory but are not physical particles.
However, a mass less vector field has only two degrees of freedom and the extra
degrees of freedom are not physical and cannot be in eqUilibrium with a heat bath
nor, of course, can the Fadeev-Popov ghosts. Fonunately there exist gauges in
which each gauge field has only two degrees of freedom and in which there are no
Fadeev-Popov ghosts and the partition function can be related to the Lagrangian
density in such a gauge. In any other gauge, it can be shown that one may continue
to use this expression for Z but with the form of the Lagrangian appropriate for
that gauge. In an arbitrary gauge the contribution of the two non-physical degrees
of freedom of the gauge field cancels the contribution from the Fadeev-Popov
ghosts. Then, the contribution to the free energy density from a massless vector
gauge field is found to be
:F = _ 2]f2T4
(2.16)
90 .
In the case of Dirac fields 1/1, the corresponding development at finite
temperature involves fields 1/I(r, x) that are anti-periodic in r in the interval
(0, fJ),
1/I(r = 0, x) = -1/I(r = fJ, x)
(2.17)
and the contribution to the free energy density is
:F = _ 7]f2T4
when T» m.
(2.18)
180
Phase transitions in the early universe
and N (fJ) is a temperature-dependent normalization. The integral f V4> is a path
integral. Such integrals may be thought of as the generalization of an integration
f~oo dYI f~ dY2 ... f~oo dYn over the finite number of components of an ncomponent column vector y to an integration over the continuous infinity of
components of a function 4>(r, x). Evaluation of the path integral gives for the
contribution of a real scalar field to the free energy
-fJF = InZ
= - f d 3 x f (~~3 (~/p2+m2+ln[l-exp(_p/p2+ m2)]).
(2.14)
When the mass of the scalar field is negligible compared with the temperature (an
ideal ultra relativistic gas of scalar particles), the free energy density simplifies to
]f2T4
: F = - -
when T» m.
(2.15)
90
For gauge vector bosons, there are some subtleties because, for a typical
choice of gauge, the Lagrangian involves all four degrees of freedom of the gauge
field AIL(X) and also involves the Fadeev-Popov ghost fields which occur in the
construction of a consistent renormalizable theory but are not physical particles.
However, a mass less vector field has only two degrees of freedom and the extra
degrees of freedom are not physical and cannot be in eqUilibrium with a heat bath
nor, of course, can the Fadeev-Popov ghosts. Fonunately there exist gauges in
which each gauge field has only two degrees of freedom and in which there are no
Fadeev-Popov ghosts and the partition function can be related to the Lagrangian
density in such a gauge. In any other gauge, it can be shown that one may continue
to use this expression for Z but with the form of the Lagrangian appropriate for
that gauge. In an arbitrary gauge the contribution of the two non-physical degrees
of freedom of the gauge field cancels the contribution from the Fadeev-Popov
ghosts. Then, the contribution to the free energy density from a massless vector
gauge field is found to be
:F = _ 2]f2T4
(2.16)
90 .
In the case of Dirac fields 1/1, the corresponding development at finite
temperature involves fields 1/I(r, x) that are anti-periodic in r in the interval
(0, fJ),
1/I(r = 0, x) = -1/I(r = fJ, x)
(2.17)
and the contribution to the free energy density is
:F = _ 7]f2T4
when T» m.
(2.18)
180