Using the property , it is easy to see that the kernel
is symmetric and
is antisymmetric in its arguments; that
is,
and
.
The physical meanings of these kernels can be
extracted if we write the renormalized CTP effective action at
finite temperature (169) in an influence
functional form [32
, 111
, 161
, 162
].
, the imaginary part of the CTP effective action can
be identified with the noise kernel and
, the antisymmetric
piece of the real part with the dissipation kernel. Campos and
Hu [54
, 55
] have shown that
these kernels identified as such indeed satisfy a thermal
fluctuation-dissipation relation.
If we denote the difference and the sum of the
perturbations defined along each branch
of the complex time path of integration
by
and
, respectively, the
influence functional form of the thermal CTP effective action may
be written to second order in
as
In the above and subsequent equations, we denote
the coupling parameter in four dimensions by
, and consequently
means
evaluated at
.
is the complete contribution
of a free massless quantum scalar field to the thermal graviton
polarization tensor [249
, 250
, 72
, 27
], and it is
responsible for the instabilities found in flat spacetime at finite
temperature [116, 249
, 250
, 72
, 27
]. Note that the
addition of the contribution of other kinds of matter fields to the
effective action, even graviton contributions, does not change the
tensor structure of these kernels, and only the overall factors are
different to leading order [249
, 250
]. Equation (177
) reflects the fact
that the kernel
has thermal as well as
non-thermal contributions. Note that it reduces to the first term
in the zero temperature limit (
),
Finally, as defined above, is the noise kernel representing the random
fluctuations of the thermal radiance and
is the dissipation kernel, describing the
dissipation of energy of the gravitational field.