3.13 Renormalization in theories with a modified propagator
Let us comment briefly on the behavior of the entropy in theories described by a wave operator
, which is a function of the standard Laplace operator
. In flat space this was analyzed
in Section 2.12. As is shown in [184
] there is a precise relation between the small
expansion of the heat
kernel of operator
and that of the Laplace operator
. The latter heat kernel has the
standard decomposition
The heat kernel of operator
then has the decomposition [184
]
where
In even dimension
the term
. This decomposition is valid both for regular manifolds and
manifolds with a conical singularity. If a conical singularity is present, the coefficients
have the
standard decomposition into regular
and surface
parts as in Eq. (68). The surface term for
is just the area of the surface
, while the surface terms with
contain surface integrals of
-th power of the Riemann curvature. Thus, Eq. (121) is a decomposition in powers of the
curvature of the spacetime.
The functions
are defined in Eq. (40). In particular, if
(
) one finds that
The terms with
in decomposition (121) produce the UV divergent terms in the effective action
and entropy. The term
gives rise to the logarithmic UV divergence. In
dimensions the area
term in the entropy is the same as in flat spacetime (see Eq.(41)). In four dimensions (
) the UV
divergent terms in the entropy are
We note that an additional contribution to the logarithmic term may come from the first term in Eq. (124)
(for instance, this is so for the Laplace operator modified by the mass term,
).
In the theory with operator
Newton’s constant is renormalized as [184
]
while the higher curvature couplings
,
in the effective action are renormalized in the same
way as in Eq. (115). The renormalization of
and
then makes both the effective action
and the entropy finite in the exact same way as in the case of the Laplace operator
.
Thus, the renormalization statement generalizes to the theories with modified wave operator
.