### 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
.