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 š’Ÿ = F (− ∇2 ), which is a function of the standard Laplace operator ∇2. In flat space this was analyzed in Section 2.12. As is shown in [184Jump To The Next Citation Point] there is a precise relation between the small s expansion of the heat kernel of operator 2 F (− ∇ ) and that of the Laplace operator 2 − ∇. The latter heat kernel has the standard decomposition
Tres∇2 = ---1---∑ a sn−dāˆ•2. (120 ) (4π)dāˆ•2n=0 n
The heat kernel of operator 2 F (− ∇ ) then has the decomposition [184Jump To The Next Citation Point]
∑ Tre− sF (−∇2 ) = --1---- an š’Æn(s), (121 ) (4 π)dāˆ•2 n=0
({ P (s) n < dāˆ•2 š’Æn(s) = d−2n n− dāˆ•2 − sF (q)|| . (122 ) ( (− ∂q) e |q=0 n ≥ dāˆ•2
In even dimension d the term š’Ædāˆ•2(s) = 1. This decomposition is valid both for regular manifolds and manifolds with a conical singularity. If a conical singularity is present, the coefficients an have the standard decomposition into regular areg n and surface aΣ n parts as in Eq. (68View Equation). The surface term for n = 1 is just the area of the surface Σ, while the surface terms with n ≥ 2 contain surface integrals of (n − 1)-th power of the Riemann curvature. Thus, Eq. (121View Equation) is a decomposition in powers of the curvature of the spacetime.

The functions Pn are defined in Eq. (40View Equation). In particular, if F (q) = qk (k > 0) one finds that

− n-Γ ( n2k) Pn (s) = s 2k--n---. (123 ) Γ ( 2)k
The terms with n ≤ d āˆ•2 in decomposition (121View Equation) produce the UV divergent terms in the effective action and entropy. The term n = d āˆ•2 gives rise to the logarithmic UV divergence. In d dimensions the area term in the entropy is the same as in flat spacetime (see Eq.(41View Equation)). In four dimensions (d = 4) the UV divergent terms in the entropy are
A(Σ )∫ ∞ ds 1 ∫ ( 1 ) S = 48-π-- 2 s-P2 (s) − 144π- R − 5(Rii − 2Rijij) ln šœ–. (124 ) šœ– Σ
We note that an additional contribution to the logarithmic term may come from the first term in Eq. (124View Equation) (for instance, this is so for the Laplace operator modified by the mass term, F (q) = q + m2).

In the theory with operator F (− ∇2 ) Newton’s constant is renormalized as [184Jump To The Next Citation Point]

∫ -1--- -1-- -1-- ∞ ds- Gren = GB + 12π šœ–2 s P2(s). (125 )
while the higher curvature couplings ci, i = 1, 2,3 in the effective action are renormalized in the same way as in Eq. (115View Equation). The renormalization of G and {ci} then makes both the effective action and the entropy finite in the exact same way as in the case of the Laplace operator − ∇2. Thus, the renormalization statement generalizes to the theories with modified wave operator F (− ∇2).
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