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By Whittaker E.T.

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These polarization vectors do not depend on x or i. Therefore they commute with the free photon propagator D (x − y) = g − (1 − ) d 4 k eik(x−y) (2 )4 −k 2 − i 9x 9x 92x ( ¿ 0). 99) We proceed with deriving a special representation of the scalar propagator. It is obtained by performing the integration over k3 in Eq. 100) k02 − k12 − k22 + i . 102) st has been introduced. 101) we have achieved a mode decomposition of the operator K st on the surface . 94), is now reduced to the algebraic problem of inverting the (2 × 2)-matrix hij .

In the cuto regularization the parameter (it has the dimension of an inverse mass) can be multiplied by any ÿnite, positive number. For physical reasons, all regularizations must deliver the same end result. How to achieve this is subject to the normalization condition to be discussed below. , that it delivers a mathematically correct deÿned expression. M. Bordag et al. / Physics Reports 353 (2001) 1–205 41 We want to study the divergent part of the ground state energy by making use of the heat kernel expansion.

56) K(t) = (4 t)d=2 n=0;1=2;1;::: where d is the dimension of the manifold M . Eq. 56) is an asymptotic expansion for t → 0 and it is in general not a converging series. The coe cients an are called heat kernel coe cients. There is, of course, an analogous expansion for the local heat kernel. , without potential or boundary conditions. The operator is P = + m2 . In that case Eq. 57) 4t (4 t)d=2 where m is the mass of the ÿeld. This solution can be easily veriÿed by inserting into Eq. 52). 53) is also satisÿed because for t → 0, the right hand side of Eq.

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