By Andrei A. Bytsenko, G. Cognola, E. Elizalde, V. Moretti, S. Zerbini

ISBN-10: 9812383646

ISBN-13: 9789812383648

One of many goals of this publication is to provide an explanation for in a uncomplicated demeanour the likely tough problems with mathematical constitution utilizing a few particular examples as a advisor. In all the circumstances thought of, a understandable actual challenge is approached, to which the corresponding mathematical scheme is utilized, its usefulness being duly verified. The authors attempt to fill the space that often exists among the physics of quantum box theories and the mathematical equipment most fitted for its formula, that are more and more tough at the mathematical skill of the physicist.

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**Example text**

4 Under the general hypotheses on M and A, a smooth heat 40 The Zeta-Function Regularization Method kernel K(t, x, y\A) of A exists. Let Iu be a sufficiently small neighborhood of any u £ M, equipped with a coordinate system. If x,y 6 / „ , for any pair of multiindices a, (3, an integer N > D/2 + 2\a\ + 2\/3\ and a real rj £ (0,1), then D5D$K{t,x,y\A) [ P-v(x,y)/2t N \ = flgltf j {^t)D/2 x(

45) A;v being the Laplace-Beltrami operator on the iV-dimensional hypersurface x° = T = const, m and £ arbitrary parameters and R the scalar curvature of the manifold. For the sake of completeness, we consider the combination dT — /J,, which is relevant in the finite-temperature theories with chemical potential and for the sake of simplicity we will limit ourselves to the simplest cases, namely when the zeta function is regular at the origin and the manifolds are without boundary. The ultrastatic metric g' can be related to the static one by the conformal transformation ^ ( x ) = e2-(xV(x) , a :=

Hawking (1977)]. The limit case of vanishing temperature (ground state) may also be studied. Further external fields, different from gravity/gauge fields, may be intro duced in the functional integral above replacing A by a more complicated operator. Moreover

### Analytic Aspects of Quantum Fields by Andrei A. Bytsenko, G. Cognola, E. Elizalde, V. Moretti, S. Zerbini

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