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

ISBN-10: 9812383646

ISBN-13: 9789812383648

One of many goals of this ebook is to give an explanation for in a uncomplicated demeanour the probably tricky problems with mathematical constitution utilizing a few particular examples as a consultant. In all the situations thought of, a understandable actual challenge is approached, to which the corresponding mathematical scheme is utilized, its usefulness being duly validated. The authors try and fill the space that usually exists among the physics of quantum box theories and the mathematical equipment most fitted for its formula, that are more and more not easy at the mathematical skill of the physicist.

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**Extra info for Analytic Aspects of Quantum Fields**

**Example text**

P. V. Roman'kov (1987)]. 49) and finally logZ = log Z'-log J[g,g']. 50) Then we see that in principle the knowledge of the partition function Z' in the ultrastatic manifold, and the heat coefficient a£>/2(x,x\Aq), are suffi cient in order to get the partition function in the static manifold. We know that the heat coefficients depend on invariant quantities build up with cur vature (field strength) and their derivatives. As a consequence, they do not depend on the parameter fi, since we may regard such a parameter as the temporal component of a (pure gauge) Abelian potential.

The existence of heat kernel for suitable operators A, also dropping the hypothesis of M. B. Davies (1989)] whenever A > 0 (or more weakly A > c for some constant c £ l ) . Holding our general hypotheses on M and A, we have several important theorems. 1 Under the stated conditions on M. , where * denote the complex conjugation. Proof. Define L= -A-d/dt. Let u1 v be complex functions defined on M x (0, t), t > 0 being any constant and such that they are C2 in the first variable and C 1 in the second.

It is useful to build up a heat kernel, taking, in a certain sense, the limit as N -> oo [I. Chavel (1984)]. -nt)-Dl'1e-^-^lu -> 5{x - y) in the weak sense as t —> 0 + in RD, one finds that FN(t, x, y) ->■ S(x, y) as t -> 0+ . 18) We can now state our theorem on the differentiated expansion of the heat kernel. 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.

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

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