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

One of many goals of this booklet is to provide an explanation for in a easy demeanour the doubtless 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 try and fill the distance that often exists among the physics of quantum box theories and the mathematical tools most suitable for its formula, that are more and more hard at the mathematical skill of the physicist.

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**Example text**

E. the gravitational field), X is a real vector field on M (for instance the electromagnetic field), V is any other real external potential, and dpbg is the natural measure induced by the metric. Operators A are supposed to act on the space of compactly-supported smooth functions. Actually, as it will be clear later, the functional integral depends on the chosen self-adjoint extension (in a suitable Hilbert space) of A, A instead of A itself. ] is strongly dependent from boundary conditions. W.

Parker et al. L. L. Hu (1983)]. D. Linde (1983)] and the expansion should be nearly adiabatic. Phase transitions in de Sitter spacetime have been first considered in [B. Allen (1983)] with the important result that the critical behavior strongly depends on curvature. Very little is known about those in anti-de Sitter spacetime, mainly because de Sitter space is more relevant than anti-de Sitter in inflationary scenarios. Now we would like to present a survey of finite-temperature quantum field theory and discuss some useful representations of thermodynamical quantities.

Moreover, by this result, taking into account that (x,y) i-» d(x,y) is everywhere continuous in M. x M. for ^d{p,q) is defined as inf {€(7)} for all 7 : [0,1] —> M. differentiable such that 7(0) = p, 7(1) = q, £(7) denoting the length of 7. 32 The Zeta-Function Regularization Method Euclidean metrics and employing Sobolev's Lemma [W. Rudin (1997)], one finds that the function (x, y) i-> a(x, y) belongs to the space of smooth func tions C°°({(x,y) £ M x M | d(x,y) < r}). P. do Carmo (1992)]. In particular, a strictly positive upper bound fco for the sectional curvature of a compact manifold is sufficient to have r > 0, indeed, in this case r > Tr/\/k~o.