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

ISBN-10: 9812383646

ISBN-13: 9789812383648

One of many goals of this booklet is to provide an explanation for in a simple demeanour the doubtless tough problems with mathematical constitution utilizing a few particular examples as a advisor. In all of the situations thought of, a understandable actual challenge is approached, to which the corresponding mathematical scheme is utilized, its usefulness being duly proven. The authors try and fill the space that usually exists among the physics of quantum box theories and the mathematical tools most fitted for its formula, that are more and more challenging at the mathematical skill of the physicist.

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**Sample text**

1992)], we define Ev = - lim dp log Z0. 54), one immediately gets (here S[

The "regularized" function ^r\s0\A) is defined by C{r)(so\A) = lim C(s\A) ResC(sol^) s—>so so + (2-21og2)ResC(s 0 |A). 56) Of course, when the zeta function is regular at so, C'rHso|^l) coincide with C(so\A). 55) not only define the finite-temperature properties of quantum fields but, as we shall see in Sec. 5, they will be our starting point for the computation of the regularized vacuum energy. 57) where Fp represents the temperature dependent part (statistical sum) and so Eqs. 55) give different representations of FQ.

A few comments on these hypotheses are in order. First of all, a countable base of the topology is required in order to endow the manifold with a par tition of the unity and allow the use of Hilbert-Schmidt's theory (which is Survey of the Chapter, Notation and Conventions 31 fundamental in our dealing with the heat-kernel theory [I. Chavel (1984)]). R. Halamos (1969)] and this allows one to de scribe Hilbert-Schmidt operators in terms of integral kernels. Anyhow, the requirement of compactness implies completeness (equivalent to geodesic completeness), paracompactness (the manifold being Hausdorff) and thus the presence of a countable base of the topology being automatically as sured by our general hypotheses.

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

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