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 uncomplicated demeanour the possible tricky problems with mathematical constitution utilizing a few particular examples as a advisor. In all the instances 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 often exists among the physics of quantum box theories and the mathematical tools most suitable for its formula, that are more and more difficult at the mathematical skill of the physicist.

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**Sample text**

68), but now, the one-loop partition function has to be defined using the "generalized zeta function" OJ(S\LD), as in Eq. 27). In the case of Bose fields one has periodic boundary conditions in the imaginary time and so, in ultrastatic manifolds, the eigenvalues of the Laplace-like operator LD are given in terms of the eigenvalues uJ1 of Ljf by Eq. 59). Using Eq. l, we obtain y J ' ViiFr(s) + -&ks) £ / ^*S~3/2 e-"2/32/4t Ke-tL». 70) For the operators we are dealing with, one can see that, in the latter equa tion, the integral is exponentially vanishing in the limit /3 —> oo and so, as in the standard case, it does not give contributions to the vacuum energy.

57) where Fp represents the temperature dependent part (statistical sum) and so Eqs. 55) give different representations of FQ. 4 The thermodynamic potential The generalization to the more general case of a charged (scalar) field with a non-vanishing chemical potential /x, in thermal equilibrium by some unspec ified process at finite temperature T = /? _ 1 is quite immediate. The grand canonical partition function has the path-integral representation [B. Freedman and L. I. Kapusta (1981); A. Actor (1985); A.

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.

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

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