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 give an explanation for in a simple demeanour the probably tough problems with mathematical constitution utilizing a few particular examples as a consultant. In all of the instances thought of, a understandable actual challenge is approached, to which the corresponding mathematical scheme is utilized, its usefulness being duly established. The authors attempt to fill the distance 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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**Extra info for Analytic Aspects of Quantum Fields**

**Example text**

Aj(t) > . . -» 0 and a possible continuousspectrum part containing 0 at most, finally the set of eigenvectors 4>j{t) forms a Hilbert base. Notice that, in particular, the dimension of every eigenspace is finite and Kt is strictly positive. 4 in Chapter 2 of [A. Pazy (1983)]). 38 in [W. Rudin (1997)]). Moreover as Kt is self-adjoint, it is simply proven that B must be self-adjoint too. The spectral theorem implies that Spec(f£T4) = {e~tx | A G Spec(iift)} and thus the spectrum of B must be discrete (in particular each eigenspace has finite dimension) with elements A0 < Ai < A2 < .

A spacetime is said to be curved if it is not Minkowski spacetime. D. W. A. M. Wald (1994)]. M. Wald (1994)] requires that the D-dimensional spacetime is globally hyperbolic (roughly speaking this means that there must exist {D — 1)dimensional spacelike submanifolds E, called Cauchy surfaces, such that the assignment of Cauchy data on E determine the evolution of a field in 25 The Zeta-Function 26 Regularization Method the whole spacetime). All the physical quantities we shall consider in this chapter arise from a Euclidean functional integral which plays a crucial role in quantum field theory in a curved spacetime.

The physical reason for this behavior is the occurrence of Bose-Einstein condensation. Of course, in the limit fi -> 0, Eqs. 63) reduces to Eqs. 55) which has been discussed in the previous section. Since Trexp(-fLjv) = ]T). ), Eq. )Ax) = ft0+I^log(l-e-^^))+-i^log(l-e-^-^) j 3 20 Survey of Path Integral Quantization and Regularization Techniques = fi0 + -„ TV log ( l - e - w + ) + | Tr log ( l - e " ^ " ) . 65) Above, we have introduced the two pseudo-differential operators Q^ = 1 ii £JY ±fi, whose eigenvalues are ujj ±/i.

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

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