Analytic aspects of quantum fields / A.A. Bytsenko [and others].

Includes bibliographical references (pages 327-339) and index.

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One of the aims of this volume is to explain in a basic manner the seemingly difficult issues of mathematical structure using some specific examples as a guide. In each of the cases considered, a comprehensible physical problem is approached, to which the corresponding mathematical scheme is applied, its usefulness being duly demonstrated. The authors try to fill the gap that always exists between the physics of quantum field theories and the mathematical methods best suited for its formulation, which are increasingly demanding on the mathematical ability of the physicist.

Ch. 1. Survey of path integral quantization and regularization techniques. 1.1. Path integral and regularization techniques for functional determinants. 1.2 Schwinger-like regularizations and heat-kernel expansion. 1.3 Logarithmic terms in the heat-kernel expansion. 1.4 One-loop renormalization group equations. 1.5 Static spacetimes: thermodynamic effects -- ch. 2. The zeta-function regularization method. 2.1. Survey of the chapter, notation and conventions. 2.2. Heat-kernel expansion and coefficients. 2.3. Local and global spectral zeta functions on compact manifolds. 2.4. Effective action, effective Lagrangian and Green functions. 2.5. Noncompact manifolds and manifolds with a boundary. 2.6. The stress-energy tensor and field-fluctuation regularization -- ch. 3. Generalized spectra and spectral functions on non-commutative spaces. 3.1 Extended Chowla-Selberg formulae and arbitrary spectral forms. 3.2. Barnes and related zeta functions. 3.3. Spectral zeta functions for scalar and vector fields on a spacetime with a non-commutative toroidal part. 3.4. Applications to quantum field theory in non-commutative space -- ch. 4. Spectral functions of laplace operator on locally symmetric spaces. 4.1. Locally symmetric spaces of rank one. 4.2. The spectral zeta function. 4.3. Asymptotics of the heat kernel. 4.4. Product of Einstein manifolds. 4.5. Real hyperbolic manifolds -- ch. 5. Spinor fields. 5.1. The Dirac operator and spectral invariants. 5.2. The massive Dirac operator. 5.3. One-dimensional example. 5.4. The one-loop effective action. 5.5. Dirac bundle and the Ray-Singer norm. 5.6. The determinant line bundles. 5.7. The Dirac index of hyperbolic manifolds -- ch. 6. Field fluctuations and related variances. 6.1. The first variation of the effective action. 6.2. The second variation of the effective action. 6.3. Some examples. 6.4. Remarks -- ch. 7. The multiplicative anomaly. 7.1. Introduction. 7.2. Zeta trace, determinant and the multiplicative anomaly. 7.3. Perturbative derivation of the multiplicative anomaly. 7.4. The multiplicative anomaly formula. 7.5. Multiplicative anomaly for locally symmetric spaces -- ch. 8. Applications of the multiplicative anomaly. 8.1. Anomalies for Dirac-like operators. 8.2. The massive Dirac operator. 8.3. Consistent, covariant and multiplicative anomalies. 8.4. Interacting charged scalar model. 8.5. Concluding remarks -- ch. 9. The Casimir effect. 9.1. Introduction. 9.2. The Casimir energy. 9.3. The Casimir energy in the ball. 9.4. A braneworld computation.

English.

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