Path integrals and anomalies in curved space.

*(English)*Zbl 1120.81057
Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press (ISBN 0-521-84761-3/hbk; 0-511-21772-2/ebook). xvii, 379 p. (2006).

The book is a monograph on the path integral approach for quantum field theory (QFT) in curved space-times. It consists of two parts.

Part I introduces path integral techniques for quantum mechanics in a curved space. Three schemes are presented for regularizing path integrals: 1) time slicing; 2) mode regularization (integration over the coefficients corresponding to the various modes) — the method used by Feynman and Hibbs for harmonic oscillator and further developed in instanton and soliton physics; 3) dimensional regularization (calculating path integrals for a convenient choice of the parameter of space dimension and analytical continuation in this parameter).

In Part II it is shown how this technique may be applied for evaluating anomalies (symmetries of a classical theory which are broken by quantum corrections) arising in QFT and quantum gravity. The derivation of the chiral (corresponding to the analogue of the phase transformtaiton but with the matrix \(\gamma_5\) in the phase) and gravitational anomalies as first given by Alvarez-GaumĂ© and Witten and later by P. Nieuwenhuizen are given in detail. Path integrals provide an alternative method in perturbative aspects of QFT, but they are essential for nonperturbative aspects, particularly in the study of instantons and solitons. In these problems the choice of the measure of path integrals is an important element of the theory. Another area where the path integral measure plays an important role is quantum gravity. Anomalies constitute one of the most important features of QFT. Requirement of the absence of the anomalies constrains the theory enormously.

The book provides an advanced text for researchers and graduate students specializing in QFT in curved spaces and string theory.

Part I introduces path integral techniques for quantum mechanics in a curved space. Three schemes are presented for regularizing path integrals: 1) time slicing; 2) mode regularization (integration over the coefficients corresponding to the various modes) — the method used by Feynman and Hibbs for harmonic oscillator and further developed in instanton and soliton physics; 3) dimensional regularization (calculating path integrals for a convenient choice of the parameter of space dimension and analytical continuation in this parameter).

In Part II it is shown how this technique may be applied for evaluating anomalies (symmetries of a classical theory which are broken by quantum corrections) arising in QFT and quantum gravity. The derivation of the chiral (corresponding to the analogue of the phase transformtaiton but with the matrix \(\gamma_5\) in the phase) and gravitational anomalies as first given by Alvarez-GaumĂ© and Witten and later by P. Nieuwenhuizen are given in detail. Path integrals provide an alternative method in perturbative aspects of QFT, but they are essential for nonperturbative aspects, particularly in the study of instantons and solitons. In these problems the choice of the measure of path integrals is an important element of the theory. Another area where the path integral measure plays an important role is quantum gravity. Anomalies constitute one of the most important features of QFT. Requirement of the absence of the anomalies constrains the theory enormously.

The book provides an advanced text for researchers and graduate students specializing in QFT in curved spaces and string theory.

Reviewer: Michael B. Mensky (Moskva)

##### MSC:

81S40 | Path integrals in quantum mechanics |

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

81T20 | Quantum field theory on curved space or space-time backgrounds |

81T50 | Anomalies in quantum field theory |

81T13 | Yang-Mills and other gauge theories in quantum field theory |

81T16 | Nonperturbative methods of renormalization applied to problems in quantum field theory |

83C47 | Methods of quantum field theory in general relativity and gravitational theory |