# Path Integrals and Quantum Anomalies

## Kazuo Fujikawa and Hiroshi Suzuki

### Abstract

This book provides an introduction to the path integral formulation of quantum field theory and its applications to the analyses of symmetry breaking by the quantization procedure. This symmetry breaking is commonly called the ‘quantum anomaly’ or simply the ‘anomaly’, and this naming shows that the effect first appeared as an exceptional phenomenon in field theory. However, it is shown that this effect has turned out to be very fundamental in modern field theory. In the path integral formulation, it has been recognized that this effect arises from a non-trivial Jacobian in the change of path ... More

This book provides an introduction to the path integral formulation of quantum field theory and its applications to the analyses of symmetry breaking by the quantization procedure. This symmetry breaking is commonly called the ‘quantum anomaly’ or simply the ‘anomaly’, and this naming shows that the effect first appeared as an exceptional phenomenon in field theory. However, it is shown that this effect has turned out to be very fundamental in modern field theory. In the path integral formulation, it has been recognized that this effect arises from a non-trivial Jacobian in the change of path integral variables, namely, the path integral measure breaks certain symmetries. The study of the quantum anomaly attempts to bring about a better understanding of the basis of quantum theory and, consequently, it is a basic notion which could influence the entire quantum theory beyond field theory. The quantum anomaly is located at the border of divergence and convergence, though the quantum anomaly itself is perfectly finite, and thus closely related to the presence of an infinite number of degrees of freedom.

*Keywords: *
path integral formulation,
quantum field theory,
symmetry breaking,
modern field theory,
non-trivial Jacobian,
path integral measure,
quantum theory

### Bibliographic Information

Print publication date: 2004 |
Print ISBN-13: 9780198529132 |

Published to Oxford Scholarship Online: September 2007 |
DOI:10.1093/acprof:oso/9780198529132.001.0001 |

### Authors

#### Affiliations are at time of print publication.

Kazuo Fujikawa, *author*

Dept of Physics, University of Tokyo, Japan

Hiroshi Suzuki, *author*

Dept of Mathematical Sciences, Ibaraki University, Japan

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