# Quantum anomaly and geometric phase: Their basic differences

## Abstract

It is sometimes stated in the literature that the quantum anomaly is regarded as an example of the geometric phase. Though there is some superficial similarity between these two notions, I here show that the differences between these two notions are more profound and fundamental. As an explicit example, I analyze in detail a quantum mechanical model proposed by M. Stone, which is supposed to show the above connection. I show that the geometric term in the model, which is topologically trivial for any finite time interval T, corresponds to the so-called 'normal naive term' in field theory and has nothing to do with the anomaly-induced Wess-Zumino term. In the fundamental level, the difference between the two notions is stated as follows: The topology of gauge fields leads to level crossing in the fermionic sector in the case of chiral anomaly, and the failure of the adiabatic approximation is essential in the analysis, whereas the (potential) level crossing in the matter sector leads to the topology of the Berry phase only when the precise adiabatic approximation holds.

- Authors:

- Institute of Quantum Science, College of Science and Technology, Nihon University, Chiyoda-ku, Tokyo 101-8308 (Japan)

- Publication Date:

- OSTI Identifier:
- 20795761

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. D, Particles Fields; Journal Volume: 73; Journal Issue: 2; Other Information: DOI: 10.1103/PhysRevD.73.025017; (c) 2006 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ADIABATIC APPROXIMATION; CHIRALITY; FERMIONS; LAGRANGIAN FIELD THEORY; POTENTIALS; QUANTUM MECHANICS; TOPOLOGY

### Citation Formats

```
Fujikawa, Kazuo.
```*Quantum anomaly and geometric phase: Their basic differences*. United States: N. p., 2006.
Web. doi:10.1103/PHYSREVD.73.0.

```
Fujikawa, Kazuo.
```*Quantum anomaly and geometric phase: Their basic differences*. United States. doi:10.1103/PHYSREVD.73.0.

```
Fujikawa, Kazuo. Sun .
"Quantum anomaly and geometric phase: Their basic differences". United States.
doi:10.1103/PHYSREVD.73.0.
```

```
@article{osti_20795761,
```

title = {Quantum anomaly and geometric phase: Their basic differences},

author = {Fujikawa, Kazuo},

abstractNote = {It is sometimes stated in the literature that the quantum anomaly is regarded as an example of the geometric phase. Though there is some superficial similarity between these two notions, I here show that the differences between these two notions are more profound and fundamental. As an explicit example, I analyze in detail a quantum mechanical model proposed by M. Stone, which is supposed to show the above connection. I show that the geometric term in the model, which is topologically trivial for any finite time interval T, corresponds to the so-called 'normal naive term' in field theory and has nothing to do with the anomaly-induced Wess-Zumino term. In the fundamental level, the difference between the two notions is stated as follows: The topology of gauge fields leads to level crossing in the fermionic sector in the case of chiral anomaly, and the failure of the adiabatic approximation is essential in the analysis, whereas the (potential) level crossing in the matter sector leads to the topology of the Berry phase only when the precise adiabatic approximation holds.},

doi = {10.1103/PHYSREVD.73.0},

journal = {Physical Review. D, Particles Fields},

number = 2,

volume = 73,

place = {United States},

year = {Sun Jan 15 00:00:00 EST 2006},

month = {Sun Jan 15 00:00:00 EST 2006}

}