Path integral for gauge theories with fermions

Journal Article · Thu May 15 00:00:00 EDT 1980 · Phys. Rev., D; (United States)

DOI:https://doi.org/10.1103/PhysRevD.21.2848· OSTI ID:5317892

Fujikawa, K

The Atiyah-Singer index theorem indicates that a naive unitary transformation of basis vectors for fermions interacting with gauge fields is not allowed in general. On the basis of this observation, it was previously shown that the path-integral measure of a gauge-invariant fermion theory is transformed nontrivially under the chiral transformation, and thus leads to a simple derivation of ''anomalous'' chiral Ward-Takahashi identities. We here clarify some of the technical aspects associated with the discussion. It is shown that the Jacobian factor in the path-integral measure, which corresponds to the Adler-Bell-Jackiw anomaly, is independent of any smooth regularization procedure of large eigenvalues of D in Euclidean theory; this property holds in any even-dimensional space-time and also for the gravitational anomaly. The appearance of the anomaly and its connection with the index theorem are thus related to the fact that the primary importance is attached to the Lorentz-covariant ''energy'' operator D and that D and ..gamma../sub 5/ do not commute. The abnormal behavior of the path-integral measure at the zero-frequency sector in the presence of instantons and its connection with spontaneous symmetry breaking is also clarified. We comment on several other problems associated with the anomaly and on the Pauli-Villars regularization method.

• Research Organization: Institute for Nuclear Study, University of Tokyo, Tanashi, Tokyo 188, Japan
• OSTI ID: 5317892
• Journal Information: Phys. Rev., D; (United States), Journal Name: Phys. Rev., D; (United States) Vol. 21:10; ISSN PRVDA
• Country of Publication: United States
• Language: English

Related Subjects

645301* -- High Energy Physics-- Particle Invariance Principles & Symmetries-- General-- (-1987)
645400 -- High Energy Physics-- Field Theory
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
EIGENVALUES
EUCLIDEAN SPACE
FERMIONS
FEYNMAN PATH INTEGRAL
GAUGE INVARIANCE
GRAVITATIONAL FIELDS
INTEGRALS
INVARIANCE PRINCIPLES
LIE GROUPS
MATHEMATICAL SPACE
P INVARIANCE
RIEMANN SPACE
SPACE
SPACE-TIME
SU GROUPS
SYMMETRY BREAKING
SYMMETRY GROUPS
WARD IDENTITY
YANG-MILLS THEORY