New Formulation of Anomaly,Anomaly Formula and Graphical Representation

TL;DR

This paper introduces a new general approach to calculating anomalies in quantum field theory using heat-kernel methods, providing systematic formulas and a graphical tensor representation for clarity and simplicity.

Contribution

It develops a novel formulation of anomalies via propagator theory and heat-kernel equations, including a graphical tensor method for better visualization and calculation.

Findings

Derived general anomaly formulas applicable to various dimensions.

Systematic calculation of anomalies in 2D and 4D theories.

Introduced a graphical tensor representation for invariants.

Abstract

A general approach to anomaly in quantum field theory is newly formulated by use of the propagator theory in solving the heat-kernel equation. We regard the heat-kernel as a sort of the point-splitting regularization in the space(-time) manifold. Fujikawa's general standpoint that the anomalies come from the path-integral measure is taken. We obtain some useful formulae which are valid for general anomaly calculation. They turn out to be the same as the 1-loop counter-term formulae except some important total derivative terms. Various anomalies in 2 and 4 dimensional theories are systematically calculated. Some important relations between them are concretely shown. As for the representation of general (global SO(n)) tensors, we introduce a graphical one. It makes the tensor structure of invariants very transparent and makes the tensor calculation so simple.