# Mathematical Melody in Quantum Anomaly via the Path Integral Approach: A Lesson from the Transverse Current Anomalies in QED

## TL;DR

This paper uses the path integral method to analyze transverse current anomalies in QED, revealing a regulator-dependent anomaly term and clarifying a paradox in anomaly calculations.

## Contribution

It introduces a novel approach to calculating anomalies in QED using infinite-dimensional Grassmann integration, challenging conventional perturbative results.

## Key findings

- Identifies a regulator-dependent anomaly term in the path integral calculation.
- Explains the paradox between divergent anomalies and null results in perturbation theory.
- Provides insights into the conceptual understanding of anomalies in quantum field theory.

## Abstract

I address and solve the natural problem of calculating the transverse current anomalies in quantum electrodynamics by means of the path-integral method. An explicitly divergent and regulator-dependent anomaly term is produced for the vector current, in apparent contradiction with the null-result prediction of the one-loop perturbative evaluation. This paradox is carefully explained using the concept of infinite-dimensional Grassmann functional integration, signifying a modification to the conventional wisdom of understanding anomalies in field theory.