The Definition of Double Commutators and Consistency in Free Field Theory

TL;DR

This paper establishes a consistent definition of double commutators in free field theory using generalized functions, demonstrating the limitations of common regularization methods like BJL and point-splitting.

Contribution

It provides a new, regularization-independent definition of double commutators that preserves the Jacobi identity in free field theory.

Findings

Double commutators are correctly defined within the generalized functions framework.

BJL and point-splitting methods fail to produce correct double commutators in free field theory.

The new definition ensures consistency even when regularization is removed.

Abstract

Within the framework of generalized functions a general consistent definition of double commutators is given. This definition respects the Jacobi identity even if the regularization is removed. The double commutator of fermionic currents is calculated in this limit. We show that BJL--type prescriptions and point--splitting prescriptions for calculating double commutators fail to give correct results in free field theory.