Anomalous commutators for energy-momentum tensors

TL;DR

This paper investigates anomalous contributions to energy-momentum tensor commutators across even-dimensional spacetimes, using a non-perturbative approach, and finds anomalies only in two dimensions, with implications for field theories.

Contribution

It provides a covariant calculation of anomalous commutators in even dimensions and extends the analysis to general Minkowskian spacetime, clarifying when anomalies occur.

Findings

Anomalous contributions exist only in two-dimensional spacetime.

No anomalies are found in higher even dimensions for free fields.

The study confirms the absence of 3-cocycles in two dimensions for both Abelian and non-Abelian cases.

Abstract

Anomalous contributions to the energy-momentum commutators are calculated for even dimensions, by using a non-perturbative approach that combines operator product expansion and Bjorken-Johnson-Low limit techniques. We first study the two dimensional case and give the covariant expression for the commutators. The expression in terms of light-cone coordinates is then calculated and found to be in perfect agreement with the results in the literature. The particular scenario of the light-cone frame is revisited using a reformulation of the BJL limit in such a frame. The arguments used for $n=2$ are then generalized to the case of any even dimensional Minkowskian spacetime and it is shown that there are no anomalous contributions to the commutators for $n\not=2$. These results are found to be valid for both fermionic and bosonic free fields. A generalization of the BJL-limit is later used to obtain double commutators of energy-momentum tensors and to study the Jacobi identity. The two dimensional case is studied and we find no existance of 3-cocycles in both the Abelian and non-Abelian case.