A Note on Commutation Relation in Conformal Field Theory

TL;DR

This paper investigates the commutation relations of scalar fields in conformal field theories, revealing that finite, well-defined commutators require smearing in both space and time, especially for non-free theories.

Contribution

It explicitly computes vacuum expectation values of scalar field commutators in conformal field theories and shows the necessity of smearing in both space and time for well-defined equal-time commutators.

Findings

Equal-time commutators are not well-defined for non-free conformal fields.

Smearing in both space and time is required for finite commutators.

Results extend from cylinder to Minkowski space via small distance limit.

Abstract

In this note, we explicitly compute the vacuum expectation value of the commutator of scalar fields in a d-dimensional conformal field theory on the cylinder. We find from explicit calculations that we need smearing not only in space but also in time to have finite commutators except for those of free scalar operators. Thus the equal time commutators of the scalar fields are not well-defined for a non-free conformal field theory, even if which is defined from the Lagrangian. We also have the commutator for a conformal field theory on Minkowski space, instead of the cylinder, by taking the small distance limit. For the conformal field theory on Minkowski space, the above statements are also applied.