The conformal symmetry generated by equal-time commutators

Summary

At the classical level we derive naively from the Ward identity for the conformal symmetry, treated as a diffeomorphism, the equal-time commutator between the improved energy-momentum tensor θ _mn (y) and the θ _k0(x)-components. The metric field is introduced as an external source for the stress tensor. The analysis is done in the weak-field approximation for the metric field. It is further shown that these equal-time commutators imply the correct conformal symmetry properties for the energy-momentum tensor and the desired algebra of the generators for the conformal symmetry group. A simple redefinition of the non-covariant time ordering operator allows to define a «symmetric» equal-time commutator. Our result reproduces an older result, obtained some time ago by Schwinger. The investigations are done in an arbitraryd-dimensional Minkowski space.