Stress-Tensor-Current Commutators, Electromagnetic and Weak Corrections to Current Commutators, and Sum Rules

Abstract

Equal-time commutation relations between selected components of the energy-momentum tensor and selected components of a current, arising from internal transformations, are derived in a model-independent fashion. These commutators are then used to establish the following three results: (1) It is shown that current-current commutators do not have the standard form in the presence of electromagnetic and weak interactions. Specifically, it is demonstrated that the space components of an isospin current do not transform as isospin vectors. (2) Weinberg's second sum rule is shown to follow from further assumptions about our commutators, and it is argued that kaon mass corrections must be expected in the $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$ generalization of this sum rule. (3) A relation between decay constants in broken $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$ is established. It is $F_{\kappa }{{\mu }_{\kappa }}^2+F_K{{\mu }_K}^2=F_{\pi }{{\mu }_{\pi }}^2$.