Stress-Tensor Commutators and Schwinger Terms

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Abstract
We investigate, in local field theory, general properties of commutators involving Poincaré generators or stress‐tensor components, particularly those of local commutators among the latter. The spectral representation of the vacuum stress commutator is given, and shown to require the existence of singular ``Schwinger terms'' at equal times, similar to those present in current commutators. These terms are analyzed and related to the metric dependence of the stress tensor in the presence of a prescribed of a prescribed gravitational field and some general results concerning this dependence presented. The resolution of the Schwinger paradox for the Tμν commutators is discussed together with some of its implications, such as ``nonclassical'' metric dependence of Tμν. A further paradox concerning the vacuum self‐stress—whether the stress tensor or its vacuum‐subtracted value should enter in the commutators—is related to the covariance of the theory, and partially resolved within this framework.