Stress‐Tensor Commutators and Schwinger Terms

TLDR: In this paper, the spectral representation of the vacuum stress commutator is given, and it is shown that the existence of singular ''Schwinger terms'' at equal times, similar to those present in current commutators, is required.

Abstract: We investigate, in local field theory, general properties of commutators involving Poincare 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.

Frequently asked questions

Q1. What is the definition of the singular operator Til V?

the singular operator Til V must be redefined, in analogy with the procedure for currents, as the limit of a nonlocal TIlV in which the constituent field operators are separated by a space like distance and the commutators evaluated before taking the limit. 

Q2. What are the main conditions that are examined in the paper?

The authors have examined a number of consistency conditions on the commutation relation among the Poincare generators and the stress-tensor components in local field theory. 

Q3. What is the simplest symmetric second-rank tensor?

For an arbitrary conserved symmetric second-rank tensor, there are two independent weight functions specifying the vacuum commutator;(01 [PV(x), T"IJ(x')] 

Q4. What is the dependence on the four components needed to evaluate the right sides of Eqs.?

The dependence on the four components gov' needed to evaluate the right sides of Eqs. (7), is explicitly exhibited for fields of spin ::;; 1, and seen to be in accord with the requirements for a Hamiltonian formulation of the coupled matter and gravitational fields. 

Q5. What is the definite metric for the radIation gauge?

Phys. 7, 10 (1966). '10 This condition includes the radIation gauge formulation of electrodynamics which possesses a positive definite metric and a gauge invariant stress tensor. 

Q6. What is the problem with the Lorentz covariance requirement?

In particular, the apparent difficulty that, while the right sides of such relations should vanish in vacuum, they actually involve the un subtracted (nonvanishing in vacuum) stresses or their integrals, was resolved by the Lorentz covariance requirement that (P") = -Atr. 

Q7. What is the metric dependent tensor density?

The authors begin with the definition of the stress tensor of a dynamical system as the coefficient of the variation of an external metric in the generally covariant form of its action13 according to oWM = f dxlogll.(x)lr(x), where 'bIlV(x) is the metric dependent symmetric tensor density.