The Energy Density Tensor in Gauge-Independent Quantum Electrodynamics

In Heisenberg representation two different definitions of an energy density tensor are given for gauge-independent quantum electrodynamics, by Eqs. (1)-(4) and Eqs. (20)-(21), respectively. Both tensors lead to the same total energy and momentum, if we assume the interaction to vanish at $t=-\infty$. They both satisfy conservation laws. The tensor character of the first one is proved, and the tensor character of the second one is manifest. The first tensor is obtained by analogy with the result of a derivation of the energy density tensor as source of the gravitational field from general-relativistic considerations in manifestly covariant quantum electrodynamics, with subsequent omission of the "phantom terms" containing the redundant variables of this theory. The second tensor has the advantage of admitting a simple covariant subtraction of its vacuum value, and of simplifying even further by use of the new covariant auxiliary condition proposed recently by the author. Its disadvantage, though, is the impossibility of direct physical interpretation, as in Heisenberg representation it is not expressed in terms of field variables in Heisenberg representation. The inconclusiveness of an argument for possible equality of the two tensors is discussed. Both tensors contain the usual self-interaction effects, and the problem is posed of how to eliminate these effects.