ar X iv : g r - qc / 0 30 40 95 v 1 2 4 A pr 2 00 3 BRX TH - 517 A Note on Matter

We consider Bel–Robinson-like higher derivative conserved two-index tensors H µν in simple matter models, following a recently suggested Maxwell field version. In flat space, we show that they are essentially equivalent to the true stress-tensors. In curved Ricci-flat backgrounds it is possible to redefine H µν so as to overcome non-commutativity of covariant derivatives, and maintain conservation, but they become model-and dimension-dependent, and generally lose their simple " BR " form. Historically, the nonexistence of a local stress tensor in generally covariant theories such as Einstein's led to a successful search for the next-best thing, the covariantly conserved but four-index and higher derivative Bel–Robinson (BR) tensor [1], quadratic in curvature. Being traceless in D=4, it has no " T µν-like " 2-index contraction. However, this discovery led ineluctably to an equally successful search [2] for matter analogs of BR, despite the presence of perfectly good T µν there. These quantities resemble BR in being of higher derivative order and quadratic in the " curvatures " of the corresponding fields. In particular, it has recently been shown [3] that there is a natural 2-index conserved BR-version of the Maxwell tensor. In flat space QFT, operators H µν whose matrix elements behave like those of the stress-tensor are essentially proportional to it [4]: In momentum space, one expects a conserved symmetric 2-tensor to have the form f (q 2)T µν (q), up to (trivial) identically conserved terms. By continuity, one might expect some similar property to hold, at least for test fields, i.e., in Ricci-flat spaces. The results we obtain here confirm these expectations, at least for H µν of simple free field models. Using the simplest – scalar and vector – free field models we will first investigate their H µν in flat space and immediately verify the expectation that they (and their obvious generalizations) are indeed related to the corresponding T µν by form factors. In gravitational backgrounds we find that while H µν can be redefined to survive non-commutation of derivatives at least in Ricci-flat spaces, generically they lose their flat space attributes and become model-and dimension-dependent.