# Topological Quantum Field Theory Seminar

### Stress-Energy-Momentum Tensors

J. Marsden and I present a new method of constructing a stress-energy-momentum tensor for a classical field theory based on covariance considerations and Noether theory. Our stress-energy-momentum tensor ${T}^{\mu }{}_{\nu }$ is defined using the (multi)momentum map associated to the spacetime diffeomorphism group. The tensor ${T}^{\mu }{}_{\nu }$ is uniquely determined as well as gauge-covariant, and depends only upon the divergence equivalence class of the Lagrangian. It satisfies a generalized version of the classical Belinfante-Rosenfeld formula, and hence naturally incorporates both the canonical stress-energy-momentum tensor and the "correction terms" that are necessary to make the latter well behaved. Furthermore, in the presence of a metric on spacetime, our ${T}^{\mu \nu }$ coincides with the Hilbert tensor and hence is automatically symmetric.

References:
1. Gotay, M. J. and J. E. Marsden [1992], Stress-energy-momentum tensors and the Belinfante-Rosenfeld formula, Contemp. Math. 132, 367-391.
2. Forger, M. and H. Römer [2004], Currents and the energy-momentum tensor in classical field theory: A fresh look at an old problem, Ann. Phys. 309, 306-389.