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Geometric origin of the energy-momentum tensor improvement terms

Damianos Iosifidis1,*, Manthos Karydas2,†, Anastasios Petkou3,‡, and Konstantinos Siampos3,§

  • *Contact author: damianos.iosifidis@ut.ee
  • †Contact author: mkarydas@berkeley.edu
  • ‡Contact author: petkou@auth.gr
  • §Contact author: ksiampos@auth.gr
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Phys. Rev. D 112, 025007 – Published 8 July, 2025

DOI: https://doi.org/10.1103/wby2-d33f

Abstract

In a flat background, the canonical energy momentum tensor of Lorentz and conformally invariant matter field theories can be improved to a symmetric and traceless tensor that gives the same conserved charges. We argue that the geometric origin of this improvement process is unveiled when the matter theory is coupled to metric-affine gravity. In particular, we show that the Belinfante-Rosenfeld improvement terms correspond to the matter theory’s hypermomentum. The improvement terms in conformally invariant matter theories are also related to the hypermomentum; however, a general proof would require an extended investigation. We demonstrate our results through various examples, such as the free massless scalar, the Maxwell field, Abelian p-forms, the Dirac field, and a nonunitary massless scalar field. Possible applications of our method for theories that break Lorentz or special conformal invariance are briefly discussed.

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Phys. Rev. D 112, 025007– Published 8 July, 2025
  • Received 10 April 2025
  • Accepted 25 June 2025
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