Noether symmetries, energy-momentum tensors and conformal invariance in classical field theory

TL;DR

This paper reviews Noether symmetries in classical field theory, compares different energy-momentum tensors, and explores conditions for scale and conformal invariance, providing tools to identify additional spacetime symmetries.

Contribution

It clarifies the roles of Belinfante and Hilbert energy-momentum tensors in relation to spacetime symmetries and offers criteria to detect conformal invariance in Poincaré invariant theories.

Findings

Belinfante tensor generates Poincaré symmetries.

Hilbert tensor tests for additional spacetime symmetries.

Scale invariance uniquely determines the Hilbert tensor.

Abstract

In the framework of classical field theory, we first review the Noether theory of symmetries, with simple rederivations of its essential results, with special emphasis given to the Noether identities for gauge theories. Will this baggage on board, we next discuss in detail, for Poincar\'e invariant theories in flat spacetime, the differences between the Belinfante energy-momentum tensor and a family of Hilbert energy-momentum tensors. All these tensors coincide on shell but they split their duties in the following sense: Belinfante's tensor is the one to use in order to obtain the generators of Poincar\'e symmetries and it is a basic ingredient of the generators of other eventual spacetime symmetries which may happen to exist. Instead, Hilbert tensors are the means to test whether a theory contains other spacetime symmetries beyond Poincar\'e. We discuss at length the case of scale and conformal symmetry, of which we give some examples. We show, for Poincar\'e invariant Lagrangians, that the realization of scale invariance selects a unique Hilbert tensor which allows for an easy test as to whether conformal invariance is also realized. Finally we make some basic remarks on metric generally covariant theories and classical field theory in a fixed curved bakground.