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

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 Poincaré 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 Poincaré 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 Poincaré. We discuss at length the case of scale and conformal symmetry, of which we give some examples. We show, for Poincaré 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.

31 pag

Cited by in corpus (13)

• Scale without Conformal Invariance at Three Loops
• Covariant-differential formulation of Lagrangian field theory
• Overconnections and the energy-tensors of gauge and gravitational fields
• Supercurrent, Supervirial and Superimprovement
• Noether and Hilbert (metric) energy-momentum tensors are not, in general, equivalent
• Covariant Differential Identities and Conservation Laws in Metric-Torsion Theories of Gravitation. I. General Consideration
• Energy-momentum tensor and duality symmetry of linearized gravity in the Fierz formalism
• Splitting the source term for the Einstein equation to classical and quantum parts
• Gauge Coupling Field, Currents, Anomalies and N=1 Super-Yang-Mills Effective Actions
• Noether symmetries for fields and branes in backgrounds with Killing vectors
• Proposal for the proper gravitational energy-momentum tensor
• Nonlocal Lagrangian fields and the second Noether theorem. Non-commutative $U(1)$ gauge theory
• Equality of the Hilbert Hamiltonian and the canonical Hamiltonian for gauge theories in a static spacetime