Noether's theorem, the stress-energy tensor and Hamiltonian constraints

TL;DR

This paper reviews Noether's theorem, focusing on linear transformations, and explores how symmetries relate to conserved currents, stress-energy tensors, and Hamiltonian constraints, especially in the context of general relativity and covariant theories.

Contribution

It clarifies the role of linear and local symmetries in deriving conserved quantities and stress-energy tensors, with explicit derivations in second order theories and insights into Hamiltonian constraints.

Findings

The stress-energy tensor is identically zero in certain covariant theories.

Explicit form of the canonical stress-energy tensor in second order theories.

Linear symmetry allows expressing currents as four divergences.

Abstract

Noether's theorem is reviewed with a particular focus on an intermediate step between global and local gauge and coordinate transformations, namely linear transformations. We rederive the well known result that global symmetry leads to charge conservation (Noether's first theorem), and show that linear symmetry allows for the current to be expressed as a four divergence. Local symmetry leads to identical conservation of the current and allows for the expression of the charge as two dimensional surface integral (Noether's second theorem). In the context of coordinate transformations, an additional step (Poincare symmetry) is of physical interest and leads to the definition of the symmetric Belinfante stress-energy tensor, which is then shown to be identically zero in generally covariant first order theories. The intermediate step of linear symmetry turns out to be important in general relativity when the customary first order Lagrangian is used, which is covariant only under affine transformations. In addition, we derive explicitely the canonical stress-energy tensor in second order theories in its identically conserved form. Finally, we analyze the relations between the generators of local transformations, the corresponding currents and the Hamiltonian constraints.