Symmetry of Minkowksi Metric -> Conserved Current?

Question

My understanding of the Minkowski Metric is that we have the freedom to choose whether to place the negative sign on the time-component or on the spatial-components. That is, either basis should yield the same physics when dealing with Lorentz invariant Terms. Thus, if we have a Lorentz invariant Lagrangian, we should be able to take $\eta_{\mu \nu} \rightarrow - \eta_{\mu \nu}$ without changing the action.

What is the associated conserved current with this symmetry?

N.B. This transformation looks similar to a TP transformation. Is it identical?

Answer 1

Actually what you have to transform to define a symmetry for a classical or quantum system are the dynamical variables describing the system and appearing in the action rather than the metric (moreover time reversal could need a further complex conjugation).

In any cases here you are thinking of a discrete symmetries. Noether theorem instead implies the existence of dynamically conserved quantities provided the symmetries of the action are continuous: There is a dynamically conserved quantity for each continuous (differentiable actually) one-parameter group of symmetries of the action.

Passing to quantum systems (fields in particular), dynamically conserved quantities may arise also for discrete symmetries, provided they are described by simultaneously unitary and self-adjoint operators.

Parity operator can be taken of this type, but time reversal one cannot (if the Hamiltonian is bounded below as is physically necessary for the stability of the system), as it is an anti unitary operator (there are the only two possibilities permitted by Kadison-Wigner theorem).