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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?

mcFreid
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1 Answers1

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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).

Valter Moretti
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  • Ah, thank you. I forgot about the continuous clause in Noether's Theorem. As for the symmetry having to affect the dynamical variables... What variables are you referring to? For example, we can consider the fields to be the dynamical variables of the Lagrangian (which seems most reasonable) and then we see symmetries like the simple global U(1) transformation. But we can also look at the spatial coordinates and see continuous symmetries like spatial translation. – mcFreid Jan 24 '14 at 14:04
  • For dynamical variables I mean the variables describing the physical system: coordinates $q, \dot{q}$ in classical systems, fields $\phi^A$ for systems of fields, quantum field operators $\hat{\phi}^A$ for systems of quantum fields. Yes you can act by means of various transformations on these objects, if the Functional Action is preserved you have a symmetry. – Valter Moretti Jan 24 '14 at 14:24
  • Coordinate displacements (in Minkowski spacetime), $x \to x' = x+ v$ for instance, act on scalar field like this $\phi \to \phi'$ with $\phi'(x)= \phi(x-v)$... – Valter Moretti Jan 24 '14 at 14:28