Is the combination of parity $P: (x,y,t)\to (-x,y,t)$ (sometimes also called reflection $R$) and time reversal $T: (x,y,t)\to (x,y,-t)$ always a symmetry in (2+1)D theories with Lorentz or Galilean invariance?

By Lorentz invariance I mean of course invariance under only the connected proper orthochronous Lorentz group $SO^+(1,d)$, which does not contain $P$ or $T$. Since $PT\notin SO^+(1,d)$, parity has no reason to imply time-reversal symmetry in a Lorentz invariant theory. This is true for any $d\geq 1$.

In $d=3$ (real life), it is easy to write down field theories that are $T$ invariant but not $P$ invariant, the weak interactions in the standard model is an example: take a Dirac fermion and a gauge field $A_\mu$; if the action contains both of the following terms

$\bar\psi\require{cancel}\cancel A \psi$ and $\bar\psi\require{cancel}\cancel A \gamma_5\psi$,

then it is impossible to realize parity on the fermions such that both terms are invariant (however, both terms are time-reversal invariant).

What is the status in $d=2$? In all the examples I can think of (massive fermions, Chern-Simons term, background magnetic field) $P$ and $T$ violation go hand in hand, in such a way that $PT$ is conserved. Does anyone have a counterexample?

  • $\begingroup$ If we go to the Euclidean spacetime, then $PT$ is simply a $\pi$ rotation which is an element of the $SO(3)$ symmetry, the Euclidean version of Lorentz invariance. This of course does not answer the question in Minkowski spacetime, but may be helpful. $\endgroup$ – Meng Cheng Jan 10 '16 at 4:16
  • $\begingroup$ Yes I agree. This also applies to any other dimension though, so I don't think it's likely to help. $\endgroup$ – user157321 Jan 10 '16 at 4:20

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