Question

Time Reversal in Euclidean Spacetime - unitary or antiunitary?

Answer 1

pre-request) We know that time reversal operator T is an anti-unitary operator in Minkowsi Spacetime. i.e.  Tz=z∗T where the complex number z becomes its complex conjugate. See, for example, Peskin and Schroeder ``An Introduction To Quantum Field Theory'' p.67 Eq (3.133).  (Question) Is Euclidean time reversal operator TE an unitary operator, i.e.  TEz=zTE(?) or an anti-unitary operator in Euclidean Spacetime? Why is necessarily that or why is it necessarily not that? Or should TE be an unitary operator like Parity P instead?  However, here see the attempt of a PRL paper, Euclidean continuation of the Dirac fermion where time reversal TE is given by, on page 3: TEz=z∗TE(?) TE is still an anti-unitary operator!  It seems to me if one consider Euclidean spacetime, the Euclidean signature is the same sign, say (−,−,−,−,…), then Parity P acts on the space is equivalent to the Euclidean time reversal operator TE acts on Euclidean time (which is now like one of the spatial dimensions). So shouldn't TE be an unitary operator as Parity P

Answer 2

Starting with Tz=z∗TTz=z∗T, and assuming T=iTET=iTE, this would give (iTE)z=z∗(iTE)(iTE)z=z∗(iTE), that is i(TEz)=(z∗i)TEi(TEz)=(z∗i)TE, that is (TEz)=−i(z∗i)TE(TEz)=−i(z∗i)TE, that is TEz=z∗TETEz=z∗TE