(pre-request) We know that time reversal operator $T$ is an anti-unitary operator in Minkowsi Spacetime. i.e.
$$ T z=z^*T $$ 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 $T_E$ an unitary operator, i.e.
$$ T_E z=z T_E \;\;\;(?) $$
or an anti-unitary operator in Euclidean Spacetime? Why is necessarily that or why is it necessarily not that? Or should $T_E$ 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 $T_E$ is given by, on page 3: $$ T_E z=z^*T_E \;\;\;(?) $$ $T_E$ is still an anti-unitary operator!
It seems to me if one consider Euclidean spacetime, the Euclidean signature is the same sign, say $(-,-,-,-,\dots)$, then Parity $P$ acts on the space is equivalent to the Euclidean time reversal operator $T_E$ acts on Euclidean time (which is now like one of the spatial dimensions). So shouldn't $T_E$ be an unitary operator as Parity $P$?
i. Please, you may read, an earlier Phys.SE question here, probably is poorly formulated, so cannot draw the efforts of people to answer the question. Here let me try an easier way and focus on one issue only.
ii. Osterwalder-Schrader (OS) approach, and ``A continuous Wick rotation for spinor fields and supersymmetry in Euclidean space'' by Peter van Nieuwenhuizen, Andrew Waldron.