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The "complex conjugation" operator is basis dependent and so ill defined. A vector that has real components in one basis may be have complex components in another. For example the $x\leftrightarrow …

The algebra here is, as far as I can see correct, but there is a conceptual issue that has confused me in the past. In @Vokaylop's construction we find a $U_T$ that is constructed for this particula …

the operation $X\mapsto A^{-1}XA$ is usually called conjugation of $X$ by $A$ because this is the language used in group theory. I assume that by $\dagger$ the author means inverse, because antilinear …

I find that your $S(\Lambda_T)$ should be $\gamma^1\gamma^2\gamma^3$ as that flips the $\gamma^0$ three times, but flips the other three gammas twice. But be careful the actual action of $T$ on spin …

If P and T are good symmetries in any one specific reference frame then they are good symmetries in any frame in a Lorentz invariant theory.

It usually just means that the material is magnetic, since magnetization ${\bf M}$ changes to $-{\bf M}$ under time reversal.

1) Time reversal takes ${\bf k}\to -{\bf k}$. 2) Time reversal takes the Berry curvature ${\bf \Omega}\to -{\bf \Omega}$. Therefore $\Omega(-{\bf k})=-\Omega({\bf k})$. I think it really is this sim …

There are two definitions of time reversal, one of which changes particle to antiparticles. The second, the Wigner definition, does not and is the one usually used these days.

I think that you are confusing the unitary matrix $O$ that appears in the first quantized formula $$ OH^*O^{-1} =H $$ with the antilinear operator $T$ that acts as on the second quantised field operat …

It's best to avoid basis dependent formulae such as the one you use. The time reversal matrix is defined in all space-time signatures by $$ T \gamma^\mu T^{-1}= {(\gamma^\mu)}^T $$ where $(\gamma^\m …