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As John Rennie wrote in his answer, what one should consider is not a generic $\Delta t$ but the period $\tau$ which is a positive number. However positivity also arises form coherence with other rela …

You forgot the crucial fact that the observables are selfadjoint and that the relevant matrix elements have the same entries. Only those matrix elements are physically measurable since they are expec …

The point is that the symmetries in QM (bijective operations sending states to states preserving the transition probability) can be represented by either unitary or antiunitary operators. This is the …

Notice that the vectors $k$ belong to the reciprocal lattice and the label $k$ is by definition ambiguous: if you change $k$ by adding a generator of the reciprocal lattice, the wave function changes …

There is a mistake in your definition of time reversal as $x$ is fixed under that transformation, the remaining transformations being correct. With this correct version of T, the Hamiltonian you study …

(I am not sure to understand what the question is actually.) Yes it is possible, if ignoring the orbital part of the state describing the relative position of the two fermions (or the relative momen …

We would like to understand whether or not the time arrow is consequence of fundamental laws or, better, of the initial conditions of the evolution of a single system or the entire universe. It is qui …

I am not sure to understand the question. Are you asking why exponent do not change as a consequence of $\Theta S_k \Theta^{-1} = -S_k$? You are forgetting that $\Theta$ is anti linear, so $$\Theta …

The statement is true if the Hamiltonian operator has spectrum bounded below ( due to thermodynamical reasons) but not above as it happens in many relevant physical systems. Here is the proof. The ti …