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The generic hamiltonian for a particle that interacts with an electromagnetic field can be written as:

$$H=\frac{1}{2M}\sum_{i}\left(P_i-\frac{q}{c}A_{i}(X_j)\right)^2+V(X_j)+q\phi (X_j)$$

Where $(\phi,\vec{A})$ are the terms of the electromagnetic potential. From inspection, we can see that this hamiltonian is invariant under time reversal (i.e. the vector potential and momentum change sign with time reversal, but since they multiply by each other these signs cancel). And thus, since the system emanates from this hamiltonian, the system must also be invariant under time reversal (correct me if I'm wrong on this one).

However, I wanted to know if there's a more formal way to prove that this hamiltonian (or in general, any hamiltonian) is invariant under the time reversal operation.

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