Quantum Correlations under Time Reversal and Incomplete Parity Transformations in the Presence of a Constant Magnetic Field

We derive the quantum analogues of some recently discovered symmetry relations for time correlation functions in systems subject to a constant magnetic field. The symmetry relations deal with the effect of time reversal and do not require—as in the formulations of Casimir and Kubo—that the magnetic field be reversed. It has been anticipated that the same symmetry relations hold for quantum systems. Thus, here we explicitly construct the required symmetry transformations, acting upon the relevant quantum operators, which conserve the Hamiltonian of a system of many interacting spinless particles, under time reversal. Differently from the classical case, parity transformations always reverse the sign of both the coordinates and of the momenta, while time reversal only of the latter. By implementing time reversal in conjunction with ad hoc “incomplete” parity transformations (i.e., transformations that act upon only some of the spatial directions), it is nevertheless possible to achieve the construction of the quantum analogues of the classical maps. The proof that the mentioned symmetry relations apply straightforwardly to quantal time correlation functions is outlined.