Unitary time-reversal on non-orientable spacetimes
Time reversal symmetry occupies a distinctive role in quantum mechanics, fundamentally requiring an anti-unitary operator to ensure a physically consistent representation. As such, the time reversal operator combines a unitary transformation with complex conjugation, enabling the necessary inversion of the imaginary unit that appears in quantum commutation relations and dynamical equations. Attempts to represent time reversal as a purely unitary operation encounter fundamental contradictions, including violations of canonical commutation relations and issues with the positivity of energy spectra. However, recent advances in quantum gravity and black hole physics reveal that in spacetimes with non-orientable topology - where a global temporal orientation is not well defined - time reversal may be realized by a purely unitary operator. Such non-orientable geometries connect two asymptotically spacetimes with opposite time directions, thereby encoding time reversal topologically and removing the need for complex conjugation. In this work, we explore the deep connection between spacetime orientability and the nature of the time reversal operator, demonstrating that orientable spacetimes require anti-unitary time reversal consistent with conventional quantum theory, while non-orientable spacetimes allow unitary time reversal operators consistent with negative energy states.