Geometric Aspects of Space-Time Reflection Symmetry in Quantum Mechanics

For nearly two decades, much research has been carried out on properties of physical systems described by Hamiltonians that are not Hermitian in the conventional sense, but are symmetric under space-time reflection; that is, they exhibit PT symmetry. Such Hamiltonians can be used to model the behavior of closed quantum systems, but they can also be replicated in open systems for which gain and loss are carefully balanced, and this has been implemented in laboratory experiments for a wide range of systems. Motivated by these ongoing research activities, we investigate here a particular theoretical aspect of the subject by unraveling the geometric structures of Hilbert spaces endowed with the parity and time-reversal operations, and analyze the characteristics ofPT -symmetric Hamiltonians. A canonical relation between aPT -symmetric operator and a Hermitian operator is established in a geometric setting. The quadratic form corresponding to the parity operator, in particular, gives rise to a natural partition of the Hilbert space into two halves corresponding to states having positive and negative PT norm. Positive definiteness of the norm can be restored by introducing a conjugation operator C ; this leads to a positive-definite inner product in terms of CPT conjugation.