Antilinearity Rather than Hermiticity as a Guiding Principle for Quantum Theory

Abstract

Currently there is much interest in Hamiltonians that are not Hermitian but instead possess an antilinear $PT$ symmetry, since such Hamiltonians can still lead to the time-independent evolution of scalar products, and can still have an entirely real energy spectrum. However, such theories can also admit of energy spectra in which energies come in complex conjugate pairs, and can even admit of Hamiltonians that cannot be diagonalized at all. Hermiticity is just a particular realization of $PT$ symmetry, with $PT$ symmetry being the more general. These $PT$ theories are themselves part of an even broader class of theories, theories that can be characterized by possessing some general antilinear symmetry, as that requirement alone is a both necessary and sufficient condition for the time-independent evolution of scalar products, with all the different realizations of the $PT$ symmetry program then being obtained. Use of complex Lorentz invariance allows us to show that the antilinear symmetry is uniquely specified to be $CPT$, with the $CPT$ theorem thus being extended to the non-Hermitian case. For theories that are separately charge conjugation invariant, the results of the $PT$-symmetry program then follow. We show that in order to construct the correct classical action needed for a path integral quantization one must impose $CPT$ symmetry on each classical path, a requirement that has no counterpart in any Hermiticity condition since Hermiticity of a Hamiltonian is only definable after the quantization has been performed and the quantum Hilbert space has been constructed. We show that whether or not a $CPT$-invariant Hamiltonian is Hermitian is a property of the solutions to the theory and not of the Hamiltonian itself. Thus Hermiticity never needs to be postulated at all.