Time-Conjugation in a Unified Quantum Theory for Hermitian and Non-Hermitian Electronic Systems under Time-Reversal Symmetry

Abstract

We propose a reformulation of the mathematical formalism of many-electron quantum theory that rests entirely on the physical properties of the electronic system under investigation, rather than conventional mathematical assumption of Hermitian operators in Hilbert space. The formalism is based on a modified dot-product that replaces the familiar complex-conjugation in Hilbert space ℌ (fixed for all physical systems) by time-conjugation in T-space (as generated by the specific spin, magnetic field, or other explicit t-dependence of the system Hamiltonian ℋ of interest), yielding different spatial structure for different systems. The usual Hermitian requirement for physical operators is thereby generalized to a self-t-adjoint (“t-reversible”) character, leading to correspondingly generalized theorems of virial and hypervirial type. The T-space reformulation preserves the real values of measurable properties and the Born-probabilistic interpretations of state functions that underlie the present quantum theory of measurement, while also properly distinguishing “temporal” behavior of internal decay (tunneling-type) phenomena from that of applied fields with parametric t-dependence on an external clock. The t-product represents a further generalization of the “c-product” that was previously found useful in complex coordinate-rotation studies of autoionizing resonances.