Biorthogonal Quantum Mechanics

TL;DR

This paper develops a finite-dimensional biorthogonal quantum mechanics framework by relaxing Hermiticity, exploring its implications for states, observables, and dynamics, with a brief discussion on extending to infinite dimensions.

Contribution

It introduces and formalizes biorthogonal quantum mechanics in finite dimensions, extending traditional Hermitian-based quantum theory.

Findings

Biorthogonal eigenstates replace orthogonality in the theory.

Probability rules and state properties are characterized within this framework.

The approach is extended with a discussion on infinite-dimensional systems.

Abstract

The Hermiticity condition in quantum mechanics required for the characterisation of (a) physical observables and (b) generators of unitary motions can be relaxed into a wider class of operators whose eigenvalues are real and whose eigenstates are complete. In this case, the orthogonality of eigenstates is replaced by the notion of biorthogonality that defines the relation between the Hilbert space of states and its dual space. The resulting quantum theory, which might appropriately be called 'biorthogonal quantum mechanics', is developed here in some detail in the case for which the Hilbert space dimensionality is finite. Specifically, characterisations of probability assignment rules, observable properties, pure and mixed states, spin particles, measurements, combined systems and entanglements, perturbations, and dynamical aspects of the theory are developed. The paper concludes with a brief discussion on infinite-dimensional systems.