Biorthogonal Renormalization

TL;DR

This paper investigates the scaling ambiguity in the biorthogonal formalism of non-Hermitian quantum mechanics, proposing an inner product that resolves this ambiguity and clarifies the physical interpretation of such systems.

Contribution

It introduces a basis-independent inner product to eliminate the scaling ambiguity and explores its implications for the physicality of quantum representations.

Findings

The biorthogonal inner product depends on eigenvector normalization in some cases.

A new gauge-independent inner product is proposed to resolve the ambiguity.

The formalism clarifies when physical quantities are basis-dependent or independent.

Abstract

The biorthogonal formalism extends conventional quantum mechanics to the non-Hermitian realm. It has, however, been pointed out that the biorthogonal inner product changes with the scaling of the eigenvectors, an ambiguity whose physical significance is still being debated. Here, we revisit this issue and argue when this choice of normalization is of physical importance. We illustrate in which settings quantities such as expectation values and transition probabilities depend on the scaling of eigenvectors, and in which settings the biorthogonal formalism remains unambiguous. To resolve the apparent scaling ambiguity, we introduce an inner product independent of the gauge choice of basis and show that its corresponding mathematical structure is consistent with quantum mechanics. Using this formalism, we identify a deeper problem relating to the physicality of Hilbert space representations, which we illustrate using the position basis. Apart from increasing the understanding of the mathematical foundations upon which many physical results rely, our findings also pave the way towards consistent comparisons between systems described by non-Hermitian Hamiltonians.