[Paper Review] Is PT-symmetric quantum mechanics just quantum mechanics in a non-orthogonal basis?

Damien Martin|ArXiv.org|Jan 29, 2007

Quantum Mechanics and Non-Hermitian Physics11 references5 citations

TL;DR

This paper argues that PT-symmetric quantum mechanics is equivalent to standard quantum mechanics formulated in a non-orthogonal basis, meaning non-Hermitian Hamiltonians in PT-symmetric theories are merely unitarily equivalent to Hermitian ones under a basis transformation. The key result is that no experimental distinction exists between PT-symmetric and ordinary quantum mechanics in finite-dimensional systems, and claims of faster quantum evolution in PT-symmetric systems are shown to be coordinate artifacts.

ABSTRACT

One of the postulates of quantum mechanics is that the Hamiltonian is Hermitian, as this guarantees that the eigenvalues are real. Recently there has been an interest in asking if $H^\dagger = H$ is a necessary condition, and has lead to the development of PT-symmetric quantum mechanics. This note shows that any finite physically acceptable non-Hermitian Hamiltonian is equivalent to doing ordinary quantum mechanics in a non-orthogonal basis. In particular, this means that there is no experimental distinction between PT-symmetric quantum mechanics and ordinary quantum mechanics for finite systems. In particular, the claim that PT-symmetric quantum mechanics allows for faster evolution than Hermitian quantum mechanics is shown to be a problem of physical interpretation.

Motivation & Objective

• To determine whether PT-symmetric quantum mechanics represents a genuine generalization of standard quantum mechanics or is equivalent to it under basis transformations.
• To resolve the foundational confusion about whether non-Hermitian Hamiltonians in PT-symmetric theories describe new physics or merely a different mathematical formulation.
• To show that the claim of faster quantum evolution in PT-symmetric systems is not a physical distinction but a consequence of basis choice.
• To clarify the role of inner products and unitarity in non-orthogonal bases, particularly in relation to probability conservation and observables.

Proposed method

• The paper constructs a unitary transformation that maps a non-orthogonal basis to an orthonormal one, showing that a non-Hermitian Hamiltonian in the new basis corresponds to a Hermitian one in the standard basis.
• It demonstrates that the physical inner product in PT-symmetric quantum mechanics is dynamically determined by the Hamiltonian and corresponds to a metric operator C, which ensures unitary time evolution.
• The method relies on the fact that observables remain Hermitian under the physical inner product, even if their matrix representations are non-Hermitian in the computational (unphysical) basis.
• It analyzes the quantum brachistochrone proposal by showing that the apparent faster evolution arises from using a non-orthogonal basis, not from any physical speedup.
• The analysis is restricted to finite-dimensional Hilbert spaces, where the equivalence between PT-symmetric and standard quantum mechanics is rigorously established.
• It distinguishes between the mathematical representation of operators (which may be non-Hermitian in a given basis) and their physical properties (which remain Hermitian under the physical inner product).

Experimental results

Research questions

• RQ1 Is PT-symmetric quantum mechanics physically distinct from standard quantum mechanics, or is it merely a reformulation in a non-orthogonal basis?
• RQ2 Can non-Hermitian Hamiltonians in PT-symmetric theories be unitarily equivalent to Hermitian ones in finite-dimensional systems?
• RQ3 Does the claimed faster quantum evolution in PT-symmetric systems represent a genuine physical advantage or a basis-dependent artifact?
• RQ4 How do inner products and unitarity behave in non-orthogonal bases, and what is the role of the C operator in ensuring physical consistency?

Key findings

• Any finite-dimensional non-Hermitian PT-symmetric Hamiltonian is unitarily equivalent to a Hermitian Hamiltonian when the system is described in an orthonormal basis.
• The physical inner product in PT-symmetric quantum mechanics is not fundamental but is determined by the Hamiltonian, ensuring unitary time evolution and real eigenvalues.
• The claim that PT-symmetric systems allow faster evolution than Hermitian systems is a coordinate artifact, not a physical distinction, as the same evolution time is obtained in the standard formulation.
• Observables in PT-symmetric quantum mechanics are still represented by Hermitian operators under the physical inner product, even if their matrix representations appear non-Hermitian in the computational basis.
• The theory of PT-symmetric quantum mechanics does not extend standard quantum mechanics but rather provides an alternative, mathematically equivalent formulation in a non-orthogonal basis.
• There is no experimental test that can distinguish between PT-symmetric quantum mechanics and standard quantum mechanics in finite-dimensional systems.