[Paper Review] The construction of a general inner product in non-Hermitian quantum theory and some explanation for the nonuniqueness of the C operator in PT quantum mechanics

Frieder Kleefeld|ArXiv.org|Jun 4, 2009

Quantum Mechanics and Non-Hermitian Physics2 references8 citations

TL;DR

This paper resolves the nonuniqueness of the 𝒞 operator in PT-symmetric quantum mechanics by showing it arises from arbitrary normalization of left and right eigenvectors of the Hamiltonian. By carefully choosing eigenvector normalization, the metric operator—and thus the 𝒞 operator—can be made unique and independent of Hamiltonian parameters, particularly in a 2×2 PT-symmetric matrix model.

ABSTRACT

Most recently it has been observed e.g. by Bender and Klevansky (arXiv:0905.4673 [hep-th]) that the C-operator related to a PT-symmetric non-Hermitian Hamilton operator is not unique. Moreover it has been remarked by Shi and Sun (arXiv:0905.1771 [hep-th]) very recently that there seems to exist a well defined inner product in the context of the Hamilton operator of the PT-symmetric non-Hermitian Lee model yielding a different C-operator as compared to the one previously derived by Bender et al.. The puzzling observations of both manuscripts are reconciled and explained in the present manuscript as follows: the actual form of the metric operator (and the induced C-operator) related to some non-Hermitian Hamilton operator constructed along the lines of Shi and Su depends on the chosen normalization of the left and right eigenvectors of the Hamilton operator under consideration and is therefore ambiguous. For a specific PT-symmetric 2x2-matrix Hamilton operator it is shown that - by a suitable choice of the norm of its eigenvectors - the metric operator yielding a positive semi-definite inner product can be made even independent of the parameters of the considered Hamilton operator. This surprising feature makes in turn the obtained metric operator rather unique and attractive. For later convenience the metric operator for the Bosonic and Fermionic (anti)causal harmonic oscillator is derived.

Motivation & Objective

• To resolve the observed nonuniqueness of the 𝒞 operator in PT-symmetric quantum mechanics, which has been reported by Bender, Klevansky, and Shi-Sun.
• To clarify that the ambiguity in the 𝒞 operator stems from the freedom in choosing the normalization of left and right eigenvectors of the non-Hermitian Hamiltonian.
• To demonstrate that a unique, parameter-independent metric operator can be constructed through a specific eigenvector normalization choice in a 2×2 PT-symmetric matrix model.
• To derive the metric operator and positive semi-definite inner product for the bosonic and fermionic (anti)causal harmonic oscillators as a foundation for field-theoretic applications.
• To support the construction of a consistent, unitary, and causal quantum theory based on quasi-Hermitian and non-Hermitian Hamiltonians using a well-defined inner product.

Proposed method

• Construct the metric operator via the similarity transformation involving the square root of the parity operator and a unitary transformation, following the framework of pseudo-Hermitian quantum mechanics.
• Use left and right eigenvectors of the non-Hermitian Hamiltonian as the basis for defining the metric, with normalization chosen to eliminate parameter dependence.
• Apply the relation 𝒞 = e^𝒬𝒫 to connect the metric operator to the 𝒞 operator, ensuring a positive semi-definite inner product.
• Diagonalize the parity operator 𝒫 to compute its square root √𝒫, enabling the construction of the metric operator in a 2×2 matrix model.
• Derive the metric operator and inner product for the (anti)causal harmonic oscillator, generalizing to Klein-Gordon and Dirac fields.
• Ensure consistency with Lorentz invariance and renormalizability by selecting operator ordering that minimizes anomalies and preserves physical symmetries.

Experimental results

Research questions

• RQ1 Why is the 𝒞 operator in PT-symmetric quantum mechanics nonunique, and what underlies this ambiguity?
• RQ2 Can the nonuniqueness of the 𝒞 operator be resolved by fixing the normalization of left and right eigenvectors of the Hamiltonian?
• RQ3 Is it possible to construct a metric operator that is independent of the parameters of the Hamiltonian through appropriate eigenvector normalization?
• RQ4 How can a consistent, positive semi-definite inner product be defined for non-Hermitian Hamiltonians in a way that supports unitary time evolution?
• RQ5 What is the role of operator ordering and anomalies in the construction of the metric and 𝒞 operator, and how do they affect the physical consistency of the theory?

Key findings

• The nonuniqueness of the 𝒞 operator arises from the freedom in choosing the normalization of left and right eigenvectors of the non-Hermitian Hamiltonian.
• For a 2×2 PT-symmetric matrix Hamiltonian, a specific eigenvector normalization leads to a metric operator that is independent of the Hamiltonian's parameters.
• This parameter-independent metric operator is unique and highly attractive for physical applications, as it removes arbitrary choices in the inner product construction.
• The metric operator and positive semi-definite inner product are explicitly derived for the bosonic and fermionic (anti)causal harmonic oscillators.
• The construction method generalizes to field theories, providing a foundation for causal, local, and unitary quantum theories based on non-Hermitian Hamiltonians.
• The results are consistent with Lorentz invariance and renormalizability, with anomalies—when present—remaining invariant under the choice of metric operator.