Consistency of PT-symmetric quantum mechanics

Abstract

In recent reports, suggestions have been put forward to the effect that parity and time-reversal (PT) symmetry in quantum mechanics is incompatible with causality. It is shown here, in contrast, that PT-symmetric quantum mechanics is fully consistent with standard quantum mechanics. This follows from the surprising fact that the much-discussed metric operator on Hilbert space is not physically observable. In particular, for closed quantum systems in finite dimensions there is no statistical test that one can perform on the outcomes of measurements to determine whether the Hamiltonian is Hermitian in the conventional sense, or PT-symmetric-the two theories are indistinguishable. Nontrivial physical effects arising as a consequence of PT symmetry are expected to be observed, nevertheless, for open quantum systems with balanced gain and loss.Since the realisation by Bender and Boettcher [1] that complex-non-Hermitian-Hamiltonians admitting space-time reflection (parity and time-reversal) symmetry can possess entirely real eigenvalues, considerable amount of research has been carried out into identifying properties of physical systems described by PTsymmetric Hamiltonians. It was subsequently observed that such Hamiltonians, although not Hermitian, can nevertheless be used to generate unitary time evolutions for the characterisation of closed quantum systems, provided that one works with a Hilbert space equipped with a preferentially selected inner product [2,3]. In fact, the idea of modifying the Hilbert space inner product in quantum mechanics has previously been proposed in [4], but the work of [2, 3] has triggered extensive research into the identification of appropriate inner products-the so-called metric operators-for a wide range of complex PT-symmetric Hamiltonians. Nevertheless, the physical significance of the metric operator has hitherto remained elusive, leading to a variety of controversial claims concerning what might be achievable by altering the inner product in a laboratory experiment.The purpose of the present paper is to unambiguously settle this issue by showing that the degrees of freedom in the Hamiltonian associated with the choice of the Hilbert space inner product are not observable. Putting the matter differently, the lack of Hermiticity of the Hamiltonian, or equivalently the Petermann factor (see, e.g., [5]), is not observable in closed systems-in contrast to open systems. The significance of this result is that not all parametric degrees of freedom in a complex Hamiltonian can be perturbed by an experimentalist in a laboratory. Our finding thus answers the open question raised in [6] regarding the possible physical constraints that prohibit experimentalists modifying the inner product (or equivalently, switching between different PT-symmetric Hamiltonians) in a laboratory. For the same token, the results here invalidate the claim in [7] that local PTsymmetric Hamiltonians violate the no-signalling condition, and the claim in [8] that local PT-symmetric Hamiltonians can be used to increase...

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