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arXiv:quant-ph/0310164

A Critique of PT-Symmetric Quantum Mechanics

Mostafazadeh, Ali

Published Oct 28, 2003• 8 citations
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Abstract

We study the physical content of the PT-symmetric complex extension of quantum mechanics as proposed in Bender et al, Phys. Rev. Lett. 80, 5243 (1998) and 89, 270401 (2002), and show that as a fundamental probabilistic physical theory it is neither an alternative to nor an extension of ordinary quantum mechanics. We demonstrate that the definition of a physical observable given in the above papers is inconsistent with the dynamical aspect of the theory and offer a consistent notion of an observable.

DOI
10.48550/arxiv.quant-ph/0310164

Related papers

Comment on Complex Extension of Quantum Mechanics

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In their Erratum [Phys. Rev. Lett. {\bf 92}, 119902 (2004), quant-ph/0208076], written in reaction to [quant-ph/0310164], Bender, Brody and Jones propose a revised definition for a physical observable in PT-symmetric quantum mechanics. We show that although this definition avoids the dynamical inconsistency revealed in quant-ph/0310164, it is still not a physically viable definition. In particular, we point out that a general proof that this definition is consistent with the requirements of the quantum measurement theory is lacking, give such a proof for a class of PT-symmetric systems by establishing the fact that this definition implies that the observables are pseudo-Hermitian operators, and show that for all the cases that this definition is consistent with the requirements of measurement theory it reduces to a special case of a more general definition given in [quant-ph/0310164]. The latter is the unique physically viable definition of observables in PT-symmetric quantum mechanics.

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Comment on ``PT-Symmetric versus Hermitian Formulations of Quantum Mechanics [hep-th/0511229]''

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We explain why the main conclusion of Bender et al, hep-th/0511229 [J. Phys. A 39 (2006) 1657] regarding the practical superiority of the non-Hermitian description of PT-symmetric quantum systems over their Hermitian description is not valid. Recalling the essential role played by the Hermitian description in the characterization and interpretation of the physical observables, we maintain that as far as the physical aspects of the theory are concerned the Hermitian description is not only unavoidable but also indispensable.

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Abstract PT-symmetric quantum theory was originally proposed with the aim of extending standard quantum theory by relaxing the Hermiticity constraint on Hamiltonians. However, no such extension has been formulated that consistently describes states, transformations, measurements and composition, which is a requirement for any physical theory. We aim to answer the question of whether a consistent physical theory with PT-symmetric observables extends standard quantum theory. We answer this question within the framework of general probabilistic theories, which is the most general framework for physical theories. We construct the set of states of a system that result from imposing PT-symmetry on the set of observables, and show that the resulting theory allows only one trivial state. We next consider the constraint of quasi-Hermiticity on observables, which guarantees the unitarity of evolution under a Hamiltonian with unbroken PT-symmetry. We show that such a system is equivalent to a standard quantum system. Finally, we show that if all observables are quasi-Hermitian as well as PT-symmetric, then the system is equivalent to a real quantum system. Thus our results show that neither PT-symmetry nor quasi-Hermiticity constraints are sufficient to extend standard quantum theory consistently.

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IS PT -SYMMETRIC QUANTUM THEORY FALSE AS A FUNDAMENTAL THEORY?

Miloslav Znojil

Yi-Chan Lee et al. claim (cf. Phys. Rev. Lett. 112, 130404 (2014)) that the “recent extension of quantum theory to non-Hermitian Hamiltonians” (which is widely known under the nickname of “PT-symmetric quantum theory”) is “likely false as a fundamental theory”. By their opinion their results “essentially kill any hope of PT-symmetric quantum theory as a fundamental theory of nature”. In our present text we explain that their toy-model-based considerations are misleading and that they do not imply any similar conclusions.

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PT-Symmetric Quantum Mechanics: A Precise and Consistent Formulation

Mostafazadeh, A

The physical condition that the expectation values of physical observables are real quantities is used to give a precise formulation of PT-symmetric quantum mechanics. A mathematically rigorous proof is given to establish the physical equivalence of PT-symmetric and conventional quantum mechanics. The results reported in this paper apply to arbitrary PT-symmetric Hamiltonians with a real and discrete spectrum. They hold regardless of whether the boundary conditions defining the spectrum of the Hamiltonian are given on the real line or a complex contour.

•Jan 1, 2004•26 citations

Can PT-Symmetric Quantum Mechanics be a Viable Alternative Quantum Theory?

Sungwook Lee, Lawrence R. Mead

Update: A time-independent $n\times n$ PT-symmetric (and symmetric) Hamiltonian is diagonalizable since it has all distinct real eigenvalues and the resulting diagonal matrix is a real symmetric matrix. The diagonalization results an isometry so there shouldn't be any issue with unitarity and unfortunately this very elementary mathematical fact somehow did not draw the authors' attention. However, PT-symmetric quantum mechanics is not out of trouble. For time-dependent PT-symmetric (and symmetric) Hamiltonians (even $2\times 2$ ones) the authors observed that there is a violation of unitarity. Moreover, the first named author showed in his recent article arXiv:1312.7738 that PT-symmetric quantum mechanics is indeed a certain kind of Hermitian quantum mechanics and that in order for time-evolution to be unitary with respect to $J$-inner product (one that gives rise to a Hilbert space structure on the space of state functions), the potential energy operator $V(x)$ must be real. This means that those complex PT-symmetric Hamiltonians that have been studied by physicists are unfortunately unphysical. The first named author discussed in a subsequent article arXiv:1401.5149 that while finite-state PT-symmetric quantum mechanics with time-independent Hamiltonians is not physically any different from Hermitian quantum mechanics, PT-symmetric quantum mechanics exhibits a distinctive symmetry from that of Hermitian quantum mechanics.

arXiv:1303.6339•Mar 25, 2013•0 citations
We study the physical content of the PT-symmetric complex extension of quantum mechanics as proposed in Bender et al, Phys. Rev. Lett. 80, 5243 (1998) and 89, 270401 (2002), and show that as a fundamental probabilistic physical theory it is neither an alternative to nor an extension of ordinary quantum mechanics. We demonstrate that the definition of a physical observable given in the above papers is inconsistent with the dynamical aspect of the theory and offer a consistent notion of an observable.
In their Erratum [Phys. Rev. Lett. {\bf 92}, 119902 (2004), quant-ph/0208076], written in reaction to [quant-ph/0310164], Bender, Brody and Jones propose a revised definition for a physical observable in PT-symmetric quantum mechanics. We show that although this definition avoids the dynamical inconsistency revealed in quant-ph/0310164, it is still not a physically viable definition. In particular, we point out that a general proof that this definition is consistent with the requirements of the quantum measurement theory is lacking, give such a proof for a class of PT-symmetric systems by establishing the fact that this definition implies that the observables are pseudo-Hermitian operators, and show that for all the cases that this definition is consistent with the requirements of measurement theory it reduces to a special case of a more general definition given in [quant-ph/0310164]. The latter is the unique physically viable definition of observables in PT-symmetric quantum mechanics.
We explain why the main conclusion of Bender et al, hep-th/0511229 [J. Phys. A 39 (2006) 1657] regarding the practical superiority of the non-Hermitian description of PT-symmetric quantum systems over their Hermitian description is not valid. Recalling the essential role played by the Hermitian description in the characterization and interpretation of the physical observables, we maintain that as far as the physical aspects of the theory are concerned the Hermitian description is not only unavoidable but also indispensable.
Abstract PT-symmetric quantum theory was originally proposed with the aim of extending standard quantum theory by relaxing the Hermiticity constraint on Hamiltonians. However, no such extension has been formulated that consistently describes states, transformations, measurements and composition, which is a requirement for any physical theory. We aim to answer the question of whether a consistent physical theory with PT-symmetric observables extends standard quantum theory. We answer this question within the framework of general probabilistic theories, which is the most general framework for physical theories. We construct the set of states of a system that result from imposing PT-symmetry on the set of observables, and show that the resulting theory allows only one trivial state. We next consider the constraint of quasi-Hermiticity on observables, which guarantees the unitarity of evolution under a Hamiltonian with unbroken PT-symmetry. We show that such a system is equivalent to a standard quantum system. Finally, we show that if all observables are quasi-Hermitian as well as PT-symmetric, then the system is equivalent to a real quantum system. Thus our results show that neither PT-symmetry nor quasi-Hermiticity constraints are sufficient to extend standard quantum theory consistently.
Yi-Chan Lee et al. claim (cf. Phys. Rev. Lett. 112, 130404 (2014)) that the “recent extension of quantum theory to non-Hermitian Hamiltonians” (which is widely known under the nickname of “PT-symmetric quantum theory”) is “likely false as a fundamental theory”. By their opinion their results “essentially kill any hope of PT-symmetric quantum theory as a fundamental theory of nature”. In our present text we explain that their toy-model-based considerations are misleading and that they do not imply any similar conclusions.
The physical condition that the expectation values of physical observables are real quantities is used to give a precise formulation of PT-symmetric quantum mechanics. A mathematically rigorous proof is given to establish the physical equivalence of PT-symmetric and conventional quantum mechanics. The results reported in this paper apply to arbitrary PT-symmetric Hamiltonians with a real and discrete spectrum. They hold regardless of whether the boundary conditions defining the spectrum of the Hamiltonian are given on the real line or a complex contour.
Update: A time-independent n×nn\times nn×n PT-symmetric (and symmetric) Hamiltonian is diagonalizable since it has all distinct real eigenvalues and the resulting diagonal matrix is a real symmetric matrix. The diagonalization results an isometry so there shouldn't be any issue with unitarity and unfortunately this very elementary mathematical fact somehow did not draw the authors' attention. However, PT-symmetric quantum mechanics is not out of trouble. For time-dependent PT-symmetric (and symmetric) Hamiltonians (even 2×22\times 22×2 ones) the authors observed that there is a violation of unitarity. Moreover, the first named author showed in his recent article arXiv:1312.7738 that PT-symmetric quantum mechanics is indeed a certain kind of Hermitian quantum mechanics and that in order for time-evolution to be unitary with respect to JJJ-inner product (one that gives rise to a Hilbert space structure on the space of state functions), the potential energy operator V(x)V(x)V(x) must be real. This means that those complex PT-symmetric Hamiltonians that have been studied by physicists are unfortunately unphysical. The first named author discussed in a subsequent article arXiv:1401.5149 that while finite-state PT-symmetric quantum mechanics with time-independent Hamiltonians is not physically any different from Hermitian quantum mechanics, PT-symmetric quantum mechanics exhibits a distinctive symmetry from that of Hermitian quantum mechanics.