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arXiv:1610.09396 · Physical Review A

Problem of the coexistence of several non-Hermitian observables in PT -symmetric quantum mechanics

Miloslav Znojil, Iveta Semorádová, František Růžička, Hafida Moulla, Ilhem Leghrib

Published Apr 18, 2017• 26 citations
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Abstract

During recent developments in quantum theory it has been clarified that observable quantities (such as energy and position) may be represented by operators $\mathrm{\ensuremath{\Lambda}}$ (with real spectra) that are manifestly non-Hermitian in a preselected friendly Hilbert space ${\mathcal{H}}^{(F)}$. The consistency of these models is known to require an upgrade of the inner product, i.e., mathematically speaking, a transition ${\mathcal{H}}^{(F)}\ensuremath{\rightarrow}{\mathcal{H}}^{(S)}$ to another, standard, Hilbert space. We prove that whenever we are given more than one candidate for an observable (i.e., say, two operators ${\mathrm{\ensuremath{\Lambda}}}_{0}$ and ${\mathrm{\ensuremath{\Lambda}}}_{1}$) in advance, such an upgrade need not exist in general.

DOI
10.1103/physreva.95.042122

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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.

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During recent developments in quantum theory it has been clarified that observable quantities (such as energy and position) may be represented by operators \ensuremathΛ\mathrm{\ensuremath{\Lambda}}\ensuremathΛ (with real spectra) that are manifestly non-Hermitian in a preselected friendly Hilbert space H(F){\mathcal{H}}^{(F)}H(F). The consistency of these models is known to require an upgrade of the inner product, i.e., mathematically speaking, a transition H(F)\ensuremath→H(S){\mathcal{H}}^{(F)}\ensuremath{\rightarrow}{\mathcal{H}}^{(S)}H(F)\ensuremath→H(S) to another, standard, Hilbert space. We prove that whenever we are given more than one candidate for an observable (i.e., say, two operators \ensuremathΛ0{\mathrm{\ensuremath{\Lambda}}}_{0}\ensuremathΛ0​ and \ensuremathΛ1{\mathrm{\ensuremath{\Lambda}}}_{1}\ensuremathΛ1​) in advance, such an upgrade need not exist in general.
Two non-Hermitian PT-symmetric Hamiltonian systems are reconsidered by means of the algebraic method which was originally proposed for the pseudo-Hermitian Hamiltonian systems rather than for the PT-symmetric ones. Compared with the way converting a non-Hermitian Hamiltonian to its Hermitian counterpart, this method has the merit that keeps the Hilbert space of the non-Hermitian PT-symmetric Hamiltonian unchanged. In order to give the positive definite inner product for the PT-symmetric systems, a new operator V, instead of C, can be introduced. The operator V has the similar function to the operator C adopted normally in the PT-symmetric quantum mechanics, however, it can be constructed, as an advantage, directly in terms of Hamiltonians. The spectra of the two non-Hermitian PT-symmetric systems are obtained, which coincide with that given in literature, and in particular, the Hilbert spaces associated with positive definite inner products are worked out.
Based on the PT-symmetric quantum theory, the concepts of PT-frame, PT-symmetric operator and CPT-frame on a Hilbert space K and for an operator on K are proposed. It is proved that the spectrum and point spectrum of a PT-symmetric linear operator are both symmetric with respect to the real axis and the eigenvalues of an unbroken PT-symmetric operator are real. For a linear operator H on C d , it is proved that H has unbroken PT-symmetry if and only if it has d different eigenvalues and the corresponding eigenstates are eigenstates of PT. Given a CPT-frame on K, a new positive inner product on K is induced and called CPT-inner product. Te relationship between the CPT-adjoint and the Dirac adjoint of a densely defined linear operator is derived, and it is proved that an operator which has a bounded CPT-frame is CPT-Hermitian if and only if it is T-symmetric, in that case, it is similar to a Hermitian operator. The existence of an operator C consisting of a CPT-frame is discussed. These concepts and results will serve a mathematical discussion about PT-symmetric quantum mechanics.
A possible method to investigate non-Hermitian Hamiltonians is suggested through finding a Hermitian operator η+ and defining the annihilation and creation operators to be η+ -pseudo-Hermitian adjoint to each other. The operator η+ represents the η+ -pseudo-Hermiticity of Hamiltonians. As an example, a non-Hermitian and non-PT-symmetric Hamiltonian with imaginary linear coordinate and linear momentum terms is constructed and analyzed in detail. The operator η+ is found, based on which, a real spectrum and a positive-definite inner product, together with the probability explanation of wave functions, the orthogonality of eigenstates, and the unitarity of time evolution, are obtained for the non-Hermitian and non-PT-symmetric Hamiltonian. Moreover, this Hamiltonian turns out to be coupled when it is extended to the canonical noncommutative space with noncommutative spatial coordinate operators and noncommutative momentum operators as well. Our method is applicable to the coupled Hamiltonian. Then the first and second order noncommutative corrections of energy levels are calculated, and in particular the reality of energy spectra, the positive-definiteness of inner products, and the related properties (the probability explanation of wave functions, the orthogonality of eigenstates, and the unitarity of time evolution) are found not to be altered by the noncommutativity.
Based on the P T-symmetric quantum theory,the concepts of P T-frame,P T-symmetric operator and CPT-frame on a Hilbert space K and for an operator on K are proposed.It is proved that the spectrum and point spectrum of a P T-symmetric linear operator are both symmetric with respect to the real axis and the eigenvalues of an unbroken P T-symmetric operator are real.For a linear operator H on Cd,it is proved that H has unbroken P T symmetry if and only if it has d diferent eigenvalues and the corresponding eigenstates are eigenstates of P T.Given a C P T-frame on K,a new positive inner product on K is induced and called C P T-inner product.Te relationship between the CP T-adjoint and the Dirac adjoint of a densely defined linear operator is derived,and it is proved that an operator which has a bounded CP T-frame is CP T-Hermitian if and only if it is T-symmetric,in that case,it is similar to a Hermitian operator.The existence of an operator C consisting of a CP T-frame is discussed.These concepts and results will serve a mathematical discussion about P T-symmetric quantum mechanics.
In the conventional Schrödinger's formulation of quantum mechanics the unitary evolution of a state ψψψ is controlled, in Hilbert space L{\cal L}L, by a Hamiltonian h\mathfrak{h}h which must be self-adjoint. In the recent, ``quasi-Hermitian'' reformulation of the theory one replaces h\mathfrak{h}h by its isospectral but non-Hermitian avatar H=Ω−1hΩH = Ω^{-1}\mathfrak{h}ΩH=Ω−1hΩ with Ω†Ω=Θ≠IΩ^\daggerΩ= Θ\neq IΩ†Ω=Θ=I. Although acting in another, manifestly unphysical Hilbert space H{\cal H}H, the amended Hamiltonian H≠H†H \neq H^\daggerH=H† can be perceived as self-adjoint with respect to the amended inner-product metric ΘΘΘ. In our paper motivated by a generic technical ``user-unfriendliness'' of the non-Hermiticity of HHH we introduce and describe a specific new family of Hamiltonians HHH for which the metrics ΘΘΘ become available in closed form.
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.