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arXiv:1401.5149

On Finite JJJ-Hermitian Quantum Mechanics

Sungwook Lee

Published Jan 21, 2014• 0 citations
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Abstract

In his recent paper arXiv:1312.7738, the author discussed $J$-Hermitian quantum mechanics and showed that $PT$-symmetric quantum mechanics is essentially $J$-Hermitian quantum mechanics. In this paper, the author discusses finite $J$-Hermitian quantum mechanics which is derived naturally from its continuum one and its relationship with finite $PT$-symmetric quantum mechanics.

DOI
10.48550/arxiv.1401.5149

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The author discusses a different kind of Hermitian quantum mechanics, called $J$-Hermitian quantum mechanics. He shows that $PT$-symmetric quantum mechanics is indeed $J$-Hermitian quantum mechanics, and that time evolution (in the Krein space of states) is unitary if and only if Hamiltonian is $J$-Hermitian (or equivalently $PT$-symmetric). An issue with unitarity comes up when time evolution is considered in the Hilbert space of states rather than in the Krein space of states. The author offers possible scenarios with this issue.

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In his recent paper arXiv:1312.7738, the author discussed JJJ-Hermitian quantum mechanics and showed that PTPTPT-symmetric quantum mechanics is essentially JJJ-Hermitian quantum mechanics. In this paper, the author discusses finite JJJ-Hermitian quantum mechanics which is derived naturally from its continuum one and its relationship with finite PTPTPT-symmetric quantum mechanics.
PTPTPT-symmetric quantum mechanics is a Hermitian quantum mechanics
The author discusses a different kind of Hermitian quantum mechanics, called JJJ-Hermitian quantum mechanics. He shows that PTPTPT-symmetric quantum mechanics is indeed JJJ-Hermitian quantum mechanics, and that time evolution (in the Krein space of states) is unitary if and only if Hamiltonian is JJJ-Hermitian (or equivalently PTPTPT-symmetric). An issue with unitarity comes up when time evolution is considered in the Hilbert space of states rather than in the Krein space of states. The author offers possible scenarios with this issue.
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†=HH^\dagger = HH†=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.
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.
A series of recent papers ``Faster than Hermitian Quantum Mechanics'' and related articles made a point of the possibility of a non-Hermitian, but PT-symmetric, operator to play the role of a Hamiltonian. In particular, they show that with an appropriate choice of an inner product, the evolution generated by such an operator will conserve the norm and scalar product. Here we observe that if one chooses such an inner product then the Hamiltonian in question is actually Hermitian, and the whole exercise is to a certain degree redundant.
In the seminal paper (Bender & Boettcher, 1998) a new view of quantum mechanics was proposed. This new view differs from the old one in that the restriction on the Hamiltonian to be Hermitian is relaxed: now the Hamiltonian is PT -symmetric. Here P is parity and T is time reversal. Since 1998, PT -symmetric Hamiltonians have been analyzed intensively by many authors. In Mostafazadeh (2002) PT -symmetry was embedded into the more general mathematical framework of pseudo-Hermiticity or, what is the same, self-adjoint operators in Kre˘ın spaces, see (Langer & Tretter, 2004; Azizov & Trunk, 2012; Hassi & Kuzhel, 2013; Leben & Trunk, 2019). For a general introduction to PT -symmetric quantum mechanics we refer to the overview paper of Mostafazadeh (2010) and to the books of Moiseyev (2011) and Bender (2019).
A physical theory of quantum mechanics can be based on a complex Hamiltonian that is not Hermitian but instead satisfies the physical condition of space-time reflection symmetry ( PTP\mathcal{T}PT symmetry).Thus, there are infinitely many new Hamiltonians that one can construct that might explain experimental data.One would think that a non-Hermitian Hamiltonian would give a quantum theory that violates unitarity.However, if PTP\mathcal{T}PT symmetry is not broken, it is possible to use a physical symmetry of the Hamiltonian to construct an inner product whose associated norm is positive definite.This construction is general and works for any PTP\mathcal{T}PT -symmetric Hamiltonian.The dynamics is governed by unitary time evolution.This formulation does not conflict with the requirements of conventional quantum mechanics.There are many possible observable and experimental consequences of extending quantum mechanics into the complex domain, both in particle physics and in solid state physics.