A New Class of non-Hermitian Quantum Hamiltonians with PT Symmetry (original) (raw)
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Non-Hermitian quantum Hamiltonians with PT symmetry
Physical Review A, 2010
We formulate quantum mechanics for non-Hermitian Hamiltonians that are invariant under PT , where P is the parity and T denotes time reversal, for the case that time reversal symmetry is odd (T 2 = −1), generalizing prior work for the even case (T 2 = 1). We discover an analogue of Kramer's theorem for PT quantum mechanics, present a prototypical example of a PT quantum system with odd time reversal, and discuss potential applications of the formalism.
PT-symmetric quantum mechanics
Journal of Mathematical Physics, 1999
This paper proposes to broaden the canonical formulation of quantum mechanics. Ordinarily, one imposes the condition H † ϭH on the Hamiltonian, where † represents the mathematical operation of complex conjugation and matrix transposition. This conventional Hermiticity condition is sufficient to ensure that the Hamiltonian H has a real spectrum. However, replacing this mathematical condition by the weaker and more physical requirement H ‡ ϭH, where ‡ represents combined parity reflection and time reversal PT, one obtains new classes of complex Hamiltonians whose spectra are still real and positive. This generalization of Hermiticity is investigated using a complex deformation Hϭ p 2 ϩx 2 (ix) ⑀ of the harmonic oscillator Hamiltonian, where ⑀ is a real parameter. The system exhibits two phases: When ⑀у0, the energy spectrum of H is real and positive as a consequence of PT symmetry. However, when Ϫ1Ͻ⑀Ͻ0, the spectrum contains an infinite number of complex eigenvalues and a finite number of real, positive eigenvalues because PT symmetry is spontaneously broken. The phase transition that occurs at ⑀ϭ0 manifests itself in both the quantum-mechanical system and the underlying classical system. Similar qualitative features are exhibited by complex deformations of other standard real Hamiltonians Hϭp 2 ϩx 2N (ix) ⑀ with N integer and ⑀ϾϪN; each of these complex Hamiltonians exhibits a phase transition at ⑀ϭ0. These PTsymmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space.
Time-Reversal Symmetry in Non-Hermitian Systems
Progress of Theoretical Physics, 2012
For ordinary hermitian Hamiltonians, the states show the Kramers degeneracy when the system has a half-odd-integer spin and the time reversal operator obeys Θ 2 = −1, but no such a degeneracy exists when Θ 2 = +1. Here we point out that for non-hermitian systems, there exists a degeneracy similar to Kramers even when Θ 2 = +1. It is found that the new degeneracy follows from the mathematical structure of split-quaternion, instead of quaternion from which the Kramers degeneracy follows in the usual hermitian cases. Furthermore, we also show that particle/hole symmetry gives rise to a pair of states with opposite energies on the basis of the split-quaternion in a class of non-hermitian Hamiltonians. As concrete examples, we examine in detail N × N Hamiltonians with N = 2 and 4 which are non-hermitian generalizations of spin 1/2 Hamiltonian and quadrupole Hamiltonian of spin 3/2, respectively.
P-, T-, PT-, and CPT-invariance of Hermitian Hamiltonians
Physics Letters A, 2003
Currently, it has been claimed that certain Hermitian Hamiltonians have parity (P) and they are PT-invariant. We propose generalized definitions of time-reversal operator (T) and orthonormality such that all Hermitian Hamiltonians are P, T, PT, and CPT invariant. The PT-norm and CPT-norm are indefinite and definite respectively. The energy-eigenstates are either E-type (e.g., even) or O-type (e.g., odd). C mimics the charge-conjugation symmetry which is recently found to exist for a non-Hermitian Hamiltonian. For a Hermitian Hamiltonian it coincides with P.
On the time-reversal symmetry in pseudo-Hermitian systems
Progress of Theoretical and Experimental Physics, 2014
In a recent paper [M. Sato, K. Hasebe, K. Esaki, and M. Kohmoto, Prog. Theor. Phys. 127, 937 (2012)] Sato and his collaborators established a generalization of the Kramers degeneracy structure to pseudo-Hermitian Hamiltonian systems, admitting even time-reversal symmetry, T 2 = 1. This extension is achieved using the mathematical structure of split-quaternions instead of quaternions, usually adopted in the case of Hermitian Hamiltonians with odd time-reversal symmetry, T 2 = −1. Here we find that the metric operator for the pseudo-Hermitian Hamiltonian H that allows the realization of the generalized Kramers degeneracy is necessarily indefinite. We show that such H with real spectrum also possesses odd antilinear symmetry induced from the existing odd time-reversal symmetry of its Hermitian counterpart h, so that the generalized Kramers degeneracy of H is in fact crypto-Hermitian Kramers degeneracy. We study in greater detail a new example of the pseudo-Hermitian split-quaternionic four-level Hamiltonian system, which admits an indefinite metric operator and time-reversal symmetry and, as a consequence, a generalized Kramers degeneracy structure. We provide a complete solution of the eigenvalue problem, construct pseudo-Hermitian ladder operators closing the normal and abnormal pseudo-fermionic algebras, and show that this system fulfills a crypto-Hermitian degeneracy.
Geometry of PT-symmetric quantum mechanics
2007
Recently, much research has been carried out on Hamiltonians that are not Hermitian but are symmetric under space-time reflection, that is, Hamiltonians that exhibit PT symmetry. Investigations of the Sturm-Liouville eigenvalue problem associated with such Hamiltonians have shown that in many cases the entire energy spectrum is real and positive and that the eigenfunctions form an orthogonal and complete basis. Furthermore, the quantum theories determined by such Hamiltonians have been shown to be consistent in the sense that the probabilities are positive and the dynamical trajectories are unitary. However, the geometrical structures that underlie quantum theories formulated in terms of such Hamiltonians have hitherto not been fully understood. This paper studies in detail the geometric properties of a Hilbert space endowed with a parity structure and analyses the characteristics of a PT -symmetric Hamiltonian and its eigenstates. A canonical relationship between a PT -symmetric operator and a Hermitian operator is established. It is shown that the quadratic form corresponding to the parity operator, in particular, gives rise to a natural partition of the Hilbert space into two halves corresponding to states having positive and negative PT norm. The indefiniteness of the norm can be circumvented by introducing a symmetry operator C that defines a positive definite inner product by means of a CPT conjugation operation.
The physics and the philosophy of time reversal in standard quantum mechanics
Synthese, 2021
A widespread view in physics holds that the implementation of time reversal in standard quantum mechanics must be given by an anti-unitary operator. In foundations and philosophy of physics, however, there has been some discussion about the conceptual grounds of this orthodoxy, largely relying on either its obviousness or its mathematicalphysical virtues. My aim in this paper is to substantively change the traditional structure of the debate by highlighting the philosophical commitments underlying the orthodoxy. I argue the persuasive force of the orthodoxy greatly depends on a relationalist metaphysics of time and a by-stipulation view of time-reversal invariance. Only with such philosophical background can the orthodoxy of time reversal in standard quantum mechanics succeed and be properly justified.
Solving the Puzzle of Time Reversal in Quantum Mechanics: A New Approach
Why does time reversal involve two operations, a temporal reflection and the operation of complex conjugation? Why is it that time reversal preserves position and reverses momentum and spin? This puzzle of time reversal in quantum mechanics has been with us since Wigner's first presentation. In this paper, I propose a new approach to solving this puzzle. First, I argue that the standard account of time reversal can be derived from the requirement that the continuity equation in quantum mechanics is time reversal invariant. Next, I analyze the physical meaning of the continuity equation and explain why it should be time reversal invariant. Finally, I discuss how the new analysis help solve the puzzle of time reversal in quantum mechanics.
International Journal of Theoretical Physics, 2017
We show in the present paper that pseudo-Hermitian Hamiltonian systems with even PT-symmetry (P 2 = 1, T 2 = 1) admit a degeneracy structure. This kind of degeneracy is expected traditionally in the odd PT-symmetric systems (P 2 = 1, T 2 = −1) which is appropriate to the fermions as shown by Jones-Smith and Mathur [1] who extended PT-symmetric quantum mechanics to the case of odd time-reversal symmetry. We establish that the pseudo-Hermitian Hamiltonians with even PTsymmetry admit a degeneracy structure if the operator PT anticommutes with the metric operator η which is necessarily indefinite. We also show that the Krein space formulation of the Hilbert space is the convenient framework for the implementation of unbroken PT-symmetry. These general results are illustrated with great details for four-level pseudo-Hermitian Hamiltonian with even PT-symmetry.
symmetric versus Hermitian formulations of quantum mechanics
Journal of Physics A: Mathematical and General, 2006
A non-Hermitian Hamiltonian that has an unbroken PT symmetry can be converted by means of a similarity transformation to a physically equivalent Hermitian Hamiltonian. This raises the following question: In which form of the quantum theory, the non-Hermitian or the Hermitian one, is it easier to perform calculations? This paper compares both forms of a non-Hermitian ix 3 quantum-mechanical Hamiltonian and demonstrates that it is much harder to perform calculations in the Hermitian theory because the perturbation series for the Hermitian Hamiltonian is constructed from divergent Feynman graphs. For the Hermitian version of the theory, dimensional continuation is used to regulate the divergent graphs that contribute to the ground-state energy and the one-point Green's function. The results that are obtained are identical to those found much more simply and without divergences in the non-Hermitian PT -symmetric Hamiltonian. The O(g 4 ) contribution to the ground-state energy of the Hermitian version of the theory involves graphs with overlapping divergences, and these graphs are extremely difficult to regulate. In contrast, the graphs for the non-Hermitian version of the theory are finite to all orders and they are very easy to evaluate.