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arXiv:quant-ph/0211123 · Journal of Physics A Mathematical and General

All Hermitian Hamiltonians have parity

Carl M. Bender, Peter N Meisinger, Qinghai Wang

Published Jan 15, 2003• 31 citations
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Abstract

It is shown that if a Hamiltonian H is Hermitian, then there always exists an operator having the following properties: (i) is linear and Hermitian; (ii) commutes with H ; (iii) 2 = 1; (iv) the n th eigenstate of H is also an eigenstate of with eigenvalue (−1) n . Given these properties, it is appropriate to refer to as the parity operator and to say that H has parity symmetry, even though may not refer to spatial reflection. Thus, if the Hamiltonian has the form H = p 2 + V ( x ), where V ( x ) is real (so that H possesses time-reversal symmetry), then it immediately follows that H has symmetry. This shows that symmetry is a generalization of Hermiticity: all Hermitian Hamiltonians of the form H = p 2 + V ( x ) have symmetry, but not all -symmetric Hamiltonians of this form are Hermitian.

DOI
10.1088/0305-4470/36/4/312

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It is shown that if a Hamiltonian HHH is Hermitian, then there always exists an operator calP{\\cal P}calP having the following properties: (i) calP{\\cal P}calP is linear and Hermitian; (ii) calP{\\cal P}calP commutes with HHH; (iii) calP2=1{\\cal P}^2=1calP2=1; (iv) the nnnth eigenstate of HHH is also an eigenstate of calP{\\cal P}calP with eigenvalue (−1)n(-1)^n(−1)n. Given these properties, it is appropriate to refer to calP{\\cal P}calP as the parity operator and to say that HHH has parity symmetry, even though calP{\\cal P}calP may not refer to spatial reflection. Thus, if the Hamiltonian has the form H=p2+V(x)H=p^2+V(x)H=p2+V(x), where V(x)V(x)V(x) is real (so that HHH possesses time-reversal symmetry), then it immediately follows that HHH has calPT{\\cal PT}calPT symmetry. This shows that calPT{\\cal PT}calPT symmetry is a generalization of Hermiticity: All Hermitian Hamiltonians of the form H=p2+V(x)H=p^2+V(x)H=p2+V(x) have calPT{\\cal PT}calPT symmetry, but not all calPT{\\cal PT}calPT-symmetric Hamiltonians of this form are Hermitian.
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. The Hamiltonians which are invariant under the joint transformation of Parity (x → −x) and Time-reversal (i → −i) are called PT-invariant. It has been conjectured [1] that such Hamiltonians possess real discrete energy-eigenvalues provided the PT symmetry is unbroken. PT-symmetry is called unbroken or exact if the energy-eigenstates are also simultaneous eigenstates of PT. On the other hand when PT-symmetry is spontaneously broken the energy-eigenvalues are complex conjugate pairs. Multipronged investigations supporting
We discuss Hamiltonian symmetries and invariants for quantum systems driven by non-Hermitian Hamiltonians. For time-independent Hermitian Hamiltonians, a unitary or antiunitary transformation AHA† that leaves the Hamiltonian H unchanged represents a symmetry of the Hamiltonian, which implies the commutativity [H,A]=0 and, if A is linear and time-independent, a conservation law, namely the invariance of expectation values of A. For non-Hermitian Hamiltonians, H† comes into play as a distinct operator that complements H in generalized unitarity relations. The above description of symmetries has to be extended to include also A-pseudohermiticity relations of the form AH=H†A. A superoperator formulation of Hamiltonian symmetries is provided and exemplified for Hamiltonians of a particle moving in one-dimension considering the set of A operators that form Klein’s 4-group: parity, time-reversal, parity&time-reversal, and unity. The link between symmetry and conservation laws is discussed and shown to be richer and subtler for non-Hermitian than for Hermitian Hamiltonians.
In recent years a generalization of Hermiticity has been investigated using a complex deformation H = p 2 + x 2 (i x ) of the harmonic oscillator Hamiltonian, where is a real parameter. These complex Hamiltonians, possessing symmetry (the product of parity and time reversal), can have a real spectrum. We will consider the most simple case: even. In this paper we describe all self-adjoint (Hermitian) and at the same time symmetric operators associated with H = p 2 + x 2 (i x ) . Surprisingly, it turns out that there is a large class of self-adjoint operators associated with H = p 2 + x 2 (i x ) which are not symmetric.
Operators in quantum mechanics which represent observables are of a special type: They are linear and Hermitian. Such operators not only form computation tools, but also form the most effective language in terms of which the theory can be formulated. This chapter presents important properties of linear operators and their eigenfunctions. It discusses the features of various operators which are necessary for studying quantum mechanical systems. The chapter shows how rich and beautiful the mathematical structure of quantum mechanics is. There exists a class of eigenvalue problems, the so-called parity-time reversal (PT) -symmetric non-Hermitian Hamiltonian problems. They have arisen in recent years in a number of physical contexts such as localization transitions, vortex pinning and population biology. Such non-Hermitian Hamiltonians can be used to describe certain experimental phenomena. PT-symmetry has been noticed in certain optical systems, such as optical couplers, microwave billiard and large-scale temporal lattices.
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.