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arXiv:quant-ph/0209054 · Journal of Optics B Quantum and Semiclassical Optics

-symmetry and its spontaneous breakdown explained by anti-linearity

Stefan Weigert

Published Jun 1, 2003• 27 citations
arXivHTMLPDF

Abstract

The impact of an anti-unitary symmetry on the spectrum of non-Hermitian operators is studied. Wigner’s normal form of an anti-unitary operator accounts for the spectral properties of non-Hermitian, 𝒫𝒯-symmetric Hamiltonians. The occurrence of either single real or complex conjugate pairs of eigenvalues follows from this theory. The corresponding energy eigenstates span either one-or two-dimensional irreducible representations of the symmetry 𝒫𝒯. In this framework, the concept of a spontaneously broken 𝒫𝒯-symmetry is not needed.

DOI
10.1088/1464-4266/5/3/380

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The impact of an anti-unitary symmetry on the spectrum of non-Hermitian operators is studied. Wigner’s normal form of an anti-unitary operator accounts for the spectral properties of non-Hermitian, 𝒫𝒯-symmetric Hamiltonians. The occurrence of either single real or complex conjugate pairs of eigenvalues follows from this theory. The corresponding energy eigenstates span either one-or two-dimensional irreducible representations of the symmetry 𝒫𝒯. In this framework, the concept of a spontaneously broken 𝒫𝒯-symmetry is not needed.
In most introductory courses on quantum mechanics one is taught that the Hamiltonian operator must be Hermitian in order that the energy levels be real and that the theory be unitary (probability conserving). To express the Hermiticity of a Hamiltonian, one writes H = H †, where the symbol † denotes the usual Dirac Hermitian conjugation; that is, transpose and complex conjugate. In the past few years it has been recognized that the requirement of Hermiticity, which is often stated as an axiom of quantum mechanics, may be replaced by the less mathematical and more physical requirement of space – time reflection symmetry (𝒫𝒯 symmetry) without losing any of the essential physical features of quantum mechanics. Theories defined by non-Hermitian 𝒫𝒯-symmetric Hamiltonians exhibit strange and unexpected properties at the classical as well as at the quantum level. This paper explains how the requirement of Hermiticity can be evaded and discusses the properties of some non-Hermitian 𝒫𝒯-symmetric quantum theories.
Abstract Currently there is much interest in Hamiltonians that are not Hermitian but instead possess an antilinear PT symmetry. Here we seek to put such PT symmetric theories into as general a context as possible. After providing a brief overview of the PT symmetry program, we show that having an antilinear symmetry that acts on a well-defined Hilbert space is the most general condition that one can impose on a quantum theory for which one can have a well-defined inner product that is time independent, have a Hamiltonian that is self-adjoint, and have energy eigenvalues that are all real. For each of these properties Hermiticity is only a sufficient condition but not a necessary one, with Hermiticity thus being the special case in which the Hamiltonian has both antilinearity and Hermiticity. As well as being the necessary condition for the reality of energy eigenvalues, antilinearity in addition allows for the physically interesting cases of manifestly non-Hermitian but nonetheless self-adjoint Hamiltonians that have energy eigenvalues that appear in complex conjugate pairs, or that are Jordan block and cannot be diagonalized at all. We show that one can extend these ideas to quantum field theory, with the dual requirements of the existence of time independent inner products and invariance under complex Lorentz transformations forcing the antilinear symmetry to uniquely be CPT . We thus extend the CPT theorem to non-Hermitian Hamiltonians. For theories that are separately charge conjugation invariant, PT symmetry then follows, with the case for the physical relevance of the PT -symmetry program thus being advanced. While CPT symmetry can be defined at the classical level for every classical path in a path integral quantization procedure, in contrast, in such a path integral there is no reference at all to the Hermiticity of the Hamiltonian or the quantum Hilbert space on which it acts, as they are strictly quantum-mechanical concepts that can only be defined after the path integral quantization has been performed and the quantum Hilbert space has been constructed. CPT symmetry thus goes beyond Hermiticity and has primacy over it, with our work raising the question of how Hermiticity ever comes into quantum theory at all. To this end we show that whether or not a CPT -invariant theory has a Hamiltonian that is Hermitian is a property of the solutions to the theory and not of the Hamiltonian itself. Hermiticity thus never needs to be postulated at all.
Anti-PT\mathcal{PT}PT symmetry for a non-Hermitian Hamiltonian
Abstract Anti-PT\mathcal{PT}PT symmetry, (PT)H=−H(PT)(\mathcal{PT})H=-H(\mathcal{PT})(PT)H=−H(PT), is a plausible variant of PT\mathcal{PT}PT symmetry. Of particular interest is the situation when all the eigenstates of an anti-PT\mathcal{PT}PT-symmetric non-Hermitian Hamiltonian HHH are also eigenstates of the PT\mathcal{PT}PT operator; then, the quasi-energies are purely imaginary, which implies that the Hermitian conjugate H+=−HH^{+}=-HH+=−H, and thus they are connected via the relation (PT)H=H+PT(\mathcal{PT})H=H^{+}\mathcal{PT}(PT)H=H+PT, similar to the quasi-Hermiticity relation. Therefore, the eigenfunctions of the anti-PT\mathcal{PT}PT-symmetric HHH form a complete orthonormal set with positive definite norms, and moreover the time evolution is unitary.
This paper proposes to broaden the canonical formulation of quantum mechanics. Ordinarily, one imposes the condition H†=H 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‡=H, 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 H=p2+x2(ix)ε of the harmonic oscillator Hamiltonian, where ε is a real parameter. The system exhibits two phases: When ε⩾0, the energy spectrum of H is real and positive as a consequence of 𝒫𝒯 symmetry. However, when −1<ε<0, 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 ε=0 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 H=p2+x2N(ix)ε with N integer and ε>−N; each of these complex Hamiltonians exhibits a phase transition at ε=0. These 𝒫𝒯-symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space.
The spectrum of non-Hermitian Hamiltonians is studied using the point-group symmetries of the Hermitian and non-Hermitian parts of the Hamiltonian. We show that, in principle, the symmetry properties of the Hamiltonian are responsible for the appearance of real eigenvalues in the spectrum of the non-Hermitian Hamiltonian. In practice, however, one must take into account the ``strength'' of the non-Hermitian part of the Hamiltonian. An effective energy-dependent Hermitian Hamiltonian is constructed that has the same real spectrum as the non-Hermitian Hamiltonian. Several properties, including the self-orthogonality of the states at branch points of the spectrum, are proven using this effective representation. An interesting possibility of complex eigenvalues returning to the real axis is exemplified along with several examples of non-Hermitian Hamiltonians with real eigenvalues.