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arXiv:hep-th/0605066 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Equivalence of a complex PT -symmetric quartic Hamiltonian and a Hermitian quartic Hamiltonian with an anomaly

Carl M. Bender, Dorje C. Brody, Junhua Chen, Hugh F Jones, Kimball A. Milton, Michael C. Ogilvie

Published Jul 24, 2006• 95 citations
arXivHTMLPDF

Abstract

In a recent paper Jones and Mateo used operator techniques to show that the non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric wrong-sign quartic Hamiltonian $H=\frac{1}{2}{p}^{2}\ensuremath{-}g{x}^{4}$ has the same spectrum as the conventional Hermitian Hamiltonian $\stackrel{\texttildelow{}}{H}=\frac{1}{2}{p}^{2}+4g{x}^{4}\ensuremath{-}\sqrt{2g}x$. Here, this equivalence is demonstrated very simply by means of differential-equation techniques and, more importantly, by means of functional-integration techniques. It is shown that the linear term in the Hermitian Hamiltonian is anomalous; that is, this linear term has no classical analog. The anomaly arises because of the broken parity symmetry of the original non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian. This anomaly in the Hermitian form of a $\mathcal{P}\mathcal{T}$-symmetric quartic Hamiltonian is unchanged if a harmonic term is introduced into $H$. When there is a harmonic term, an immediate physical consequence of the anomaly is the appearance of bound states; if there were no anomaly term, there would be no bound states. Possible extensions of this work to $\ensuremath{-}{\ensuremath{\phi}}^{4}$ quantum field theory in higher-dimensional space-time are discussed.

DOI
10.1103/physrevd.74.025016

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arXiv:0704.2959•Apr 23, 2007•2 citations
In a recent paper Jones and Mateo used operator techniques to show that the non-Hermitian PT\mathcal{P}\mathcal{T}PT-symmetric wrong-sign quartic Hamiltonian H=12p2\ensuremath−gx4H=\frac{1}{2}{p}^{2}\ensuremath{-}g{x}^{4}H=21​p2\ensuremath−gx4 has the same spectrum as the conventional Hermitian Hamiltonian H\texttildelow=12p2+4gx4\ensuremath−2gx\stackrel{\texttildelow{}}{H}=\frac{1}{2}{p}^{2}+4g{x}^{4}\ensuremath{-}\sqrt{2g}xH\texttildelow​=21​p2+4gx4\ensuremath−2g​x. Here, this equivalence is demonstrated very simply by means of differential-equation techniques and, more importantly, by means of functional-integration techniques. It is shown that the linear term in the Hermitian Hamiltonian is anomalous; that is, this linear term has no classical analog. The anomaly arises because of the broken parity symmetry of the original non-Hermitian PT\mathcal{P}\mathcal{T}PT-symmetric Hamiltonian. This anomaly in the Hermitian form of a PT\mathcal{P}\mathcal{T}PT-symmetric quartic Hamiltonian is unchanged if a harmonic term is introduced into HHH. When there is a harmonic term, an immediate physical consequence of the anomaly is the appearance of bound states; if there were no anomaly term, there would be no bound states. Possible extensions of this work to \ensuremath−\ensuremathϕ4\ensuremath{-}{\ensuremath{\phi}}^{4}\ensuremath−\ensuremathϕ4 quantum field theory in higher-dimensional space-time are discussed.
It can be shown using operator techniques that the non-Hermitian PTPTPT-symmetric quantum mechanical Hamiltonian with a ``wrong-sign'' quartic potential $\ensuremath{-}g{x}^{4}$ is equivalent to a Hermitian Hamiltonian with a positive quartic potential together with a linear term. A na\"{\i}ve derivation of the same result in the path-integral approach misses this linear term. In a recent paper by Bender et al. it was pointed out that this term was in the nature of a parity anomaly and a more careful, discretized treatment of the path integral appeared to reproduce it successfully. However, on re-examination of this derivation we find that a yet more careful treatment is necessary, keeping terms that were ignored in that paper. An alternative, much simpler derivation is given using the additional potential that has been shown to appear whenever a change of variables to curvilinear coordinates is made in a functional integral.
Six years ago, by using operator techniques and path-integral methods, it was shown that the complex non-Hermitian PTPTPT-symmetric Hamiltonian ${p}^{2}\ensuremath{-}g{x}^{4}$ is equivalent to a conventional Hermitian Hamiltonian ${p}^{2}+4g{x}^{4}\ensuremath{-}2\ensuremath{\hbar}\sqrt{g}x$. Further it was revealed that the linear term in the Hermitian Hamiltonian is anomalous in the sense that it has no classical analog. In this paper we show that the complex non-Hermitian PT\mathit{PT}PT-symmetric Hamiltonian ${p}^{2}\ensuremath{-}g{x}^{4}+4i\ensuremath{\hbar}\sqrt{g}x$ and the conventional Hermitian Hamiltonian ${p}^{2}+4g{x}^{4}+6\ensuremath{\hbar}\sqrt{g}x$ have the same eigenspectra. In this case, the anomalous terms in both Hamiltonians are different from the previous one and vanish in the semiclassical limit. Further these equivalent Hamiltonians have zero-energy ground states. The exact ground-state wave functions and supersymmetric partner potentials are derived.
The potential $V(x)=\ensuremath{-}{x}^{4}$, which is unbounded below on the real line, can give rise to a well-posed bound state problem when xxx is taken on a contour in the lower-half complex plane. It is then PTPTPT-symmetric rather than Hermitian. Nonetheless it has been shown numerically to have a real spectrum, and a proof of reality, involving the correspondence between ordinary differential equations and integrable systems, was subsequently constructed for the general class of potentials $\ensuremath{-}(ix{)}^{N}$. For such Hamiltonians the natural PTPTPT metric is not positive definite, but a dynamically-defined positive-definite metric can be defined, depending on an operator QQQ. Further, with the help of this operator an equivalent Hermitian Hamiltonian hhh can be constructed. This programme has been carried out exactly for a few soluble models, and the first few terms of a perturbative expansion have been found for the potential m2x2+igx3{m}^{2}{x}^{2}+ig{x}^{3}m2x2+igx3. However, until now, the $\ensuremath{-}{x}^{4}$ potential has proved intractable. In the present paper we give explicit, closed form expressions for QQQ and hhh, which are made possible by a particular parametrization of the contour in the complex plane on which the problem is defined. This constitutes an explicit proof of the reality of the spectrum. The resulting equivalent Hamiltonian has a potential with a positive quartic term together with a linear term.
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.
A new method to work out the Hermitian correspondence of a PT\mathcal{PT}PT-symmetric quantum mechanical Hamiltonian is proposed. In contrast to the conventional method, the new method ends with a local Hamiltonian of the form 12p2+12m2x2+v(x)\frac{1}{2}{p}^{2}+\frac{1}{2}{m}^{2}{x}^{2}+v(x)21​p2+21​m2x2+v(x) without any higher-derivative terms. This method is available for Hamiltonians with general antilinear symmetries in the perturbative regime. Possible extensions to multivariable quantum mechanics and quantum field theories are discussed.
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.