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arXiv:quant-ph/0509034 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Semiclassical analysis of a complex quartic Hamiltonian

Carl M. Bender, Dorje C. Brody, Hugh F Jones

Published Jan 5, 2006• 8 citations
arXivHTMLPDF

Abstract

It is necessary to calculate the $\mathcal{C}$ operator for the non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian $H=\frac{1}{2}{p}^{2}+\frac{1}{2}{\ensuremath{\mu}}^{2}{x}^{2}\ensuremath{-}\ensuremath{\lambda}{x}^{4}$ in order to demonstrate that $H$ defines a consistent unitary theory of quantum mechanics. However, the $\mathcal{C}$ operator cannot be obtained by using perturbative methods. Including a small imaginary cubic term gives the Hamiltonian $H=\frac{1}{2}{p}^{2}+\frac{1}{2}{\ensuremath{\mu}}^{2}{x}^{2}+ig{x}^{3}\ensuremath{-}\ensuremath{\lambda}{x}^{4}$, whose $\mathcal{C}$ operator can be obtained perturbatively. In the semiclassical limit all terms in the perturbation series can be calculated in closed form and the perturbation series can be summed exactly. The result is a closed-form expression for $\mathcal{C}$ having a nontrivial dependence on the dynamical variables $x$ and $p$ and on the parameter $\ensuremath{\lambda}$.

DOI
10.1103/physrevd.73.025002

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It is necessary to calculate the C\mathcal{C}C operator for the non-Hermitian PT\mathcal{P}\mathcal{T}PT-symmetric Hamiltonian H=12p2+12\ensuremathμ2x2\ensuremath−\ensuremathλx4H=\frac{1}{2}{p}^{2}+\frac{1}{2}{\ensuremath{\mu}}^{2}{x}^{2}\ensuremath{-}\ensuremath{\lambda}{x}^{4}H=21​p2+21​\ensuremathμ2x2\ensuremath−\ensuremathλx4 in order to demonstrate that HHH defines a consistent unitary theory of quantum mechanics. However, the C\mathcal{C}C operator cannot be obtained by using perturbative methods. Including a small imaginary cubic term gives the Hamiltonian H=12p2+12\ensuremathμ2x2+igx3\ensuremath−\ensuremathλx4H=\frac{1}{2}{p}^{2}+\frac{1}{2}{\ensuremath{\mu}}^{2}{x}^{2}+ig{x}^{3}\ensuremath{-}\ensuremath{\lambda}{x}^{4}H=21​p2+21​\ensuremathμ2x2+igx3\ensuremath−\ensuremathλx4, whose C\mathcal{C}C operator can be obtained perturbatively. In the semiclassical limit all terms in the perturbation series can be calculated in closed form and the perturbation series can be summed exactly. The result is a closed-form expression for C\mathcal{C}C having a nontrivial dependence on the dynamical variables xxx and ppp and on the parameter \ensuremathλ\ensuremath{\lambda}\ensuremathλ.
It has recently been shown that a non-Hermitian Hamiltonian H possessing an unbroken PT\mathcal{PT}PT symmetry (i) has a real spectrum that is bounded below, and (ii) defines a unitary theory of quantum mechanics with positive norm. The proof of unitarity requires a linear operator C,\mathcal{C},C, which was originally defined as a sum over the eigenfunctions of H. However, using this definition to calculate C\mathcal{C}C is cumbersome in quantum mechanics and impossible in quantum field theory. An alternative method is devised here for calculating C\mathcal{C}C directly in terms of the operator dynamical variables of the quantum theory. This method is general and applies to a variety of quantum mechanical systems having several degrees of freedom. More importantly, this method is used to calculate the C\mathcal{C}C operator in quantum field theory. The C\mathcal{C}C operator is a time-independent observable in PT\mathcal{PT}PT-symmetric quantum field theory.
Nonuniqueness of the C\mathcal {C}C operator in PT\mathcal {P}\mathcal {T}PT-symmetric quantum mechanics
The operator in -symmetric quantum mechanics satisfies a system of three simultaneous algebraic operator equations, , , and . These equations are difficult to solve exactly, so perturbative methods have been used in the past to calculate . The usual approach has been to express the Hamiltonian as H = H 0 + H 1 , and to seek a solution for in the form , where Q = Q ( q , p ) is odd in the momentum p , even in the coordinate q , and has a perturbation expansion of the form Q = Q 1 + 3 Q 3 + 5 Q 5 + ⋅⋅⋅. (In previous work it has always been assumed that the coefficients of even powers of in this expansion would be absent because their presence would violate the condition that Q ( p , q ) is odd in p .) In an earlier paper it was argued that the operator is not unique because the perturbation coefficient Q 1 is nonunique. Here, the nonuniqueness of is demonstrated at a more fundamental level: it is shown that the perturbation expansion for Q actually has the more general form Q = Q 0 + Q 1 + 2 Q 2 + ⋅⋅⋅ in which all powers and not just odd powers of appear. For the case in which H 0 is the harmonic-oscillator Hamiltonian, Q 0 is calculated exactly and in closed form and it is shown explicitly to be nonunique. The results are verified by using powerful summation procedures based on analytic continuation. It is also shown how to calculate the higher coefficients in the perturbation series for Q .
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.
In my talk at the Seventh QCD Workshop held in Villefranche in January 2003 I showed that a non-Hermitian Hamiltonian H possessing an unbroken [Formula: see text] symmetry (i) has a real spectrum that is bounded below, and (ii) defines a unitary theory of quantum mechanics with positive norm. The proof of unitarity requires a linear operator [Formula: see text], which was originally defined as a sum over the eigenfunctions of H. However, using this definition to calculate [Formula: see text] is cumbersome in quantum mechanics and impossible in quantum field theory. I describe here an alternative method for calculating [Formula: see text] directly in terms of the operator dynamical variables of the quantum theory. This new method is general and applies to a variety of quantum mechanical systems having several degrees of freedom. More importantly, this method gives the [Formula: see text] operator in quantum field theory. The [Formula: see text] operator is a new time-independent observable in [Formula: see text]-symmetric quantum field theory.
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 $H=\frac{1}{2}{p}^{2}\ensuremath{-}g{x}^{4}$ 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 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 $\ensuremath{-}{\ensuremath{\phi}}^{4}$ quantum field theory in higher-dimensional space-time are discussed.
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.