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Journal of Physics Conference Series

Upside-down potentials

Carl M. Bender

Published Feb 8, 2012• 1 citation
Publisher

Abstract

A quantum-mechanical theory is -symmetric if it is described by a Hamiltonian that commutes with , where represents space reflection and represents time reversal. Often, but not always, a -symmetric Hamiltonian is non-Hermitian, and thus its eigenvalues may not be all real. However, -symmetric Hamiltonians are interesting because they usually have a parametric region of unbroken symmetry in which the eigenvalues are indeed all real and a region of broken symmetry in which the some of the eigenvalues are complex. These regions are separated by a phase transition that has been repeatedly observed in laboratory experiments. This paper focuses on the properties of a particular -symmetric Hamiltonian, H = p 2 − x 4 , which has an upside-down potential. The spectrum of this -symmetric Hamiltonian is rigorously known to be entirely real, and the purpose of this paper is to present an intuitive explanation of why the spectrum of this Hamiltonian is real, positive, and discrete.

DOI
10.1088/1742-6596/343/1/012014

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A quantum-mechanical theory is -symmetric if it is described by a Hamiltonian that commutes with , where represents space reflection and represents time reversal. Often, but not always, a -symmetric Hamiltonian is non-Hermitian, and thus its eigenvalues may not be all real. However, -symmetric Hamiltonians are interesting because they usually have a parametric region of unbroken symmetry in which the eigenvalues are indeed all real and a region of broken symmetry in which the some of the eigenvalues are complex. These regions are separated by a phase transition that has been repeatedly observed in laboratory experiments. This paper focuses on the properties of a particular -symmetric Hamiltonian, H = p 2 − x 4 , which has an upside-down potential. The spectrum of this -symmetric Hamiltonian is rigorously known to be entirely real, and the purpose of this paper is to present an intuitive explanation of why the spectrum of this Hamiltonian is real, positive, and discrete.
The average quantum physicist on the street would say that a quantum-mechanical Hamiltonian must be Dirac Hermitian (invariant under combined matrix transposition and complex conjugation) in order to guarantee that the energy eigenvalues are real and that time evolution is unitary. However, the Hamiltonian H = p 2 + ix 3 , which is obviously not Dirac Hermitian, has a positive real discrete spectrum and generates unitary time evolution, and thus it defines a fully consistent and physical quantum theory. Evidently, the axiom of Dirac Hermiticity is too restrictive. While H = p 2 + ix 3 is not Dirac Hermitian, it is PT symmetric; that is, invariant under combined parity P (space reflection) and time reversal T . The quantum mechanics defined by a PT -symmetric Hamiltonian is a complex generalization of ordinary quantum mechanics. When quantum mechanics is extended into the complex domain, new kinds of theories having strange and remarkable properties emerge. In the past few years, some of these properties have been verified in laboratory experiments. A particularly interesting PT-symmetric Hamiltonian is H = p 2 − x 4 , which contains an upside-down potential. This potential is discussed in detail, and it is explained in intuitive as well as in rigorous terms why the energy levels of this potential are real, positive, and discrete. Applications of PT -symmetry in quantum field theory are also discussed.
Most quantum physicists believe that a quantum-mechanical Hamiltonian must be Dirac Hermitian (invariant under matrix transposition and complex conjugation) to be sure that the energy eigenvalues are real and that time evolution is unitary. However, the non-Dirac-hermitian Hamiltonian H = p2+ix3 has a real positive discrete spectrum and generates unitary time evolution and defines a fully consistent and physical quantum theory. Evidently, Dirac Hermiticity is too restrictive. While H = p2+ix3 is not Dirac Hermitian, it is PT symmetric (invariant under combined space reflection P and time reversal T). Another PT-symmetric Hamiltonian whose energy levels are real, positive and discrete is H = p2−x4, which contains an upside-down potential. The quantum mechanics defined by a PT-symmetric Hamiltonian is a complex generalization of ordinary quantum mechanics. When quantum mechanics and quantum field theory are extended into the complex domain, new kinds of theories having strange and remarkable properties emerge. In the past two years some of these properties have been verified in laboratory experiments. Here, we first discuss PT-symmetric Hamiltonians at a simple intuitive level and explain why the energy levels of such Hamiltonians may be real, positive, and discrete. Second, we describe a recent experiment in which the PT phase transition was observed. Third, we briefly mention that PT-symmetric theories can be useful at a fundamental level. While the double-scaling limit of an O(N)-symmetric gφ4 quantum field theory appears to be inconsistent because the critical value of g is negative, this limit is in fact not inconsistent because the critical theory is PT symmetric.
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 = p 2 + x 2 (ix) ǫ of the harmonic oscillator Hamiltonian, where ǫ is a real parameter.The system exhibits two phases: When ǫ ≥ 0, the energy spectrum of H is real and positive as a consequence of PT symmetry.However, when -1 < ǫ < 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 ǫ = 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 = p 2 + x 2N (ix) ǫ with N integer and ǫ > -N ; each of these complex Hamiltonians exhibits a phase transition at ǫ = 0.These PT -symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space.
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.
If a quantum-mechanical Hamiltonian is PT symmetric, there are two possibilities: either all of the eigenvalues are real, in which case the Hamiltonian is said to be in an unbroken-PT-symmetric phase, or else some of the eigenvalues are real and some are complex, in which case the Hamiltonian is said to be in a broken-PT-symmetric phase. As one varies the parameters of the Hamiltonian, one can pass through the phase transition that separates the unbroken and broken phases. This transition has recently been observed in a variety of laboratory experiments. This paper explains the phase transition in a simple and intuitive fashion and then describes an elementary experiment in which the phase transition is easily observed.
Large families of Hamiltonians that are non-Hermitian in the conventional sense have been found to have all eigenvalues real, a fact attributed to an unbroken symmetry. The corresponding quantum theories possess an unconventional scalar product. The eigenvalues are determined by differential equations with boundary conditions imposed in wedges in the complex plane. For a special class of such systems, it is possible to impose the -symmetric boundary conditions on the real axis, which lies on the edges of the wedges. The -symmetric spectrum can then be obtained by imposing the more transparent requirement that the potential be reflectionless.