Making Sense of Non-Hermitian Hamiltonians

Abstract

The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A standard axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose + complex conjugate) is replaced by the physically transparent condition of space-time reflection (PT) symmetry. If H has an unbroken PT symmetry, then the spectrum is real. Examples of PT-symmetric non-Hermitian quantum-mechanical Hamiltonians are H=p^2+ix^3 and H=p^2-x^4. Amazingly, the energy levels of these Hamiltonians are all real and positive! In general, if H has an unbroken PT symmetry, then it has another symmetry represented by a linear operator C. Using C, one can construct a time-independent inner product with a positive-definite norm. Thus, PT-symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution. The Lee Model is an example of a PT-symmetric Hamiltonian. The renormalized Lee-model Hamiltonian has a negative-norm "ghost" state because renormalization causes the Hamiltonian to become non-Hermitian. For the past 50 years there have been many attempts to find a physical interpretation for the ghost, but all such attempts failed. Our interpretation of the ghost is simply that the non-Hermitian Lee Model Hamiltonian is PT-symmetric. The C operator for the Lee Model is calculated exactly and in closed form and the ghost is shown to be a physical state having a positive norm. The ideas of PT symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models.

78 pages, 26 figures, invited review article to be published in Rep. Prog. Phys

References in corpus (1)

• MHD alpha^2-dynamo, Squire equation and PT-symmetric interpolation between square well and harmonic oscillator

Cited by in corpus (24)

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• PT-symmetric extensions of the supersymmetric Korteweg-de Vries equation
• The S-matrix of the Faddeev-Reshetikhin Model, Diagonalizability and PT Symmetry
• Identification of observables in quantum toboggans
• A complex periodic QES potential and exceptional points
• Conjecture on the analyticity of PT-symmetric potentials and the reality of their spectra
• Optimal Covariant Measurement of Momentum on a Half Line in Quantum Mechanics
• PT symmetry and large-N models
• Non-local PT-symmetric potentials in the one-dimensional Dirac equation
• Quantum knots
• Conditional observability versus self-duality in a schematic model