Introduction to PT-Symmetric Quantum Theory

Abstract

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.

References in corpus (2)

• Extension of PT-Symmetric Quantum Mechanics to Quantum Field Theory with Cubic Interaction
• Semiclassical Calculation of the C Operator in PT-Symmetric Quantum Mechanics

Cited by in corpus (288)

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• Observation of PT phase transition in a simple mechanical system
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• PT-symmetric quartic anharmonic oscillator and position-dependent mass in a perturbative approach
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• WKB Analysis of PT-Symmetric Sturm-Liouville problems. II
• WKB Analysis of PT-Symmetric Sturm-Liouville problems
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• Floquet topological phase in a generalized PT-symmetric lattice
• Emergent Phenomena with Broken Parity-Time Symmetry: Odd-order vs. Even-order Effects
• Complexified Synchrony
• Temporal correlation beyond quantum bounds in non-hermitian dynamics
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• Conjecture on the analyticity of PT-symmetric potentials and the reality of their spectra
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• Global solution of the cubic oscillator
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• The counting of Nambu-Goldstone bosons in a non-Hermitian field theory
• Real eigenvalues of a non-self-adjoint perturbation of the self-adjoint Zakharov-Shabat operator
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• Faster than Hermitian Time Evolution
• Momentum relation and classical limit in the future-not-included complex action theory
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• Cosmological perturbation theory with York time
• Some Theoretical Aspects of Quantum Mechanical Equations in Rindler Space
• The Large-N Limit of PT-Symmetric O(N) Models
• Robust preparation of ground state phases under noisy imaginary time evolution
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