Must a Hamiltonian be Hermitian?

Abstract

A consistent physical theory of quantum mechanics can be built on a complex Hamiltonian that is not Hermitian but instead satisfies the physical condition of space-time reflection symmetry (PT symmetry). Thus, there are infinitely many new Hamiltonians that one can construct that might explain experimental data. One would think that a quantum theory based on a non-Hermitian Hamiltonian violates unitarity. However, if PT symmetry is not broken, it is possible to use a previously unnoticed physical symmetry of the Hamiltonian to construct an inner product whose associated norm is positive definite. This construction is general and works for any PT-symmetric Hamiltonian. The dynamics is governed by unitary time evolution. This formulation does not conflict with the requirements of conventional quantum mechanics. There are many possible observable and experimental consequences of extending quantum mechanics into the complex domain, both in particle physics and in solid state physics.

Revised version to appear in American Journal of Physics

References in corpus (10)

• Complex Extension of Quantum Mechanics
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• Non-Hermitian Hamiltonians with real and complex eigenvalues in a Lie-algebraic framework
• Calculation of the Hidden Symmetry Operator in PT-Symmetric Quantum Mechanics
• Bound States of Non-Hermitian Quantum Field Theories
• All Hermitian Hamiltonians Have Parity
• Two-Point Green's Function in PT-Symmetric Theories

Cited by in corpus (179)

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• Physical Aspects of Pseudo-Hermitian and $PT$-Symmetric Quantum Mechanics
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• Extension of PT-Symmetric Quantum Mechanics to Quantum Field Theory with Cubic Interaction
• Optical solitons in $\mathcal{P}\mathcal{T}$-symmetric nonlinear couplers with gain and loss
• Exactly solvable PT-symmetric Hamiltonian having no Hermitian counterpart
• Quantum noise and self-sustained radiation of PT-symmetric systems
• Observation of Fast Evolution in Parity-Time-Symmetric System
• Scalar Quantum Field Theory with Cubic Interaction
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• Time evolution of non-Hermitian Hamiltonian systems
• Dirac Quantization of the Pais-Uhlenbeck Fourth Order Oscillator
• Equivalence of a Complex $\cP\cT$-Symmetric Quartic Hamiltonian and a Hermitian Quartic Hamiltonian with an Anomaly
• Non-hermitian quantum thermodynamics
• Ghost-free higher-derivative theory
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• Optical waveguide arrays: quantum effects and PT symmetry breaking
• Non-Hermitian physics and master equations
• Jarzynski equality in $\mathcal{P}\mathcal{T}$-symmetric quantum mechanics
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• Quasinormal modes and Purcell factors of coupled loss-gain resonators and index-modulated ring resonators near exceptional points
• Complexified Dynamical Systems
• PT-Symmetric Quantum Electrodynamics
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• Non-linear Supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians: I. General properties
• Conceptual Aspects of PT-Symmetry and Pseudo-Hermiticity: A status report
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• Spectral Singularities of a General Point Interaction
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• PT-Invariant Periodic Potentials with a Finite Number of Band Gaps
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• Completeness of non-normalizable modes
• Recursive Calculation of Effective Potential and Variational Resummation
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• Harmonic oscillators from displacement operators and thermodynamics
• Distinguishability Transitions in Non-Unitary Boson Sampling Dynamics
• Fock Space Representations for Non-Hermitian Hamiltonians
• Stability boundaries of a Mathieu equation having PT symmetry
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• Euclidean Quantum Mechanics and Universal Nonlinear Filtering
• New solutions to the $s{\ell }_q(2)$-invariant Yang-Baxter equations at roots of unity
• Spherical-separablility of non-Hermitian Hamiltonians and pseudo-PT-symmetry
• Dynamics of ion channels via non-Hermitian quantum mechanics
• Canonical Expansion of PT-Symmetric Operators and Perturbation Theory
• Spectra of PT-symmetric Hamiltonians on Tobogganic Contours
• Spherical-separablility of non-Hermitian Dirac Hamiltonians and pseudo-PT-symmetry
• Conserved quantities in non-Hermitian systems via vectorization method
• Nonparaxial electromagnetic Bragg scattering in periodic media with PT symmetry
• Quantumness and its hierarchies in PT-symmetric down-conversion models
• Perpetual emulation threshold of PT-symmetric Hamiltonians
• Exact solution for the time dependent non-Hermitian generalized Swanson oscillator
• Remarks on the preservation of no-signaling principle in parity-time-symmetric quantum mechanics
• Fall-to-the-centre as a $\mathcal{P}\mathcal{T}$ symmetry breaking transition
• $\mathcal{P}\mathcal{T}$-symmetric mapping of three states and its implementation on a cloud quantum processor
• Non-Hermitian topological superconductivity with symmetry-enriched spectral and eigenstate features
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• Exploring Metamaterial Lasers through Non-Hermitian Scattering Formalism
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• Exploring Spectral Singularities in Dirac Semimetals: The Role of Non-Hermitian Physics and Dichroism