We discuss Hamiltonian symmetries and invariants for quantum systems driven by non-Hermitian Hamiltonians. For time-independent Hermitian Hamiltonians, a unitary or antiunitary transformation AHA† that leaves the Hamiltonian H unchanged represents a symmetry of the Hamiltonian, which implies the commutativity [H,A]=0 and, if A is linear and time-independent, a conservation law, namely the invariance of expectation values of A. For non-Hermitian Hamiltonians, H† comes into play as a distinct operator that complements H in generalized unitarity relations. The above description of symmetries has to be extended to include also A-pseudohermiticity relations of the form AH=H†A. A superoperator formulation of Hamiltonian symmetries is provided and exemplified for Hamiltonians of a particle moving in one-dimension considering the set of A operators that form Klein’s 4-group: parity, time-reversal, parity&time-reversal, and unity. The link between symmetry and conservation laws is discussed and shown to be richer and subtler for non-Hermitian than for Hermitian Hamiltonians.
We discuss Hamiltonian symmetries and invariants for quantum systems driven by non-Hermitian Hamiltonians. For time-independent Hermitian Hamiltonians, a unitary or antiunitary transformation
AHA† that leaves the Hamiltonian
H unchanged represents a symmetry of the Hamiltonian, which implies the commutativity
[H,A]=0, and a conservation law, namely the invariance of expectation values of
A. For non-Hermitian Hamiltonians,
H† comes into play as a distinct operator that complements
H in generalized unitarity relations. The above description of symmetries has to be extended to include also
A-pseudohermiticity relations of the form
AH=H†A. A superoperator formulation of Hamiltonian symmetries is provided and exemplified for Hamiltonians of a particle moving in one-dimension considering the set of
A operators forming Klein's 4-group: parity, time-reversal, parity\&time-reversal, and unity. The link between symmetry and conservation laws is discussed and shown to be more subtle for non-Hermitian than for Hermitian Hamiltonians.
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.
In nonrelativistic quantum mechanics and in relativistic quantum field theory, the time coordinate
t is a parameter and thus the time-reversal operator
T does not actually reverse the sign of
t. In contrast, in the five-dimensional approach to relativistic quantum mechanics introduced by Feynman, time
t is a quantum-mechanical operator. In this paper it is shown how one can use this five-dimensional approach to extend
T and
PT symmetry from nonrelativistic to relativistic quantum mechanics and implement time-reversal as an operation that effects $\mathcal{T}\text{ }t\mathcal{T}=\ensuremath{-}t$ just as
P effects $\mathcal{P}x\mathcal{P}=\ensuremath{-}x$, with
PT thus effecting $\mathcal{P}\mathcal{T}\text{ }{x}_{\ensuremath{\mu}}\mathcal{P}\mathcal{T}=\ensuremath{-}{x}_{\ensuremath{\mu}}$. Some illustrative relativistic quantum-mechanical models are constructed whose associated Hamiltonians are non-Hermitian but
PT symmetric, and it is shown that for each such Hamiltonian the energy eigenvalues are all real.
A physical theory of quantum mechanics can be based 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 non-Hermitian Hamiltonian would give a quantum theory that violates unitarity.However, if
PT symmetry is not broken, it is possible to use a 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.