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arXiv:0908.4255 · Physical Review A

Non-Hermitian quantum Hamiltonians with PT symmetry

Katherine Jones-Smith, Harsh Mathur

Published Oct 6, 2010• 77 citations
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Abstract

We formulate quantum mechanics for non-Hermitian Hamiltonians that are invariant under $\mathcal{P}\mathcal{T}$, where $\mathcal{P}$ is the parity and $\mathcal{T}$ denotes time reversal, for the case that time-reversal symmetry is odd (${\mathcal{T}}^{2}=\ensuremath{-}1$), generalizing prior work for the even case (${\mathcal{T}}^{2}=1$). We discover an analog of Kramer's theorem for $\mathcal{P}\mathcal{T}$ quantum mechanics, present a prototypical example of a $\mathcal{P}\mathcal{T}$ quantum system with odd time reversal, and discuss potential applications of the formalism.

DOI
10.1103/physreva.82.042101

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We formulate quantum mechanics for non-Hermitian Hamiltonians that are invariant under PT\mathcal{P}\mathcal{T}PT, where P\mathcal{P}P is the parity and T\mathcal{T}T denotes time reversal, for the case that time-reversal symmetry is odd (T2=\ensuremath−1{\mathcal{T}}^{2}=\ensuremath{-}1T2=\ensuremath−1), generalizing prior work for the even case (T2=1{\mathcal{T}}^{2}=1T2=1). We discover an analog of Kramer's theorem for PT\mathcal{P}\mathcal{T}PT quantum mechanics, present a prototypical example of a PT\mathcal{P}\mathcal{T}PT quantum system with odd time reversal, and discuss potential applications of the formalism.
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†AHA^\daggerAHA† that leaves the Hamiltonian HHH unchanged represents a symmetry of the Hamiltonian, which implies the commutativity [H,A]=0[H,A]=0[H,A]=0, and a conservation law, namely the invariance of expectation values of AAA. For non-Hermitian Hamiltonians, H†H^\daggerH† comes into play as a distinct operator that complements HHH in generalized unitarity relations. The above description of symmetries has to be extended to include also AAA-pseudohermiticity relations of the form AH=H†AAH=H^\dagger AAH=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 AAA 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.
The basic structure of quantum mechanics was delineated in the early days of the theory and has not been modified since.One of the fundamental assumptions used in formulating the theory is that operators are represented by Hermitian matrices.In recent years it has been shown that quantum mechanics can be formulated consistently without making this assumption, using instead a combination of the parity (P) and time-reversal (T) operators and a number of other requirements related to P and T .Only the case of even T has been analyzed in the literature; here we generalize the principles to include odd time-reversal.We use this generalization to construct a non-Hermitian version of the Dirac equation, and in doing so discover a new type of particle not allowed within the (Hermitian) Standard Model.Finally we present a potential application of the ideas of non-Hermitian quantum mechanics to the unsolved problems of quantum magnetism and high temperature superconductivity.
Currently, it has been claimed that certain Hermitian Hamiltonians have parity (P) and they are PT-invariant. We propose generalized definitions of time-reversal operator (T) and orthonormality such that all Hermitian Hamiltonians are P, T, PT, and CPT invariant. The PT-norm and CPT-norm are indefinite and definite respectively. The energy-eigenstates are either E-type (e.g., even) or O-type (e.g., odd). C mimics the charge-conjugation symmetry which is recently found to exist for a non-Hermitian Hamiltonian. For a Hermitian Hamiltonian it coincides with P. The Hamiltonians which are invariant under the joint transformation of Parity (x → −x) and Time-reversal (i → −i) are called PT-invariant. It has been conjectured [1] that such Hamiltonians possess real discrete energy-eigenvalues provided the PT symmetry is unbroken. PT-symmetry is called unbroken or exact if the energy-eigenstates are also simultaneous eigenstates of PT. On the other hand when PT-symmetry is spontaneously broken the energy-eigenvalues are complex conjugate pairs. Multipronged investigations supporting
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.
For ordinary hermitian Hamiltonians, the states show the Kramers degeneracy when the system has a half-odd-integer spin and the time reversal operator obeys \Theta^2=-1, but no such a degeneracy exists when \Theta^2=+1. Here we point out that for non-hermitian systems, there exists a degeneracy similar to Kramers even when \Theta^2=+1. It is found that the new degeneracy follows from the mathematical structure of split-quaternion, instead of quaternion from which the Kramers degeneracy follows in the usual hermitian cases. Furthermore, we also show that particle/hole symmetry gives rise to a pair of states with opposite energies on the basis of the split quaternion in a class of non-hermitian Hamiltonians. As concrete examples, we examine in detail NxN Hamiltonians with N=2 and 4 which are non-hermitian generalizations of spin 1/2 Hamiltonian and quadrupole Hamiltonian of spin 3/2, respectively.
This dissertation concerns the concept of PT symmetry as a replacement of the conventional symmetry of the Dirac Hermiticity of the operators, in particular, for the Hamiltonian. After solving a controversial issue surrounding the claimed violation of the special relativity by PT-symmetric quantum mechanics, we move further to discuss PT symmetry for the systems whose associated time reversal symmetry is odd, i.e., T^2 = −1. Our results, as published papers, are incorporated in the subsequent chapters. Then we conclude by suggestions for future work.