Abstract
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.
Highlights
The intimate link between invariance and symmetry is well studied and understood for Hermitian Hamiltonians but non-Hermitian Hamiltonians pose some interesting conceptual and formal challenges
For non-Hermitian Hamiltonians the symmetry Equation (8) applied to Equation (13) gives
The eight superoperators form the elementary abelian group of order eight [12], with a minimal set of three generators L†, LΠ, LΘ, from which all elements may be formed by multiplication, i.e., successive application. (Θ is the antilinear time-reversal operator acting as Θa| x i = a∗ | x i in coordinate representation, and as Θa| pi = a∗ | − pi in momentum representation.) The eight superoperators may be found from the generating set {L A }, L†, where A is one of the elements of Klein’s 4-group {1, Θ, Π, ΠΘ}
Summary
The intimate link between invariance and symmetry is well studied and understood for Hermitian Hamiltonians but non-Hermitian Hamiltonians pose some interesting conceptual and formal challenges. That and A, which we assume to be time-independent unless stated otherwise, represents a conserved quantity when A is unitary (and linear), hψ(t), Aψ(t)i = hψ(0), Aψ(0)i, Mathematics 2018, 6, 111; doi:10.3390/math6070111. For antilinear operators expectation values are ambiguous since multiplication of the state by a unit modulus phase factor eiφ changes the expectation value by e−2iφ This ambiguity does not mean at all that antilinear symmetries do not have physical consequences. Time-independent linear operators A satisfying (2), fullfill (3) without the need to be unitary, and represent invariant quantities. If |φj i is a right eigenstate of H with eigenvalue Ej , Equation (7) implies that A|φj i is a right eigenstate of H, with the same eigenvalue if A is linear, and with the complex conjugate eigenvalue E∗j if A is antilinear. As right and left eigenvectors must be treated on equal footing, since both are needed for the resolution (9), this argument points at a similar importance of the relations (7) and (8)
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