Geometry of the complex extension of Wigner's theorem

Abstract

Wigner’s theorem asserts that an isometric (probability conserving) transformation on a quantum state space must be generated by a Hamiltonian that is Hermitian. It is shown that when the hermiticity condition on the Hamiltonian is relaxed, we obtain the following complex generalization of Wigner’s theorem: a holomorphically projective (complex geodesic curves preserving) transformation on a quantum state space must be generated by a Hamiltonian that is not necessarily Hermitian.