Alternative Algebraic Structures From Bi-Hamiltonian Quantum Systems

Abstract

We discuss the alternative algebraic structures on the manifold of quantum states arising from alternative Hermitian structures associated with quantum bi-Hamiltonian systems. We also consider the consequences at the level of the Heisenberg picture in terms of deformations of the associative product on the space of observables.

A Cartesian classical second-quantized many-electron Hamiltonian, for use with the semiclassical initial value representation

“…28 Recently there have been many applications and discussions for the quantum dynamical systems on quaternionic Hilbert space. 29,30 We make use of only the most elementary properties of quaterions, the essential step being to use vector cross products to represent the quaterions. The new classical many-electron Hamiltonian that is obtained incorporates the anti-commutation properties of the fermionic operators in a simpler and more natural way than the MW Hamiltonian in action-angle variables.…”