Observables and density matrices embedded in dual Hilbert spaces

Abstract

The introduction of operator states and of observables in various fields of quantum physics has raised questions about the mathematical structures of the corresponding spaces. In the framework of third quantization it had been conjectured that we deal with Hilbert spaces although the mathematical background was not entirely clear, particularly, when dealing with bosonic operators. This in turn caused some doubts about the correct way to combine bosonic and fermionic operators or, in other words, regular and Grassmann variables. In this paper we present a formal answer to the problems on a simple and very general basis. We illustrate the resulting construction by revisiting the Bargmann transform and finding the known connection between L 2 (R) and the Bargmann-Hilbert space. We pursue this line of thinking one step further an discuss the representations of complex extensions of linear canonical transformations as isometries between dual Hilbert spaces. We then use the formalism to give an explicit formulation for Fock spaces involving both fermions and bosons thus solving the problem at the origin of our considerations.

Extending third quantization with commuting observables: a dissipative spin-boson model

“…Some time ago, one of the authors developed canonical formalism of quantization over operator spaces for the diagonalization of the generator of the many-body Lindblad equationthe so-called Liouvillian superoperator, or completely solving the Lindbladian initial value problem, for a general quadratic Hamiltonian and a set of Lindblad jump operators which are linear in canonical creation/annihilation operators. The original proposal for fermionic systems [9,10] was later extended to bosonic systems [11] (see also [12] for a more abstract discussion of quantization over operator spaces), and further developed by other authors [13][14][15]. Specifically, the latter reference extended the technique to include quadratic Hermitian jump operators, allowing the analytical treatment of nonequilibrium phase transitions [16] and crossovers between ballistic and diffusive as well as quantum and classical transport [17,18].…”