Observables and density matrices embedded in dual Hilbert spaces

TL;DR

This paper clarifies the mathematical structure of operator states and observables in quantum physics by embedding them in dual Hilbert spaces, resolving issues related to bosonic and fermionic operators and their combination.

Contribution

It provides a formal framework for combining bosonic and fermionic operators within Hilbert spaces, addressing foundational questions in quantum operator theory.

Findings

Revisited the Bargmann transform and its connection to L^2(R)

Developed an explicit formulation for mixed fermionic and bosonic Fock spaces

Resolved longstanding doubts about the mathematical structure of operator states

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 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.