2014/04/25 by Tomaž Prosen, T. Prosen, Prosen, T. +5
Computer Science · Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #FOS: Physical sciences #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Spectral Theory in Mathematical Physics #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.1404.6336
10 pages, no figures
openalex publication_date 2014/04/25 · arxiv created 2015/02/27 · arxiv updated 2015/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 L2(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.