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Stochastics

Probabilistic analysis of the ( q , 2)-Fock space: vacuum distribution and moments of the field operator

Yun Gang Lu

Published Aug 15, 2024• 0 citations
Publisher

Abstract

We study some probabilistic aspects of the (q,2)-Fock space over a given Hilbert space. Specifically, our focus is on determining the vacuum distribution of the field operator. We ascertain the precise expression for the vacuum expectation of any arbitrary product of creation and annihilation operators, with particular emphasis on the vacuum moments of the field operator. Utilizing these results, we calculate the moment-generating function of the field operator and subsequently determine the corresponding distribution.

DOI
10.1080/17442508.2024.2390095

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We study some probabilistic aspects of the (q,2)-Fock space over a given Hilbert space. Specifically, our focus is on determining the vacuum distribution of the field operator. We ascertain the precise expression for the vacuum expectation of any arbitrary product of creation and annihilation operators, with particular emphasis on the vacuum moments of the field operator. Utilizing these results, we calculate the moment-generating function of the field operator and subsequently determine the corresponding distribution.
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We survey some results obtained in the last 25 years to illustrate the following probabilistic quantization statement: quantum probability is not a generalization of classical probability but a deeper level of it. Generalizing the classical theory of orthogonal polynomials, one can show that any random field XXX admitting all the moments can be represented as the sum of three operators, which are natural extensions of the creation, annihilation, and preservation operators in the usual boson Fock quantum theory. These operators generate a noncommutative ∗*∗-algebra on which the quantum extension of the expected value relative to the probability distribution of the field XXX induces a quantum state. So, any classical algebraic probability space generates a quantum space with its own commutation relations. The Heisenberg commutation relation characterizes the classical fields, while the new type of commutation relations (of type II) appears in non-Gaussian cases. The same machinery, but applied to Bernoulli fields, leads to Fermi-Dirac anticommutation relations (see the introduction).
Abstract This chapter formally presents the representation of the quantum density operator by phase space distribution functionals of phase space fields, leading to expressions for quantum correlation functions in terms of phase space functional integrals, where the field functions—complex fields for bosons, Grassmann fields for fermions—replace field annihilation and creation operators. The emphasis is on double-phase-space normally ordered (positive P type) normalised distribution functionals, with distinct fields for annihilation and creation field operators; however, symmetrically ordered (Wigner type) distribution functionals are also considered, along with unnormalised B distribution functionals, which lead to phase space functional integrals for Fock position state populations and coherences. Characteristic functionals are first defined and then shown to be related to distribution functionals via phase space functional integrals. The existence of the distribution functionals is demonstrated from previous results for equivalent distribution functions by using mode expansions.
We introduce tensor products, which are the appropriate tool to construct the state space for systems that can be decomposed into non-interacting simpler components. The boson Fock space, built from such tensor products, is the appropriate state space to describe collections of any number of identical bosons. The annihilation and creation operators act on this Fock space. We then introduce the idea of quantum fields, which are operator-valued distributions, not trying yet to incorporate ideas from special relativity, and we argue that these quantum fields can be viewed as a quantized version of certain spaces of functions.