Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:hep-th/9708069 · Infinite Dimensional Analysis Quantum Probability and Related Topics

Functional Representations for Fock Superalgebras

Joachim Kupsch, Oleg Georgievich Smolyanov

Published Apr 1, 1998• 13 citations
arXivHTMLPDF

Abstract

The Fock space of bosons and fermions and its underlying superalgebra are represented by algebras of functions on a superspace. We define Gaussian integration on infinite-dimensional superspaces, and construct super-analogs of the classical function spaces with a reproducing kernel — including the Bargmann–Fock representation — and of the Wiener–Segal representation. The latter representation requires the investigation of Wick ordering on Z2-graded algebras. As application we derive a Mehler formula for the Ornstein–Uhlenbeck semigroup on the Fock space.

DOI
10.1142/s0219025798000168

Related papers

Bosonic kernels

Paul Lee Robinson

We show that the bosonic Fock representation of a complex Hilbert space admits a purely algebraic kernel calculus; as an illustration, we use it to reproduce the standard integral kernel formulae for metaplectic operators within the complex-wave representation.

arXiv:1204.3693•Apr 17, 2012•0 citations

GAUSSIAN QUANTUM MARKOV SEMIGROUPS IN THE FOCK-ANTI-FOCK REPRESENTATION OF WEYL ALGEBRA

Ameur Dhahri, Franco Fagnola, D Poletti +1 more

We construct Gaussian quantum Markov semigroups on the representation of the Weyl algebra in the tensor product of the Fock space and its conjugate Fock space by the minimal semigroup method. We prove the explicit formula for the action on Weyl operators and study some properties.

Journal of Stochastic Analysis•Mar 1, 2025•0 citations

A PROBABILISTIC FORMULATION OF BOSONIC AND FERMIONIC INTEGRATION

Joachim Kupsch

A unified picture of bosonic and fermionic integration is presented. The Wiener isomorphism between the bosonic Fock space and the space of functions which are square integrable with respect to the canonical Gaussian promeasure of the basic Hilbert space is extended to the fermionic Fock space. Bosonic and fermionic functionals can be identified with Gaussian integrals on the same probability space.

Reviews in Mathematical Physics•Jan 1, 1990•8 citations

Stochastic Integration on the Full Fock Space with the Help of a Kernel Calculus

Roland Speicher

We develop a stochastic integration theory with respect to creation, annihilation and gauge operators on the full Fock space. This is done by using a kernel representation for a large class of bounded operators on the full Fock space. It is shown that the kernels form a Banach algebra. Having established the definition of processes and stochastic integrals we go on to prove an Ito formula and use this for examining stochastic evolutions and constructing dilations of special completely positive semigroups. Explicit solutions of the corresponding stochastic differential equations are given.

Publications of the Research Institute for Mathematical Sciences•Feb 28, 1991•21 citations

Coherent states in the fermionic Fock space

Robert Oeckl

We construct the coherent states in the sense of Gilmore and Perelomov for the fermionic Fock space. Our treatment is from the outset adapted to the infinite-dimensional case. The fermionic Fock space becomes in this way a reproducing kernel Hilbert space of continuous holomorphic functions.

arXiv:1408.2760•Journal of Physics A Mathematical and Theoretical•Dec 19, 2014•2 citations

ANALYTIC CHARACTERIZATION OF ONE-MODE INTERACTING FOCK SPACE

Nobuhiro Asai

In this paper, we shall give the Segal–Bargmann representation space of the one-mode interacting Fock space. For this purpose, we shall define the generalized coherent vector associated with the Szegö–Jacobi parameters and introduce the associated integral transformation.

Infinite Dimensional Analysis Quantum Probability and Related Topics•Sep 1, 2001•13 citations
The Fock space of bosons and fermions and its underlying superalgebra are represented by algebras of functions on a superspace. We define Gaussian integration on infinite-dimensional superspaces, and construct super-analogs of the classical function spaces with a reproducing kernel — including the Bargmann–Fock representation — and of the Wiener–Segal representation. The latter representation requires the investigation of Wick ordering on Z2-graded algebras. As application we derive a Mehler formula for the Ornstein–Uhlenbeck semigroup on the Fock space.
We develop a stochastic integration theory with respect to creation, annihilation and gauge operators on the full Fock space. This is done by using a kernel representation for a large class of bounded operators on the full Fock space. It is shown that the kernels form a Banach algebra. Having established the definition of processes and stochastic integrals we go on to prove an Ito formula and use this for examining stochastic evolutions and constructing dilations of special completely positive semigroups. Explicit solutions of the corresponding stochastic differential equations are given.