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:1805.00758

Composition of states and observables in Fock spaces

Laurent Amour, Lisette Jager, Jean Nourrigat

Published May 2, 2018• 1 citation
arXivHTMLPDF

Abstract

This article is concerned with compositions in the context of three standard quantizations in the Fock space framework, namely, anti-Wick, Wick and Weyl quantizations. The first one is a composition of states and is closely related to the standard scattering identification operator encountered in Quantum Electrodynamics for time dynamics issues. Anti-Wick quantization and Segal Bargmann transforms are implied there for that purpose. The other compositions are for observables (operators in some specific classes) for the Wick and Weyl symbols. For the Wick symbol of the composition of two operators, we obtain an absolutely converging series, and for the Weyl symbol, the remainder term of the asymptotic expansion is absolutely converging, still in the Fock spaces framework.

DOI
10.48550/arxiv.1805.00758

Related papers

Compositions of states and observables in Fock spaces

Laurent Amour, Lisette Jager, Jean Nourrigat

This article is concerned with compositions in the context of three standard quantizations in the framework of Fock spaces, namely, anti-Wick, Wick and Weyl quantizations. The first one is a composition of states also known as a Wick product and is closely related to the standard scattering identification operator encountered in Quantum Electrodynamics for issues on time dynamics (see [ 29 , 13 ]). Anti-Wick quantization and Segal–Bargmann transforms are implied here for that purpose. The other compositions are for observables (operators in some specific classes) for the Wick and Weyl symbols. For the Wick and Weyl symbols of the composition of two operators, we obtain an absolutely converging series and for the Weyl symbol, the remainder terms up to any orders of the expansion are controlled, still in the Fock space framework.

Reviews in Mathematical Physics•Oct 18, 2019•1 citation

Torus as phase space: Weyl quantization, dequantization, and Wigner formalism

Marilena Ligabò

The Weyl quantization of classical observables on the torus (as phase space) without regularity assumptions is explicitly computed. The equivalence class of symbols yielding the same Weyl operator is characterized. The Heisenberg equation for the dynamics of general quantum observables is written through the Moyal brackets on the torus and the support of the Wigner transform is characterized. Finally, a dequantization procedure is introduced that applies, for instance, to the Pauli matrices. As a result we obtain the corresponding classical symbols.

arXiv:1409.3345•Journal of Mathematical Physics•Aug 1, 2016•17 citations

Wick product of white noise operators and quantum stochastic differential equations

Nobuaki Obata

The Wick product of operators on Fock space is introduced on the basis of the analytic characterization theorem for operator symbols established within the framework of white noise distribution theory. Existence and uniqueness of solutions are proved for a certain class of ordinary differential equations for Fock space operators. Quantum stochastic differential equations of Itô type and their generalizations involving higher powers of quantum white noises enter into our consideration.

Journal of the Mathematical Society of Japan•Jul 1, 1999•29 citations

COVARIANT AND CONTRAVARIANT SYMBOLS OF OPERATORS

Felix Alexandrovich Berezin

We consider the covariant and contravariant symbols of operators that are a generalization of the Wick and anti-Wick symbols introduced earlier. With the help of these symbols we investigate the spectral characteristics of operators, and construct a method of successive approximations to calculate the exponential of an operator. In an application to anti-Wick symbols this method provides a basis of one of the variants of the Feynman functional integral along trajectories in phase space.

Mathematics of the USSR-Izvestiya•Oct 31, 1972•481 citations

Complex and real Hermite polynomials and related quantizations

Nicolae Cotfas, Jean‐Pierre Gazeau, K. Górska

It is known that the anti-Wick (or standard coherent state) quantization of the complex plane produces both canonical commutation rule and quantum spectrum of the harmonic oscillator (up to the addition of a constant). In this work, we show that these two issues are not necessarily coupled: there exists a family of separable Hilbert spaces, including the usual Fock–Bargmann space, and in each element in this family there exists an overcomplete set of unit-norm states resolving the unity. With the exception of the Fock–Bargmann case, they all produce non-canonical commutation relation whereas the quantum spectrum of the harmonic oscillator remains the same up to the addition of a constant. The statistical aspects of these non-equivalent coherent state quantizations are investigated. We also explore the localization aspects in the real line yielded by similar quantizations based on real Hermite polynomials.

arXiv:1001.3248•Journal of Physics A Mathematical and Theoretical•Jun 28, 2010•58 citations

Functional Representations for Fock Superalgebras

Joachim Kupsch, Oleg Georgievich Smolyanov

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.

arXiv:hep-th/9708069•Infinite Dimensional Analysis Quantum Probability and Related Topics•Apr 1, 1998•13 citations
This article is concerned with compositions in the context of three standard quantizations in the Fock space framework, namely, anti-Wick, Wick and Weyl quantizations. The first one is a composition of states and is closely related to the standard scattering identification operator encountered in Quantum Electrodynamics for time dynamics issues. Anti-Wick quantization and Segal Bargmann transforms are implied there for that purpose. The other compositions are for observables (operators in some specific classes) for the Wick and Weyl symbols. For the Wick symbol of the composition of two operators, we obtain an absolutely converging series, and for the Weyl symbol, the remainder term of the asymptotic expansion is absolutely converging, still in the Fock spaces framework.
This article is concerned with compositions in the context of three standard quantizations in the framework of Fock spaces, namely, anti-Wick, Wick and Weyl quantizations. The first one is a composition of states also known as a Wick product and is closely related to the standard scattering identification operator encountered in Quantum Electrodynamics for issues on time dynamics (see [ 29 , 13 ]). Anti-Wick quantization and Segal–Bargmann transforms are implied here for that purpose. The other compositions are for observables (operators in some specific classes) for the Wick and Weyl symbols. For the Wick and Weyl symbols of the composition of two operators, we obtain an absolutely converging series and for the Weyl symbol, the remainder terms up to any orders of the expansion are controlled, still in the Fock space framework.
The Weyl quantization of classical observables on the torus (as phase space) without regularity assumptions is explicitly computed. The equivalence class of symbols yielding the same Weyl operator is characterized. The Heisenberg equation for the dynamics of general quantum observables is written through the Moyal brackets on the torus and the support of the Wigner transform is characterized. Finally, a dequantization procedure is introduced that applies, for instance, to the Pauli matrices. As a result we obtain the corresponding classical symbols.
It is known that the anti-Wick (or standard coherent state) quantization of the complex plane produces both canonical commutation rule and quantum spectrum of the harmonic oscillator (up to the addition of a constant). In this work, we show that these two issues are not necessarily coupled: there exists a family of separable Hilbert spaces, including the usual Fock–Bargmann space, and in each element in this family there exists an overcomplete set of unit-norm states resolving the unity. With the exception of the Fock–Bargmann case, they all produce non-canonical commutation relation whereas the quantum spectrum of the harmonic oscillator remains the same up to the addition of a constant. The statistical aspects of these non-equivalent coherent state quantizations are investigated. We also explore the localization aspects in the real line yielded by similar quantizations based on real Hermite polynomials.
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.