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

Tsukuba Journal of Mathematics

The complex Weyl calculus as a Stratonovich-Weyl correspondence for the real diamond group

Benjamin Cahen

Published Jul 1, 2020• 1 citation
Publisher

Abstract

We revisit the problem of constructing a Stratonovich-Weyl correspondence for each generic representation of a the real diamond group which was already considered in [B. Cahen, Riv. Mat. Univ. Parma 4 (2013), 197-213]. Here we use the complex Weyl calculus on the Fock space. This allows us, in particular, to obtain some closed formulas for the Stratonovich-Weyl symbols of the representation operators.

DOI
10.21099/tkbjm/20204401121

Related papers

Complex Weyl correspondence for a generalized diamond group

Benjamin Cahen

The generalized diamond group is the semi-direct product $G$ of the abelian group ${\mathbb R}^m$ by the $(2n+1)$-dimensional Heisenberg group $H_n$. We construct the generic representations of $G$ on the Fock space by extending those of $H_n$. Then we study the Berezin correspondence and the complex Weyl correspondence in connection with a generic representation $π$ of $G$, proving in particular that these correspondences are covariant with respect to $π$. We give also some explicit formulas for the Berezin symbols and the complex Weyl symbols of the representation operators $π(g)$ for $g\in G$. These results are applied to recover various formulas involving the Moyal product. Moreover, we relate $π$ to a coadjoint orbit of $G$ in the spirit of the Kirillov-Kostant method of orbits. This allows us to establish that the complex Weyl correspondence is a Stratonovich-Weyl correspondence for $π$.

arXiv:2509.08082•Sep 9, 2025•0 citations

Weyl calculus on the Fock space and Stratonovich-Weyl correspondence for Heisenberg motion groups

Benjamin Cahen

We construct a Stratonovich-Weyl correspondence for each generic representation of a Heisenberg motion group by using the Weyl calculus on the Fock space.

arXiv:2012.14398•Dec 28, 2020•2 citations

Stratonovich-Weyl correspondence via Berezin quantization

Benjamin Cahen

Benjamin Cahen, "Stratonovich-Weyl correspondence via Berezin quantization", in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 46 (2014), pp.157-180

•Dec 23, 2014•4 citations

Simple Approach for Deriving the Weyl Correspondence Product Formula

Hong-Yi Fan, Hai-Guang Weng

By making use of the coherent state representation of the Wigner operator we present a simple approach for deriving the Weyl correspondence product formula in complex phase space. The formula tells us how to express the Weyl classical function corresponding to an operator in terms of the Weyl functions corresponding to and .

Communications in Theoretical Physics•Oct 1, 1992•3 citations

The Stratonovich–Weyl correspondence for one-dimensional kinematical groups

Manuel Gadella, Miguel A. Martín, Luis Miguel Nieto +1 more

The Stratonovich–Weyl correspondence is a restatement of the Moyal quantization where the phase space is a manifold and where a group of transformations acts on it transitively. The first and most important step is to define a mapping from the manifold into the set of self-adjoint operators on a Hilbert space, under suitable conditions. This mapping is called a Stratonovich–Weyl kernel. The construction of this mapping is discussed on coadjoint orbits of the one-dimensional Galilei, Poincaré, and Newton–Hooke groups as well as the two-dimensional Euclidean group.

Journal of Mathematical Physics•May 1, 1991•43 citations

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
We revisit the problem of constructing a Stratonovich-Weyl correspondence for each generic representation of a the real diamond group which was already considered in [B. Cahen, Riv. Mat. Univ. Parma 4 (2013), 197-213]. Here we use the complex Weyl calculus on the Fock space. This allows us, in particular, to obtain some closed formulas for the Stratonovich-Weyl symbols of the representation operators.
The generalized diamond group is the semi-direct product GGG of the abelian group Rm{\mathbb R}^mRm by the (2n+1)(2n+1)(2n+1)-dimensional Heisenberg group HnH_nHn​. We construct the generic representations of GGG on the Fock space by extending those of HnH_nHn​. Then we study the Berezin correspondence and the complex Weyl correspondence in connection with a generic representation πππ of GGG, proving in particular that these correspondences are covariant with respect to πππ. We give also some explicit formulas for the Berezin symbols and the complex Weyl symbols of the representation operators π(g)π(g)π(g) for g∈Gg\in Gg∈G. These results are applied to recover various formulas involving the Moyal product. Moreover, we relate πππ to a coadjoint orbit of GGG in the spirit of the Kirillov-Kostant method of orbits. This allows us to establish that the complex Weyl correspondence is a Stratonovich-Weyl correspondence for πππ.
The Stratonovich–Weyl correspondence is a restatement of the Moyal quantization where the phase space is a manifold and where a group of transformations acts on it transitively. The first and most important step is to define a mapping from the manifold into the set of self-adjoint operators on a Hilbert space, under suitable conditions. This mapping is called a Stratonovich–Weyl kernel. The construction of this mapping is discussed on coadjoint orbits of the one-dimensional Galilei, Poincaré, and Newton–Hooke groups as well as the two-dimensional Euclidean group.