Infinite degrees of freedom Weyl representation: Characterization and application

By means of infinite-dimensional nuclear spaces, we generalize important results on the representation of the Weyl commutation relations. For this purpose, we construct a new nuclear Lie group generalizing the groups introduced by Parthasarathy [An Introduction to Quantum Stochastic Calculus (Birkhäuser, 1992)] and Gelfand–Vilenkin [Generalized Functions (Academic Press, 1964)] (see Ref. 15). Then we give an explicit construction of Weyl representations generated from a non-Fock representation. Moreover, we characterize all these Weyl representations in quantum white noise setting.

Annular Khovanov homology and knotted Schur–Weyl representations

Let $\mathbb{L}\subset A\times I$ be a link in a thickened annulus. We show that its sutured annular Khovanov homology carries an action of $\mathfrak{sl}_{2}(\wedge )$, the exterior current algebra of $\mathfrak{sl}_{2}$. When $\mathbb{L}$ is an $m$-framed $n$-cable of a knot $K\subset S^{3}$, its sutured annular Khovanov homology carries a commuting action of the symmetric group $\mathfrak{S}_{n}$. One therefore obtains a ‘knotted’ Schur–Weyl representation that agrees with classical $\mathfrak{sl}_{2}$ Schur–Weyl duality when $K$ is the Seifert-framed unknot.

Weyl representation of the canonical commutation relations algebras in a Krein space

In this paper, the existence of the Weyl representation for the canonical commutation relations algebras was proved in a Krein space.

A NOTE ON LONGITUDINAL FIELDS IN THE WEYL EXPANSION OF THE ELECTROMAGNETIC GREEN TENSOR

Many calculations of dispersion interactions between atoms and macroscopic bodies or between two bodies make use of Green tensors. The expansion of this tensor in polarizations allows for an anatomic interpretation of the interaction. In planar systems with partial translation invariance, the Weyl representation of the Green tensor is often applied. Although it is transverse in this representation, we argue that the field it describes contains nonradiative parts as well. This can be seen by calculating the Green tensor in the momentum representation and observing certain cancellations among longitudinal and transverse contributions.