Reviews in Mathematical Physics

Let $H$ be a separable Hilbert space and $T$ be a self-adjoint bounded linear operator on $H^{\otimes 2}$ with norm $\le 1$, satisfying the Yang–Baxter equation. Bożejko and Speicher ([10]) proved that the operator $T$ determines a $T$-deformed Fock space ${ℱ}(H)={\oplus }_{n=0}^{\infty }{{ℱ}}_n(H)$. We start with reviewing and extending the known results about the structure of the $n$-particle spaces ${{ℱ}}_n(H)$ and the commutation relations satisfied by the corresponding creation and annihilation operators acting on ${ℱ}(H)$. We then choose $H=L^2(X\to V)$, the $L^2$-space of $V$-valued functions on $X$. Here $X:={ℝ}^d$ and $V:={ℂ}^m$ with $m\ge 2$. Furthermore, we assume that the operator $T$ acting on $H^{\otimes 2}=L^2(X^2\to V^{\otimes 2})$ is given by $(T{f}^{(2)})(x,y)=C_{x,y}{f}^{(2)}(y,x)$. Here, for a.a. $(x,y)\in X^2$, $C_{x,y}$ is a linear operator on $V^{\otimes 2}$ with norm $\le 1$ that satisfies $C_{x,y}^{\ast }=C_{y,x}$ and the spectral quantum Yang–Baxter equation. The corresponding creation and annihilation operators describe a multicomponent quantum system. A special choice of the operator-valued function $C_{xy}$ in the case $d=2$ determines non-Abelian anyons (also called plektons). For a multicomponent system, we describe its $T$-deformed Fock space and the available commutation relations satisfied by the corresponding creation and annihilation operators. Finally, we consider several examples of multicomponent quantum systems.

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 [, ]). 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.

We construct the moduli space of Spin(7)-instantons on a hermitian complex vector bundle over a closed 8-dimensional manifold endowed with a (possibly non-integrable) Spin(7)-structure. We find suitable perturbations that achieve regularity of the moduli space, so that it is smooth and of the expected dimension over the irreducible locus.

We study the existence and the asymptotic behavior of the solution of an abstract viscoelastic system submitted to non-local initial data.