Wigner–Weyl formalism for the lattice models

We discuss application of Wigner–Weyl formalism to the lattice models of condensed matter physics and relativistic quantum field theory. For the noninteracting models our technique relates Wigner transformation of the fermionic Green’s function with the Weyl symbol [Formula: see text] of lattice Dirac operator. We take as an example of the model defined on rectangular lattice the model of Wilson fermions. It represents the regularization of relativistic quantum field theory and, in addition, describes qualitatively certain topological materials in condensed matter physics. In this model we derive expression for [Formula: see text] in the presence of the most general case of external [Formula: see text] gauge field. Next, we solve the Groenewold equation to all orders in powers of the derivatives of [Formula: see text].

Spectral properties of Landau Hamiltonians with non-local potentials

We consider the Landau Hamiltonian H 0 , self-adjoint in L 2 ( R 2 ), whose spectrum consists of an arithmetic progression of infinitely degenerate positive eigenvalues Λ q , q ∈ Z + . We perturb H 0 by a non-local potential written as a bounded pseudo-differential operator Op w ( V ) with real-valued Weyl symbol V, such that Op w ( V ) H 0 − 1 is compact. We study the spectral properties of the perturbed operator H V = H 0 + Op w ( V ). First, we construct symbols V, possessing a suitable symmetry, such that the operator H V admits an explicit eigenbasis in L 2 ( R 2 ), and calculate the corresponding eigenvalues. Moreover, for V which are not supposed to have this symmetry, we study the asymptotic distribution of the eigenvalues of H V adjoining any given Λ q . We find that the effective Hamiltonian in this context is the Toeplitz operator T q ( V ) = p q Op w ( V ) p q , where p q is the orthogonal projection onto Ker ( H 0 − Λ q I ), and investigate its spectral asymptotics.

Compositions of states and observables in Fock spaces

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.

METAPLECTIC REPRESENTATION, CONLEY–ZEHNDER INDEX, AND WEYL CALCULUS ON PHASE SPACE

We define and study a metaplectically covariant class of pseudo-differential operators acting on functions on symplectic space and generalizing a modified form of the usual Weyl calculus. This construction requires a precise calculation of the twisted Weyl symbol of a class of generators of the metaplectic group and the use of a Conley–Zehnder type index for symplectic paths, defined without restrictions on the endpoint. Our calculus is related to the usual Weyl calculus using a family of isometries of L2(ℝn) on closed subspaces of L2(ℝ2n) and to an irreducible representation of the Heisenberg algebra distinct from the usual Schrödinger representation.