Weyl Symbol Research Articles

Symmetry breaking operators for the reductive dual pair (U[formula omitted],U[formula omitted])

We consider the dual pair (G,G′)=(Ul,Ul′) in the symplectic group Sp2ll′(R). Fix a Weil representation of the metaplectic group Sp˜2ll′(R). Let G˜ and G˜′ be the preimages of G and G′ in Sp˜2ll′(R), and let Π⊗Π′ be a genuine irreducible representation of G˜×G˜′. We study the Weyl symbol fΠ⊗Π′ of the (unique up to a possibly zero constant) symmetry breaking operator (SBO) intertwining the Weil representation with Π⊗Π′. This SBO coincides with the orthogonal projection of the space of the Weil representation onto its Π-isotypic component and also with the orthogonal projection onto its Π′-isotypic component. Hence fΠ⊗Π′ can be computed in two different ways, one using Π and the other using Π′. By matching the results, we recover Weyl’s theorem stating that Π⊗Π′ occurs in the Weil representation with multiplicity at most one and we also recover the complete list of the representations Π⊗Π′ occurring in Howe’s correspondence.

Complex Weyl symbols of the extended metaplectic representation operators

Complex Weyl symbols of the extended metaplectic representation operators

Boundedness of metaplectic Toeplitz operators and Weyl symbols

We study Toeplitz operators on the Bargmann space, whose Toeplitz symbols are exponentials of complex inhomogeneous quadratic polynomials. Extending a result by Coburn–Hitrik–Sjöstrand [7], we show that the boundedness of such Toeplitz operators implies the boundedness of the corresponding Weyl symbols, thus completing the proof of the Berger–Coburn conjecture in this case. We also show that a Toeplitz operator is compact precisely when its Weyl symbol vanishes at infinity in this case.

Lipschitz continuity of spectra of pseudodifferential operators in a weighted Sjöstrand class and Gabor frame bounds

We study one-parameter families of pseudodifferential operators whose Weyl symbols are obtained by dilation and a smooth deformation of a symbol in a weighted Sjöstrand class. We show that their spectral edges are Lipschitz continuous functions of the dilation or deformation parameter. Suitably local estimates hold also for the edges of every spectral gap. These statements extend Bellissard's seminal results on the Lipschitz continuity of spectral edges for families of operators with periodic symbols to a large class of symbols with only mild regularity assumptions.

Discrete Wigner–Weyl calculus for the finite lattice

We develop the approach of Felix Buot to construction of Wigner–Weyl calculus for the lattice models. We apply this approach to the tight-binding models with finite number of lattice cells. For simplicity we restrict ourselves to the case of rectangular lattice. We start from the original Buot definition of the symbol of operator. This definition is corrected in order to maintain self-consistency of the algebraic constructions. It appears, however, that the Buot symbol for simple operators does not have a regular limit when the lattice size tends to infinity. Therefore, using a more dense auxiliary lattice we modify the Buot symbol of operator in order to build our new discrete Weyl symbol. The latter obeys several useful identities inherited from the continuum theory. Besides, the limit of infinitely large lattice becomes regular. We formulate Keldysh technique for the lattice models using the proposed Weyl symbols of operators. Within this technique the simple expression for the electric conductivity of a two dimensional non-equilibrium and non-homogeneous system is derived. This expression smoothly approaches the topological one in the limit of thermal equilibrium at small temperature and large system area.

Characterizing Boundedness of Metaplectic Toeplitz Operators

Abstract We study Toeplitz operators on the Bargmann space, with Toeplitz symbols given by exponentials of complex quadratic forms. We show that the boundedness of the corresponding Weyl symbols is necessary for the boundedness of the operators, thereby completing the proof of the Berger–Coburn conjecture in this case. We also show that the compactness of such Toeplitz operators is equivalent to the vanishing of their Weyl symbols at infinity.

A Quantum-Classical Model of Brain Dynamics.

The study of the human psyche has elucidated a bipartite structure of logic reflecting the quantum-classical nature of the world. Accordingly, we posited an approach toward studying the brain by means of the quantum-classical dynamics of a mixed Weyl symbol. The mixed Weyl symbol can be used to describe brain processes at the microscopic level and, when averaged over an appropriate ensemble, can provide a link to the results of measurements made at the meso and macro scale. Within this approach, quantum variables (such as, for example, nuclear and electron spins, dipole momenta of particles or molecules, tunneling degrees of freedom, and so on) can be represented by spinors, whereas the electromagnetic fields and phonon modes can be treated either classically or semi-classically in phase space by also considering quantum zero-point fluctuations. Quantum zero-point effects can be incorporated into numerical simulations by controlling the temperature of each field mode via coupling to a dedicated Nosé-Hoover chain thermostat. The temperature of each thermostat was chosen in order to reproduce quantum statistics in the canonical ensemble. In this first paper, we introduce a general quantum-classical Hamiltonian model that can be tailored to study physical processes at the interface between the quantum and the classical world in the brain. While the approach is discussed in detail, numerical calculations are not reported in the present paper, but they are planned for future work. Our theory of brain dynamics subsumes some compatible aspects of three well-known quantum approaches to brain dynamics, namely the electromagnetic field theory approach, the orchestrated objective reduction theory, and the dissipative quantum model of the brain. All three models are reviewed.

Canonical mean-field molecular dynamics derived from quantum mechanics

Canonical quantum correlation observables can be approximated by classical molecular dynamics. In the case of low temperature theab initiomolecular dynamics potential energy is based on the ground state electron eigenvalue problem and the accuracy has been proven to beO(M-1), provided the first electron eigenvalue gap is sufficiently large compared to the given temperature andMis the ratio of nuclei and electron masses. For higher temperature eigenvalues corresponding to excited electron states are required to obtainO(M-1) accuracy and the derivations assume that all electron eigenvalues are separated, which for instance excludes conical intersections. This work studies a mean-field molecular dynamics approximation where the mean-field Hamiltonian for the nuclei is the partial traceh := Tr(He−βH)/Tr(e−βH) with respect to the electron degrees of freedom andHis the Weyl symbol corresponding to a quantum many body Hamiltonian ̂H. It is proved that the mean-field molecular dynamics approximates canonical quantum correlation observables with accuracyO(M-1+tϵ2), for correlation timetwhereϵ2is related to the variance of mean value approximationh. Furthermore, the proof derives a precise asymptotic representation of the Weyl symbol of the Gibbs density operator using a path integral formulation. Numerical experiments on a model problem with one nuclei and two electron states show that the mean-field dynamics has similar or better accuracy than standard molecular dynamics based on the ground state electron eigenvalue.

Pseudo-differential operators with isotropic symbols, Wick and anti-Wick operators, and hypoellipticity

We study the link between Ψdos and Wick operators via the Bargmann transform. We deduce a formula for the symbol of the Wick operator in terms of the short-time Fourier transform of the Weyl symbol. This gives characterizations of Wick symbols of Ψdos of Shubin type and of infinite order, and results on composition. We prove a series expansion of Wick operators in terms of anti-Wick operators which leads to a sharp Gårding inequality and transition of hypoellipticity between Wick and Shubin symbols. Finally we show continuity results for anti-Wick operators, and estimates for the Wick symbols of anti-Wick operators.

Global regularity of second order ordinary differential operators with polynomial coefficients

Let Lu=u″+2Au′+Bu be an ordinary differential operator with A and B are polynomials of any degree. In this paper we show that L is globally regular, i.e. Lu∈S implies u∈S, for all tempered distribution u∈S′, if and only if the complex roots ξ±(x)=iA(x)±B(x)−A′(x)−(A(x))2 of the Weyl symbol σL(x,ξ)=−ξ2+2iA(x)ξ+B(x)−A′(x) satisfy the condition lim|x|→+∞⁡|xImξ±(x)|=+∞.

The response field and the saddle points of quantum mechanical path integrals

In quantum statistical mechanics, Moyal’s equation governs the time evolution of Wigner functions and of more general Weyl symbols that represent the density matrix of arbitrary mixed states. A formal solution to Moyal’s equation is given by Marinov’s path integral. In this paper we demonstrate that this path integral can be regarded as the natural link between several conceptual, geometric, and dynamical issues in quantum mechanics. A unifying perspective is achieved by highlighting the pivotal role which the response field, one of the integration variables in Marinov’s integral, plays for pure states even. The discussion focuses on how the integral’s semiclassical approximation relates to its strictly classical limit; unlike for Feynman type path integrals, the latter is well defined in the Marinov case. The topics covered include a random force representation of Marinov’s integral based upon the concept of “Airy averaging”, a related discussion of positivity-violating Wigner functions describing tunneling processes, and the role of the response field in maintaining quantum coherence and enabling interference phenomena. The double slit experiment for electrons and the Bohm–Aharonov effect are analyzed as illustrative examples. Furthermore, a surprising relationship between the instantons of the Marinov path integral over an analytically continued (“Wick rotated”) response field, and the complex instantons of Feynman-type integrals is found. The latter play a prominent role in recent work towards a Picard–Lefschetz theory applicable to oscillatory path integrals and the resurgence program.

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].

Weyl symbols and boundedness of Toeplitz operators

Weyl symbols and boundedness of Toeplitz operators

Continuity Properties for Born-Jordan Operators with Symbols in Hörmander Classes and Modulation Spaces

We show that the Weyl symbol of a Born-Jordan operator is in the same class as the Born-Jordan symbol, when Hormander symbols and certain types of modulation spaces are used as symbol classes. We use these properties to carry over continuity, nuclearity and Schatten-von Neumann properties to the Born-Jordan calculus.

Characterization of the Moyal Multiplier Algebras for the Generalized Spaces of Type S

We investigate the properties of the Moyal multiplier algebras for the generalized Gelfand-Shilov spaces $$S_{{a_k}}^{{b_n}}$$ . We prove that these algebras contain Palamodov spaces of type $${\mathscr{E}}$$ , and establish continuity properties of the operators with Weyl symbols in this class. Analogous results are obtained for the projective version of the spaces of type S and are extended to the multiplier algebras for various translation-invariant star products.

Hall conductivity as topological invariant in phase space

It is well known that the quantum Hall conductivity in the presence of constant magnetic field is expressed through the topological TKNN invariant. The same invariant is responsible for the intrinsic anomalous quantum Hall effect (AQHE), which, in addition, may be represented as one in momentum space composed of the two point Green’s functions. We propose the generalization of this expression to the QHE in the presence of non-uniform magnetic field. The proposed expression is the topological invariant in phase space composed of the Weyl symbols of the two-point Green’s function. It is applicable to a wide range of non-uniform tight-binding models, including the interacting ones.

Precise Wigner-Weyl calculus for lattice models

We propose a new version of Wigner-Weyl calculus for tight-binding lattice models. It allows to express various physical quantities through Weyl symbols of operators and Green's functions. In particular, Hall conductivity in the presence of varying and arbitrarily strong magnetic field is represented using the proposed formalism as a topological invariant.

Global regularity of second order twisted differential operators

In this paper we characterize global regularity in the sense of Shubin of twisted partial differential operators of second order in dimension 2. These operators form a class containing the twisted Laplacian, and in bi-unique correspondence with second order ordinary differential operators with polynomial coefficients and symbol of degree 2. This correspondence is established by a transformation of Wigner type. In this way the global regularity of twisted partial differential operators turns out to be equivalent to global regularity and injectivity of the corresponding ordinary differential operators, which can be completely characterized in terms of the asymptotic behavior of the Weyl symbol. In conclusion we observe that we have obtained a new class of globally regular partial differential operators which is disjoint from the class of hypo-elliptic operators in the sense of Shubin.

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.

Positivity, complex FIOs, and Toeplitz operators

We establish a characterization of complex linear canonical transformations that are positive with respect to a pair of strictly plurisubharmonic quadratic weights. As an application, we show that the boundedness of a class of Toeplitz operators on the Bargmann space is implied by the boundedness of their Weyl symbols.