Weyl Calculus Research Articles

SOME APPLICATIONS OF WEYL CALCULUS TO BURCHNALL-CHAUNDY THEORY, III

We show how Weyl calculus can be applied to the following problem. Suppose that are differential operators with polynomial coefficients and they commute. Then there is a polynomial such that . We use Weyl calculus, but Kohn-Nirenberg calculus also can be used. In this paper, we give the proof of existence of in general case, prove that the ideal of all such is the principal ideal, and give an algorithm for finding generator of in some particular cases.

Precise Wigner–Weyl Calculus for the Honeycomb Lattice

Precise Wigner–Weyl Calculus for the Honeycomb Lattice

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.

Wigner–Weyl calculus in description of nondissipative transport phenomena

Application of Wigner–Weyl calculus to the investigation of nondissipative transport phenomena is reviewed. We focus on the quantum Hall effect, Chiral Magnetic effect and Chiral separation effect, and discuss the role of interactions, inhomogeneity and deviations from equilibrium.

A calculus for magnetic pseudodifferential super operators

This work develops a magnetic pseudodifferential calculus for super operators OpA(F); these map operators onto operators (as opposed to Lp functions onto Lq functions). Here, F could be a tempered distribution or a Hörmander symbol. An important example is Liouville super operators L̂=−iopA(h),⋅ defined in terms of a magnetic pseudodifferential operator opA(h). Our work combines ideas from the magnetic Weyl calculus developed by Măntoiu and Purice [J. Math. Phys. 45, 1394–1417 (2004)]; Iftimie, Măntoiu, and Purice [Publ. Res. Inst. Math. Sci. 43, 585–623 (2007)]; and Lein (Ph.D. thesis, 2011) and the pseudodifferential calculus on the non-commutative torus from the work of Ha, Lee, and Ponge [Int. J. Math. 30, 1950033 (2019)]. Thus, our calculus is inherently gauge-covariant, which means that all essential properties of OpA(F) are determined by properties of the magnetic field B = dA rather than the vector potential A. There are conceptual differences to ordinary pseudodifferential theory. For example, in addition to an analog of the (magnetic) Weyl product that emulates the composition of two magnetic pseudodifferential super operators on the level of functions, the so-called semi-super product describes the action of a pseudodifferential super operator on a pseudodifferential operator.

Influence of interactions on Integer Quantum Hall Effect

Conductivity of Integer Quantum Hall Effect (IQHE) may be expressed as the topological invariant composed of the two-point Green function. Such a topological invariant is known both for the case of homogeneous systems with intrinsic Anomalous Quantum Hall Effect (AQHE) and for the case of IQHE in the inhomogeneous systems. In the latter case we may speak, for example, of the AQHE in the presence of elastic deformations and of the IQHE in presence of magnetic field. The topological invariant for the general case of inhomogeneous systems is expressed through the Wigner transformed Green functions and contains Moyal product. When it is reduced to the expression for the IQHE in the homogeneous systems the Moyal product is reduced to the ordinary one while the Wigner transformed Green function (defined in phase space) is reduced to the Green function in momentum space. Originally the mentioned above topological representation has been derived for the non-interacting systems. We demonstrate that in a wide range of different cases in the presence of interactions the Hall conductivity is given by the same expression, in which the noninteracting two-point Green function is substituted by the complete two-point Green function with the interactions taken into account. Several types of interactions are considered including the contact four-fermion interactions, Yukawa and Coulomb interactions. We present the complete proof of this statement up to the two loops, and argue that the similar result remains to all orders of perturbation theory. It is based on the incorporation of Wigner–Weyl calculus to the perturbation theory. We, therefore, formulate Feynmann rules of diagram technique in terms of the Wigner transformed propagators.

Quantizations and Global Hypoellipticity for Pseudodifferential Operators of Infinite Order in Classes of Ultradifferentiable Functions

We study the change of quantization for a class of global pseudodifferential operators of infinite order in the setting of ultradifferentiable functions of Beurling type. The composition of different quantizations as well as the transpose of a quantization are also analysed, with applications to the Weyl calculus. We also compare global \(\omega \)-hypoellipticity and global \(\omega \)-regularity of these classes of pseudodifferential operators.

Stochastic extensions of symbols in Wiener spaces and heat operator

The construction, in [1], of a pseudodifferential calculus analogous to the Weyl calculus, in an infinite dimensional setting, required the introduction of convenient symbol classes. The symbols are functions defined on an infinite dimensional Hilbert space H, which compels us to investigate, in particular, their stochastic extensions to a Wiener space linked with H.

Generalization of Fourier Transform and Weyl Calculus

In this paper, a surjective morphism of the topological groups from the real line R to the p -curve Cp is introduced, this function maps from the real line to the p -curve on the complex and when p = 2 then coincide with a classical exponent. The properties of p -Fourier transform is studied. The generalization of the Weyl functional calculus is considered.

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

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 Weyl calculus for group generators satisfying the canonical commutation relations

We generalise the classical Weyl pseudo-differential calculus on Rd to the setting of two d-tuples of operators A=(A1,…,Ad) and B=(B1,…,Bd) acting on a Banach space generating bounded C0-groups satisfying the Weyl canonical commutation relations. We show that the resulting Weyl calculus extends to symbols in the standard symbol class S0 provided appropriate bounds can be established. Using transference techniques we obtain boundedness of the H∞-functional calculus (and even the H\"ormander calculus), for the abstract harmonic oscillator L=12d∑j=1(A2j+B2j)−12d.

Pseudodifferential Weyl Calculus on (Pseudo-)Riemannian Manifolds

One can argue that on flat space {mathbb {R}}^d, the Weyl quantization is the most natural choice and that it has the best properties (e.g., symplectic covariance, real symbols correspond to Hermitian operators). On a generic manifold, there is no distinguished quantization, and a quantization is typically defined chart-wise. Here we introduce a quantization that, we believe, has the best properties for studying natural operators on pseudo-Riemannian manifolds. It is a generalization of the Weyl quantization—we call it the balanced geodesic Weyl quantization. Among other things, we prove that it maps square-integrable symbols to Hilbert–Schmidt operators, and that even (resp. odd) polynomials are mapped to even (resp. odd) differential operators. We also present a formula for the corresponding star product and give its asymptotic expansion up to the fourth order in Planck’s constant.

The weyl calculus for group generators satisfying the canonical commutation relations

We generalise the classical Weyl pseudo-differential calculus on Rd to the setting of two d-tuples of operators A = (A1,..., Ad) and B = (B1,..., Bd) acting on a Banach space generating bounded C0-groups satisfying the Weyl canonical commutation relations. We show that the resulting Weyl calculus extends to symbols in the standard symbol class S0 provided appropriate bounds can be established. Using transference techniques we obtain boundedness of the H∞-functional calculus (and even the Hormander calculus), for the abstract harmonic oscillator.

Wave kinetic equation for inhomogeneous drift-wave turbulence beyond the quasilinear approximation

The formation of zonal flows from inhomogeneous drift-wave (DW) turbulence is often described using statistical theories derived within the quasilinear approximation. However, this approximation neglects wave–wave collisions. Hence, some important effects such as the Batchelor–Kraichnan inverse-energy cascade are not captured within this approach. Here we derive a wave kinetic equation that includes a DW collision operator in the presence of zonal flows. Our derivation makes use of the Weyl calculus, the quasinormal statistical closure and the geometrical-optics approximation. The obtained model conserves both the total enstrophy and energy of the system. The derived DW collision operator breaks down at the Rayleigh–Kuo threshold. This threshold is missed by homogeneous-turbulence theory but expected from a full-wave quasilinear analysis. In the future, this theory might help better understand the interactions between drift waves and zonal flows, including the validity domain of the quasilinear approximation that is commonly used in the literature.

Weyl calculus in Wiener spaces and in QED

The concern of this article is a semiclassical Weyl calculus on an infinite dimensional Hilbert space H. If (i, H, B) is a Wiener triplet associated to H, the quantum state space will be the space of $$L^2$$ functions on B with respect to a Gaussian measure with h / 2 variance, where h is the semiclassical parameter. We prove the boundedness of our pseudodifferential operators (PDO) in the spirit of Calderon–Vaillancourt with an explicit bound, a Beals type characterization, and metaplectic covariance. An application to a model of quantum electrodynamics is added in the last section (Sect. 7), for fixed spin 1 / 2 particles interacting with the quantized electromagnetic field (photons). We prove that some observable time evolutions, the spin evolutions, the magnetic and electric evolutions when subtracting their free evolutions, are PDO in our class.

Spectral asymptotics for infinite order pseudo-differential operators

We study spectral properties of a class of global infinite order pseudo-differential operators and obtain the asymptotic behaviour of the spectral counting functions of such operators. Unlike their finite order counterparts, their spectral asymptotics are not of power-log-type but of log-type. The ultradistributional setting of such operators of infinite order makes the theory more complex so that the standard finite order global Weyl calculus cannot be used in this context.

The Weyl calculus with respect to the Gaussian measure and restricted Lp-Lq boundedness of the Ornstein-Uhlenbeck semigroup in complex time

The authors gratefully acknowledge financial support by the ARC discovery Grant DP 160100941.

Ponderomotive dynamics of waves in quasiperiodically modulated media

Similarly to how charged particles experience time-averaged ponderomotive forces in high-frequency fields, linear waves also experience time-averaged refraction in modulated media. Here we propose a covariant variational theory of this "ponderomotive effect on waves" for a general nondissipative linear medium. Using the Weyl calculus, our formulation accommodates waves with temporal and spatial period comparable to that of the modulation (provided that parametric resonances are avoided). Our theory also shows that any wave is, in fact, a polarizable object that contributes to the linear dielectric tensor of the ambient medium. The dynamics of quantum particles is subsumed as a special case. As an illustration, ponderomotive Hamiltonians of quantum particles and photons are calculated within a number of models. We also explain a fundamental connection between these results and the commonly known expression for the electrostatic dielectric tensor of quantum plasmas.

Weyl calculus in QED I. The unitary group

In this work, we consider fixed 1/2 spin particles interacting with the quantized radiation field in the context of quantum electrodynamics. We investigate the time evolution operator in studying the reduced propagator (interaction picture). We first prove that this propagator belongs to the class of infinite dimensional Weyl pseudodifferential operators recently introduced in Amour et al. [J. Funct. Anal. 269(9), 2747–2812 (2015)] on Wiener spaces. We give a semiclassical expansion of the symbol of the reduced propagator up to any order with estimates on the remainder terms. Next, taking into account analyticity properties for the Weyl symbol of the reduced propagator, we derive estimates concerning transition probabilities between coherent states.

Symbol calculus of square-integrable operator-valued maps

We develop an abstract framework for the investigation of quantization and dequantization procedures based on orthogonality relations that do not necessarily involve group representations. To illustrate the usefulness of our abstract method, we show that it behaves well with respect to infinite tensor products. This construction subsumes examples from the study of magnetic Weyl calculus, magnetic pseudo-differential Weyl calculus, metaplectic representation on locally compact abelian groups, irreducible representations associated with finite-dimensional coadjoint orbits of some special infinite-dimensional Lie groups, and square-integrability properties shared by arbitrary irreducible representations of nilpotent Lie groups.