Starting with the Wigner distribution formulation for beam wave propagation in Hölder continuous non-Gaussian random refractive index fields we show that the wave beam regime naturally leads to the white-noise scaling limit and converges to a Gaussian white-noise model which is characterized by the martingale problem associated to a stochastic differential-integral equation of the Itô type. In the simultaneous geometrical optics the convergence to the Gaussian white-noise model for the Liouville equation is also established if the ultraviolet cutoff or the Fresnel number vanishes sufficiently slowly. The advantage of the Gaussian white-noise model is that its $n$-point correlation functions are governed by closed form equations.
Discussed are some geometric aspects of the phase space formalism in quantum mechanics in the sense of Weyl, Wigner, Moyal and Ville. We analyze the relationship between this formalism and geometry of the Galilei group, classical momentum mapping, theory of unitary projective representations of groups, and theory of groups algebras. Later on we present some generalization to quantum mechanics on locally compact Abelian groups. It is based on Pontryagin duality. Indicated are certain physical aspects in quantum dynamics of crystal lattices, including the phenomenon of "Umklapp-Prozessen"
The Weyl-Wigner-Moyal formalism is developed for spin by means of a correspondence between spherical harmonics and spherical harmonic tensor operators. The analogue of the Moyal expansion is developed for the Weyl symbol of the product of two operators in terms of the symbols for the individual operators, and it is shown that in the classical limit, the Weyl symbol for a commutator equals $i$ times the Poisson bracket of the corresponding Weyl symbols. It is also found that, to the same order, there is no correction in the symbol for the anticommutator.
A system of coupled kinetic transport equations for the Wigner distributions of a free variable mass Klein-Gordon field is derived. This set of equations is formally equivalent to the full wave equation for electromagnetic waves in nonlinear dispersive media, thus allowing for the description of broadband radiation-matter interactions and the associated instabilities. The standard results for the classical wave action are recovered in the short wavelength limit of the generalized Wigner-Moyal formalism for the wave equation.
A conjecture in quantum mechanics states that any quantum canonical transformation can decompose into a sequence of three basic canonical transformations; gauge, point and interchange of coordinates and momenta. It is shown that if one attempts to construct the three basic transformations in star-product form, while gauge and point transformations are immediate in star-exponential form, interchange has no correspondent, but it is possible in an ordinary exponential form. As an alternative approach, it is shown that all three basic transformations can be constructed in the ordinary exponential …
A full consideration of classical and quantum systems with radiation (electromagnetic/gravitational) requires the involvement of a mathematical description in the generalized phase space of high kinematical values. Based on the dispersion chain of equations of quantum mechanics, we construct a generalization of the von Neumann equation for the density matrix in the phase space of fourth-order kinematical values. The paper introduces a new extended definition of the fourth rank Wigner function, which is constructed from the wave functions of the second rank. A new extended Moyal equation (PSI-M…
A relativistic phase-space representation for a class of observables with matrix-valued Weyl symbols proportional to the identity matrix (charge-invariant observables)is proposed. We take into account the nontrivial charge structure of the position and momentum operators. The evolution equation coincides with its analog in relativistic quantum mechanics with nonlocal Hamiltonian under conditions where particle-pair creation does not take place (free particle and constant magnetic field). The differences in the equations are connected with peculiarities of the constraints on the initial conditi…
In this paper, we study the classical limit and unitary evolution of quantum cosmology by applying the Weyl--Wigner--Groenewold--Moyal formalism of deformation quantization to quantum cosmology of a homogeneous and isotropic universe with positive spatial curvature and conformally coupled scalar field. The corresponding quantum cosmology (similar to the Schrödinger interpretation in canonical quantization scheme of quantum cosmology) is described by the Moyal--Wheeler--DeWitt equation which has an exact solution in Moyal phase space, resulting in Wigner quasiprobability distribution function, …
Starting from the quantum Liouville equation for the density operator and applying the Weyl quantization, Wigner equations for the longitudinal and transversal optical and acoustic phonons are deduced. The equations are valid for any solid, including 2D crystals like graphene. With the use of Moyal's calculus and its properties the pseudo-differential operators are expanded up to the second order in $\hbar$. The phonon-phonon collision operators are modelled in a BGK form and describe the relaxation of the Wigner functions to a local equilibrium function, depending on a local equilibrium tempe…
The relationship between two different asymptotic techniques developed in order to describe the propagation of waves beyond the standard geometrical optics approximation, namely, the Wigner-Weyl kinetic formalism and the complex geometrical optics method, is addressed. More specifically, a solution of the wave kinetic equation, relevant to the Wigner-Weyl formalism, is obtained which yields the same wavefield intensity as the complex geometrical optics method. Such a relationship is also discussed on the basis of the analytical solution of the wave kinetic equation specific to Gaussian beams o…
We consider the Cauchy-problem for a class of scalar linear dispersive equations with rapidly oscillating initial data. The problem of high-frequency asymptotics of such models is reviewed,in particular we highlight the difficulties in crossing caustics when using (time-dependent) WKB-methods. Using Wigner measures we present an alternative approach to such asymptotic problems. We first discuss the connection of the naive WKB solutions to transport equations of Liouville type (with mono-kinetic solutions) in the prebreaking regime. Further we show that the Wigner measure approach can be used t…
We introduced few years ago a new notion of causality for noncommutative spacetimes directly related to the Dirac operator and the concept of Lorentzian spectral triple. In this paper, we review in a non-technical way the noncommutative causal structure of many toy models as almost commutative spacetimes and the Moyal-Weyl spacetime. We show that those models present some unexpected physical interpretations as a geometrical explanation of the Zitterbewegung trembling motion of a fermion as well as some geometrical constraints on translations and energy jumps of wave packets on the Moyal spacet…
We present the application of the variational-wavelet analysis to the quasiclassical calculations of the solutions of Wigner/von Neumann/Moyal and related equations corresponding to the nonlinear (polynomial) dynamical problems. (Naive) deformation quantization, the multiresolution representations and the variational approach are the key points. We construct the solutions via the multiscale expansions in the generalized coherent states or high-localized nonlinear eigenmodes in the base of the compactly supported wavelets and the wavelet packets. We demonstrate the appearance of (stable) locali…
In this thesis, we construct an approximate series solution of the Wigner equation in terms of Airy functions, which are semiclassically concentrated on certain Lagrangian curves in two-dimensional phase space. These curves are defined by the eigenvalues and the Hamiltonian function of the associated one-dimensional Schrödinger operator, and they play a crucial role in the quantum interference mechanism in phase space. We assume that the potential of the Schrödinger operator is a single potential well, such that the spectrum is discrete. The construction starts from an eigenfunction series exp…
In order to realize supersymmetric quantum mechanics methods on a four dimensional classical phase-space, the complexified Clifford algebra of this space is extended by deforming it with the Moyal star-product in composing the components of Clifford forms. Two isospectral matrix Hamiltonians having a common bosonic part but different fermionic parts depending on four real-valued phase space functions are obtained. The Hamiltonians are doubly intertwined via matrix-valued functions which are divisors of zero in the resulting Moyal-Clifford algebra. Two illustrative examples corresponding to Jay…
We show that the short-time Fourier transform of the pointwise product of two functions $f$ and $h$ can be written as a suitable product of the short-time Fourier transforms of $f$ and $h$. The same result is then shown to be valid for the Wigner wave-packet transform. We study the main properties of the new products. We then use these products to derive integro-differential equations on the time-frequency space equivalent to, and generalizing, the cubic nonlinear Schrödinger equation. We also obtain the Weyl-Wigner-Moyal equation satisfied by the Wigner-Ville function associated with the solu…
We present new asymptotic formulas for the Wigner $15j$-symbol with two, three, or four small quantum numbers, and provide numerical evidence of their validity. These formulas are of the WKB form and are of a similar nature as the Ponzano-Regge formula for the Wigner $6j$-symbol. They are expressed in terms of edge lengths and angles of geometrical figures associated with angular momentum vectors. In particular, the formulas for the $15j$-symbol with two, three, and four small quantum numbers are based on the geometric figures of the $9j$-, $6j$-, and $3j$-symbols, respectively, The geometric …
We consider 6 types of scaling limits for the Wigner-Moyal equation of the parabolic waves in random media, the limiting cases of which include the radiative transfer limit, the diffusion limit and the white-noise limit. We show under fairly general assumptions on the random refractive index field that sufficient amount of medium diversity (thus excluding the white-noise limit) leads to statistical stability or self-averaging in the sense that the limiting law is deterministic and is governed by various transport equations depending on the specific scaling involved. We obtain 6 different radia…
Exploring the concept of the extended Galilei group $\mathcal{G}$, a representation for the symplectic quantum mechanics in the manifold of $\mathcal{G}$, written in the light-cone of a five-dimensional De Sitter space-time, is derived consistently with the method of the Wigner function. A Hilbert space is constructed endowed with a simplectic structure, studying unitary operators describing rotations and translations, whose generators satisfy the Lie algebra of $\mathcal {G}$. This representation gives rise to the Schrödinger (Klein-Gordon-like) equation for the wave functions in phase-space,…
