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Lectures on Mechanics

In previous papers, a generalization of the Weyl calculus was introduced in connection with the quantization of a particle moving in R n under the influence of a variable magnetic field B. It incorporates phase factors defined by B and reproduces the usual Weyl calculus for B = 0. In the present article we develop the classical pseudodifferential theory of this formalism for the standard symbol classes S m .A mong others, we obtain properties and asymptotic developments for the magnetic symbol multiplication, existence of parametrices, boundedness and positivity results, properties of the magnetic Sobolev spaces. In the case when the vector potential A has all the derivatives of order ≥ 1 bounded, we show that the resolvent and the fractional powers of an elliptic magnetic pseudodifferential operator are also pseudodifferential. As an application, we get a limiting absorption principle and detailed spectral results for self-adjoint operators of the form H = h(Q, Π A ), where h is an elliptic symbol, Q denotes multiplication with the variables Π A = D −A, D is the operator of derivation and A is the vector potential corresponding to a short-range magnetic field.

Magnetic Pseudodifferential Operators

http://math.univ-lyon1.fr/~richard/download/Framework.pdf

Spectral and Propagation Results for Magnetic Schrodinger Operators; a C -Algebraic Framework

A gauge invariant quantization in a closed integral form is developed over a linear phase space endowed with an inhomogeneous Faraday electromagnetic tensor. An analog of the Groenewold product formula (corresponding to Weyl ordering) is obtained via a membrane magnetic area, and extended to the product of N symbols. The problem of ordering in quantization is related to different configurations of membranes: A choice of configuration determines a phase factor that fixes the ordering and controls a symplectic groupoid structure on the secondary phase space. A gauge invariant solution of the quantum evolution problem for a charged particle in an electromagnetic field is represented in an exact continual form and in the semiclassical approximation via the area of dynamical membranes.

Symplectic areas, quantization, and dynamics in electromagnetic fields

Based on the f-oscillator formalism, we introduce a nonlinear Jaynes–Cummings model (NJCM) which is constructed from the standard JCM by deforming the single-mode field operators. Such a generalization of the JCM describes the interaction of a two-level atom with a single mode of the electromagnetic field in the presence of a nonlinear Kerr-like medium. Since the medium is modelled as an f-oscillator, it is possible to consider the field f-coherent states (nonlinear coherent states) and their evolution.

https://iopscience.iop.org/article/10.1088/0953-4075/45/1/015502/pdf

The f-deformed Jaynes?Cummings model and its nonlinear coherent states

The symplectic geometry of the phase space associated with a charged particle is determined by the addition of the Faraday 2-form to the standard dp ∧ dq structure on  In this paper, we describe the corresponding algebra of Weyl-symmetrized functions in operators satisfying nonlinear commutation relations The multiplication in this algebra generates an associative product of functions on the phase space This product is given by an integral kernel whose phase is the symplectic area of a groupoid-consistent membrane A symplectic phase space connection with non-trivial curvature is extracted from the magnetic reflections associated with the Stratonovich quantizer Zero and constant curvature cases are considered as examples The quantization with both static and time-dependent electromagnetic fields is obtained The expansion of the product by the deformation parameter , written in the covariant form, is compared with the known deformation quantization formulae

https://iopscience.iop.org/article/10.1088/0305-4470/37/6/025/pdf

Quantum magnetic algebra and magnetic curvature

Semiclassical Moyal quantum mechanics for atomic systems

For manifolds of noncompact type endowed with an affine connection (for example, the Levi–Civita connection) and a closed 2-form (magnetic field), we define a Hilbert algebra structure in the space and construct an irreducible representation of this algebra in . This algebra is automatically extended to polynomial in momenta functions and distributions. Under some natural conditions, this algebra is unique. The non-commutative product over is given by an explicit integral formula. This product is exact (not formal) and is expressed in invariant geometrical terms. Our analysis reveals that this product has a front, which is described in terms of geodesic triangles in . The quantization of δ-functions induces a family of symplectic reflections in and generates a magneto-geodesic connection Γ on . This symplectic connection entangles, on the phase space level, the original affine structure on and the magnetic field. In the classical approximation, the 2-part of the quantum product contains the Ricci curvature of Γ and a magneto-geodesic coupling tensor.

Cotangent bundle quantization : entangling of metric and magnetic field

A new and computationally viable full quantum version of line shape theory is obtained in terms of a mixed Weyl symbol calculus. The basic ingredient in the collision-broadened line shape theory is the time-dependent dipole autocorrelation function of the radiator-perturber system. The observed spectral intensity is the Fourier transform of this correlation function. A modified form of the Wigner-Weyl isomorphism between quantum operators and phase space functions (Weyl symbols) is introduced in order to describe the quantum structure of this system. This modification uses a partial Wigner transform in which the radiator-perturber relative motion degrees of freedom are transformed into a phase space dependence, while operators associated with the internal molecular degrees of freedom are kept in their original Hilbert space form. The result of this partial Wigner transform is called a mixed Weyl symbol. The star product, Moyal bracket and asymptotic expansions native to the mixed Weyl symbol calculus are determined. The correlation function is represented as the phase space integral of the product of two mixed symbols: one corresponding to the initial configuration of the system, the other being its time evolving dynamical value. There are, in this approach, two semiclassical expansions - one associated with the perturber scattering process, the other with the mixed symbol star product. These approximations are used in combination to obtain representations of the autocorrelation that are sufficiently simple to allow numerical calculation. The leading O(0) approximation recovers the standard classical path approximation for line shapes. The higher-order O(1) corrections arise from the noncommutative nature of the star product.

T. A. Osborn

Papers

Symplectic areas, quantization, and dynamics in electromagnetic fields

Quantum magnetic algebra and magnetic curvature

Semiclassical Moyal quantum mechanics for atomic systems

Cotangent bundle quantization : entangling of metric and magnetic field

Mixed Weyl symbol calculus and spectral line shape theory