scholarly journals Variations à la Fourier-Weyl-Wigner on Quantizations of the Plane and the Half-Plane

2018 ◽  
Author(s):  
Hervé Bergeron ◽  
Jean-Pierre Gazeau
Keyword(s):  
Phase Space ◽  
Large Family ◽  
Integral Calculus ◽  
Integral Means ◽  
Weyl Transform ◽  
Quantization Maps ◽  
Affine Groups ◽  
Singular Functions ◽  
Unitary Transformations ◽  
Symmetric Operators

Any quantization maps linearly functions on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary transformations) symmetries. Integral means that we use all ressources of integral calculus, in order to implement the method when we apply it to singular functions, or distributions, for which the integral calculus is an essential ingredient. In this paper we emphasize the deep connection between Fourier transform and covariant integral quantization when the Weyl-Heisenberg and affine groups are involved. We show with our generalisations of the Wigner-Weyl transform that many properties of the Weyl integral quantization, commonly viewed as optimal, are actually shared by a large family of integral quantizations.

scholarly journals Variations à la Fourier-Weyl-Wigner on Quantizations of the Plane and the Half-Plane

Entropy ◽  
2018 ◽  
Vol 20 (10) ◽  
pp. 787 ◽  
Author(s):  
Hervé Bergeron ◽  
Jean-Pierre Gazeau
Keyword(s):  
Phase Space ◽  
Large Family ◽  
Quantization Scheme ◽  
Integral Calculus ◽  
Integral Means ◽  
Weyl Transform ◽  
Quantization Maps ◽  
Singular Functions ◽  
Unitary Transformations ◽  
Symmetric Operators

Any quantization maps linearly function on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with the symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary transformations) symmetries. Integral means that we use all resources of integral calculus, in order to implement the method when we apply it to singular functions, or distributions, for which the integral calculus is an essential ingredient. We first review this quantization scheme before revisiting the cases where symmetry covariance is described by the Weyl-Heisenberg group and the affine group respectively, and we emphasize the fundamental role played by Fourier transform in both cases. As an original outcome of our generalisations of the Wigner-Weyl transform, we show that many properties of the Weyl integral quantization, commonly viewed as optimal, are actually shared by a large family of integral quantizations.

scholarly journals Phase space formulation of density operator for non-Hermitian Hamiltonians and its application in quantum theory of decay

2018 ◽  
Vol 32 (25) ◽  
pp. 1850276 ◽  
Author(s):  
Ludmila Praxmeyer ◽  
Konstantin G. Zloshchastiev
Keyword(s):  
Phase Space ◽  
Density Operator ◽  
Quantum Systems ◽  
Matrix Approach ◽  
Weyl Transform ◽  
Energy Eigenvalues ◽  
Decay Constants ◽  
Dissipative Effects ◽  
Mean Lifetime ◽  
Space Formulation

The Wigner–Weyl transform and phase space formulation of a density matrix approach are applied to a non-Hermitian model which is quadratic in positions and momenta. We show that in the presence of a quantum environment or reservoir, mean lifetime and decay constants of quantum systems do not necessarily take arbitrary values, but could become functions of energy eigenvalues and have a discrete spectrum. It is demonstrated also that a constraint upon mean lifetime and energy appears, which is used to derive the resonance conditions at which long-lived states occur. The latter indicate that quantum dissipative effects do not always lead to decay but, under certain conditions, can support stability of a system.

scholarly journals MINISUPERSPACES: OBSERVABLES AND QUANTIZATION

1993 ◽  
Vol 02 (01) ◽  
pp. 15-50 ◽  
Author(s):  
ABHAY ASHTEKAR ◽  
RANJEET TATE ◽  
CLAES UGGLA
Keyword(s):  
Quantum Theory ◽  
Phase Space ◽  
Canonical Transformation ◽  
Parameter Family ◽  
Hamiltonian Constraint ◽  
Canonical Coordinates ◽  
Quantum Theories ◽  
Canonical Variables ◽  
Unitary Transformations ◽  
Homogeneous Cosmologies

A canonical transformation is performed on the phase space of a number of homogeneous cosmologies to simplify the form of the scalar (or Hamiltonian) constraint. Using the new canonical coordinates, it is then easy to obtain explicit expressions of Dirac observables, i.e. phase-space functions which commute weakly with the constraint. This, in turn, enables us to carry out a general quantization program to completion. We are also able to address the issue of time through “deparametrization” and discuss physical questions such as the fate of initial singularities in the quantum theory. We find that they persist in the quantum theory in spite of the fact that the evolution is implemented by a one-parameter family of unitary transformations. Finally, certain of these models admit conditional symmetries which are explicit already prior to the canonical transformation. These can be used to pass to the quantum theory following an independent avenue. The two quantum theories — based, respectively, on Dirac observables in the new canonical variables and conditional symmetries in the original ADM variables — are compared and shown to be equivalent.

scholarly journals QUANTUM-DEFORMED CANONICAL TRANSFORMATIONS, W∞ ALGEBRAS AND UNITARY TRANSFORMATIONS

1994 ◽  
Vol 09 (32) ◽  
pp. 5801-5820 ◽  
Author(s):  
E. GOZZI ◽  
M. REUTER
Keyword(s):  
Phase Space ◽  
Pseudodifferential Operators ◽  
Star Product ◽  
Canonical Transformations ◽  
Symbol Calculus ◽  
Algebra Of Functions ◽  
Unitary Transformations ◽  
Symbols Of Operators ◽  
Left And Right ◽  
Gns Construction

We investigate the algebraic properties of the quantum counterpart of the classical canonical transformations using the symbol calculus approach to quantum mechanics. In this framework we construct a set of pseudodifferential operators which act on the symbols of operators, i.e. on functions defined over phase space. They act as operatorial left and right multiplication and form a W∞×W∞ algebra which contracts to its diagonal subalgebra in the classical limit. We also describe the Gel’fand-Naimark-Segal (GNS) construction in this language and show that the GNS representation space (a doubled Hilbert space) is closely related to the algebra of functions over phase space equipped with the star product of the symbol calculus.

Symbol Correspondence for Euclidean Systems

2021 ◽  
Vol 62 ◽  
pp. 67-84
Author(s):  
Laarni B. Natividad ◽  
◽  
Job A. Nable
Keyword(s):  
Phase Space ◽  
Wigner Distribution ◽  
Star Product ◽  
Wigner Distribution Function ◽  
Symbol Calculus ◽  
Euclidean Motion Group ◽  
Weyl Transform ◽  
C Algebra ◽  
Foundation Of Quantum Mechanics ◽  
Euclidean Motion

The three main objects that serve as the foundation of quantum mechanics on phase space are the Weyl transform, the Wigner distribution function, and the $\star$-product of phase space functions. In this article, the $\star$-product of functions on the Euclidean motion group of rank three, $\mathrm{E}(3)$, is constructed. $C^*$-algebra properties of $\star_s$ on $\mathrm{E}(3)$ are presented, establishing a phase space symbol calculus for functions whose parameters are translations and rotations. The key ingredients in the construction are the unitary irreducible representations of the group.

scholarly journals Vectorlike deformations of relativistic quantum phase-space and relativistic kinematics

2017 ◽  
Vol 26 (11) ◽  
pp. 1750123 ◽  
Author(s):  
Niccoló Loret ◽  
Stjepan Meljanac ◽  
Flavio Mercati ◽  
Danijel Pikutić
Keyword(s):  
Phase Space ◽  
Hopf Algebra ◽  
Relativistic Quantum ◽  
Star Product ◽  
Large Family ◽  
Momentum Dependence ◽  
Large Set ◽  
Lorentz Transformations ◽  
Relativistic Kinematics ◽  
First Order

We study a family of noncommutative spacetimes constructed by one four-vector. The large set of coordinate commutation relations described in this way includes many cases that are widely studied in the literature. The Hopf-algebra symmetries of these noncommutative spacetimes, as well as the structures of star product and twist are introduced and considered at first order in the deformation, described by four parameters. We also study the deformations to relativistic kinematics implied by this framework, and calculate the most general expression for the momentum dependence of the Lorentz transformations on momenta, which is an effect that is required by consistency. At the end of the paper we analyse the phenomenological consequences of this large family of vectorlike deformations on particles propagation in spacetime. This leads to a set of characteristic phenomenological effects.

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