The Weyl Transform and L p Functions on Phase Space

1992 ◽  
Vol 116 (4) ◽  
pp. 1045 ◽  
Author(s):  
Barry Simon
Keyword(s):  
Phase Space ◽  
Weyl Transform

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 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.

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 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.

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

Phase space of mechanical systems with a gauge group

1991 ◽  
Vol 161 (2) ◽  
pp. 13-75 ◽  
Author(s):  
Lev V. Prokhorov ◽  
Sergei V. Shabanov
Keyword(s):  
Phase Space ◽  
Gauge Group ◽  
Mechanical Systems
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