June 23, 1999
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December 31, 2020
Recently, it was shown that when reference frames are associated to quantum systems, the transformation laws between such quantum reference frames need to be modified to take into account the quantum and dynamical features of the reference frames. This led to a relational description of the phase space variables of the quantum system of which the quantum reference frames are part of. While such transformations were shown to be symmetries of the system's Hamiltonian, the quest...
June 23, 2003
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 pseudo-differential 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 $W_{\infty}\times W_{\infty}$- algebra which contracts to its diagonal subalgebra in the cla...
September 1, 2010
The paper develop the alternative formulation of quantum mechanics known as the phase space quantum mechanics or deformation quantization. It is shown that the quantization naturally arises as an appropriate deformation of the classical Hamiltonian mechanics. More precisely, the deformation of the point-wise product of observables to an appropriate noncommutative $\star$-product and the deformation of the Poisson bracket to an appropriate Lie bracket is the key element in int...
May 3, 1999
We observe that, within the effective generating function formalism for the implementation of canonical transformations within wave mechanics, non-trivial canonical transformations which leave invariant the form of the Hamilton function of the classical analogue of a quantum system manifest themselves in an integral equation for its stationary state eigenfunctions. We restrict ourselves to that subclass of these dynamical symmetries for which the corresponding effective gener...
April 14, 2005
Complex numbers appear in the Hilbert space formulation of quantum mechanics, but not in the formulation in phase space. Quantum symmetries are described by complex, unitary or antiunitary operators defining ray representations in Hilbert space, whereas in phase space they are described by real, true representations. Equivalence of the formulations requires that the former representations can be obtained from the latter and vice versa. Examples are given. Equivalence of the t...
June 27, 2006
The basic ideas in the theory of quantum mechanics on phase space are illustrated through an introduction of generalities, which seem to underlie most if not all such formulations and follow with examples taken primarily from kinematical particle model descriptions exhibiting either Galileian or Lorentzian symmetry. The structures of fundamental importance are the relevant (Lie) groups of symmetries and their homogeneous (and associated) spaces that, in the situations of inte...
September 9, 2012
We present a phase space formulation of quantum mechanics in the Schr\"odinger representation and derive the associated Weyl pseudo-differential calculus. We prove that the resulting theory is unitarily equivalent to the standard "configuration space" formulation and show that it allows for a uniform treatment of both pure and mixed quantum states. In the second part of the paper we determine the unitary transformation (and its infinitesimal generator) that maps the phase spa...
May 27, 2013
We propose a modification of a recently introduced generalized translation operator, by including a $q$-exponential factor, which implies in the definition of a Hermitian deformed linear momentum operator $\hat{p}_q$, and its canonically conjugate deformed position operator $\hat{x}_q$. A canonical transformation leads the Hamiltonian of a position-dependent mass particle to another Hamiltonian of a particle with constant mass in a conservative force field of a deformed phase...
March 30, 1999
In this lecture, a limited introduction of gauge invariance in phase-space is provided, predicated on canonical transformations in quantum phase-space. Exact characteristic trajectories are also specified for the time-propagating Wigner phase-space distribution function: they are especially simple - indeed, classical - for the quantized simple harmonic oscillator. This serves as the underpinning of the field theoretic Wigner functional formulation introduced. Scalar field the...
June 18, 2010
Unitary representations of the Galilei group are studied in phase space, in order to describe classical and quantum systems. Conditions to write in general form the generator of time translation and Lagrangians in phase space are then established. In the classical case, Galilean invariance provides conditions for writing the Liouville operator and Lagrangian for non-linear systems. We analyze, as an example, a generalized kinetic equation where the collision term is local and...