Related papers: Canonical Transformations in Quantum Mechanics
The Fourier transform, known in classical analysis, and generalized in abstract harmonic analysis, can also be considered in the theory of locally compact quantum groups. In this note, I discuss some aspects of this more general Fourier…
We address the issue of the relation between the canonical functional Schr\"odinger representation in quantum field theory and the approach of precanonical field quantization proposed by the author, which requires neither a distinguished…
Recently, a new choice of variables was identified to understand how the quantum group structure appeared in three-dimensional gravity [1]. These variables are introduced via a canonical transformation generated by a boundary term. We show…
Classical mechanics, in the Koopman-von Neumann formulation, is described in Hilbert space. It is shown here that classical canonical transformations are generated by Hermitian operators that are in general noncommutative. This naturally…
The standard C*-algebraic version of the algebra of canonical commutation relations, the Weyl algebra, frequently causes difficulties in applications since it neither admits the formulation of physically interesting dynamical laws nor does…
Cirelli, Mani\`{a} and Pizzocchero generalized quantum mechanics by K\"{a}hler geometry. Furthermore they proved that any unital C$^{*}$-algebra is represented as a function algebra on the set of pure states with a noncommutative…
A general classical theorem is presented according to which all invariant relations among the space time metric scalars, when turned into functions on the Phase Space of full Pure Gravity (using the Canonical Equations of motion), become…
Canonical quantization of gravity in general relativity is greatly simplified by the artificial decomposition of space and time into a 3+1 formalism. Such a simplification may appear to come at the cost of general covariance. This requires…
Quantum computations operate in the quantum world. For their results to be useful in any way, there is an intrinsic necessity of cooperation and communication controlled by the classical world. As a consequence, full formal descriptions of…
We consider scalar-tensor theories of gravity defined in Weyl integrable space-time and show that in the ADM formalism Weyl transformations corresponding to change of frames induce canonical transformations between different representations…
We are taught that gauge transformations in classical and quantum mechanics do not change the physics of the problem. Nevertheless here we discuss three broad scenarios where under gauge transformations: (i) conservation laws are not…
We construct a canonical basis for quantum generalized Kac-Moody algebra via semisimple perverse sheaves on varieties of representations of quivers. We compare this basis with the one recently defined purely algebraically by Jeong, Kang and…
While real Hamiltonian mechanics and Hermitian quantum mechanics can both be cast in the framework of complex canonical equations, their complex generalisations have hitherto been remained tangential. In this paper quaternionic and…
Fourier transforms are ubiquitous mathematical tools in basic and applied sciences. We here report classical and quantum optical realizations of the discrete fractional Fourier transform, a generalization of the Fourier transform. In the…
We address the phase space formulation of a noncommutative extension of quantum mechanics in arbitrary dimension, displaying both spatial and momentum noncommutativity. By resorting to a covariant generalization of the Weyl-Wigner transform…
By means of a canonical transformation it is shown how it is possible to recast the equations for molecular nonlinear optics to completely eliminate ground-state static dipole coupling terms. Such dipoles can certainly play a highly…
The act of describing how a physical process changes a system is the basis for understanding observed phenomena. For quantum-mechanical processes in particular, the affect of processes on quantum states profoundly advances our knowledge of…
The mathematical formulation of Quantum Mechanics is derived from purely operational axioms based on a general definition of "experiment" as a set of transformations. The main ingredient of the mathematical construction is the postulated…
Some precisions are given about the definition of the Hamiltonian operator H and its transformation properties, for a linear wave equation in a general spacetime. In the presence of time-dependent unitary gauge transformations, H as an…
Ambiguities arising in different approaches (canonical, quasiclassical, path integration) to quantization are discussed by an example of the mechanics of a point-like particle in the Riemannian space (the geodesic dynamics). A way to select…