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Register a new user2-D Covariant Affine Integral Quantization(s)
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Covariant affine integral quantization is studied and applied to the motion of a particle in a punctured plane Pp, for which the phase space is Pp X plane. We examine the consequences of different quantizer operators built from weight functions on this phase space. To illustrate the procedure, we examine two examples of weights. The first one corresponds to 2-D coherent state families, while the second one corresponds to the affine inversion in the punctured plane. The later yields the usual canonical quantization and a quasi-probability distribution (2-D affine Wigner function) which is real, marginal in both position and momentum.
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Covariant affine integral quantization of the half-plane is studied and applied to the motion of a particle on the half-line. We examine the consequences of different quantizer operators built from weight functions on the half-plane. To illustrate the procedure, we examine two particular choices of the weight function, yielding thermal density operators and affine inversion respectively. The former gives rise to a temperature-dependent probability distribution on the half-plane whereas the later yields the usual canonical quantization and a quasi-probability distribution (affine Wigner function) which is real, marginal in both momentum p and position q.
A covariant phase space observable is uniquely characterized by a positive operator of trace one and, in turn, by the Fourier-Weyl transform of this operator. We study three properties of such observables, and characterize them in terms of the zero set of this transform. The first is informational completeness, for which it is necessary and sufficient that the zero set has dense complement. The second is a version of informational completeness for the Hilbert-Schmidt class, equivalent to the zero set being of measure zero, and the third, known as regularity, is equivalent to the zero set being empty. We give examples demonstrating that all three conditions are distinct. The three conditions are the special cases for $p=1,2,infty$ of a more general notion of $p$-regularity defined as the norm density of the span of translates of the operator in the Schatten-$p$ class. We show that the relation between zero sets and $p$-regularity can be mapped completely to the corresponding relation for functions in classical harmonic analysis.
Using the method of canonical group quantization, we construct the angular momentum operators associated to configuration spaces with the topology of (i) a sphere and (ii) a projective plane. In the first case, the obtained angular momentum operators are the quantum version of Poincares vector, i.e., the physically correct angular momentum operators for an electron coupled to the field of a magnetic monopole. In the second case, the obtained operators represent the angular momentum operators of a system of two indistinguishable spin zero quantum particles in three spatial dimensions. We explicitly show how our formalism relates to the one developed by Berry and Robbins. The relevance of the proposed formalism for an advance in our understanding of the spin-statistics connection in non-relativistic quantum mechanics is discussed.
The classical quantization of a Lienard-type nonlinear oscillator is achieved by a quantization scheme (M.C. Nucci. Theor. Math. Phys., 168:997--1004, 2011) that preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schrodinger equation. This method straightforwardly yields the correct Schrodinger equation in the momentum space (V. Chithiika Ruby, M. Senthilvelan, and M. Lakshmanan. J. Phys. A: Math. Gen., 45:382002, 2012), and sheds light into the apparently remarkable connection with the linear harmonic oscillator.
The classical quantization of the motion of a free particle and that of an harmonic oscillator on a double cone are achieved by a quantization scheme [M.C. Nucci, Theor. Math. Phys. 168 (2011) 994], that preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schroedinger equation. The result is different from that given in [K. Kowalski, J.Rembielnski, Ann. Phys. 329 (2013) 146]. A comparison of the different outcomes is provided.
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