TITLE:
About Classical to Quantum Weyl Correspondence
AUTHORS:
Alfred Wünsche
KEYWORDS:
Wigner Quasiprobability, Symmetrical (Weyl) Ordering, Nonclassicality of Steates, Distance of States, Sub- and Super-Poissonian Statistics, Phase Operator, Laguerre 2D Polynomials, Jacobi Polynomials
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.7 No.10,
October
30,
2017
ABSTRACT: After developing the mathematical means for the correspondence of classical phase-space function to quantum-mechanical operators with symmetrical ordering of the basic canonical operators in the sense of Weyl the approach is applied to an infinite series of classical monomial functions of the canonical variables. These include as well as pure powers of the amplitude as also basic periodic functions of the phase φ with their quantum-mechanical correspondence. In the representation by number states, all the considered operators involve the Jacobi polynomials as the essential formative element. Whereas the quantity in normal ordering due to its indeterminacy leads to the introduction of the notions of sub- and super-Poissonian statistics the analogous quantity in (Weyl) symmetrical orderingis positive definite and satisfies an inequality. The notions of sub- and super-Poissonian statistics are problematic when they are used for the definition of nonclassicality of states since the mentioned measure in normal ordering does not determine the Poisson statistics in their middle in unique way but determines only a large set of statistics which may be very far in the sense of the Hilbert-Schmidt distance from a Poisson statistics that is discussed.