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arXiv:quant-ph/0603082 · Foundations of Physics

Remark on a Group-Theoretical Formalism for Quantum Mechanics and the Quantum-to-Classical Transition

J. K. Korbicz, Maciej Lewenstein

Published May 25, 2007• 3 citations
arXivHTMLPDF
DOI
10.1007/s10701-007-9130-z

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Let N be the space of Gaussian distribution functions over ℝ, regarded as a 2-dimensional statistical manifold parameterized by the mean μ and the deviation σ. In this paper, we show that the tangent bundle of N, endowed with its natural Kähler structure, is the Siegel-Jacobi space appearing in the context of Number Theory and Jacobi forms. Geometrical aspects of the Siegel-Jacobi space are discussed in detail (completeness, curvature, group of holomorphic isometries, space of Kähler functions, and relationship to the Jacobi group), and are related to the quantum formalism in its geometrical form, i.e., based on the Kähler structure of the complex projective space. This paper is a continuation of our previous work [M. Molitor, “Remarks on the statistical origin of the geometrical formulation of quantum mechanics,” Int. J. Geom. Methods Mod. Phys. 9(3), 1220001, 9 (2012); M. Molitor, “Information geometry and the hydrodynamical formulation of quantum mechanics,” e-print arXiv (2012); M. Molitor, “Exponential families, Kähler geometry and quantum mechanics,” J. Geom. Phys. 70, 54–80 (2013)], where we studied the quantum formalism from a geometric and information-theoretical point of view.
From a deterministic point of view of quantum theory the Schrodinger equation, considered as an equation of motion, is defined within an area of instability formed by two potential minima.This view can be related to the renormalization group theory of critical phenomena, and the location of two nontrivial fixed points can be defined with respect to those two potential minima.The renormalization group theory parameter: d, the dimensionality of space, defines the transition between classical and quantum theory.For a system in which d 2 4, classical mechanics applies, and for d < 4, quantum mechanics applies.Whereas the classical potential defining an equation of motion has one potential minimum (at the Gaussian fixed point), the quantum potential defining an equation of motion has two potential minima (at the nontrivial fixed points).An analogy exists between systems based on wave mechanics and those based on lattice systems.The underlying dynamics of that potential, the minimization of which results in an equation formally equivalent to the Schrodinger equation, can, in this formal analogy, be identified with the underlying dynamics of the exchange energy at the vertex of a lattice.There is thus a mathematical global similarity underlying the behavior of these two distinct theoretical frameworks which describes both.The principal and angular momentum quantum numbers of electron systems in this analogy are equivalent to the dimensionality of space, d, and the dimensionality of spin space, n, in the general theory of critical phenomena.The general theory describes the phase transitions of electron (wave mechanical) systems, and also a "Mendeleev's table" based on the d and n parameters constructed for those electron systems, in analogy with such a table for lattice systems.