The Transition Point Between Classical and Quantum Mechanical Systems Deterministically Defined

Abstract

From a deterministic point of view of quantum theory the Schrödinger 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 ≧ 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 Schrödinger 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.