Physical Framework of Quantization Problem

The paper presents shortly the geometric approach to the problem of a general quantization formalism, both physically meaningful and mathematically consistent. The notion of quantization has appeared at the beginning of the century in the theory of heat radiation, since M. Planck has formulated the hypothesis of the energy quanta [1]. This hypothesis assumed a finite number of ways to distribute the energy of the ν-frequency radiation over a given number of oscillators, every distribution assigning to each oscillator integer multiples of the " energy quantum " ǫ = hν, h = 6.626 × 10 −34 J·s. Later on, the Planck's hypothesis was interpreted as a consequence of a " quantum constraint " acting in general on the harmonic oscillator considered as classical system. Thus, the problem of quantization was to " select an infinite , discrete number of quantum possible real motions, from the continuous manifold of all mechanically possible motions " [2], by an appropriate system of constraints. The action of these constraints on the classical motion as a whole, and not on the physical state at a given time, has suggested to express them as restrictions on the geometrical quantities associated with the trajec-tory in the space of the classical states. These states were identified with the points of the phase space, locally parameterized by generalized coordinates and momenta. The dimensional equality between the constant h and the classical action has allowed to select the quantum trajectories for the separable systems with multiple periodic motions by using the conditions of Bohr, Wilson and Sommerfeld (BWS), requiring the phase integrals J k = p k dq k , k = 1, 2, ...n, (n is the number of freedom degrees) to be integer multiples of 1 first printed as preprint 102-(1990) by the University of Timisoara, Romania.