Universal Journal of Physics and Application Vol. 1(1-2), pp. 11 - 50
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ALTERNATIVE HAMILTONIANS OF CLASSICAL MECHANICS AND NONCANONICAL QUANTIZATION

AUTHOR(S) INFORMATION: Horzela A.

ABSTRACT

Noncanonical quantization schemes have been proposed in quantum physics long time ago. Recalling some of them I associate these investigations to the quantum mechanical implications of the canonically inequivalent Hamilton formulations of classical Newtonian dynamics. As the simplest illustration of such alternative Hamiltonian descriptions I consider one dimensional system of two particles interacting harmonically. Description of the motion of their center of mass is standard and guarantees the Galilean covariance but the internal motion is described as a noncanonical oscillator for which the Hamiltonian and the total energy are not related by a simple proportionality. Alternative Hamiltonians lead to alternative Poisson structures which may be considered as classical analogues of the noncanonical quantum commutation rules. Generated alternative quantizations are consistent with the same Heisenberg equations of motion, i.e., they satisfy the Wigner principle of quantization. For the one dimensional harmonic oscillator the simplest noncanonical algebraic description is given in terms of the Lie algebra so(p, q), p + q = 3. In such a case the Hamiltonian is still a linear function while the energy is a quadratic polynomial of the main quantum number. This leads to nonstandard termodynamical properties obeyed by such a noncanonical oscillator. The 3-dimensional harmonic oscillator also has a noncanonical description given in terms of the simple Lie algebra, so(p, q), p + q = 5 in the case under consideration. The most important physical consequence of such a description is the noncommutativity of the coordinate operators, interpreted to be an implication of the noncommutative geometry of the underlying spacetime.