Studies of particles statistics in one and two dimensions, based on the quantization methods of Heisenberg, Schroedinger and Feynman

Jon Magne Leinaas, Jan Myrheim

We show that the algebraic quantization method of Heisenberg and the analytical method of Schrödinger are not necessarily equivalent when applied to systems of identical particles. Heisenberg quantization is a natural approach, but inherently more ambiguous and difficult than Schrödinger quantization. We apply the Heisenberg method to the examples of two identical particles in one and two dimensions, and relate the results to the so-called fractional statistics known from Schrödinger quantization. For two particles in d dimensions we look for linear, Hermitian representations of the symplectic Lie algebra sp(d, R). The boson and fermion representations are special cases, but there exist other representations. In one dimension there is a continuous interpolation between boson and fermion systems, different from the interpolation found in Schrödinger quantization. In two dimensions we find representations that can be realized in terms of multicomponent wave functions on a three-dimensional space, but we have no clear physical interpretation of these representations, which include extra degrees of freedom compared to the classical system.

Shoucheng Zhang

SERGEI B. ISAKOV

We show that Heisenberg quantization of a system of two identical particles in one dimension recently proposed by Leinaas and Myrheim leads to the family of 1D fractional statistics introduced by Polychronakos in terms of Schrödinger quantization.

S. Kamefuchi

A. B. Govorkov

Eno E. Ituen

With Feynman's path- integral method we can obtain the quantum mechanics of a quantum system like a free particle outside Schroedinger's method of differential equations and Heisenberg's method of algebra. The work involves obtaining the quantum propagator Kf, of the system which leads to summation over infinite number of paths. With Van Vleck's formula in one dimension, the classical propagator Kcl for a free particle is computed as the analytical result. This then serves as a yardstick for justifying the theoretical method used to compute the quantum propagator, kf, by direct path summation. The graphical display of the results shows that the Feynman – Schulman's checkerboard model used to enumerate the paths is reliable. Furthermore, this work shows that by windowing off a large number of paths and weighting the rest non uniformly we can compute the required propagator. The weights used in this case are the random and exponential window functions, wr and we respectively, which yield kwr and kwe to compare with kf. Key words: Propagator, action, path - integral, window function, free particle. (Global Journal of Pure and Applied Sciences: 2003 9(4): 561-566)