HEISENBERG QUANTIZATION FOR SYSTEMS OF IDENTICAL PARTICLES

We show that the algebraic quantization method of Heisenberg and the a nalytical method of Schrodinger are not necessarily equivalent when ap plied to systems of identical particles. Heisenberg quantization is a natural approach, but inherently more ambiguous and difficult than Sch rodinger quantization. We apply the Heisenberg method to the examples of two identical particles in one and two dimensions, and relate the r esults to the so-called fractional statistics known from Schrodinger q uantization. For two particles in d dimensions we look for linear, Her mitian representations of the symplectic Lie algebra sp(d, R). The bos on and fermion representations are special cases, but there exist othe r representations. In one dimension there is a continuous interpolatio n between boson and fermion systems, different from the interpolation found in Schrodinger quantization. In two dimensions we find represent ations that can be realized in terms of multicomponent wave functions on a three-dimensional space, but we have no clear physical interpreta tion of these representations, which include extra degrees of freedom compared to the classical system.