Jm. Leinaas et J. Myrheim, HEISENBERG QUANTIZATION FOR SYSTEMS OF IDENTICAL PARTICLES, International journal of modern physics A, 8(21), 1993, pp. 3649-3695
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.