Generalized Ray Spaces for Paraparticles

Paraparticles of order p = 2 must be pair produced, so the least massive are absolutely stable. Consequently, paraparticles are excellent candidates to be associated with dark matter and/or dark energy. For a fixed number of paraparticles, in a "generalized ray space" there are simple orthonormal bracket "ray representatives" constructed from paraparticle creation operators. Each such ray representative is associated with a Young Diagram with a unique "P-bar-sum" eigenvalue. "P-bar-sum" is the sum of the particle-exchange operators which exchange two identical particles. In this paper, by using totally symmetric and totally antisymmetric brackets, generalized ray spaces for order p=2 are constructed for parabosons (pB), for parafermions (pB), and for possible para-Families (pFam). Para-Families occur in the case of non-commuting pB and pF creation operators. Explicit arbitrary order p ray representatives for a few paraparticles may be useful to establish the order p if, indeed, dark matter and/or dark energy is discovered to exhibit parastatistic properties. Up to p identical pB's (pF's) can occupy a totally antisymmetric (symmetric) state, unlike for normal boson (fermion) statistics.