Selection Rules for Parafields and the Absence of Para Particles in Nature

Jacobson generators, Fock representations and statistics of sl(n+1)

“…, leads to a representation with an order of the (para)statistics p [22]. Then the energy of the vacuum is also zero.…”

Theories of violation of statistics

“…The case p = 1 corresponds to the usual bose or fermi statistics. Later, Messiah and I [11] proved that Green's ansatz gives all Fock-like solutions of Green's commutation rules. Local observables have a form analogous to the usual ones; for example, the local current for a spin-1/2 theory is This is all well and good; however, the violations of statistics provided by parastatistics are gross.…”

“…This condition imposes "consevation of statistics" rules, the simplest of which is that only an even number of fermi fields can appear in any term of the Hamiltonian. For parafields, which have local observables, Messiah and I [11] showed that parafields must occur in even degree, except that, for p odd, p parafields can occur. I defined paragrassmann and quongrassmann numbers which must be used in coupling para and quon operators to external sources and showed that for external parasources there are analogous restrictions [27].…”

Parasupersymmetry in quantum mechanics

“…For example, in two space dimensions, one can have a oneparameter family of statistics interpolating between Bose and Fermi statistics [232]. On the other hand, in three and higher space dimensions parafermi and parabose statistics [233,234,235,236] are the natural extensions of the usual Fermi and Bose statistics. In particular, whereas Fermi and Bose statistics describe the two one-dimensional representations of the permutation group, parafermi and parabose statistics describe the higher dimensional representations of the same group.…”