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Additional info for Astrophysics and general relativity, - (lectures by Layzer, Lin & Shu only)
The minimum number of elements in a cell for our simplification is six: four for space-time and two for a complex or symplectic plane. 11 N hexads define a unit mode space V1 = N60 . There are two possibilities here as to how to proceed. We can either use the Hexad Lemma described in Appendix E and work with hexadic cells whose variables commute with one another, or we can work directly with 6N anticommuting Clifford units of Cliff(3N, 3N). In any case the external variables of the quantum probe related to different hexads as defined below will commute.
Using this statistics I proposed a simple model for the grand canonical ensemble of the carriers in the theory of fractional quantum Hall effect. The model led to a temperature limit associated with the permutational degrees of freedom of such an ensemble. As was pointed out before, building on the work on nonabelions of Read and Moore [36, 43], Nayak and Wilczek [39, 57, 58] proposed a spinorial statistics for the fractional quantum Hall effect (FQHE) carriers. The prototypical example was furnished by a so-called Pfaffian mode (occuring at filling fraction ν = 1/2), in which 2n quasiholes form an 2n−1 -dimensional irreducible multiplet of the corresponding braid group.
Maxwell-Boltzmann statistics then reduces to Fermi, Bose, and all the para- statistics. This seems adequate for much of physics. The construction of this isomorphism resembles the well-known Jordan-Wigner construction of higher-dimensional spin representations. We repeat it in the present context for convenient reference. For definiteness we exhibit the construction for a neutral hexad 60 . We label the 6N generators iω (n) of Cliff(N60 ), Cliff(N60 ) = Cliff[i5 (N), . . , i0 (N), . . , i5 (1), .