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29 SIMPLIFYING EXTERNAL VARIABLES The compound symmetry group for the Dirac equation is the covering group of the Poincar´e group ISO(M). We represent this as the contraction of a simple group SO(3, 3) acting on the spinor pseudo-Hilbert (ket) space of 6N Clifford generators γ ω (n) (ω = 0, . . , 5; n = 1, . . , N) of the orthogonal group SO(3N, 3N). The size of the experiment fixes the parameter N. As in Dirac one-electron theory (where the spin generators are represented by second-degree elements h ¯ h ¯ (66) Sˆµν := [γ µ , γ ν ] ≡ γ µν , µ, ν = 0, .
Unfortunately in most cases, including this one, the method just shifts the problem of consistency of a geometry to the problem of consistency of the model itself. If the geometry is such that a finite model can be built whose incidence classes contain a finite number of elements, we may try to establish the consistency of the geometry by direct inspection of the model and determining whether its elements satisfy the original axioms. However for most of the axiomatic systems that are important in mathematics and used by physicists finite consistency bases cannot be constructed.
Thus, we start with an N = 2n-quasiparticle effective Hamiltonian whose only relevant to our problem energy level E2n is 2n−1 -fold degenerate. The degeneracy of the ground mode with no quasiparticles present is taken to be g(E0 ) = 1. Assuming that adding a pair of quasiparticles to the composite increases the total energy by ε, and ignoring all the external degrees of freedom, we can tabulate the resulting many-body energy spectrum as follows: Number of Quasiparticles, N = 2n 0 2 4 6 8 10 12 · · · Degeneracy, g(E2n ) = 2n−1 0 1 2 4 8 16 32 · · · Composite Energy, E2n 0ε 1ε 2ε 3ε 4ε 5ε 6ε · · · (170) Notice that the energy levels so defined furnish irreducible multiplets for projective representations of permutation groups in Schur’s theory , as was first pointed out by Wilczek 61 [57, 58].