Cliffordons

Abstract

At higher energies the present complex quantum theory with its unitary group might expand into a real quantum theory with an orthogonal group, broken by an approximate i operator at lower energies. Implementing this possibility requires a real quantum double-valued statistics. A Clifford statistics, representing a swap (12) by a difference γ1−γ2 of Clifford units, is uniquely appropriate. Unlike the Maxwell–Boltzmann, Fermi–Dirac, Bose–Einstein, and para-statistics, which are tensorial and single-valued, and unlike anyons, which are confined to two dimensions, Clifford statistics are multivalued and work for any dimensionality. Nayak and Wilczek such Clifford statistics for the fractional quantum Hall effect. We apply them to toy quanta here. A complex-Clifford example has the energy spectrum of a system of spin-1/2 particles in an external magnetic field. This supports the proposal that the double-valued rotations—spin—seen at current energies might arise from double-valued permutations—swap—to be seen at higher energies. Another toy with real Clifford statistics illustrates how an effective imaginary unit i can arise naturally within a real quantum theory.