Cliffordons

TL;DR

This paper explores a real quantum theory framework using Clifford statistics, proposing it as a higher-energy extension of complex quantum theory, with implications for understanding spin and the emergence of the imaginary unit.

Contribution

It introduces Clifford statistics as a novel multivalued, dimension-independent quantum statistics and applies it to toy models demonstrating higher-energy behaviors.

Findings

Clifford statistics are multivalued and applicable in any dimension.

A real quantum model can naturally produce an effective imaginary unit.

Supports the idea that spin and complex structures emerge at lower energies from higher-energy real theories.

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 $\gamma_1-\gamma_2$ 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 proposed a 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.