Schizosymmetry: A New Paradigm for Superfield Expansions

Abstract

A new principle of ‘schizosymmetry’ is proposed for interpreting superfield expansions over Grassmann parameters in quantum field theory. Symmetry generators T phys , determining component field transformation properties, are allowed to depend on the grading via T phys =Tℙ e +T′ℙ o , where ℙ e and ℙ o are the projections onto even and odd components in the Grassmann coordinates, respectively, and T and T′ correspond to different representations. Examples are given for both internal and space-time schizosymmetry. In the latter case, superfield expansions have components with both integer and half-integer spin. A program for the Lagrangian realization of schizosymmetry, including unified model building, is outlined.

The Relativity of Space–time–property

“…To carry out this programme I need to introduce some basic notational niceties first. 3 With the extended coordinate X M , let M = m (Roman) correspond to spacetime x and let M = µ (Greek) correspond to property ζ,ζ. Set [m]=0 (no grading) and [µ]=1 (grading).…”

The Relativity of Space–time-Property

“…From these extended coordinates, superfields (functions of space-time and property) may be constructed. [4][5][6] Since the product of two a-nos. is a (nilpotent) c-no., a Bose superfield Φ should be a Taylor series in even powers of ζ,ζ and a Fermi superfield Ψ α a series in odd powers of ζ,ζ -up to the fifth:…”

Hilbert space for quantum mechanics on superspace

“…Schrödinger equations in superspace were considered first as a method to incorporate spin, see e.g. [13,14]. The quantum (an-)harmonic oscillator ( [12,14,17]), the Kepler problem ( [27]), the delta potential ( [9]) and the CM S-model ( [16]) have already been studied in superspace.…”