Factorial Unitary Representations of the Translational Group and the Supersymmetric Graviton

TL;DR

This paper explores unitary representations of the translational group in quantum gravity, revealing a natural emergence of supersymmetry that predicts a massless graviton and gravitino partner.

Contribution

It introduces a new group-theoretic approach to unitary representations of the translation subgroup, leading to a minimal form of supersymmetry in quantum gravity.

Findings

Unitary representations are shown to be norm continuous and implementable by a unitary operator.

A minimal form of supersymmetry naturally emerges from these representations.

The approach predicts the existence of a massless graviton and its gravitino partner.

Abstract

Unitary Representations corresponding to local shifts in reference frames are a key topic for progress in Quantum Gravity. The fibre bundle construct defined in our previous work in which quantum fields become liftings of; or sections through; a fibre bundle with base space curved space-time, continues to be the context for this paper. We investigate in more depth the subgroup T of the Poincare group consisting of translations of space-time as a gauge group of automorphisms. We define to be a representation of T as such automorphisms of the local fibre algebra A(x) which we assume to be isomorphic to a von Neumann algebra with trivial centre acting on a separable Hilbert space. Provided this group representation is weakly measurable, then we have previously proved that it is also norm continuous, and is implemented by a norm, hence weakly and strongly, continuous unitary representation. We discuss the dependence of this key result on the strange properties of Stonean spaces and extend our exploitation of Mackey Theory to give a more straightforward group theoretic proof. From such unitary representations a minimal form of Supersymmetry naturally emerges, which predicts the existence of both the mass zero graviton, and its gravitino partner.