At the boundary of Minkowski space

TL;DR

This paper explores a compactification of Minkowski space using the Cayley transform, linking it to unitary groups and quantum field theory backgrounds, and extends models to more general spin four-manifolds.

Contribution

It introduces a novel compactification of Minkowski space via the Cayley transform and extends quantum field theory models to general spin four-manifolds.

Findings

Compactification of Minkowski space as the unitary group U(2).

Identification of the Brauer-Wall group structure for U(2).

Extension of models to general spin four-manifolds.

Abstract

The Cayley transform compactifies Minkowski space $\M$, realized as self-adjoint $2\times2$ complex matrices following Penrose, as the unitary group $\U(2)$. Its complement is a compactification of a copy of a light-cone as it is usually drawn, constructed by adjoining a bubble or $\CP_1$ of unitary matrices with eigenvalue $\pm 1$ at the ends of a lightcone at infinity. The Brauer-Wall group of $\U(2)$ (i.e. of fields of certain kinds of graded $\Cs$-algebras, up to projective equivalence) is $\Z_2 \times \Z$, defining an interesting class of nontrivial examples of Araki-Haag-Kastler backgrounds for quantum field theories on compactified Minkowski space. The second part of this paper extends such models to link presentations of more general spin four-manifolds.