Holography and the Future Tube

TL;DR

This paper explores the mathematical relationship between the Future Tube in Minkowski spacetime and the phase space of particles in Anti-de-Sitter space, revealing deep geometric and conformal structures.

Contribution

It establishes an isomorphism between the Future Tube and the reduced phase space of particles in Anti-de-Sitter space, linking conformal geometry with complex analysis.

Findings

Future Tube is isomorphic to a bounded homogeneous domain in complex space.

Shilov boundary corresponds to conformally compactified Minkowski spacetime.

Provides a geometric framework connecting Minkowski and Anti-de-Sitter spacetimes.

Abstract

The Future Tube $T^+_n$ of n-dimensional Minkowski spacetime may be identified with the reduced phase space or "space of motions" of a particle moving in (n+1)-dimensional Anti-de-Sitter spacetime. Both are isomorphic to a bounded homogeneous domain in ${\bf C}^{n}$ whose Shilov boundary may be identified with $n$-dimensional conformally compactified Minkowski spacetime.