On Conformally Compactified Phase Space

TL;DR

This paper explores the concept of conformally compactified phase space as a dual framework for classical and quantum mechanics, highlighting its potential to unify geometrical descriptions and address quantum field theory divergences.

Contribution

It introduces a novel perspective on phase space via conformal duality, linking classical and quantum mechanics through conformal reflections and spinor geometry.

Findings

Fermion multiplets emerge naturally from higher symmetries.

Euclidean geometry from spinors may prevent ultraviolet divergences.

Conformal reflections could have significant implications for physics.

Abstract

Conformally compactified phase space is conceived as an automorphism space for the global action of the extended conformal group. Space time and momentum space appear then as conformally dual, that is conjugate with respect to conformal reflections. If now the former, as generally agreed, is appropriate for the description of classical mechanics in euclidean geometrical form, then the latter results appropriate for the description of quantum mechanics in spinor geometrical form. In such description, fermion multiplets will naturally appear as consequence of higher symmetries and furthermore, the euclidean geometry, bilinearly resulting from that of spinors, will a priori guarantee the absence of ultraviolet divergences when dealing with quantum field theories. Some further possible consequences of conformal reflections of interest for physics, are briefly outlined.