Quantum reference frames and triality

TL;DR

This paper explores the role of dynamical reference frames in background independent theories, proposing a triality symmetry that unifies classical and quantum mechanics through extended symplectic geometry and matrix models.

Contribution

It introduces a novel triality symmetry extending Born duality, linking reference frames, and demonstrates how matrix models with cubic actions reveal spontaneous symmetry breaking and emergence of time.

Findings

Discovery of a triality symmetry involving reference frames and phase space variables

Extension of symplectic geometry to a higher-dimensional cubic invariant structure

Matrix models illustrating symmetry breaking and emergence of temporal reference frames

Abstract

In a background independent theory without boundary, physical observables may be defined with respect to dynamical reference systems. However, I argue here that there may be a symmetry that exchanges the degrees of freedom of the physical frame of reference with the other degrees of freedom which are measured relative to that frame. This symmetry expresses the fact that the choice of frame of reference is arbitrary, but the same laws apply to all, including observer and observed. It is then suggested that, in a canonical description, this leads to an extension of the Born duality, which exchanges coordinate and momentum variables to a triality that mixes both with the temporal reference frame. This can also be expressed by extending 2n dimensional symplectic geometry to a d= 2n+1 dimensional geometry with a cubic invariant. The choice of a temporal reference frame breaks the triality of the cubic invariant to the duality represented by the canonical two form. We discover that a very elegant way to display this structure which encompasses both classical and quantum mechanics, is in terms of matrix models based on a cubic action. There we see explicitly in either case how a spontaneous symmetry breaking leads to the emergence of a temporal reference frame.