Towards a Quantum Erlangen Program

The classical Erlangen Program sought to classify metric spaces entirely in terms of their symmetries. In physical spacetimes, these symmetries define transformations between classical reference frames, yielding a one-to-one correspondence between frame transformations and the underlying geometry. More recently, the classical notion of an ideal frame has been extended to the quantum regime, by considering observers as embodied physical systems, subject to the laws of quantum mechanics. Here, we build on this approach, but outline an alternative definition of the term ‘quantum reference frame’, which differs somewhat from the mainstream view. We then show how the new definition can be used to construct a simple model of Planck-scale spacetime, which makes contact with existing quantum gravity phenomenology. Finally, we show how classical spacetime symmetries can be ‘mathematically preserved but operationally broken’ using the new model, suggesting that quantum spacetime may be classified, at least locally, in terms of transformations between quantised frames of reference. This work is based on a talk given at the 13th Bolyai–Gauss–Lobachevsky Conference on Non-Euclidean Geometry in Oujda, Morocco, in May 2025.