Towards a Quantum Erlangen Program

Physics > General Physics
[Submitted on 23 Jul 2026]
Title:Towards a Quantum Erlangen Program
View PDF HTML (experimental)Abstract: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 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'', in the new model, suggesting that {\it quantum} spacetime may be classified, at least locally, in terms of transformations between quantised frames of reference.
Submission history
From: Matthew J. Lake Dr [view email][v1] Thu, 23 Jul 2026 08:37:33 UTC (119 KB)
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* The classical Erlangen Program sought to classify metric spaces using symmetries.
* Classical symmetries define transformations between classical reference frames in physical spacetimes.
* The notion of ideal frame has been extended to the quantum regime by considering observers as physical systems subject to quantum mechanics.
* The authors outline an alternative definition for 'quantum reference frame'.
* The new definition is intended to construct a simple model of Planck-scale spacetime relevant to quantum gravity phenomenology.
* Classical spacetime symmetries can be "mathematically preserved but operationally broken" in the new model.

The work proposes an alternative definition for the term 'quantum reference frame' by extending the classical Erlangen Program, which classifies metric spaces via symmetries. The authors suggest that observers in quantum physics should be considered embodied physical systems subject to quantum mechanics, and they seek a framework where spacetime symmetries are mathematically preserved but operationally broken at the Planck scale. This new framework aims to construct a simple model of Planck-scale spacetime relevant to quantum gravity phenomenology. The central claim is that spacetime may be classified locally by transformations between quantised frames of reference.

The proposal taps into a deep tension between classical geometric intuition and the constraints imposed by quantum mechanics when applied to spacetime. The core implication is that fundamental notions of symmetry governing classical physics must be re-evaluated when dealing with quantized frames of reference. This suggests a pattern where emergent physical descriptions (like classical geometry) are approximations whose underlying symmetries are masked or broken at extremely small scales, requiring a transition in the mathematical classification system. The challenge lies in operationalizing this distinction—defining precisely what constitutes a 'quantum reference frame' that respects both quantum mechanical constraints and geometrical structure. Further investigation is needed into how these broken symmetries manifest observationally at the Planck scale, and whether this framework offers novel, testable predictions for quantum gravity phenomena beyond existing phenomenology.

Towards a Quantum Erlangen Program

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Towards a Quantum Erlangen Program

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Executive Summary

Full Take

Sentinel — Human