Towards a Quantum Erlangen Program
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.
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BibTeX
@misc{lake2026towards,
title = {{Towards a Quantum Erlangen Program}},
author = {Matthew J. Lake},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09167},
eprint = {2609.09167},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09167},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Towards a Quantum Erlangen Program
AU - Matthew J. Lake
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.09167
UR - https://arxiv.org/abs/2609.09167
AB - 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.
ER -