Quantum Reference Frames in terms of Tetrads

@misc{butterfield2026quantum,
  title = {{Quantum Reference Frames in terms of Tetrads}},
  author = {Jeremy Butterfield and Daniel A. Turolla Vanzella},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.17225},
  eprint = {2610.17225},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.17225},
}

TY  - UNPB
TI  - Quantum Reference Frames in terms of Tetrads
AU  - Jeremy Butterfield
AU  - Daniel A. Turolla Vanzella
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.17225
UR  - https://arxiv.org/abs/2610.17225
AB  - We adapt to the context of quantum theory the treatment, familiar in relativity theory, of a reference frame as a tetrad. Since a tetrad is a set of four orthonormal vectors at a spacetime point, it encodes a timelike direction, three orthogonal spatial directions, and units; and so it gives a configuration in spacetime (a spatial orientation and state of motion) of an idealized infinitesimal apparatus (``rulers and a clock"). So a natural ``first guess" is that the state of a quantum reference frame should be an assignment of complex amplitudes to a set of tetrads. We develop this idea, in terms of bundles over spacetime of bases of the tangent spaces. This paper is a companion to two other papers. It emphasises the main conceptual aspects of our proposal, while the other two papers give physical details. It also explains the main contrast between those two papers. Namely: (i) the first considers a single spacetime geometry, so that a quantum reference frame, on our proposal, gives a superposition of perspectives on that geometry; while (ii) the second considers the spacetime manifold with no geometry given a priori, which will imply that a quantum reference frame, on our proposal, can describe a superposition of perspectives on a superposition of geometries---not ``just" a superposition of perspectives on a single geometry. The main mathematical idea for representing this contrast will be choosing between (i) the orthonormal frame bundle, for a given spacetime geometry; (ii) the frame bundle in the usual sense of containing all bases of all spacetime points' tangent spaces.
ER  -