Relational dynamics and Page-Wootters formalism in group field theory

Andrea Calcinari; Steffen Gielen

doi:10.22331/q-2025-01-27-1610

Relational dynamics and Page-Wootters formalism in group field theory

Andrea Calcinari^1,2 and Steffen Gielen¹

¹School of Mathematical and Physical Sciences, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, United Kingdom
²Departamento de Física Teórica and IPARCOS, Facultad de Ciencias Físicas, Universidad Complutense de Madrid, Plaza de las Ciencias 1, Madrid 28040, Spain

Published:	2025-01-27, volume 9, page 1610
Editor:	Maximilian Lock
Eprint:	arXiv:2407.03432v3
Doi:	https://doi.org/10.22331/q-2025-01-27-1610
Citation:	Quantum 9, 1610 (2025).

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Abstract

Group field theory posits that spacetime is emergent and is hence defined without any background notion of space or time; dynamical questions are formulated in relational terms, in particular using (scalar) matter degrees of freedom as time. Unlike in canonical quantisation of gravitational systems, there is no obvious notion of coordinate transformations or constraints, and established quantisation methods cannot be directly applied. As a result, different canonical formalisms for group field theory have been discussed in the literature. We address these issues using a parametrised version of group field theory, in which all (geometry and matter) degrees of freedom evolve in a fiducial parameter. There is a constraint associated to the freedom of reparametrisation and the Dirac quantisation programme can be implemented. Using the "trinity of relational dynamics", we show that the resulting "clock-neutral" theory is entirely equivalent to a deparametrised canonical group field theory, interpreted in terms of the Page-Wootters formalism. Our results not only show that the deparametrised quantisation is fully covariant and can be seen as encoding the dynamics of joint quantum matter and geometry degrees of freedom, they also appear to be the first application of the Page-Wootters formalism directly to non-perturbative quantum gravity. We show extensions to a setting in which many independent gauge symmetries are introduced, which connects to the "multi-fingered time" idea in quantum gravity and provides a somewhat novel extension of the Page-Wootters formalism.

Featured image: Left: the three-dimensional shape represents the history state $|\Psi_\text{phys}\rangle$, where the Hilbert space of the clock is visualised horizontally. The quantum geometry (GFT) state $|\psi(\chi_0)\rangle_\varphi$ at the time $\chi_0$ is obtained by conditioning $|\Psi_\text{phys}\rangle$ on the clock being in the state $|\chi_0\rangle$, and is pictorially represented by a two-dimensional slice. Right: the tetrahedra counted by the GFT number expectation value $N_{D}(\chi_0)$ can be thought as labelled by the same value of the scalar field.

Popular summary

In traditional quantum theory, time is a fixed background in which quantum systems evolve and observations are made. This starting point is often seen as unsatisfactory since time should itself be defined by quantum clocks, and it seems incompatible with Einstein's general relativity which tells us that time is in fact relative to the observer. One proposal for a relativistic time in quantum mechanics is the Page-Wootters formalism: one starts with a unified quantum description for "system" and "clock" and time is derived from quantum correlations by asking questions such as "given that the clock shows X, what is the state of the system?". Recently, the "trinity of relational dynamics" has shown that the Page-Wootters formalism is equivalent to other techniques for quantisation developed in quantum gravity, where spacetime itself is a quantum system and there is similarly no fundamental background time. Our work is the first application of this "trinity" to a (non-perturbative) theory of quantum gravity known as group field theory. We define group field theory in a new setting that allows applying the Page-Wootters formalism, and shows the distinction between "clock" (constructed from matter degrees of freedom) and "system" (discrete quantum spacetime) explicitly, shedding light on the fundamental nature of time in such theories. In particular, this allows the derivation of a Universe that expands relative to the matter clock as coming out of discrete building blocks of geometry.

► BibTeX data

@article{Calcinari2025relationaldynamics,
  doi = {10.22331/q-2025-01-27-1610},
  url = {https://doi.org/10.22331/q-2025-01-27-1610},
  title = {Relational dynamics and {P}age-{W}ootters formalism in group field theory},
  author = {Calcinari, Andrea and Gielen, Steffen},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1610},
  month = jan,
  year = {2025}
}

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[6] Leonardo Chataignier, Philipp A. Hoehn, Maximilian P. E. Lock, and Fabio M. Mele, "Relational Dynamics with Periodic Clocks", arXiv:2409.06479, (2024).

[7] Luca Marchetti and Edward Wilson-Ewing, "Relational observables in group field theory", Classical and Quantum Gravity 42 15, 155008 (2025).

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