Abstract
In canonical quantum gravity the wave function of the universe is static, leading to the so-called problem of time. We summarize here how Bohmian mechanics solves this problem.
This paper is dedicated to Detlef Dürr with whom I had the pleasure to discuss quantum gravity and the problem of time on many occasions. For Detlef conceptual clarity was always quintessential to get a firm grip on this problem.
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Notes
- 1.
The wave function itself is not an object in space-time, but in some Hilbert space.
- 2.
An ontology based on the reduced phase space is not always problematic. In other cases, it makes sense to consider such an ontology. For example, in the case of the free electromagnetic field, the reduced phase space can be parameterized by the transverse part of the field potential, together with its conjugate momentum, and one could entertain an ontology based on this field.
- 3.
Rather than considering the positions as functions of time t, we could equally well have done this discussion in terms of an arbitrary parameterization s. That there is an external time plays no role here, as long as there is change with respect to s.
- 4.
Throughout we assume units such that \(\hbar = c =1\).
- 5.
To show the equivalence as quantum theories, more work is needed, by also considering the associated Hilbert spaces.
- 6.
Just like in special relativity, this splitting is not unique.
- 7.
We used the same operator ordering as in [22].
- 8.
The papers of Padmabhan and Greensite actually predate the development of Bohmian quantum gravity, which was initiated shortly afterwards by Vink [24]. Vink also emphasized that Bohmian quantum gravity yields a generalization of the semi-classical approach to time.
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Acknowledgements
It is a pleasure to thank Detlef Dürr, Sheldon Goldstein, Stefan Teufel, Roderich Tumulka, Hans Westman and Nino Zanghì for discussions on this topic over the years. This work was presented at the OLOFOS Seminar in Leuven. I thank the audience and especially the respondent Alexandre Guay for the discussions. This work is supported by the Research Foundation Flanders (Fonds Wetenschappelijk Onderzoek, FWO), Grant No. G0C3322N.
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Struyve, W. (2024). The Bohmian Solution to the Problem of Time. In: Bassi, A., Goldstein, S., Tumulka, R., Zanghì, N. (eds) Physics and the Nature of Reality. Fundamental Theories of Physics, vol 215. Springer, Cham. https://doi.org/10.1007/978-3-031-45434-9_15
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