Abstract
In any quantum theory of gravity, it is of the utmost importance to recover the limit of quantum theory in an external spacetime. In quantum geometrodynamics (quantization of general relativity in the Schrödinger picture), this leads in particular to the recovery of a semiclassical (WKB) time which governs the dynamics of non-gravitational fields in spacetime. Here, we first review this procedure with special emphasis on conceptual issues. We then turn to an alternative theory - Weyl (conformal) gravity, which is defined by a Lagrangian that is proportional to the square of the Weyl tensor. We present the canonical quantization of this theory and develop its semiclassical approximation. We discuss in particular the extent to which a semiclassical time can be recovered and contrast it with the situation in quantum geometrodynamics.
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Notes
- 1.
For a review and reference to original articles, see [11].
- 2.
Recall that the modern time standard is based on the hyperfine transitions in caesium-133.
- 3.
English translation: “On a Natural Addition to the Foundation of the General Theory of Relativity”.
- 4.
- 5.
The \(\varphi _{\nu }\) denote the components of the electromagnetic four-potential.
- 6.
The original German reads ([8], p. 416): “Kurze Zusammenfassung: Es wird gezeigt, dass man entsprechend dem Weyl’schen Grundgedanken auf die objektive Existenz der Lichtkegel (Invarianz der Gleichung \(ds^2=0\)) alleine eine Invarianten-theorie gründen kann, die jedoch im Gegensatz zu Weyl’s Theorie keine Hypothese über Streckenübertragung enthält und in welcher die Potentiale \(\varphi _{\nu }\) nicht explizite in die Gleichungen eingehen. Ob die Theorie auf physikalische Gültigkeit Anspruch erheben kann, müssen spätere Untersuchungen ergeben.”.
- 7.
We write “seems”, because the ghosts connected with non-unitarity may be removable [21].
- 8.
For the history of such theories, see for example [28].
- 9.
In quantum GR, there exist attempts to quantize solely the conformal factor [23]. Paddy has derived from this the interesting conclusion, that the Planck length provides a lower bound to measurable physical lengths. The situation will be different here, because Weyl gravity does not contain an intrinsic length scale.
- 10.
It can be shown that terms in \(\left( \,^{\scriptscriptstyle (3)}\! R_{ij}^{\scriptscriptstyle \mathrm T}+D_{i}D_{j}\right) \bar{P}^{ij}\) which depend on a cancel, making this expression conformally invariant.
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Kiefer, C., Nikolić, B. (2017). Notes on Semiclassical Weyl Gravity. In: Bagla, J., Engineer, S. (eds) Gravity and the Quantum. Fundamental Theories of Physics, vol 187. Springer, Cham. https://doi.org/10.1007/978-3-319-51700-1_11
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