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The Problem of Time pp 511–520Cite as

Geometrical Quantization with Nontrivial . ii. Field Theories and GR

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Part of the book series: Fundamental Theories of Physics ((FTPH,volume 190))

Abstract

We now extend our treatment of geometrical quantization to the most physically desirable case of field theories exhibiting redundant groups of transformations. Quantum field theory and general relativity are both frameworks of this kind, with Lie and diffeomorphism gauge groups respectively. We first supplement Chap. 6’s account of time and background dependence in quantum field theory by outlining some further subtleties required for Part III’s more advanced discussion of Quantum Gravity, such as topological aspects and regularization. We then consider the Dirac quantization of some whole-universe models which thus exhibit the Frozen Formalism Problem too: canonical general relativity in all of plain geometrodynamical, affine geometrodynamical and loop formulations, as well as supersymmetric model counterparts. We end by arguing that while it would be technically convenient if quantization were a functorial prescription, this is unfortunately not the case in general.

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Notes

  1. 1.

    These techniques are named after physicists Carlo Becchi, Alain Rouet, Raymond Stora and Igor Tyutin, and Igor Batalin and Grigori Vilkovisky, respectively.

  2. 2.

    This is mass-weighted area, so this is as yet another analogy between \(I\) and \(\sqrt{\mbox{h}}\).

  3. 3.

    See e.g. physicists Rodolfo Gambini and Jorge Pullin’s book [330] for a conceptually if not technically detailed account of the loop transformation underlying the loop representation. This in turn rests on loop space mathematics (outlined in Appendix N.12). See also e.g. Ashtekar and physicist Jerzy Lewandowski’s account [77] for a more rigorous treatment. It has also been determined that the loop representation works just as well for whichever value of the Barbero–Immirzi parameter \(\beta \). Thus, in particular, it works for both complex-variables and real-variables forms of Nododynamics.

  4. 4.

    For instance, one might use a framing by which loops become ribbons—a further type of regularization that is possibly more amenable to theories with metric Background Independence. This is illustrative of a main approach in freeing oneself from the consequences of loops themselves being too singular.

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    Edward Anderson

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Anderson, E. (2017). Geometrical Quantization with Nontrivial . ii. Field Theories and GR. In: The Problem of Time. Fundamental Theories of Physics, vol 190. Springer, Cham. https://doi.org/10.1007/978-3-319-58848-3_43

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