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Geometrodynamics of Gauge Fields pp 161–179Cite as

BRST Quantization of Gravity

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Abstract

On the macroscopic level, Einstein’s general relativity (GR) has passed every test “with flying colors”; see Will (2006, 2014) for recent reviews. However, Einstein’s theory has thus far resisted every attempt at quantization, e.g., it is known to be perturbatively nonrenormalizable, partially due to its dimensional coupling constant.

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Notes

  1. 1.

    It has to be kept in mind that Newton’s gravitational constant G is one of the less precisely known constants of physics. In order to improve this situation, there are plans (Alexeev et al. 2001) to measure the gravitational attraction of two bodies in a spaceship (Project SEE), where the larger body will function as a shepherd for the movement of the test mass, similarly as in the rings of Saturn.

  2. 2.

    The grading permits using the direct sum \(\oplus \) of exterior forms carrying different form degree p and ghost number g, such that the graded commutator is more generally defined by \([\varPsi , \varPhi ] {:}{=} \varPsi \wedge \varPhi - (-1)^{(p_1+g_1)(p_2+g_2)} \varPhi \wedge \varPsi \), (Chang & Soo 1992).

  3. 3.

    For the CP-invariant Euler term, a similar result would hold, i.e., \(sL_\mathrm{Euler}=(-1)^{\mathrm{sig}+1}d\left( \varPsi ^{\alpha \beta }\wedge R_{\alpha \beta }^{(\star )}\right) \). However, the latter is only partially metric-free, since it involves the signature \(\mathrm{sig}\) of the metric implicitly in the definition of the Lie dual \(^{(\star )}\) (Blau & Thompson 1991), and therefore appears less well qualified as a starting point.

  4. 4.

    The instanton solutions of Yang’s theory of gravity, classified as early as (1981) by Mielke, are a special case of the Ansatz (8.6.4) for the choice \(\theta _\mathrm{L}=\theta _\mathrm{T}=0\) and \(\theta _\mathrm{L}^\star =\mp (-1)^\mathrm{sig}\). Interestingly enough, it can be regarded as a field redefinition (FR) of the linear connection \(\varGamma \) such that (8.6.4) is induced; see Mielke (2006b) for details. Such an FR was applied in Obukhov & Hehl (1996) to Euler- and Pontryagin-type terms. However, such deformations change the latter four-forms to no longer being d-exact terms, thus preventing a topological interpretation.

  5. 5.

    There occurs an interesting modification in the case that \(\theta _\mathrm{L}= 0\), since then the pseudoscalar curvature four-form \(\theta _\mathrm{T} R^{\alpha \beta }\wedge \vartheta _{\alpha }\wedge \vartheta _{\beta }/4\ell ^2\) needs to be subtracted from (8.6.12), which would induce (Mielke 1992) a partially chiral reformulation of Einsteinian gravity à la Ashtekar. However, this would violate parity P and even CP if \(\theta _\mathrm{T}\) remained real, (Mielke 2001).

  6. 6.

    One has to keep in mind that due to the Bach–Lanczos identity, the parameters \(b_{(N)}\) are not all independent; cf. Eq. (A.3.7) of Hehl et al. (1995).

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  1. Departamento de Física, CBI Iztapalapa, Universidad Autónoma Metropolitana, Mexico City, México

    Eckehard W. Mielke

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Mielke, E.W. (2017). BRST Quantization of Gravity. In: Geometrodynamics of Gauge Fields. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-29734-7_8

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