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Quantum gravity at a large number of dimensions

N. E. J. Bjerrum-Bohr

Abstract

We consider the large-$D$ limit of Einstein gravity. It is observed that a consistent leading large-$D$ graph limit exists, and that it is built up by a subclass of planar diagrams. The graphs in the effective field theory extension of Einstein gravity are investigated in the same context, and it is seen that an effective field theory extension of the basic Einstein-Hilbert theory will not upset the latter leading large-$D$ graph limit, {\it i.e.}, the same subclass of planar diagrams will dominate at large-$D$ in the effective field theory. The effective field theory description of large-$D$ quantum gravity limit will be renormalizable, and the resulting theory will thus be completely well defined up to the Planck scale at $\sim 10^{19}$ GeV. The $(\frac1D)$ expansion in gravity is compared to the successful $(\frac1N)$ expansion in gauge theory (the planar diagram limit), and dissimilarities and parallels of the two expansions are discussed. We consider the expansion of the effective field theory terms and we make some remarks on explicit calculations of $n$-point functions.

Quantum gravity at a large number of dimensions

Abstract

We consider the large- limit of Einstein gravity. It is observed that a consistent leading large- graph limit exists, and that it is built up by a subclass of planar diagrams. The graphs in the effective field theory extension of Einstein gravity are investigated in the same context, and it is seen that an effective field theory extension of the basic Einstein-Hilbert theory will not upset the latter leading large- graph limit, {\it i.e.}, the same subclass of planar diagrams will dominate at large- in the effective field theory. The effective field theory description of large- quantum gravity limit will be renormalizable, and the resulting theory will thus be completely well defined up to the Planck scale at GeV. The expansion in gravity is compared to the successful expansion in gauge theory (the planar diagram limit), and dissimilarities and parallels of the two expansions are discussed. We consider the expansion of the effective field theory terms and we make some remarks on explicit calculations of -point functions.

Paper Structure

This paper contains 9 sections, 39 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Review of the large-$D$ limit of Einstein gravity
  3. The large-$D$ limit and the effective field theory extension of gravity
  4. Considerations about the space-time integrals
  5. Comparison of the large-$N$ limit and the large-$D$ limit
  6. Other expansions of the fields
  7. Discussion
  8. Effective 3- and 4-point vertices
  9. Comments on general $n$-point functions

Figures (24)

  • Figure 1: A graphical representation of the various terms in the 3-point vertex factor. A dashed line represents a contraction of a index with a momentum line. A full line means a contraction of two index lines. The above vertex notation for the indices and momenta also apply here.
  • Figure 2: A graphical representation of the terms in the 4-point vertex factor. The notation here is the same as the above for the 3-point vertex factor.
  • Figure 3: A graphical representation of the graviton propagator, where a full line corresponds to a contraction of two indices.
  • Figure 4: A graphical representation of a propagator contraction in an amplitude. The black half dots represent an arbitrary internal structure for the amplitude, the full lines between the two half dots are internal contractions of indices, the outgoing lines represents index contractions with external sources.
  • Figure 5: Two possible types of contractions for the propagator. Whenever we have an index loop we have a contraction of indices. It is seen that none of the above contractions will generate something with a positive power of ($D$).
  • ...and 19 more figures