Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A new vacuum for Loop Quantum Gravity

Bianca Dittrich, Marc Geiller

Abstract

We construct a new vacuum for loop quantum gravity, which is dual to the Ashtekar-Lewandowski vacuum. Because it is based on BF theory, this new vacuum is physical for $(2+1)$-dimensional gravity, and much closer to the spirit of spin foam quantization in general. To construct this new vacuum and the associated representation of quantum observables, we introduce a modified holonomy-flux algebra which is cylindrically consistent with respect to the notion of refinement by time evolution suggested in [1]. This supports the proposal for a construction of a physical vacuum made in [1,2], also for $(3+1)$-dimensional gravity. We expect that the vacuum introduced here will facilitate the extraction of large scale physics and cosmological predictions from loop quantum gravity.

A new vacuum for Loop Quantum Gravity

Abstract

We construct a new vacuum for loop quantum gravity, which is dual to the Ashtekar-Lewandowski vacuum. Because it is based on BF theory, this new vacuum is physical for -dimensional gravity, and much closer to the spirit of spin foam quantization in general. To construct this new vacuum and the associated representation of quantum observables, we introduce a modified holonomy-flux algebra which is cylindrically consistent with respect to the notion of refinement by time evolution suggested in [1]. This supports the proposal for a construction of a physical vacuum made in [1,2], also for -dimensional gravity. We expect that the vacuum introduced here will facilitate the extraction of large scale physics and cosmological predictions from loop quantum gravity.

Paper Structure

This paper contains 12 sections, 23 equations, 5 figures.

Table of Contents

  1. 1. Introduction
  2. 2. Setup
  3. 3. Cylindrical consistency of closed holonomies
  4. 4. Cylindrical consistency of integrated simplicial flux operators
  5. 5. Measure
  6. 6. On the projective limit and spatial diffeomorphisms
  7. 7. Conclusions
  8. Acknowledgements

Figures (5)

  • Figure 1: Example of refinement via 1--3 and 2--2 moves. The rightmost triangulation is finer than the leftmost one, and represents the common refinement $\overline{\Delta}$ of the triangulations on the top and on the bottom.
  • Figure 2: Local change of the graph $\Gamma$ to $\Gamma'$ under the action of a 1--3 move. Notice the reversal of the edges, which is due to the fact that the 1--3 move glues a tetrahedron onto a triangle, so that one actually replaces oppositely oriented half--edges in the dual. The path $\gamma$ is changed to $\gamma'$.
  • Figure 3: Local change of the graph $\Gamma$ to $\Gamma'$ under the action of a 2--2 move, with the example of a path $\gamma$ changed to $\gamma'$.
  • Figure 4: Replacement of a portion of triangulation (dashed triangle) and its dual by a ribbon graph that contains both the holonomies and the fluxes. Our convention is such that the ribbons have a clockwise orientation, and that the pairs $(\text{holonomy},\text{flux})$ have positive orientation. A path defining an integrated flux $\mathbf{X}_{\pi^*}$ is represented with thick lines.
  • Figure 5: Replacement under a 2--2 move of an integrated flux by another flux that defines the same displacement vector in the triangulation but goes along a different path.