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Quantization of Geometry

Jan Ambjorn

Abstract

Contents: 1. Introduction 2. Bosonic propagators and random paths 3. Random surfaces and strings 4. Matrix models and two-dimensional quantum gravity 5. The mystery of $c > 1$ 6. Euclidean quantum gravity in $d > 2$ 7. Discussion

Quantization of Geometry

Abstract

Contents: 1. Introduction 2. Bosonic propagators and random paths 3. Random surfaces and strings 4. Matrix models and two-dimensional quantum gravity 5. The mystery of 6. Euclidean quantum gravity in 7. Discussion

Paper Structure

This paper contains 33 sections, 401 equations, 30 figures.

Table of Contents

  1. Quantization of Geometry
  2. Introduction
  3. Bosonic propagators and random paths
  4. Quantization
  5. One-dimensional gravity
  6. Scaling relations
  7. Smooth random walks
  8. Fermionic random walks
  9. Random surfaces and strings
  10. Definition of the model
  11. Physical observable
  12. Non-scaling of the string tension
  13. Branched polymers
  14. Extrinsic curvature terms (I)
  15. Supersymmetric random surfaces
  16. ...and 18 more sections

Figures (30)

  • Figure 1: A typical piecewise linear path between $0$ and $x$.
  • Figure 2: The geometry related to the calculation of curvature. $r(l) = 1/\kappa(l)$.
  • Figure 3: The angle $\theta(\hat{e}_1,\hat{e}_2)$ between successive tangent vectors in the piecewise linear random walk.
  • Figure 4: The phase diagram in the $(\mu,\lambda)$-plane. The theory is defined to the right of the critical line $\mu_c(\lambda)$.
  • Figure 5: The cancellation between two paths. For the first we get a phase factor $e^{i\Delta \theta/2} = 1$, while the second gives a phase factor $e^{i\Delta \theta /2} = -1$.
  • ...and 25 more figures