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Causal evolution of spin networks

Fotini Markopoulou, Lee Smolin

TL;DR

This work introduces a discrete, non-perturbative framework for quantum gravity that fuses loop-quantized spin-network kinematics with a causal-set path integral. Dynamics are defined by amplitudes assigned to a restricted class of causal sets formed from spin networks connected by labeled null edges, and are implemented via two local rules that generate spacetime nets and preserve causal structure. In 2+1 and 3+1 dimensions the amplitudes hinge on tetrahedral and 4-simplex combinatorics (e.g., $J(mnp;jkl)$ and $J^{15}$, with natural links to $6j$ and $15j$ symbols), while 1+1 dimensional realizations connect to directed percolation and potential critical behavior, offering a route to a classical limit. The framework aims to realize a continuum Einstein theory from an underlying discrete, causal, and combinatorial substrate, and it raises important questions about time asymmetry and diffeomorphism invariance in the quantum-to-classical transition.

Abstract

A new approach to quantum gravity is described which joins the loop representation formulation of the canonical theory to the causal set formulation of the path integral. The theory assigns quantum amplitudes to special classes of causal sets, which consist of spin networks representing quantum states of the gravitational field joined together by labeled null edges. The theory exists in 3+1, 2+1 and 1+1 dimensional versions, and may also be interepreted as a theory of labeled timelike surfaces. The dynamics is specified by a choice of functions of the labelings of d+1 dimensional simplices,which represent elementary future light cones of events in these discrete spacetimes. The quantum dynamics thus respects the discrete causal structure of the causal sets. In the 1+1 dimensional case the theory is closely related to directed percolation models. In this case, at least, the theory may have critical behavior associated with percolation, leading to the existence of a classical limit.

Causal evolution of spin networks

TL;DR

This work introduces a discrete, non-perturbative framework for quantum gravity that fuses loop-quantized spin-network kinematics with a causal-set path integral. Dynamics are defined by amplitudes assigned to a restricted class of causal sets formed from spin networks connected by labeled null edges, and are implemented via two local rules that generate spacetime nets and preserve causal structure. In 2+1 and 3+1 dimensions the amplitudes hinge on tetrahedral and 4-simplex combinatorics (e.g., and , with natural links to and symbols), while 1+1 dimensional realizations connect to directed percolation and potential critical behavior, offering a route to a classical limit. The framework aims to realize a continuum Einstein theory from an underlying discrete, causal, and combinatorial substrate, and it raises important questions about time asymmetry and diffeomorphism invariance in the quantum-to-classical transition.

Abstract

A new approach to quantum gravity is described which joins the loop representation formulation of the canonical theory to the causal set formulation of the path integral. The theory assigns quantum amplitudes to special classes of causal sets, which consist of spin networks representing quantum states of the gravitational field joined together by labeled null edges. The theory exists in 3+1, 2+1 and 1+1 dimensional versions, and may also be interepreted as a theory of labeled timelike surfaces. The dynamics is specified by a choice of functions of the labelings of d+1 dimensional simplices,which represent elementary future light cones of events in these discrete spacetimes. The quantum dynamics thus respects the discrete causal structure of the causal sets. In the 1+1 dimensional case the theory is closely related to directed percolation models. In this case, at least, the theory may have critical behavior associated with percolation, leading to the existence of a classical limit.

Paper Structure

This paper contains 15 sections, 4 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Kinematics of spin networks
  3. Rules for causal evolution: $2+1$ case
  4. Rule 1
  5. Rule 2
  6. Combining the two rules: the transition law
  7. Interpretation in terms of timelike surfaces
  8. The $3+1$ dimensional theory
  9. $1+1$ dimensional models
  10. A first $1+1$ dimensional model
  11. Directed percolation model
  12. Conclusions
  13. Causal sets and the classical limit
  14. Time reversal invariance and noninvariance
  15. Spacetime diffeomorphism invariance

Figures (16)

  • Figure 1: The construction of a new spin network under Rule 1.
  • Figure 2: Two steps in the evolution of a trivalent spin network, following first Rule 1, then Rule 2.
  • Figure 3: The new nodes represent events defined by meeting of causal processes.
  • Figure 4: The backwards light cone ${\cal C}_{n^\prime_{12}}^-$ of the node $n^\prime_{12}$.
  • Figure 5: The labeling of a new four valent nodes is gotten from that of the edge that gave rise to it. A dotted circle around a subgraph means that it is virtual.
  • ...and 11 more figures