[Paper Review] Quantum Graphity

Tomasz Konopka, Fotini Markopoulou+1 more|arXiv (Cornell University)|Nov 17, 2006

Black Holes and Theoretical Physics8 references46 citations

TL;DR

This paper proposes a background-independent quantum model on a complete graph where permutation symmetry among nodes gives rise to emergent spacetime geometry. At low temperatures, symmetry breaks to a lattice translation group, yielding a classical spacetime phase; this phase transition may resolve the horizon problem in cosmology.

ABSTRACT

We introduce a new model of background independent physics in which the degrees of freedom live on a complete graph and the physics is invariant under the permutations of all the points. We argue that the model has a low energy phase in which physics on a low dimensional lattice emerges and the permutation symmetry is broken to the translation group of that lattice. In the high temperature, or disordered, phase the permutation symmetry is respected and the average distance between degrees of freedom is small. This may serve as a tractable model for the emergence of classical geometry in background independent models of spacetime. We use this model to argue for a cosmological scenario in which the universe underwent a transition from the high to the low temperature phase, thus avoiding the horizon problem.

Motivation & Objective

• To develop a background-independent quantum model where spacetime geometry emerges from fundamental degrees of freedom.
• To explore how permutation symmetry on a complete graph can lead to low-dimensional spacetime lattices.
• To propose a cosmological scenario in which a phase transition from high to low temperature resolves the horizon problem.
• To provide a tractable framework for studying the emergence of classical geometry in quantum gravity.

Proposed method

• The model defines quantum degrees of freedom on a complete graph with full permutation symmetry among nodes.
• The Hamiltonian is constructed to be invariant under all permutations of the graph nodes.
• The system is analyzed using statistical mechanics techniques to study phase transitions.
• Low-energy phases are identified where permutation symmetry is spontaneously broken to a translation group of a lattice.
• The emergence of a spatial lattice is studied via order parameters and correlation functions.
• Cosmological implications are drawn by interpreting the high-to-low temperature transition as a cosmic phase transition.

Experimental results

Research questions

• RQ1 How can classical spacetime geometry emerge from a fundamentally background-independent quantum system?
• RQ2 What role does spontaneous symmetry breaking play in the emergence of a low-dimensional spacetime?
• RQ3 Can a phase transition in such a model resolve the horizon problem in cosmology?
• RQ4 What is the nature of the low-energy phase and how does it realize a lattice-like structure?
• RQ5 How does the permutation symmetry of the complete graph lead to a physical spacetime at low energies?

Key findings

• The model exhibits a phase transition from a high-temperature, symmetric phase with short average distances between nodes to a low-temperature phase with broken permutation symmetry.
• In the low-temperature phase, a discrete lattice structure emerges, with the residual symmetry matching the translation group of a spatial lattice.
• The emergence of a low-dimensional spacetime is driven by spontaneous symmetry breaking of the complete graph's permutation symmetry.
• The phase transition provides a mechanism to avoid the horizon problem by ensuring causal connectivity in the early universe.
• The model offers a tractable framework for studying the dynamical emergence of spacetime in quantum gravity.
• The complete graph structure allows for a systematic study of geometry emergence without prior geometric assumptions.