Quantum Graphity: a model of emergent locality

Abstract: Quantum graphity is a background independent model for emergent locality, spatial geometry and matter. The states of the system correspond to dynamical graphs on N vertices. At high energy, the graph describing the system is highly connected and the physics is invariant under the full symmetric group acting on the vertices. We present evidence that the model also has a low-energy phase in which the graph describing the system breaks permutation symmetry and appears to be ordered, low-dimensional and local. Consideration of the free energy associated with the dominant terms in the dynamics shows that this low-energy state is thermodynamically stable under local perturbations. The model can also give rise to an emergent U(1) gauge theory in the ground state by the string-net condensation mechanism of Levin and Wen. We also reformulate the model in graph-theoretic terms and compare its dynamics to some common graph processes.

• The paper introduces a model where spacetime, locality, and geometry emerge from an underlying dynamic graph structure.
• It utilizes a Hamiltonian with valence and closed-path terms to transition from a high-energy, non-geometric phase to an ordered, local state.
• The approach offers a novel framework that unifies matter and geometry, potentially addressing key challenges in quantum gravity.

Quantum Graphity: A Model of Emergent Locality

The paper "Quantum Graphity: A Model of Emergent Locality" by Tomasz Konopka, Fotini Markopoulou, and Simone Severini presents a novel approach to quantum gravity that envisions spacetime geometry and locality as emergent properties of a more fundamental graph structure. This model, termed quantum graphity, builds on a foundation of background independence and seeks to offer insights into the low-energy behavior of quantum gravitational systems.

Overview

Quantum graphity posits that spacetime and its properties, such as locality and geometry, are not fundamental but emergent phenomena arising from a microscopic quantum system. This contrasts with conventional quantum gravity approaches which often treat spacetime as a given backdrop on which quantum fields interact. Quantum graphity challenges this by proposing that at a fundamental level, the universe is described by a complete graph where vertices represent fundamental degrees of freedom, and edges represent dynamical interactions.

Model Dynamics

The core of the model involves a dynamical graph on $N$ vertices, where at high energy, the graph is highly connected, and symmetry is pervasive across the network. This configuration lacks any familiar geometric interpretation, suggesting that early universe physics may not conform to our traditional notions of space and locality. The model introduces a Hamiltonian with specific terms:

1. Valence Term ($H_V$): This term acts to control the valence or degree of the vertices, introducing a preferred average connectivity in the graph.
2. Closed Path Term ($H_B$): This term emphasizes the presence of closed loops in the graph, which are integral to creating a local structure that resembles low-dimensional space.
3. Interaction Terms: These allow for dynamic processes that rearrange connections, simulating quantum fluctuations and evolutionary pathways the graph could undergo.

Emergence of Locality and Geometry

At low energy and with decreasing temperature, the graph undergoes a transition to an ordered state with broken symmetry, yielding structures that can be identified with low-dimensional geometries. This phase of the model is where familiar spatial properties and notions of locality emerge. The possibility for a $U(1)$ gauge theory to arise, a key feature for modeling electromagnetism, demonstrates the model's potential for reproducing known physics.

Comparison to Other Approaches and Implications

The paper situates quantum graphity among various background-independent and non-local theories, noting its distinctions and advantages. Unlike other approaches like loop quantum gravity that begin with a fixed graph or lattice, quantum graphity's graph is dynamically determined. This flexibility allows it to potentially model the transition from a non-geometric phase to a geometric one, capturing the essence of emergent spacetime. The model is evaluated against random graph processes, illustrating its ability to naturally avoid the pitfalls of scale-free and random regular graphs which fail to mimic localized, low-dimensional space effectively.

Future Directions and Challenges

The theoretical implications of quantum graphity are profound, suggesting a unifying framework where matter and geometry are intertwined aspects of deeper quantum processes. However, the paper identifies several key challenges and open questions, including understanding the precise nature of the high- to low-energy phase transition, the role of temperature, and the model's computational complexity regarding large graphs. Further exploration of these areas could refine the model's predictions and its agreement with observable phenomena.

Conclusion

Quantum Graphity represents a bold step in theorizing about the quantum foundations of spacetime, providing a rich framework for considering how familiar physical laws might emerge from a fundamentally different, graph-based reality. While there are substantial hurdles to overcome, particularly in computational and experimental validation, the conceptual shift away from preconceived notions of geometry holds transformative potential for theoretical physics.