Causal Sets and Discrete Spacetime: A Local Relativistic Operator.

The fundamental challenge of reconciling quantum gravity with general relativity necessitates exploration of discrete approaches to spacetime, and causal set theory represents a promising avenue. This theory posits that spacetime is fundamentally discrete, composed of indivisible elements related by a partial order representing causality. A persistent difficulty within this framework lies in defining differential operators, essential tools for relativistic field theories, in a manner consistent with the inherent nonlocality of causal sets. Researchers at the Universitat de Barcelona and Northeastern University, including Marián Boguňá and Dmitri Krioukov et al, address this issue in their paper, “Local d’Alembertian for causal sets”. They present a locally-defined d’Alembertian operator, a relativistic wave operator, for causal sets, demonstrating its convergence to the standard continuum operator for fields within Minkowski spacetime. This construction, utilising recent advances in intrinsic distance measurement within causal sets, offers a potential resolution to the conflict between locality and Lorentz invariance, and provides a pathway towards constructing discrete approximations of differential operators applicable to inherently nonlocal theories.

Causal set theory proposes a fundamentally discrete structure for spacetime, challenging the conventional view of a smooth continuum. Researchers investigate this framework as a potential resolution to the longstanding inconsistencies between quantum mechanics and general relativity, positing that spacetime emerges from a partially ordered set of events, known as a causal set, where order defines causal relationships. This inherent discreteness introduces nonlocality, a characteristic that complicates the definition of fundamental differential operators, such as the d’Alembertian, essential for constructing relativistic field theories and accurately describing physical phenomena.

The d’Alembertian, a wave operator central to relativistic physics, traditionally requires well-defined distances to operate effectively. However, defining distances within a discrete causal set proves problematic due to its inherent nonlocality. Previous attempts to construct a d’Alembertian within this framework often employed nonlocal operators, aiming to preserve the causal set’s nonlocality and maintain consistency with the underlying discrete structure. Recent research demonstrates these nonlocal d’Alembertians fail to converge to the standard continuum d’Alembertian for certain fields, hindering the development of a consistent relativistic field theory on discrete spacetime and necessitating a revised approach.

To address this issue, researchers introduce a local d’Alembertian operator specifically designed for causal sets, representing a departure from previous attempts and a crucial step towards a viable theory. This construction leverages recent advances in measuring both timelike and spacelike distances within causal sets, relying solely on the intrinsic structure of the set itself and avoiding the pitfalls of nonlocal constructions. Crucially, this local operator demonstrably converges to its continuum counterpart for arbitrary fields in Minkowski spacetime, validating the approach and offering a pathway towards defining discrete approximations to differential operators even within inherently nonlocal theories.

This work successfully reconciles locality with Lorentz invariance, a cornerstone of modern physics, within the causal set framework, addressing a fundamental challenge in the development of a discrete spacetime theory. The successful construction of a converging local d’Alembertian represents a significant step forward, providing a robust mathematical tool for exploring the dynamics of discrete spacetime and potentially resolving the fundamental conflicts between quantum mechanics and general relativity. Researchers build upon connections to areas such as graph theory, random matrix theory, and manifold learning, fostering an interdisciplinary approach to quantum gravity and expanding the toolkit available for tackling this complex problem.

Researchers actively address these challenges by developing new mathematical tools and techniques specifically tailored to the discrete nature of causal sets. The local d’Alembertian operator, constructed in this work, overcomes these challenges by leveraging recent advances in the measurement of distances within causal sets, relying solely on the intrinsic structure of the set itself. This approach avoids the pitfalls of nonlocal constructions, which have been shown to lead to inconsistencies and non-convergence. The operator demonstrably converges to its continuum counterpart for arbitrary fields in Minkowski spacetime, validating the approach and opening up new avenues for research.

This work has significant implications for the development of a consistent theory of quantum gravity, providing a robust mathematical framework for exploring the dynamics of discrete spacetime. Researchers plan to extend this work to more complex spacetimes, such as those with curvature and cosmological expansion, and to investigate the implications for black hole physics and cosmology. The successful construction of the local d’Alembertian operator represents a major step forward in the quest to understand the fundamental nature of spacetime and the universe.