The causal set approach to quantum gravity (1903.11544v2)

Abstract: The causal set theory (CST) approach to quantum gravity postulates that at the most fundamental level, spacetime is discrete, with the spacetime continuum replaced by locally finite posets or "causal sets". The partial order on a causal set represents a proto-causality relation while local finiteness encodes an intrinsic discreteness. In the continuum approximation the former corresponds to the spacetime causality relation and the latter to a fundamental spacetime atomicity, so that finite volume regions in the continuum contain only a finite number of causal set elements. CST is deeply rooted in the Lorentzian character of spacetime, where a primary role is played by the causal structure poset. Importantly, the assumption of a fundamental discreteness in CST does not violate local Lorentz invariance in the continuum approximation. On the other hand, the combination of discreteness and Lorentz invariance gives rise to a characteristic non-locality which distinguishes CST from most other approaches to quantum gravity. In this review we give a broad, semi-pedagogical introduction to CST, highlighting key results as well as some of the key open questions. This review is intended both for the beginner student in quantum gravity as well as more seasoned researchers in the field.

• The paper presents the causal set theory, showing that a discreet, locally finite order can reconstruct continuum spacetime invariants like dimension, volume, and curvature.
• The paper details sequential growth models, both classical and quantum, that use stochastic processes to generate causal sets consistent with causality and Lorentz invariance.
• The paper explores cosmological implications, including the everpresent Λ hypothesis, while addressing challenges such as proving the Hauptvermutung and experimentally detecting spacetime discreteness.

The Causal Set Approach to Quantum Gravity

In the pursuit of a quantum theory of gravity, the causal set approach posits a fundamentally discrete structure of spacetime, where the continuous manifold of general relativity is replaced by a locally finite, partially ordered set known as a causal set. The paper by Sumati Surya provides a comprehensive review of this approach, discussing both kinematical and dynamical aspects, along with the implications and challenges within this framework.

The central premise of causal set theory (CST) is the replacement of spacetime continuum by discrete elements linked by a partial order that represents causality. Each element corresponds to a spacetime event, and the order relation portrays the causal structure of spacetime. This discrete model of spacetime not only retains the causal ordering of events but also encodes a discrete volume element, ensuring a fundamental Lorentz invariance that is typically challenged in discrete settings.

One of the significant strengths of CST lies in its principal conjecture, the Hauptvermutung, which posits that any causal set that provides a good approximation to a continuum manifold should be unique up to a conformal factor. This idea is supported by kinematic studies where manifold invariants—such as dimension, volume, and curvature—are reconstructed from order-theoretic notions within the causal set, thus underscoring the theorem that causal structure, when combined with cardinality, is sufficient to approximate Lorentzian geometry.

The classical sequential growth (CSG) models are a cornerstone in the dynamics of CST, outlining a manner in which causal sets can grow consistently with causality and covariance. These models generate causal sets through a stochastic process, governed by a sequential development from the empty set, reflecting the inherent indeterminism of quantum spacetime. The models are parameterized by coupling constants that control the growth process, with the transitive percolation being a notable instance that, despite its simplicity, yields causal sets that are not dominated by the entropy of non-manifold-like structures such as Kleitman-Rothschild posets. Furthermore, the interplay of renormalization and cosmic bounces offers a profound analogy to phase transitions in cosmology, potentially hinting at mechanisms for the emergence of classical spacetime from quantum origins.

In terms of quantum dynamics, the quantum sequential growth (QSG) models extend the CSG framework into the quantum field, aiming to construct a quantum measure space characterized by a decoherence functional. These models are further enriched by the Sorkin-Johnston vacuum, a state that emerges naturally in causal set theory and potentially offers a new perspective on issues like the cosmological constant problem and spacetime entanglement entropy.

The theoretical import of CST extends to phenomenology, particularly in providing insights into the cosmological constant. The {\it everpresent $\Lambda$} hypothesis, born out of causal set discreteness, suggests a fluctuating $\Lambda$ driven by cosmic volume fluctuations, yielding predictions remarkably consistent with observed values. This approach emphasizes the utility of causal set theory in addressing long-standing puzzles in cosmology, and ongoing efforts continue to explore these avenues through observational data, assessing the viability of CST-inspired cosmological models against established paradigms like $\Lambda$CDM.

However, numerous challenges remain. A rigorous proof of the Hauptvermutung is elusive, and while numerical simulations bolster the theoretical framework, the development of an effective, non-perturbative understanding of causal set dynamics remains incomplete. Moreover, bridging the gap between CST and experimental tests requires innovative strategies to detect discrete spacetime signatures in a fundamentally continuous world.

Overall, the causal set approach to quantum gravity, with its fundamental discreteness and novel dynamical structures, offers a unique and promising avenue in the search for a coherent theory of quantum spacetime. It challenges conventional paradigms while providing a fertile ground for addressing the deep-rooted issues in combining quantum theory with general relativity.