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Register a new userDimensionally Restricted Causal Set Quantum Gravity: Examples in Two and Three Dimensions
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We study dimensionally restricted non-perturbative causal set quantum dynamics in $2$ and $3$ spacetime dimensions with non-trivial global spatial topology. The causal set sample space is generated from causal embeddings into spacetime lattices with global spatial topology $S^1$ and $T^2$ in $2$ and $3$ dimensions, respectively. The quantum gravity partition function over these sample spaces is studied using Markov Chain Monte Carlo (MCMC) simulations after analytic continuation. In both $2$ and $3$ dimensions we find a phase transition that separates the dominance of the action from that of the entropy. The action dominated phase is characterised by ``layered posets with a high degree of connectivity, while the causal sets in the entropy dominated phase are manifold-like. This phase transition is similar in character to that seen for the sample space of $2$-orders, which are topologically trivial, hence suggesting that this is a generic feature of dimensionally restricted sample spaces. The simulations use a newly developed framework for causal set MCMC calculations. Ours is the first implementation of a causal set dynamics restricted to $3$ dimensions.
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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 goal of this paper is to propose an approach to the formulation of dynamics for causal sets and coupled matter fields. We start from the continuum version of the action for a Klein-Gordon field coupled to gravity, and rewrite it first using quantities that have a direct correspondent in the case of a causal set, namely volumes, causal relations, and timelike lengths, as variables to describe the geometry. In this step, the local Lagrangian density $L(f;x)$ for a set of fields $f$ is recast into a quasilocal expression $L_0(f;p,q)$ that depends on pairs of causally related points $p prec q$ and is a function of the values of $f$ in the Alexandrov set defined by those points, and whose limit as $p$ and $q$ approach a common point $x$ is $L(f;x)$. We then describe how to discretize $L_0(f;p,q)$, and use it to define a discrete action.
An important probe of quantum geometry is its spectral dimension, defined via a spatial diffusion process. In this work we study the spectral dimension of a ``spatial hypersurface in a manifoldlike causal set using the induced spatial distance function. In previous work, the diffusion was taken on the full causal set, where the nearest neighbours are unbounded in number. The resulting super-diffusion leads to an increase in the spectral dimension at short diffusion times, in contrast to other approaches to quantum gravity. In the current work, by using a temporal localisation in the causal set, the number of nearest spatial neighbours is rendered finite. Using numerical simulations of causal sets obtained from $d=3$ Minkowski spacetime, we find that for a flat spatial hypersurface, the spectral dimension agrees with the Hausdorff dimension at intermediate scales, but shows clear indications of dimensional reduction at small scales, i.e., in the ultraviolet. The latter is a direct consequence of ``discrete asymptotic silence at small scales in causal sets.
In causal set theory the gravitational path integral is replaced by a path-sum over a sample space $Omega_n$ of $n$-element causal sets. The contribution from non-manifold-like orders dominates $Omega_n$ for large $n$ and therefore must be tamed by a suitable action in the low energy limit of the theory. We extend the work of Loomis and Carlip on the contribution of sub-dominant bilayer orders to the causal set path-sum and show that the link action suppresses the dominant Kleitman-Rothschild orders for the same range of parameters.
This is the second paper in a series on the dynamics of matter fields in the causal set approach to quantum gravity. We start with the usual expression for the Lagrangian of a charged scalar field coupled to a SU(n) Yang-Mills field, in which the gauge field is represented by a connection form, and show how to write it in terms of holonomies between pairs of points, causal relations, and volumes or timelike distances, all of which have a natural correspondence in the causal set context. In the second part of the paper we present an alternative model, in which the gauge field appears as the result of a procedure inspired by the Kaluza-Klein reduction in continuum field theory, and the dynamics can be derived simply using the gravitational Lagrangian of the theory.
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