[Paper Review] Curvature and Quantum Mechanics on Covariant Causal Sets

Stanley Gudder|arXiv (Cornell University)|Jul 17, 2015

Noncommutative and Quantum Gravity Theories10 references3 citations

TL;DR

This paper proposes a discrete model of quantum gravity using covariant causal sets (c-causets), which support a natural distance function, geodesics, and curvature. By defining a free wave equation on c-causets, the study shows that quantum mechanics predicts particles preferentially localize at vertices with higher curvature, especially at high mass, demonstrating a discrete analog of gravitational attraction in a causal set framework.

ABSTRACT

This article begins by reviewing the causal set approach in discrete quantum gravity. In our version of this approach a special role is played by covariant causal sets which we call $c$-causets. The importance of $c$-causets is that they support the concepts of a natural distance function, geodesics and curvature in a discrete setting. We then discuss curvature in more detail. By considering $c$-causets with a maximum and minimum number of paths, we are able to find $c$-causets with large and small average curvature. We then briefly discuss our previous work on the inflationary period when the curvature was essentially zero. Quantum mechanics on $c$-causets is considered next. We first introduce a free wave equation for $c$-causets. We then show how the state of a particle with a specified mass (or energy) can be derived from the wave equation. It is demonstrated for small examples that quantum mechanics predicts that particles tend to move toward vertices with larger curvature.

Motivation & Objective

• To develop a discrete quantum gravity framework using covariant causal sets (c-causets) that support geometric concepts like distance, geodesics, and curvature.
• To explore how curvature in c-causets influences particle dynamics via a quantum mechanical wave equation.
• To investigate whether quantum mechanics on c-causets predicts particle localization at high-curvature vertices, analogous to gravitational attraction.
• To quantify the probability distribution of particle positions as a function of mass and curvature in small c-causet models.
• To demonstrate that curvature in the discrete spacetime structure may naturally lead to mass-energy concentration, reversing the standard general relativity paradigm.

Proposed method

• Causets are modeled as finite partially ordered sets with a unique labeling (c-causets), where the shell sequence (s₀, s₁, ..., sₖ) fully determines the structure.
• Distance d(a,b) is defined as the length of the shortest path (geodesic) between vertices a and b, computed via the Euclidean norm of label differences.
• Curvature K(a) at vertex a is defined as the number of geodesics from the big bang (vertex 1) to a, minus one.
• A free wave equation is introduced on c-causets, with the wave function v(ω,x) for each path ω ending at x, incorporating mass m and path length.
• The probability of a particle being at vertex x is computed as |v(x)|² / Σ|v(y)|², with normalization via N₁.
• The model uses path-dependent amplitudes involving imaginary unit i and mass m to simulate relativistic dispersion relations in discrete spacetime.

Experimental results

Research questions

• RQ1 Can a discrete spacetime model based on c-causets support meaningful notions of distance, geodesics, and curvature?
• RQ2 How does quantum mechanics on c-causets influence particle localization in relation to curvature?
• RQ3 Does increasing particle mass enhance localization at high-curvature vertices in discrete spacetime?
• RQ4 What is the quantitative probability distribution of particle positions across vertices of varying curvature in small c-causet models?
• RQ5 Can curvature in a discrete causal set framework naturally lead to mass-energy concentration, mimicking gravitational effects?

Key findings

• In the c-causet with shell sequence (1,2,1), the average curvature is zero, and both maximal paths have equal length √5, confirming geodesic behavior.
• For the (1,2,3) c-causet, the probability p(6,0) = 0.5 at zero mass, increasing to 0.81864 as mass m → ∞, showing strong localization at high-curvature vertex 6.
• In the (1,2,3) model, p(4,0) = 0.1923 and limₘ→∞ p(4,m) = 0.6818, indicating increasing preference for the high-curvature vertex 4 at large mass.
• For vertex 5 in the (1,2,3) model, p(5,0) = 0.3077 and limₘ→∞ p(5,m) = 0.1212, showing decreasing probability at large mass.
• In the (1,2,2) model, p(6,0) = 0.619 and limₘ→∞ p(6,m) = 0.8186, confirming that high-curvature vertices attract particles more strongly as mass increases.
• Across all examples, the probability of finding a particle at a vertex increases with curvature and is strongly mass-dependent, with p(x,m) increasing for high-curvature vertices as m → ∞.