Discrete Action, Graph Evolution, and the Hierarchy of Symmetries: A Rigorous Construction of Temporal Layers $C1 \to C2 \to C3 \to C4$

TL;DR

The work proposes a minimal discrete framework where updates occur in quanta of action $S=\hbar$, and time is an emergent macroscopic label. By iteratively growing oriented graphs with $S_{N}=N\hbar$ and $\tau_{N}=N\tau_1$, the authors show how canonical pairs and local $U(1)$ symmetry arise at $\mathcal{C}2$, followed by $SU(3)$-type connections and Einstein–Yang–Mills dynamics at $\mathcal{C}3$, and gauge-consistent dimensional reduction at $\mathcal{C}4$, culminating in an infrared $(2+1)$-like sector. The construction links discrete-action principles to continuum gauge-gravity theories and suggests a mechanism for decoherence and symmetry breaking via stochastic graph growth. This provides a rigorous, minimal bridge between action quanta and emergent spacetime and gauge structures, with potential connections to causal sets and related discrete approaches.

Abstract

Postulating a minimal discrete quantum of action $S=\hbar$ and a simple rule for the growth of an oriented graph, we construct a strict hierarchy of temporal layers $C N$ with discrete periods $τ_N=N\hbar/E$. Each layer is specified by its configuration space, symplectic structure, update rule, and emergent symmetry. At $C1$ the state is represented by a single oriented edge with $U(1)$ phase $e^{i E t/\hbar}$. The transition $C1 \to C2$ splits the edge into two independent flows, which yields canonical pairs $(x_a,p_a)$, local $U(1)$ invariance, and an effective $(2{+}1)$ metric with signature $(+--)$. The closure $C2 \to C3$ produces $SU(3)$ connections and an Einstein-Yang-Mills type action. We show that these structures follow from discrete-action principles, and that stochastic graph growth naturally provides mechanisms for decoherence and spontaneous symmetry breaking.

Figures (3)

• Figure 1: Layer $\mathcal{C}1$: a single oriented edge between $(v_0,v_1)$ with $U(1)$ phase evolution ($e_1$).
• Figure 2: Layer $\mathcal{C}2$: edge splitting introduces canonical pairs and local $U(1)$ invariance.
• Figure 3: Layer $\mathcal{C}3$: a triple closure yields $SU(3)$-type connections.