Meta Theory of the Zeckendorf-Hilbert Universe

3.2 Causal Locality Theorem: Deriving Strict Light Cone Structure from Finite Propagation Radius

In Section 3.1, we established the kinematic foundation of QCA universe and introduced the dynamical core—local unitary update operator $U$. This section will prove that it is precisely the algebraic locality (Algebraic Locality) of $U$ that strictly derives the crucial Light Cone Structure in physics on discrete graph background.

In continuous quantum field theory, causality is usually imposed as an axiom a priori (e.g., microcausality axiom: field operators at spacelike separation commute). But in the discrete ontology of this book, causality is not an a priori assumption, but a emergent theorem from microscopic discrete dynamics. We will prove that there exists a strict upper bound on information propagation speed, which manifests as light speed $c$ in the macroscopic limit.

3.2.1 Algebraic Support and Heisenberg Evolution

To mathematically describe “information propagation,” we need to examine evolution of observables (operators) over time in Heisenberg picture (Heisenberg Picture).

Definition 3.2.1 (Support Set of Operator)

For a local operator $O$ in total algebra $\mathcal{A}$, if it acts non-trivially only on subset $R\subset \Lambda$ (i.e., acts as identity operator $1$ on complement of $R$), then $R$ is called the support set of this operator, denoted $\text{supp}(O)$.

Formally, if $O\in {\mathcal{A}}_R\otimes 1_{\Lambda \setminus R}$, then $\text{supp}(O)\subseteq R$.

Definition 3.2.2 (Dynamical Mapping)

Let $U$ be the one-step update operator of QCA. For any operator $A\in \mathcal{A}$, its one-step time evolution is given by automorphism $\alpha$:

$$\alpha (A)=U^{†}AU$$

$t$-step evolution is denoted ${\alpha }^t(A)=(U^{†}{)}^{t}A{U}^t$.

3.2.2 Strict Locality Theorem

Lieb-Robinson bounds in continuous systems show that information propagation decays exponentially outside light cones, but mathematically still not strictly zero. In sharp contrast, the discrete structure of QCA guarantees Strict locality, i.e., information leakage outside light cones is strictly zero.

Theorem 3.2.3 (Finite Propagation Radius Theorem)

Let QCA update operator $U$ satisfy structural locality (Definition 3.1.2), i.e., for any single-point operator $A_x$ (supported on $x$), $\text{supp}(\alpha (A_x))\subset \mathcal{N}(x)$, where $\mathcal{N}(x)$ is the finite neighborhood of $x$.

Then for any local operator $O$ and its $t$-step evolution ${\alpha }^t(O)$, there exists a finite region ${\mathcal{C}}_t(\text{supp}(O))$ depending only on graph structure and $t$, such that:

$$\text{supp}({\alpha }^t(O))\subseteq {\mathcal{C}}_t(\text{supp}(O))$$

This region ${\mathcal{C}}_t$ grows linearly with time $t$.

Proof:

We proceed by induction on time step $t$.

1. Base Case ($t=0$): ${\alpha }^0(O)=O$, support set unchanged.

2. Inductive Step: Assume at $t=k$, $\text{supp}({\alpha }^k(O))\subseteq R_k$.

   Consider $t=k+1$:

   $${\alpha }^{k+1}(O)=\alpha ({\alpha }^k(O))$$

   Since ${\alpha }^k(O)$ can be decomposed as a linear combination of basis operators supported on $R_k$, and according to locality of $U$, operators supported on $y\in R_k$ evolve to have support within $\mathcal{N}(y)$.

   Therefore, support set of ${\alpha }^{k+1}(O)$ is contained in the neighborhood union of $R_k$:

   $$R_{k+1}=\bigcup _{y\in R_k}\mathcal{N}(y)$$

   If graph $\Lambda$ has uniform degree (e.g., lattice), and neighborhood radius is $r$, then linear scale (diameter) of $R_t$ grows at most by $2r$. This proves that expansion of support set is strictly bounded.

$\square$

3.2.3 Construction of Geometric Light Cone

Based on Theorem 3.2.3, we can define geometric light cone purely from graph-theoretic perspective.

Definition 3.2.4 (Geometric Influence Cone)

For spacetime point $(x,n)\in \Lambda \times \mathbb{Z}$, its future geometric light cone $C_{geo}^{+}(x,n)$ is defined as the set of all spacetime points that may be affected by perturbations at $x$ at time $n$:

$$C_{geo}^{+}(x,n)=\{(y,m)\in \Lambda \times \mathbb{Z}\mid m\ge n,\text{dist}(x,y)\le R\cdot (m-n)\}$$

where $R$ is the propagation radius of $U$ (i.e., maximum graph distance of neighborhood). Similarly, past geometric light cone $C_{geo}^{-}(x,n)$ is the set of all points that may influence $(x,n)$.

This geometric definition gives the Maximum Velocity (Maximal Velocity) in QCA universe:

$$v_{\max }=\frac{R}{\Delta t}(\text{set }\Delta t=1)$$

This is the “light speed” $c$ in this discrete universe.

3.2.4 Causal Structure: From Commutators to Partial Order

Physical causality is usually defined as “possibility of signal transmission.” In quantum mechanics, this corresponds to non-commutativity of two observables. If $[A,B]=0$, then measuring $A$ does not affect statistical results of $B$, i.e., no signal transmission.

Definition 3.2.5 (Causal Partial Order $⪯$)

Define relation $⪯$ on event set $E=\Lambda \times \mathbb{Z}$: we say event $(x,n)$ causally precedes $(y,m)$, denoted $(x,n)⪯(y,m)$, if there exist local operators $A\in {\mathcal{A}}_{\{x\}}$ and $B\in {\mathcal{A}}_{\{y\}}$ such that under Heisenberg evolution:

$$[{\alpha }^{m-n}(A),B]\ne 0$$

(Note: For $m<n$, defined as uncorrelated).

The following theorem establishes strict equivalence between QCA geometric structure and physical causality, which has profound significance in foundations of physics: causal structure is not imposed, but a direct consequence of local dynamics.

Theorem 3.2.6 (Causal Locality Theorem)

For any QCA universe satisfying locality conditions, its physical causal relation $⪯$ is strictly contained in geometric light cone relation ${\le }_{geo}$. That is:

$$(x,n)⪯(y,m)⟹(y,m)\in C_{geo}^{+}(x,n)$$

In other words, if $(y,m)$ lies outside the geometric light cone of $(x,n)$ (spacelike separation), then any local observables at the two points necessarily commute:

$$\text{dist}(x,y)>R|m-n|⟹[{\alpha }^{m-n}(A_x),B_y]=0$$

Proof:

Let $\Delta t=m-n$. If $(y,m)$ is outside geometric light cone, i.e., $y\notin B_{R\Delta t}(x)$ (ball centered at $x$ with radius $R\Delta t$).

According to Finite Propagation Radius Theorem (Theorem 3.2.3), support set $\text{supp}({\alpha }^{\Delta t}(A_x))$ of ${\alpha }^{\Delta t}(A_x)$ is contained within $B_{R\Delta t}(x)$.

Since $y\notin B_{R\Delta t}(x)$, $y$ does not intersect support set of ${\alpha }^{\Delta t}(A_x)$.

In quantum mechanics, operators with disjoint support sets (operators acting on different subsystems) always commute.

Therefore $[{\alpha }^{\Delta t}(A_x),B_y]=0$.

QED.

3.2.5 Physical Picture: “Hardness” of Light Cone

In continuous quantum field theory, correlation functions outside light cones usually decay exponentially $e^{-m|x|}$ (e.g., in massive fields), leading to subtle discussions about superluminal influences (such as Hegerfeldt theorem).

However, this theorem shows that in discrete ontology, light cones are Absolutely Hard. Within $N$ steps of evolution, no information, energy, or entanglement can cross graph distance $N\times R$.

This conclusion lays the foundation for emergence of special relativity at discrete level. The causal structure of Minkowski spacetime ($d{s}^2\ge 0$) is precisely the smooth approximation of the above discrete causal partial order $(E,⪯)$ in the continuous limit. We will discuss this emergence mechanism in detail in Chapter 4.