This document discusses using holographic soliton automata to model discrete spacetime. It proposes that spacetime is fundamentally discrete and composed of causal sets. This allows the examination of nonlocal phenomena through soliton cellular automata which behave as localized excitations on the discrete spacetime background. Tensor calculus is applied to model the automata as crystals undergoing time evolution according to affine quantum group symmetries and box-ball soliton dynamics. The goal is to understand if the dynamics of a combinatorial metric crystal respects the laws of physics on a discrete spacetime substrate.

1. Holographic Soliton Automata - Causal Crystal Approach
Periodic Modulation of the refractive index has been a well recorded phe-
nomena in Optics. To this day, we understand that altering certain diffraction
properties in materials, induces a non linear propagation and localization of
light. Optical Spatial Solitons are understood as pertaining to a self-phase
(self-focusing) regularity. This paper meddles specifically with a symmetric ex-
change of energy between two or more mutually coherent beams of light.
In Optics, Vortices are associated with the screw phase dislocations created
by diffracting two or more optical beams In Kerr Media. As the vortices spread,
their core becomes self-trapped, and the resulting structure is a Soliton. Ini-
tially, the background theme of our studies relied heavily on the properties of
what many physicists have labelled as ’discrete vortex solitons’, usually obtained
experimentally through light interactions with Photo-refractive Crystals.
We understand from nonlinear phase coupling that two or more mutually
coherent beams can exchange energy symmetrically. The phase coupling mech-
anism can be established as a grating effect in the refractive index induced by
real-time interference. A paradox emerges: Vortex Solitons are localized excita-
tions which carry a screw-phase dislocation; whilst Non-linear surface solitons,
which are usually found in Optical Surface Waves, exist in both the interface of
local and non-local non-linear media. We must question, ’Is there a fundamental
information exchange mechanism which gives Solitons their inherent structure?’
In Theoretical Physics, many workers of Quantum Gravity suspect, that
spacetime is fundamentally discrete, If such assumption is deemed trustworthy,
we must also ponder the validity of the continuum symmetries of Lorentz In-
variance. Can Nonlocality be expanded to such an extent to allow local physics
to emerge at large distances?
1

2. The Discreteness of Spacetime gives rise to unavoidable non locality, this
non locality we speak of should obey Lorentz Symmetry. If spacetime is ul-
timately composed of atoms, the number of each object is always one planck
time to the past of any given P , infinitely distributed along a hyperboloid
on Minkowski spacetime C ∞ . The foundations of General Relativity are built
upon non-re-normalizable infinities in a smooth spacetime manifold. Classic
Lorentzian Gravity is regarded as a Yang-Mills type of Gauge Theory (Sl (2, C))
on local Minkowskian fibre bundles p of Cartan Ω forms over a bounded region
X of spacetime M ; on this occasion, we abide to the view ’finite topological
spaces’, modelled after partially ordered sets (posets) by Sorkin [].
We question the validity of a Causal Set theoretic approach to the open prob-
lem of discrete symmetric spaces in Soliton Cellular Automata, based heavily
on the theory of quantum groups and perfect crystals. Does the dynamic of a
combinatorial crystallization of the metric tensor remain in tune with the laws
of physics?
A cellular Automaton is a dynamical system in which points in the one-
dimensional lattice are assigned discrete values which evolve in a semi-deterministic
rule. Soliton Cellular Automata (SCA) are a breed of CA which possesses stable
configurations analogous to Solitons.
Tensorial Calculus of Crystals
We select an integer n ∈Z≥2 for an arbitrarily chosen l ∈Z≥0
Bl = (v1 , v2 , ..., vl |vj ∈ 1, 2, ..., n, v1≤v2 ≤...≤vl )
In most literature on the subject [source1][source2] Bl is defined as a set of
semi-standard tableaux of shape (l) graded in 1, 2, ..., n for i = 0, 1, ..., n-1
such that
ei , fi −→Bl (0) i= 0, 1, ..., n − 1
For The action at i = 0
e0 (v1 , v2 , ... , vl ) = δv1 1 (v2 , ..., vl , n)
f0 (v1 , v2 , ... , vl ) = δvl n (v2 , ..., vl−l , n)
If fi b = b’ for b, b’ ∈Bl , then b = ei b’. Bl is therefore considered a crystal
base of an l-th symmetric tensor representation of the quantum affine algebra
Uq (SLn )
2

3. Let us now choose b ∈Bl such that
εi (b) = max (m ≥ (0) |em b = 0)
i ϕi (b) = max (m ≥ 0 |fm b= 0)
i
ei (b ⊗b ) = ei b ⊗b if αi (b) ≥εi (b’)
ei (b ⊗b ) = b⊗ei b’ if αi (b) < εi (b’)
fi (b ⊗ b ) = fi b ⊗b’ if αi (b) > εi (b’)
fi (b ⊗ b ) = b ⊗fi b if αi (b) ≤εi (b’)
We have formulated an isomorphism for Crystals Bl and Bl based on a tensorial
operation B ⊗Bl
will study this concept as we progress. We define an affinization Af f (Bl ) of the
crystal Bl . At this point we introduce an indeterminate z (spectral parameter)
and set
Af f (Bl )=z d b|d∈Z, b ∈ Bl
Thus Af f (Bl ) is an infinite set (at this point).
A combinatorial R matrix is another very important tool which we will use
extensively, if we have a map Bl Bl which is a combinatorial map R:
Af f (Bl ) Af f (Bl ) −→ Af f (Bl ) Af f (Bl )
We are in better shape to discuss the well known Box-Ball Soliton (BBS),
which is a pillar of our theoretical construct. We can imagine a discrete system
were infinitely many balls move along a one dimensional array of boxes under
strict conditions.
L
set B=B1 and consider the crystal B for a sufficiently large L. The
L
elements of B are constructed as follows
... (n) (n) ... (n) (vk )...
where v1 , v2 , ..., vk ∈ 1,2,...n. (consider this an iteration)
3

4. L
L
B −→ B
Box Ball systems in this interpretation are considered as time dependent
L
on factor B describing the current state. Where T plays the role of time
evolution (Tl ) or time-steps. This description is not so easily understood at
deeper levels of abstraction. We do understand that time in a stochastic process
(or semi-deterministic) is measured depending on the states, not on other
alternative factors.
• longer isolated solitons move faster
• the number of solitons does not change under time evolution
• if the solitons have enough distance between their initial states, then their
lengths do not change.
If B is a finite crystal of level l whose subsets are noted ...⊗B⊗...⊗B and we
call these paths, then we are allowed to willingly f ix as ref erence an element p
= ...’⊗bj ⊗...⊗b2 ⊗b1 . For any j, where ε(bj ) should have a level l, which satisfies
ϕ(bj+a ) = ε(bj )
The set
P (p,B) = p = ...⊗bj ⊗...⊗b2 ⊗b1 | bj ∈B, bj =Bj for J 1
Defines An element of P (p,B)
with energy
∞
E(p)= j=1 j(H(bj+1 ⊗bj )-H(bj+1 ⊗bj ))
and weight
∞
wtp=ϕ(b1 )+ j=1 (wtbj -wtbj ) - (E(p/a0 )δ
4

5. Causal Lorentz Manifold
A sprinkling Causal Lorentz Manifold is a random (stochastic) process that
produces what Sorkin and his team have come to call a causet - A partially
ordered set which follows the foundations of transitivity.
¸
if(M ,g ) is of finite volume, the causet at hand is surely finite.
A partial order is a relation defined on a set S which satisfies
(i)asymmetry: p and q p.
(ii)transitivity: p q and q r⇒p r
Our Causal Lorentz Manifold (M ,g) suffers a decomposition:
the metric g is an af f ine lie algebra. Or as we have discussed previously,
a Crystal
¸ r
g is a kac moody algebra or affine quantum group XN , which we define as
intelligent (behaving as an Automaton)
A crystal B is a set B= λ Bλ (wt(b) = λ if b∈Bλ equipped with a mapping
consisting of ei : Bλ Bλ+xi
ˆ 0, and fi : Bλ Bλ−xi 0
ˆ ˆ
ei , f i : B 0
In the affine Crystal literature we are mostly concerned with the isomorphic
properties of an R-matrix. More specifically, the Lorentz group SO(3,1) and
it’s double cover SL(2,C) with infinite dimensional unitary representations.
Weyl groupoid is the highest weight Demazure Crystal
P (p( λ), B) B(λ)
(
P (p λ, ..., λd )), B) B(λ1 ) ... B(λd )
there exists an isomorphism of Pcl weighted crystals
B(λ1 ) ... B(λd ) (B(σ1 ...σd λd )) B
uλ is the highest weight element in B(λ) where we can find a one dimensional
path Xl (B, λ,
q)
5

6. set
B = B1 ... Bd .
We understand all Bi s are perfect, so we let li be the level of Bi .
Assuming l1 ≥l2 ≥...≥ld ≥ld+1 =0
b∗ = b1 ... bd
for an element of B fix b2
ˆ
Ml (W , λ, q) = M l (W ,λ, q)
for l≤∞ and λ ∈
3
affine g is D4 such that q=1
3
for g=D4
Q1 2 = Q1 1 Q1 1 +Q2
J J−1 J+1 J
Q1 2 = Q2 1 Q2 1 +Q1
J J−1 J+1 J
g =ˆ such that
g
3
(D4 , G2 ) with q=1
(g, g )=(D43 ), G2 )
1
Xj =
6