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  1. 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 diffractionproperties in materials, induces a non linear propagation and localization oflight. 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 createdby 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 ofwhat many physicists have labelled as ’discrete vortex solitons’, usually obtainedexperimentally through light interactions with Photo-refractive Crystals. We understand from nonlinear phase coupling that two or more mutuallycoherent beams can exchange energy symmetrically. The phase coupling mech-anism can be established as a grating effect in the refractive index induced byreal-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 oflocal and non-local non-linear media. We must question, ’Is there a fundamentalinformation exchange mechanism which gives Solitons their inherent structure?’ In Theoretical Physics, many workers of Quantum Gravity suspect, thatspacetime 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 physicsto emerge at large distances? 1
  2. 2. The Discreteness of Spacetime gives rise to unavoidable non locality, thisnon locality we speak of should obey Lorentz Symmetry. If spacetime is ul-timately composed of atoms, the number of each object is always one plancktime to the past of any given P , infinitely distributed along a hyperboloidon Minkowski spacetime C ∞ . The foundations of General Relativity are builtupon non-re-normalizable infinities in a smooth spacetime manifold. ClassicLorentzian 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 regionX of spacetime M ; on this occasion, we abide to the view ’finite topologicalspaces’, 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 heavilyon the theory of quantum groups and perfect crystals. Does the dynamic of acombinatorial crystallization of the metric tensor remain in tune with the lawsof 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-deterministicrule. Soliton Cellular Automata (SCA) are a breed of CA which possesses stableconfigurations 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 ofsemi-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 crystalbase of an l-th symmetric tensor representation of the quantum affine algebraUq (SLn ) 2
  3. 3. Let us now choose b ∈Bl such thatεi (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 tensorialoperation B ⊗Blwill study this concept as we progress. We define an affinization Af f (Bl ) of thecrystal Bl . At this point we introduce an indeterminate z (spectral parameter)and set Af f (Bl )=z d b|d∈Z, b ∈ BlThus Af f (Bl ) is an infinite set (at this point).A combinatorial R matrix is another very important tool which we will useextensively, 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 systemwere infinitely many balls move along a one dimensional array of boxes understrict conditions. L set B=B1 and consider the crystal B for a sufficiently large L. The Lelements of B are constructed as follows ... (n) (n) ... (n) (vk )...where v1 , v2 , ..., vk ∈ 1,2,...n. (consider this an iteration) 3
  4. 4. L L B −→ B Box Ball systems in this interpretation are considered as time dependent Lon factor B describing the current state. Where T plays the role of timeevolution (Tl ) or time-steps. This description is not so easily understood atdeeper levels of abstraction. We do understand that time in a stochastic process(or semi-deterministic) is measured depending on the states, not on otheralternative 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 theirlengths do not change. If B is a finite crystal of level l whose subsets are noted ...⊗B⊗...⊗B and wecall 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. 5. Causal Lorentz Manifold A sprinkling Causal Lorentz Manifold is a random (stochastic) process thatproduces what Sorkin and his team have come to call a causet - A partiallyordered 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 rOur 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¸ rg is a kac moody algebra or affine quantum group XN , which we define asintelligent (behaving as an Automaton) A crystal B is a set B= λ Bλ (wt(b) = λ if b∈Bλ equipped with a mappingconsisting 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 isomorphicproperties of an R-matrix. More specifically, the Lorentz group SO(3,1) andit’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 )) Buλ is the highest weight element in B(λ) where we can find a one dimensionalpath Xl (B, λ,q) 5
  6. 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 ... bdfor 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 Jg =ˆ such that g 3 (D4 , G2 ) with q=1 (g, g )=(D43 ), G2 ) 1 Xj = 6