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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
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 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
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 ⊗Bl The Box-Ball Soliton (BBS) is a pillar of our theoretical construct. We canimagine a discrete system were inﬁnitely many balls move along a one dimen-sional array of boxes under strict conditions.• 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 an ﬁnite crystal of level l whose subsets are noted ...⊗B⊗...⊗B and wecall these paths. Let us ﬁx as a reference p = ...”⊗ bj ⊗...⊗b2 ⊗b1 .F oranyj,ε(bj )should have 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 3
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 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) 4