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Majorana Fermions, Braiding and the Dirac Equation (And Containing Joint

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Majorana Fermions, Braiding and the Dirac Equation (And Containing Joint MeeMeetingting: : 10102020,,CincinnatCincinnati,i, Ohio, SSSS113A,3A,SpSpeciaecial Sessionl SessionononPhPhysicalysicalKnoKttingnottingandaLndinkingLinking Majorana Fermions, Braiding and The Dirac Equation LoLouisuis H. Kauffffmman*an*(ka(kaufuffmafman@[email protected]),duUICMath), MathUIC andUIC, 85 ,1NSU85Sout1 South Morh ganMorStreet,gan Street,ChicagoChicago, IL , IL 606060607-707-7045, DanDanSandinSandin(da(dan@[email protected]), Ele), ElectrctroniconicVisuaVislizatioualization LanboraLatorboray, torUIC,y, UIC,842 W.842 W. (and containingTTaayloylorr St reet,jointRRooom omwork20203232, ,ChicagwithChicago, o,PeterILIL606607-7006 07-70Rowlands45,45and, andRobRobert erKotoKoimaoim(rlak@([email protected]), edu), ElecElecttronicronic VisuaualizatiolizationnLLababoraoratortory,yUIC,, UIC,842842W.W.TaylorTaylorStreStret, eReto,omRo20om32,20Chicag32, Chicago, IL o, IL on a nilpotent606060607-7047-7045. ExploExplo Majorana-DiracrrationsationsininKnotKnotSelf-RSelf-R Equation)epulsion.epulsion. s This talk will discuss phenomena in computer modeling of knot self-repulsion by force laws of the form F = kr− s This talk will discuss phenomena in computer modeling of knot self-repulsion by force laws of the form F = kr− for varying values of s and different configurations of joints and springs between vertex points in the piecewise linear for varying values of s and different configurations of joints and springs between vertex points in the piecewise linear representation of the knot. These phenomena include transitions from non-minimal and semi-stable forms to minimal representation of the knot. These phenomena include transitions from non-minimal and semi-stable forms to minimal energy forms of the knots and behaviour for different values of s (above). In particular, we find values of s for which the energy forms of the knots and behaviour for different values of s (above). In particular, we find values of s for which the knot tends to tighten into an apparently ideal knot form localized on a long length of string. Comparisions with ideal knoknot tendst parametottighers willten binetodisacussed.n apparen(Rectlyeiveideald Augustknot15fo,rm200lo6)calized on a long length of string. Comparisions with ideal knot parameters will be discussed. (Received August 15, 2006) 1 1 A possible sighting of Majorana states Nearly 80 years ago, the Italian physicist Ettore Majorana proposed the existence of an unusual type of particle that is its own antiparticle, the so-called Majorana fermion. The search for a free Majorana fermion has so far been unsuccessful, but bound Majorana-like collective excitations may exist in certain exotic superconductors. Nadj-Perge et al. created such a topological superconductor by depositing iron atoms onto the surface of superconducting lead, forming atomic chains (see the Perspective by Lee). They then used a scanning tunneling microscope to observe enhanced conductance at the ends of these chains at zero energy, where theory predicts Majorana states should appear. A Very Elementary Particle - Fusion Rules for a Majorana Fermion P P P P P * The “particle” P interacts with P to produce either P or *. * is neutral. For a Standard Fermion there is a an annihilation operator F and a creation operator F*. These correspond to the fact that the antiparticle is distinct from the particle. We have FF = F*F* =0 (Pauli Exclusion) and FF* + F*F = 1. Later in the talk we will see much more about this relation. Majorana and Clifford Algebra The Creation/Annihilation algebra for a Majorana Fermion is very simple. Just an element a with aa =1. If there are two Majorana Fermions, we have a,b with aa = 1, bb=1 and ab +ba = 0. A Clifford Algebra. Algebraic Justification of this Statement Follows... Majorana Fermions are their own antiparticles. An Electron’s creation and annihilation operators are combinations of Majorana Fermion operators: U = (a + ib)/2 and U* = (a - ib)/2 where ab+ba = 0 and aa =bb=1. U = (a + ib)/2 and U* = (a - ib)/2 4UU = (a+ib)(a+ib) = aa -bb +i(ab + ba) UU= 0 U*U* = 0. UU* +U*U = (U + U*)(U+U*) = aa = 1 This is the creation/annihilation algebra for an electron. Iconics This next part is motivated by G. Spencer-Brown’s invention of a ‘logical particle’ that interacts with itself to either confirm itself or to cancel itself. This interaction, combined with recursion, leads both to matrix algebra and the very elementary mathematics of a Majorana Fermion. The Mark is a logical particle (Laws of Form by G. Spencer-Brown) that interacts with itself either to annihilate itself, or to produce itself. = * = The Mark is a “logical particle” for a level of logic deeper than Boolean Logic. In this formalism the mark is seen to make a distinction in the plane. The formal language of the calculus of indications refers to the mark and is built from the mark. The language using the mark is inherently self- referential. The Calculus writes itself in terms of itself. The first distinction, the mark, and the observer are not only interchangeable, but, in the form, identical. Formally, we can distinguish the two interactions via adjacency and concentricity. * PP = * + P Flattening String Iconics For this specialization we see that the matrix F becomes 1/⇤ ⇤/Θ 1/⇤ ⇤/Θ 1/⇤ ⇤/Θ F = = = Θ/⇤2 T ⇤/Θ2 Θ/⇤2 ( Θ2/⇤2)⇤/Θ2 Θ/⇤2 1/⇤ ⇥ − ⇥ − ⇥ This version of F has square equal to the identity independent of the value of Θ, so long as ⇤2 = ⇤ + 1. The Final Adjustment. Our last version of F su⇥ers from a lack of symme- try. It is not a symmetric matrix, and hence not unitary. A final adjustment of the model gives this desired symmetry. Consider the result of replacing each trivalent vertex (with three 2-projector strands) by a multiple by a given quan- tity ⇥. Since the Θ has two vertices, it will be multiplied by ⇥2. Similarly, the tetradhedron T will be multiplied by ⇥4. The ⇤ and the δ will be unchanged. ONothtierce tphartopiterfotllioeswsofrfomthtehemsyomdelmetwirylol frtemhe daiiangruamnmchaatincgredeco.upThlingefonr-ew recoupling mulas of Figure 36 that the square of the recoupling matrix F is equal to the matrix, after such an adjustment is made, becomes identity. That is, / /⇥2 1 0 2 1/⇥ ⇥1/Θ⇤ ⇤1/⇥ Θ ⇥/Θ = F = 2 2 2 2 2 2 = 0 1 ⇥ Θ/⇥ ⇥T ⇥Θ/Θ/⇤⇥ Θ/1⇥/⇤ T⇥⇥/Θ ⇥ Notice that it follows from the symmetry of the diagrammatic−recoupling for- 2 2 3 mulas of FigureFo36r sythamt mtheetrsquary wee rofeq1/the⇥uir+reec1oupling/⇥ 1matr/Θ +ixTF⇥ is/Θequal to the 3 2 2 4 . Θ/⇥ + T/(⇥Θ) 1/⇥ + ⇥ T /Θ ⇥ identity. That is, 2 2 2 Thus we need the relation ⇤/(⇥ Θ) = ⇥ Θ/⇤ . 1 0 1/⇥ ⇥/Θ 1/⇥ ⇥/Θ = F 2 = 2 = 0 1We take Θ/⇥2 T ⇥/Θ21/⇥ +Θ1//⇥⇥2 =T1⇥. /Θ2 F⇥or this specialization we see that the matrix F⇥becomes ⇥ 2 ⇥ 3 This i1s/⇤equi⇤v/2aΘlent to sa1/⇤ying th⇤a/tΘ 2⇥ 3=1/⇤ ⇤⇤/Θ/Θ. F = 12 /⇥ +2 1=/⇥ 2 1/Θ2 +2 T ⇥2 /=Θ 2 Θ/⇤ T ⇤/Θ ⇥ Θ/⇤ ( Θ /⇤ )⇤/2Θ ⇥ Θ/⇤ 1/⇤ ⇥ For this specializa3 tion we see that the m−atrix F ⇥beco2m=es21 +4⇥, . − WiThistvhersiΘotn/h⇥oifsF ch+ahs Tosqiu/cea(re⇥eqoΘufa)l ⇥to 1thw/e ⇥eidenh+tiaty⇥vinedepTen/denΘt of⇥the value of 1/⇤ ⇤/Θ 1/⇤ ⇤/Θ 1/⇤ ⇤/Θ ΘF ,=so long as ⇤2 = ⇤ + 1=. = Thus we need thaequrelaΘda/r⇤tai2otinTc⇤eq/Θu2 ationΘwh/⇤2ose( Θso2/⇤lu2)2t⇤io/Θn2s are Θ/⇤2 1/⇤ ⇥ − ⇥ − ⇥⇥ 3 ⇥ The Final Astdjate usand atblmank spenace tfor.theOunumarkedlastastte. Thvener⇤onsie hoa/s ntwo(omo⇥fdesFof suΘ⇥er)s=from⇤a lΘack/o(f syΘmme-⇤ ) = 1/ ⇤. This version oinfterFactionhoaf asboxsqwithuitaselrf:e equal to the2identity independen⇥t of the value of try. It is not a symmetric matrix, and hence⇥not=uni(t1ary. A fin5a)l/a2dju. stment Θ, so long as ⇤1.2Ad=jacen⇤cy: +11/. ⇥ + 1/⇥ = 1. ± Henof the mceodeltghiveesand tnhiews desirsyed symmmetetry. rConsideric F theisresultgivofenreplacingby teachhe equation This is equivalenFtrTheivalentutrttFiohnaersaverlmAytexdjoi2n.r(withusNeste,gintg:mtwhthren.eatet.ekO2-prnuorwojelastctorthverastrsitoands)n of Fbysua⇥ermultiples from abylaackgivenof symquan-me- With this convention we take the adjacency interaction to yield a single box, 2 2 titytry. ⇥It. SisinnceottaahndesytheΘnmestimnhg ainettserarcttiwocn toomprvoaderutcertnioixtces,hi,ng:anidt wihenllcebenmotu⇥lutnipitl=aiedry.δbAy ⇥fin.a1Sl iamdijulastrlym, enthte 2 4 ⇥ toetf trhaedhmedodroelngTiveswitllhbisedmesiurl⇥tediplsyi=edm=mbyet1⇥ry+. ThConsider⇥e ,⇤1an/thed⇤threesultδ wi−of1ll /breplacinge un⇤chanegached. ⇤ ⇥⇤ frOtrivalenttohmer pFiroverpgertexutires(withe o3f 3tthrh.e eHenme o2-prdelceoje=wiFctorll r=emstraands)in uncbyhaangmultipleed. Thebynewa givenrecouquan-pling= We take the notational opportunity to denote nothing by an asterisk (*). The ⇥ 2 mtityatr⇥ix. ,SainftceertsusyhnetactihcaΘl raulhesnaforsaopdterwjuatinogstthvemaerstenertiskitces,areiThsusmithteaawistdere,iskllisbba ecosteand1m-inm2u/esltipli⇤ed by ⇥ . S1im/il⇤2arly, ⇥the ⇥⇤ ⇤ ⇥ a quadratic equation whfoor nosemark atsoall andlitucantbeieroasednor pslacedawherreveer4 it is conven⇥ient to = ⇥ + 1 = δ . tetradhedron Tdo so.wiThulsl be multiplied by ⇥ . The2 ⇤ and the δ wil−l be unchanged. − = 1. /⇤ ⇤/⇥ Θ Other properties of the model wi∗ ll remain unchanged. The new recoupling We shall make a recoupling theory⇥ba2sedΘo/n t⇤his p2article, but1it/is⇤worth whWeersheal⇤l notinogisowsme of spits phureleciyecombiangatolroiialzeplropdertiestenfiorst. ⇥Thtehrariteahmetticac oif o⇥seawhndere⇤ = 1/⇤. This gives the Fibonacci model. matrix, after sucombcinhing abonxes (astadn⇥djuing stfor =mactsenof d(tist1inictsionm) accoarddine,g−to5bthecoese)/rulesm2es. has been studied and formalized in [52] and correlated with Boolean algebra For symmetryandwcleassicarleqlogiuc. iHerree within and next to are±ways to refer to t2he two Using Fisidges duelineartedesby the giv3en d7istinctioan.1Fn/rom⇤dthis poi3nt o⇤f8vi/ew,⇥,⇥therwΘe=are eδh=av(1e+th⇥a5t)/t2h,e local braiding matrix for the model Furthermore, we know thtwo amodtes of relationship (adjacency an2d nest2ing) th2at arise a2t once in the2 presence of a distinction. ⇤⇥/(⇥Θ/Θ⇤) =Fibonacci⇥ 1Θ//⇤⇤ ⇥. properties (the operator is idempotent and a self-attached strand yields a zero is given by the formul2a−below withevaluaAtion) and=give diagreamm3atiπc prioo/fs o5f th.ese properties. lWFeaoer vtsyaikmnemgettrhy eweorteqhuerire ca⇥ses=foδr the1exploration of the reader. We then take P P P P Model 2 −3 ⇥ 2 = ⇥⇤ 2/Θ.
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