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. Kaumman*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 dierent configurations of joints and springs between vertex points in the piecewise linear for varying values of s and dierent 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 dierent values of s (above). In particular, we find values of s for which the energy forms of the knots and behaviour for dierent 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 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/⇤ ⇤/ 1/⇤ ⇤/ 1/⇤ ⇤/ F = = = /⇤2 T ⇤/2 /⇤2 ( 2/⇤2)⇤/2 /⇤2 1/⇤ ⇥ ⇥ ⇥ This version of F has square equal to the identity independent of the value of , so long as ⇤2 = ⇤ + 1.

The Final Adjustment. Our last version of F su⇥ers from a lack of symmetry. 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 has two vertices, it will be multiplied by ⇥2. Similarly, the tetradhedron T will be multiplied by ⇥4. The ⇤ and the 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/⇤ ⇤1/⇥ ⇥/ = F = 2 2 2 2 2 2 = 0 1 ⇥ /⇥ ⇥T ⇥//⇤⇥ /1⇥/⇤ T⇥⇥/ ⇥ Notice that it follows from the symmetry of the diagrammaticrecoupling for- 2 2 3 mulas of FigureFo36r sythamt mtheetrsquary wee rofeq1/the⇥uir+reec1oupling/⇥ 1matr/ +ixTF⇥ is/equal to the 3 2 2 4 . /⇥ + T/(⇥) 1/⇥ + ⇥ T / ⇥ identity. That is, 2 2 2 Thus we need the relation ⇤/(⇥ ) = ⇥ /⇤ . 1 0 1/⇥ ⇥/ 1/⇥ ⇥/ = F 2 = 2 = 0 1We take /⇥2 T ⇥/21/⇥ +1//⇥⇥2 =T1⇥. /2 F⇥or this specialization we see that the matrix F⇥becomes ⇥ 2 ⇥ 3 This i1s/⇤equi⇤v/2alent to sa1/⇤ying th⇤a/t 2⇥ 3=1/⇤ ⇤⇤//. F = 12 /⇥ +2 1=/⇥ 2 1/2 +2 T ⇥2 /= 2 /⇤ T ⇤/ ⇥ /⇤ ( /⇤ )⇤/2 ⇥ /⇤ 1/⇤ ⇥ For this specializa3 tion we see that the matrix F ⇥beco2m=es21 +4⇥, . WiThistvhersiotn/h⇥oifsF ch+ahs Tosqiu/cea(re⇥eqoufa)l ⇥to 1thw/e ⇥eidenh+tiaty⇥vinedepTen/dent of⇥the value of 1/⇤ ⇤/ 1/⇤ ⇤/ 1/⇤ ⇤/ F ,=so long as ⇤2 = ⇤ + 1=. = Thus we need thaequrelada/r⇤tai2otinTc⇤eq/u2 ationwh/⇤2ose( so2/⇤lu2)2t⇤io/n2s are /⇤2 1/⇤ ⇥ ⇥ ⇥⇥ 3 ⇥ The Final Astdjate usand atblmank spenace tfor.theOunumarkedlastastte. Thvener⇤onsie hoa/s ntwo(omo⇥fdesFof su⇥er)s=from⇤a lack/o(f symme-⇤ ) = 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 , 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. Sisinnceottaahndesythenmestimnhg ainettserarcttiwocn toomprvoaderutcertnioixtces,hi,ng:anidt wihenllcebenmotu⇥lutnipitl=aiedry.bAy ⇥fin.a1Sl iamdijulastrlym, enthte 2 4 ⇥ toetf trhaedhmedodroelngTiveswitllhbisedmesiurl⇥tediplsyi=edm=mbyet1⇥ry+. ThConsider⇥e ,⇤1an/thed⇤threesult wiof1ll /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. ,Sainftceertsusyhnetactihcal 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 = . tetradhedron Tdo so.wiThulsl be multiplied by ⇥ . The2 ⇤ and the will be unchanged. = 1. /⇤ ⇤/⇥ Other properties of the model wi ll remain unchanged. The new recoupling We shall make a recoupling theory⇥ba2sedo/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,gto5bthecoese)/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,⇥,⇥therwe=are eh=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. ⇤⇥/(⇥/⇤) =Fibonacci⇥ 1//⇤⇤ ⇥. properties (the operator is idempotent and a self-attached strand yields a zero is given by the formul2abelow withevaluaAtion) and=give diagreamm3atic prioo/fs o5f th.ese properties. lWFeaoer vtsyaikmnemgettrhy eweorteqhuerire ca⇥ses=for the1exploration of the reader. We then take P P P P Model 2 3 ⇥ 2 = ⇥⇤ 2/. 2 ⇤/(⇥ ) = ⇥ /⇤ . 4 3i/5 1/⇥ 4i/5 from Figure 33. Hence = With this choice of ⇥ we have AA = e0 e 0 We take 2 R P= 2 8 = . ⇥ 2 = ⇥2 + 1⇥=3 . 2i/5 so that Fi⇤gure/2(5 ⇥- *Fibona)cci=P⇥art⇤icl=eInt/⇥era(ct⇤ion3/⇤.) =01/⇥⇤.A 0 e ⇥ = 0 = ⇥ 62 2 2 We shall now spHenWiecithceathtlhiisezecnhewoitceosyomtf mh⇥etewreicach=aFvseeis gwhAivenerbyeAtheequ=ation 2cos(6⇥/5) = (1 + ⇥5)/2. The simplest2 exam⇥ple3 of⇥a braid group representation arising from this ⇤/(⇥ 1/⇤) = ⇤1/⇥/(⇤ ⇤ ) = ⇤1/ ⇥⇤⇤. Forbidden (1/⇥)⇥ F = ⇥= = 1/⇥ = = 0 Note that ⇥ =11//⇥⇤==(11/.⇤+Th⇥ u5s)⇥/⇤2, ⇤ ⇥ tHenheoce trhye newis sytmhmeetrriepc F risesengiven btyathteieqonuatioonfthe three strand braid group generated by S1 = where ⇤ is the golden ratio and ⇤ = 1/⇤. This gives2the Fibonacci model. 1/⇤ 1/⇥⇤ ⇤ ⇥⇤ = =T 1 leaving the other cases for the explorati=on(of t1h/e)rea der⇥./W =e t 1/h⇥ en1=.take RUsinagnFidgurSes 237 =aFnd=F38,RweFhave(tRhatemthe eml=ocal bbraeridintg hmatrtix Ffor t=he mFodel = F .). The matrices S1 and S2 1/⇥⇤ 1/⇤ ⇥ ⇥⇤ ⇤ ⇥ is given by the formula below withA = e3i/5. aarned both unitary,Aa=nde3thi/ey5 generate a denFigure 2se8 - The 2su-Projectbor set of the unitary group U(2), where ⇤ is the golden ratio 4and ⇤ = 1/⇤.4Thi/5 is gives the FibInFioFigngureurae 2cci299, w- eFishbmoownatohcciedessenPelartce.icloef tahse 2T-Pemroperjectley-Lorieb recoupling model A 0 2 2e 0 for the Fibonacci pa2rticle. The Fibonaccie particle is, in this mathematical Using Figures 37 and 38, we have that the local braiding mNoatetthrmaitoxidnelFi,Temperleyfogiduenrert2ifi9edtwhewiheathvethamdeo2p-potedrodjectaelsinorglietselst rf.aLiebnAsd ntohteatrieaondfoer rcathneseeparfrticlome Figure TR == ( 1/8 )=( 2) 2i/25 inter/.act2io9n, s,thwi=ertehaaresot(lwidostbraasincd icontrerrespactoion2ndsinog)fttohteh2e-pmraorjectk2edopr(awirtitclhe,itaseldof,ttoedne1giv)ing/ supplying the fi0rstApart 3oi/f05 thee trastnrand sfo(or nothingr) comrresponadingtto ithoe unmnarkedsparnticle. eedA dark vertexed for quantum computing. ⇥ ⇥ a 2-projector, the other giving nothing. This is the pattern of self-iteraction is given by the formula below with A = e . indicates either an interaction point, or it may be used to indicate the the so that ofRepresentationthe Fibonacci particle.There is a thirdpossibilit y, ofdepicted in Figure 29, single stwhranerdeitswshoo2r-pthraonjectd foorstwinoteroradctintaoryprstordaunceds.a R4-pemroemjectberor.thWatetcoheseuldarreme ark at The simp2lest examp2le of4 a braid=gr(oup4ir/ep5 1r)esen(tatio2anll sh)oar/thraeniodusitexsetnp=,ressitghaotntfrhs3efo4or-pumrnodjecterlyotirnwighblli5rbaseckzer.et opoiflywneocmhioaolsecatlhcuelbartaiocnks.et polynomial A 0 e 0 vari⇥able A = e3/5. Rather than start there, we will assume that the 4-projector = A ABraid = Representations2cos(6⇥/5) = (1 +In FiguresFibonacci30, 531, )32,/33, 234 a.nd 35 we haModelve provided complete diagram- is forbidden and deduce (below) that the theory has to be at this root of unity. theory is the represenR =tation of the8 thr=ee strand braid2gir/o5 upma.tigc caenlculerationas tofedall ofbthye relSeva1nt sm=all nets and evaluations that are 0 A ⇥ T 0 1 e ⇥useful in the two-strand theory that is being used here. The reader may wish NoR antdeSt2 h=aFtRF (RememDenseber that inF =UnitaryF = F .). Thetomskipaditrectrilycesto FiguSre 316a andnFidgureS362b6wh6 ere we determine the form of the recoupling coecients for this theory. We will discuss the resulting algebra Note that are Th1bo/tehsium=niptlaest1ry., exaThnadmtuphleyse ogfenaerGroupsbartaeida dgrenousep surepbTsetresen=ofttahteiobunelonw.2ait/rai⇥siryng2gr,frouopmUt(h2i)s, sutheopprlyyiinsgththeerepfirstresenparttatoifotnhoefttrhaensfothrreemasttiroannsdnbeedraiedd gfororuqpugaennterumatedco7mb9pyuSt1i6n7 =g. 2 T 1 frR oanmd Swh2 =iFcRh=F i(tR(emfolemlob1wser/thi)amt Fm=edF⇥ia=/telF y=.)t. hThaet m1a.trices S1 and S2 are both unitary, and they generate a dense subset of the unitary group U(2), and supplying the first part of the transform79ations neededFfo2r q=uanItu.m computing. 2 2 2 T = ( 1/) ( 2) 2/ = ( 2) 2( 1)/ ⇥ This proves that we can sa79tisfy thismodel when ⇥ = = (1 + 5)/2. = ( 1)( 2)/ = 3 5. Note that 78 T = 2/⇥2, from which it follows immediately that F 2 = I. This proves that we can satisfy this model when ⇥ = = (1 + ⇥5)/2.

78 The Majorana P is a Fibonacci Particle

P^2 = * + P P^3 = P* + PP = P + * + P = * + 2P P^4 = P + 2* + 2P = 2* + 3P P^5 = 3* + 5P P^6 = 5* + 3P P^7 = 8* + 5P

Fibonacci Processes * from Self-Interaction of P. P P* = P

P * PP = P + *

P * P PPP= 2P + * PPPP = 3P + 2* P * P P * Note that we have shown how the formalism of the mark, as logical particle is coherent with its interpretation as a Majorana Fermion.

= = *

Fermion Algebra Directly From Re-entering Mark ... +1, -1, +1, -1, +1, -1, +1, -1, ...

[-1,+1] [+1,-1]

Figure 29:

The square root of minus one is a perfect example of an eigenform that occurs in a new and wider domain than the original context in which its recursive process arose. The process has no ﬁxed point in the original domain. Looking at the oscillation between +1 and 1, we see that there are naturally two phase- − shifted viewpoints. We denote these two views of the oscillation by [+1, 1] and[ 1, +1]. These viewpoints correspond to whether one regards the oscillation at time zer−o as start−ing with +1 or with 1. See Figure 29. We shall let the word iterant stand for an undisclosed alternation or ambig−uity between +1 and 1. There are two iterant views: [+1, 1] and [ 1, +1] for the basic process we are examining. G−iven an iterant [a, b], we can think of−[b, a] as th−e same process with a shift of one time step. The two iterant views, [+1, 1] and [ 1, +1], will become the square roots of negative unity, i and i. − − − We introduce a temporal shift operator η such that

[a, b]η = η[b, a] and ηη = 1 for any iterant [a, b], so that concatenated observations can include a time step of one-half period of the process abababab . · · · · · · We combine iterant views term-by-term as in

[a, b][c, d] = [ac, bd].

We now deﬁne i by the equation i = [1, 1]η. − This makes i both a value and an operator that takes into account a step in time. We calculate

ii = [1, 1]η[1, 1]η = [1, 1][ 1, 1]ηη = [ 1, 1] = 1. − − − − − − − 35 and ηη = 1 and for any iterant [a, b], so that concatenated observations can include a time step of one-half period ηη = 1 of the process and abababab . for any iterant [a, b], so thηaηt c=on1catenated observations can include a time step of one-half period and · · · f·o·r·any iterant [a, b]o, fsothtehaptrococenscsatenated observations can include a time step of one-half period We combine iterant views term-by-term as in ηη = 1 abababab . of the process · · · · · · and for any iterant [a, bW], esoctohmatbcionneciatteernaantetdvioebwasebsravtbeaartmiboan-bbsyc-atne.rimnclausdeina time step of one-half period [a, b][c, d] = [ac,obfdt]h.e process · · · ηη = 1 · · · We combfoirnaeniyteitrearanntt v[ai,ebw],ssotethrmat -cbonyc-atteernmateadsaoibbnasebravbaatibons c.an include a time step of one-half period · · · · · · [a, b][c, d] = [ac, bd]. We now define i by the equation of the process We combine iterant views term-by-term as in i = [1, 1]η. [a, b][ca,badb]a=bab[ac,.bd]. − We now define i by th·e· ·equation· · · We combine iterant views term-by-t[ear,mb]a[cs,ind] = [ac, bd]. i = [1, 1]η. This makes i both a value and an operator that takesWinetonoawccdoeufinnteaisbteypthine etiqmuaet.ion − This makes i both a[va,aibl]u[=ce, da[1]n=,d [aa1nc],ηob.dp]e. rator that takes into account a step in time. We calculate We now define i by the equation − i = [1, 1]η. This makWeesniobwodtehfinaeviablWyuetheeacneaqdlucautniloaonteperator that t−akes into account a step in time. ii = [1, 1]η[1, 1]η = [1, 1][ 1, 1]ηη = [ 1, 1] = 1. i = [1, 1]η. − − − − TWhisemcaalkceu−slaitbeo−th a val−ue and an operator that tak−es into account a step in time. WeTchaislcmualaketes i both a value and an opieira=tor[t1h,at ta1k]eηs[1in,to a1c]cηou=nt a[1s,tep 1in][tim1e,. 1]ηη = [ 1, 1] = 1. Thus we have constructed a square root of minus one by using an iterant viewpoint. In this view− − − − − − − We calcuiliat=e [1, 1]η[1, 1]η = [1, 1][ 1, 1]ηη = [ 1, 1] = 1. i represents a discrete oscillating temporal process and it is an eigeniTifho=urms[1w,−feo1rh]aηTv[1e(,xc−)o1n]=sηtr=uc[1te,/dx−a1, ]s[qu−1a,r1e]ηrηoo=t o[ f 1m−, in1u−]s=one1b.−y using an iterant viewpoint. In this view i repiir=es−[e1n,ts1a]η[−d1i,sc1r]eηt−e=o[1s−,cil1la]−[tin1,g1]tηeηm=po[ r−a1l, p−1r]o=cess1−.and it is an eigenform for T (x) = 1/x, participating in the algebraic structure of the complexThnuusmwbeerhsa.vIencfoancstttrhuectceodraressq−puoanredirn−ogotaolgfembirn−aus −one by usi−ng a−n iter−ant viewpoint. In this view Thus we have consptarurtcitceidpatsiqnugairne trhoeotaolgf embirnauics ostnreubctyuuresiongf tahneicteormanptlveixewnupominbte.rIsn. tIhnisfavcietwthe corresponding a−lgebra structure of linear combinations [a, b]+[c, d]η is isomiorrepphreicsewTnhitutshs aw2aenddhisa2cvermectoaentrsotirsxuccaitlellgdaetaibnsrqgauatareenmrdopoioteroarfalmnpitnrsousceonses baynudsiintgiasn aitneraenitgveinewfopromint.fIonrthTis(vxi)ew= 1/x, i represents a×discsrtertuecotuscreilloaftilnignetaemr cpoomrabl ipnraotcioesnssa[and, bi]t+is[ca,nd]eηigiesnifsoormmofroprhTic(wx)ith=2 12/xm−,atrix algebra and iterants can be used to construct n n matrix algebra. We trpeaarttitchipisaitgirneepngreeirnseantlhtiszeaaatldigoisencbreertalesicoeswsctihrllueactritneug[re7te2mo,fp7othr3ae]l.cporomηcηeps=sle1axndnuitmisbaenrse.igIennffoarcmt tfhoer cTo(xrr)es=pon1d/−i×xn,g algebra × participaarttiincigpaintinthgeicnaatlnhgeebbaelrgaueibcsresadticrutsoctrtucucorteunrosetfrotufhctehtecnocmomppnlelexmxnnauutmmribxbeearrsls.g.IeInnbfarfacat.cttWhteheceotcrrroeerasrpteostnhpdioisngdiean−lnggeearbalrgaliezbartaion elsewhere [72, 73]. Now we can make contact with the algebra of thsetrMustcrautujcortueraroenfoalfifnlefioenrraemaranricyocointomemsrab.nbitLin[naeaa,ttbii]oeo, nsno=st[haaa[1t,,cb,bo]]n++c1a[[ct]ec,.n,daTdt]eη]hdηieosinb×sisseiorsvmoatmoiornopsrhcpiachniwicnicwtlhuid2teha2t2imme2astmerpixaotfarloigxneea-bhlragalfeapbnerdraioiadtenrdanitserants structuoref tohfelipnroeacerscsombinations [a, b]+[c, d]η is isomorphic with 2 2 m×atrix algebra and iterants 2 cancabnebuesuesdedtotococonnstsrturuccNttnonw wnnemcaatntrriix−mxaaalklggeeebcbroar.na.tWaWcetetwretraietthathtthitshegisaelnggeeer×nba×elrirazaaoltiifzoatnhtieolnsMeewaljhsoerwraenh[ae7r2fee, r7[m732]i.o, n7s3.].Let e = [1, 1]. Then we have e = [1, 1] = 1 and eη = [1, 1]η = [ 1, 1]η = ecηa.n Tbehuussedwtoechonasvtreuc×t n2 n matrix algebra.aWbabeatbraebat th.is generalization elsewhere [72, 73]. − − − N−ow we can mawkee choanvte×aect×w=it[h1,th1e] =alg1e·ab·nr·adoefηth=e·[M·1·,ajo1r]aηna=fe[ rm1,io1n]ηs.=Let eeη=. T[h1u, s w1]e. Thahveen Now wNeowWcaewnceommcaabnkineme ciatokerenatncaotcvntiteawwctistwtheirtmh-etbhyae-ltgaelregmberbaasraionoffththeeMMajo−rraannaafefermrmio−inosn.sL.eLt et=e [−=1, [11,]. T1h].e−nThen 2 we havewee2 h=2ave[1e,21=] =[1, 1] a=nd1 aenηd =eη =[1,[1, 11]η]η== [ 11,,11]η]η==eη.eTηh.uTs whueshawvee have − − e = 1, we have e = [1, 1] = 1 and eη = [1, 1−]η = [ −1, 1]η = −eη. Thus2we have −− [a, b−][c−, d] = [ac, b−d].− e = 1, 2 2 e2ee2===11,1, , η = 1, We now define i by the equation η2 = 1, 2 2ηi ==[11,, 1]η. and ηη2==1,1−, and This maaknesdi both a value and an operator that takes into account a step in time. eη = ηe. and eη = ηe. − and We calculate eη = ηe. eη = −ηe. − We can regard e and η as a fundamental pair of MajoranaWefecramn iroegnasrWd. eeTachnadisnηiirisae=sgaa[a1r,fduon1red]mηa[am1an,eldnct1a]ηoηlerp=ηaraesi[=r1-a−,off1uM]η[neadj1.o,ar1ma]ηneηan=ftear[ml 1ipo,anis1r.] o=TfhiMs1.iasjaorfaonrmaaflecromrreio- ns. This is a formal corre- We casnpornegdeanrdce,ebaunt dit ηis astsriakinfugnhd−oawmtehnist−aMl aprajoirraonaf−−fMer−amjoiornanaalgefeb−rrma ei−omnesr.ges−Tfhriosmisana afnoarlmysaisl ocforre- spondence, but it is striking how this Marjorana feWrmeiocnanarlgegeabrrda eemansedprgoηensdaesfnroacemf,ubnaudntamaitneiasnltysastlriispkaionirfgohfoMwatjhoirsanMaafrejormrainoansf.ermThioisn iaslgaebforarmeamlecrogrerse-from an analysis of spondethnecere, cbTuurhstuisvitewiensahtasuvtrreeickooifnsgtthruehcortewdenathesqriisunagMremraoarojrotko,rafwnmhaiinlfeuestrhomneeifobunysiuoasnlignaeglbgarenabirtearmafnoetrrvgtiheewspMforiaonjtmo.rIanantnahifasenvrmiaelwiyosnis of the recursive nature of the reentering mark, while the fusion algierbeprraestfehonetrs ratehdceiuscrMrsetievaejooscnriaallnatutainrgefetoermfmptohiroeanl rperoecnetses rainndgitmisaarnke,igwenhfoilrem tfhoreTf(uxs)io=n a1l/gxe,bra for the Majorana fermion spothnederencecumer,seirbvguestnfiartotumisrethsoetrfdiktihsitneingrcetihevnoetwperoitnphgeirstmieMasroakfr,jtohwerhaminleaarktfheietrsmeflufis.oiWonneaasleglgeeehbborrwaa tefhomerseterhegedesMsofarjtohermafn−earmnfieoarnmnailoynsis of emerges from the distinctive properties of the marktihteserlefc. uWraslgeivebesreanpealairhvttuieocriiwpeneatmoihtnhfigesertienhgxstetheeseneraedfdlregsodnembotlroeafgritcihtcnhsaetgrleudccmfoiteusnartrteiermnxkotcf,i.tohwivnehecoiplmerpoltephxenrutfmiuebsseiroosn.fIntahflagecetmbthraearckofroriertsspteohlnfed.iMnWg aeljgoserbeareanhaofwermthieosneeds of the fermion emerges fromstruthcteurdaeliosgfteilinbncertaairvcleiovmpebroiinnpaettirohtnisess[ae,oxbft]e+thn[ced,edm]dηailrsokigsioitmcseaollrfp.choiWcnwetietshxet2e. h2omwattrhixealsgeeebdrasaonfd ittheeranftesrmion algebra live in this extended logical context. emerges fromctahnebeduissetdintocctiovnestrpucrtonperntimesatroixf atlhgebrma.aWrke tiretsatetlhfi.s gWeneeras×leizeathioonwelstehwehesreee[7d2s, 7o3f]. the fermion algebra livNeoitenhthowis tehxetdeenvdeeldoplmogenicta×olfcthoenatelgxetb. ra works at this point. We have that algebra live in thNisowexwte ncadnemdalkoegcoicntaalctcwointhtethxeta.lgebra of the Majorana fermions. Let e = [1, 1]. Then − Note how the development of the algebra works at thisNpooteinhto.wwWethheeavhdeaeev2ee=lNot[hpo1,mat1et]eh=not1woafnttdhheeηad=legv[e1e,blroa(1ep]wηm)o=2er=k[nst1oa,1t1f]tηthh=ies paoelgηin.eTtb.hrWuasewheoahrvakvesetahtatthis point. We have that − −− − Note how the development of the algebra wo2rks at this point. We have that 2 e = 1, 2 (eη)2 = 1 and so regard this as a natural constructio(enηo)f th=e squ1are root of m(einηu)s on=e in t1erms of the phase synchronization of the clock that is the iteration o2f−the reentering mark. Onc−e we have the square − (eη)2η == 1,1 and so regard this asnda nsoatureraglacrodntshtriuscatsioan noafttuhreaslqcu−oanrestrrouocttioofnmoifnuthseonsqeuinarteerrmoos toof fthme ipnhuasseone in terms of the phase and so regard this as a natural construction of the square root of manidnus one in terms of the phase36 synchronization ofstyhnecchlorocnkitzhaattioisnthoef ittheeractilooncekηof=ththaetηreies. etnhteeriitnegramtiaornk.oOf nthcee rweeenhtaevreinthgemsqaurkar.eOnce we have the square synchronization of the clock that is the iteration of thaendreseontreergianrgd mthaisrka.s Oa nacteuwrael choanvsetrtuhcetisoqnuoafrethe s−quare root of minus one in terms of the phase synchronizatioWneocfanthreegacrldoeckantdhηatasisatfhuendiatmereantailopnaiorfofthMearjoereanatefreirmngionms.arTkh.isOisnacfeorwmeal hcoarvree- the square spondence, but it is striking how this Mar3jo6rana fermion algebra em3er6ges from an analysis of 36 the recursive nature of the reentering mark, while the fusion algebra for the Majorana fermion emerges from the distinctive properties of th3e6mark itself. We see how the seeds of the fermion algebra live in this extended logical context.

Note how the development of the algebra works at this point. We have that

(eη)2 = 1 − and so regard this as a natural construction of the square root of minus one in terms of the phase synchronization of the clock that is the iteration of the reentering mark. Once we have the square

36 At this point we see that it is not just the square root of minus one that has emerged from the structure of the oscillation, but the simple non-commutative algebra of the split quaternions.

In fact, we have uncovered matrix algebra.

But note that we have shown that the Fermion algebra is primordial, related just to the oscillation of a distinction. The Clifford algebra occurs at the Arithmetic level with its extended polarity of -1, 0 , +1. Note also that -0 = 0 so that at the arithmetic level, 0 becomes re-entrant. SpaceTime

ss ss = 1 ii = -1 -

At this stage we have shown how the Clifford algebra generated by e and eta (the split quaternions) emerges naturally from the discrete dynamics of the square root of minus one. Dynamics of the reentering mark. The mark embodies the fusion algebra for a Majorana Fermion. It can interact with itself to either produce itself or annihilate itself. A row of n electrons can be regarded as a row of 2n Majorana Fermions. Recent work suggests that Majorana Fermions can be detected in nanowires. arXiv:cond-mat/0010440v2 [cond-mat.mes-hall] 30 Oct 2000 N I p o n ✍ ✎ c f a o 1 H s b r a s t t i L t ) ) e e t 1 a r y e t t m e M t s j s | n o t N T U e I I t 2 = i s c h ∆ m e r w t i o n i n p u ( 2 n t ✌ ☞ h s o a a s u s a ∗ g a p n − t o e m e | t O w e t t l s e n p e m j 2 o r e w i p r n e = µ o r h t j m b o b z t 3 p t s s a g t a d o f s t s o r i y i y o h l p - n r ! i r e a u r h ) t o n e t u t e r d e r t o t s n e s d e t a l a j a n a w h c e d o t n m ✍ ✎ c a i e i u i s k s j t i r $ h h C e r v h m e c e v e n r t 3 r s r a n a h d e r e r s o m s e n e u s ( " i s t t e i u + 0 e h i d i e s n n a t r e p > t j M f o h c o e ff s a t n − r t a a a ( g . e M i r c e r t d H e o e v a a s g c s t s a p ˜ c l n s s w c o e y o r m t a o i i h p i 1 h o t 2 µ t e o t a 0 f o n d e e a a c j a a d e c r e s c a p f h ) p j y a n n i . e i t n n e M r L , s u n r c 4 i t o − a r f e m a t a a ✌ ☞ o = e b p j o e e u . r m e v s b n 2 h o b a n l | n o u s i 1 t f r − s µ D e l T e r n j n ψ n e fi d o i e p i g l l w e l y h a c r n o a - − a w a d k u . l r d - . R u b i 2 1 t d g c a : 0 t s h 2 t m w i r n f n e = t L t h 1 t w s r j o o . i i c b e ( t n F j ⟩ m o . t e n e m h d a o i a e o i u i m o b a c t l t , . i n u | c . l o d e h e v n a t n o c e n l i ∆ h o a r a ′ 2 e r s m r c n 2 2 r e a b p 0 e d r r # m b g n e o d i k j H s i t a i s y j g n b t i a a c o n ) a b ′ . s r c a s e s a r u T i q | , n a u ′ M e o o n u 1 o o ✍ ✎ + p e i c . g + y s t A e r | b o a p e r o u l n u n p n e r s p 2 a y 2 r ψ I l . d p o = 1 I t n t n h s ′ n y a p n m n n a i s a L e e n l i e ) 7 h e h P o 3 e a d a c m n s o i M n i ( e t e s r 0 d e r = O c , | T i − n r e f p t v t l i t c / c r e m w t e , x r a ⟩ ψ i f g t 2 w c i a O . i a i t r e o @ F u t o h F 1 l i 2 s r 2 o y r a a e l o n u = e f M m c e : W j n t h a A y e H l A m e 0 = j e e H u s o c o s c c s 0 t i t e t m + r l d r o n v o + e d ⟩ u o e . e r t = n r l t i 2 1 m t b s m T ✌ ☞ o 3 fi u u A Y r n a r e s a i 1 a n n e t L o t o f % c w o 1 c f ff i s e S n c n | s 2 m a r o e i i t c e m a a a ˜ o | n c c j 1 m t j ) w u r a t b i ψ e t u r r i t a = c s 0 ∆ , p e s s o t m r t a i o r r . , 9 a t g e f n n j R c . s c y e e e T a e a o o ( r o o 0 r s , o y f f h O r i | n 8 a p n e s i p T n , r e c d d i a ˜ c K h − | e s c e o l ⟩ ψ s s t n a r d i l a o a r 0 y c ) u p s t g t e 2 s l h e s r e a n , µ o † j 5 t m t w r n r 2 m e e y o y n s a c . r i o t m j 5 b ⟩ s e i e n e a m b y d c | t f , n t e r o e 0 p s i s , 2 c = a t o 2 p e a n t ψ e ′ o i v t s a e r c d p < e ! e ′ 0 j n ) s = t t m . h c 2 e r n r , o i i i M a i l e c i e c h h c w fi d 1 o n m j e 0 j c i o y e c 2 n s = a s e s o o v a o i p a r U e w d − o ⟩ 2 2 1 ) j c i h s d a n s n s 0 i e v i l n d r . f g m 0 o j ∗ r p u ✍ ✎ + n c m s a c ( c i n 1 c e − . P i i e + . h r t l c i h g c 1 c w r b o t c e S r s c t c e e 1 t i 2 c a e h m a o h b o n f o e i 2 i o 1 e e x 2 e j f n d h 2 r i y . o T q n a n o l a . i d g L n p j p A s s c u e b j a s a f n a e t + t l t p i r r b b T o r e ) u l 2 e ′ j i a h c c b h h − a n t e f s . t o y n s b e o s r . j 2 l f a h y w s o s c | h o h e r o L r j a e t d ′ + e ( s s ψ e m 2 w t u e p ′ i r f o m ✌ ☞ i e n a o i r t | − o m r c g n 1 p s t r W p d u c t n n t ψ 1 s = h h e o i y s y a i . h o r l e $ o e 2 e ⟩ s n d x t w H a a d o t a i s 1 d e x fi r n j g a i s c s s s M l p u a g m ( m p ⟩ a + e i a l i e h 1 B y j L 2 n b e e 1 n i d a s ( p ff s + t . a a = i m j r 1 n . , , r ✍ ✎ − c a p = e s − t i e t l j i q − c e y i ) = g 3 e i e h . o l 1 y s s n L n r u o 1 s s u l a e | r o 1 r . . p n o i w o ( / b o w a ∆ t c − n u t t e , l a e w . f d w l i h t f a n ˜ i − h h p a i 0 t c t n o p c i i b , n h n i o a a a e e s | ) | - j † - t e t e 2 ψ ) L s c i µ a t i a ) ˜ i v m h i j r r s c 4 c o a o ✌ ☞ i 1 j t o s h 2 $ r e b − n a ⟩ y d t f j d i e − l s r . h . e − t - d . d s j s o e 1 a e 1 p . ( u m s 1 2 c a . . p n I ( a p o 2 n ) i j † a d a n . j e . T c a d r t + r e r c ✍ ✎ c j i h t e e h c . p 2 2 t G u s h e − e r L # e y o d p p d e e − r r . . s o a a a s o e , 1 i 2 1 s a t t t T i u s c i ) n t r i i i a ( ( ( ( m t 2 n o o o ✌ ☞ h e h 8 6 9 7 o e r L = n n d n d e e e e s ) ) ) ) t ( i o d p p g ( T c w ( ( r T t M t s c c u 7 o h e a p e e u r i h h 1 1 p ) a p e n n e r s T I I T N T e r m c e . i n n n c e v r j + e + p r s h o h r h o o e b t r p a i r a o e d o o n w i s e i e t t r t a r e t i b i y s s e h h a h u e s e o c t c σ e t a n e e t n c fi e e t s o e s 2 τ τ τ 2 r f b w s z d r x t h e o o l ) s r ) a n ( ( ( . p y u e a e c n / c o p / s n r e T T T i e γ b x h a t c s a f m t 2 2 l s d s 1 3 2 d T e i a i i s t y p e s a s t , , o c ) ) ) a e r i i u e t c 1 d e r s r i s m i n m t Ψ o Ψ h N t , u n e = = = l e t c P e o a p i o b e s c p u T † i o 2 arXiv:cond-mat/0005069v2o [cond-mat.supr-con] 11 May 2000 r a f f a f s s f o e y t m e e e a l s a n y f o 2 p i e = u r e = e n f x x x s o c t , c a r c r a B o v c t d a T a A i a v p t a c v d v o s p s t b m l c o n fi s p d ˆ s i e l w s o . o t p p p n h h h m u e t t h o o a e n n s r f o o e k t n o e r a o l h i o a t t u h n a u d ( c l e o e i i n r c p o i ( a l F . L c T C A T W O f r r a 2 m d d r n t i i a s t s s a o o a u t e t f b e e n l c o n t $ $ $ t t h v m a h r o s r c e n u o h t - s f r e n i t h e e s h e h i e r n l a o n e i e o a a i m b g a m m i r i r d d h m t 1 e i i o n t r c y a ± d n t [ e 3 d r d ) e n x d r s i i e e r o t - o x p c n l c i 2 a π b y n 4 e i s a p v t m p e v t c a a o n h t r ϕ c e p . a e e e r t r o t o u f ] i o e a π π s g − d h 4 4 e ˆ t B g m s o b h a r e t l r + a r n . h e e a t o s s u o r o d n n o t t o r t s p n e i T h l r c 9 o s a a e w m e r e i m c τ r a N n o s i h n n o e u i C i t r m τ d o t s g a i T r l a b o σ σ p i f a a r t i c r t n t r r 3 u s r t b n n t n a e a w e s f t h o 3 z u p n u ( r i i a l S n o n t y i n i a W o t a t t i ( t h a t f r o o v t t y e i l o e c - T t d z z e m a t ( ( i i w x n y o s c s c o b T y a o l s l o p t s n b o a e c o i h h T d i g m e i t q h o s 2 1 i l i π a o n t p s t m 2 h o f i s t F M - n e s d y n p v r c e 2 i c b g f t n a a n i a d i o e e τ 4 i d i t s t h l i r n l n f ) ) d - a o a e ( b r n l e e i u n s % i e n e e b n - l t t f i i p a r a g s e s e e e n ) a e q e e l a a e o ) a o i Φ t o s ) e s e % % t i l c r n o - ( ) s e i e n a t d m g t y l n a t r n E y s r u t / j m t o o n h b n d e h a s c p a s t n o s b i h o / h r e t w o t n i o T t s i o e h t a a e b n v o s i o n = r x s a = l d n e s s . i n a f a , u e y fl m - o , l z t e 2 r t c e o u o i e h r l f h r m p t e n 2 = = t e i e t e t e f z o c r t n t s Ψ n a y o n h a a r o g c s r e u t e d n g g w n e a i r x e ) e m h x b p r i e T o . . r t a n t p h i c M − i M u t h t t s s s d e e t c a e e B e - x y l n v a c ± t f r e d s h u o f h r d k s i i s i t s a u u a a o t ( o h n I t r t h n i I o o h , √ e e o t a v t i o = u 1 i t c e C ⎝ ⎜ ⎛ ⎝ ⎜ ⎛ e e d t h m T u n b a o i o n h a e p s l p i t f q l i s v a a a r s n f a e e h r t b e f a t = n s p 1 i t / y e r n a e h v n e s s p i f e e o e e r e i n a e o p u S e r e i e t s F n d c m r g f ( p a t a u r v o a h v n 2 t o e n t o 2 o c l r e l e r h b n i a v ( e r n d r c a i u a i e a i p o i e t j d f e T s e c h a e m s e e n l o ) p i s t n n e e e n a a t c f e I n d l o h m l t g c n e a ϕ d s o f o n s t d t a r a w l u s i i p s ( i r e a r a r t a n ϕ a s e r p o n b o n x a b G τ y r p t v t s ϕ i s i o t − − o m u i i i p fi d t e a w r s t t i . r a a o g h d s h c 1 i l f r t s n o o i t e b i y i n u y n d t s o e s s u y m a e i ! t i l r e o m o s o e p e d n r s n e ( u l e f , a c e o a c b s a a t g a t d l : i i s e T i p d r t . n . s t c m m i y d u l ¨ u u n n o n t x s e e e . p g a e e n d e i T p π π s r s a + g a a n t r d y w t u r h e s b i t e τ a r r h f u r o s x r n h n b n a m $ s e ⎩ ⎨ ⎧ e 3 t r o o t i o s e k T o t R o w m t p c r r g b c y d ) i / / d a c ˆ w m f e " n a u i i l Φ x o u e i o w e d h R o t s T t π m ( a i c e r l s l s . e a s t M w t ϕ ( e o h o e i . o e t n 4 4 ) M s m f e e s r t p r π s v s c t t o u i e 4 s o r h | c a T 0 π : t o w u t h h w e n e i r | t a o n t i r i h ) r ↑ i u i g i i t e h r o u o e s o n o e r t s . b r s c o m n 3 e e a c b i t h E 0 f x b k i l p h t d h f e e = ↑ i i e e a c a o s h r i n y b p c a c ( a p o s n c t h d a s f a e r l t v h i d o s t a a i l y u a c i n e o c t i t v ⟩ f ϕ c e h l ⟩ n x ) t e r e e p s y l h y p a a s o l n fi u a a 2 e r s e w t a + i h o a e d o r r f e s j h r e t t f s n 2 a u s + e j e e b i , o d c j , i P e p s e o fl d - π l n d o n d [ c c o f s c i t t d o u o m r p e , s c a o t m e o i m e n q t 4 l o 2 t d v t n o c ˆ i m ) o r 1 a t o f s u h h e − i j t a d u j y r l l e t v e t u n s p y Ψ e g i o i u 1 ] u t | i n . o r i u x i e t u f s i o h π s m a a r l ) t a v d n t h x i h r c . o s ↓ n n r [ o e f t a o i m s i a s o n s r x a n g i r a G = t n i s i % a k l t v p h a u c e τ e e e r s r w e a ↓ e / a o p d o f a t a i %→ s T t i r π o f t t t y † f n i t c r n t e a e q f e f !→ !→ !→ e t - n r h o i a ffi s ⟩ d n t ( n n T r i n f o h r l 4 r w t i a v o s g e q i n i ( r t t h a n p t o s o l i t l n e o u i h e h n / e c i n n t i o a # i = | $ y l i t P p i a e t t t u e r u o e s c t h T r p i a f e e r r r i g e a e h s l o e c o a e n o o a √ 0 t g 4 r . r i ( m a c t o t a a o a o m i a m e h o s s a l t v e e l n + n w c s n f m e v a n ϕ t d d m n r n t d a r c h t t t e fi e 1 i t m π n t r f t h y 4 ⟩ p s e e h . o i { a i n t n h r e o n i t e t r a e g t e e t a a a e - o ) Ψ e s e f t o s p 2 c s d P e i e x l o v o ) h e f i i i + − c e u t a o o g u g r o y i d s n v u i q o s t l t u m B l e c i z n ] c d f p e l s l e r r f x c i . o t o p k t i O r l a . e d e c p f − p p , i r r h c s e o o m f d o a − ˆ y s g e r n a t c C o ( f a f y o n , i u o + r 1 † i n f r i e e a r & h t e e r π t - , v y a t o t e , y r j c r t p n o t g t b r l p g " t t p ffi o r + , 1 o t n g e h u e i i d m q t E i ( a o t t h m o i i r i a n m i 1 l , 1 v f e h t e [ r e 1 D h o h i b r t r π π ( i a e M s i o c u r o t h i e e . c o t r 3 a o d r | m o − ˆ T e i 0 b n c o x i a y s c i a l a & t l 1 n o e ( a o h t r e a ↑ u u n s n t g i e n n n t e h i f . ] a p / / r ) m t r c h j a . e t 1 i c o n h . o a + l u l c H o h n p v = i i ) ↑ d M c ˆ i t s i t e e t t e d g u d m i r e o y s e c c A y e a s ) 4 4 f e c n o i c o e } b r e t i f d t e t t e e ) ⟩ e t E d e o - r u ) i e o ) o e s r d d π n T a h y s n e 4 o a h o n c x s o H o a n b e r o a a e . t a . f a b h . 1 t i n − r p s % n t l p v a x t l t “ i r i s t d w v m t P e d h f f 1 p a h u p l o j a = r t c 1 a . o a p w ¨ o t r i r a i I T e h i r e e s i d o s l p l r h o ( d o 4 ( - e n s c i n s e e t h t t p , T v a a e e t s o Ψ n f t e o e | u s y c d e e n s f r e c i e n q r h a e a i a i i e 2 ↓ Ψ M f n e e r i I = s a e a i w i l i a r n 2 g o t d p o n r a m t j j m e o a h d n a v t r f r i b u a t e l ↓ m m t r t t l o s e n ( v o b i g c t 0 e . s h r 2 t t i i Ψ - n o i o a 2 † g s n n o o e s h a r y a p i l i ⟩ t c G d e a a i m f m a a c t 0 w t i r π π g e t o 2 † c e e ffi h . e f s o o e v c m , a a e f s r a r ̸= δ ) - o r t e t j o h o t k # m 0 m r t i t b v e i h i a e h e a a w r a √ t j s o e t t g / / e b i † t t e d e ˆ n i c o r t I t f r i . ) m O a o i e c n o t + o e n e f e o e t s h n l o l m c r a n o h n o P n + o 1 y e e u a a ) 4 4 y Ψ j e e g t o e . f e b o f x v h i n l e . r n t h a n n i n l i t n P t v - h t a n e s i n e n r h , n g 2 i f h r f , r f s l o e e n T i n q n e h e n b n a e e a ” e a ⎠ ⎟ ⎞ ⎠ ⎟ ⎞ t m B t f d L - t c , I e d d c T o e e f ( a c w o e s Ψ r s a o c r s u q i e g a l o g f b a ffi w n z o a − r i s w c t e e a i e c i u l l p C n 8 o o o v f ( c t i h h s i e d o u r a n i n g a M f ffi y g u " a n r o s n o t r u x o v i h n g t - h s f f n e q o n o a e 1 H t 2 r e k b , n t i ) e a e - p e p fl h , , t n q e a n o n w s t e s h g f g e n o e - | e d u o a y r k u a h n e s e t o t n t e r ) 1 r e a ↑ d u t - e y t n u d s s s t h a c l ( i e o a n a i b s e e u h e h n e 8 f + e s m t r o e t A i i c t ↓ a ( i ( c c o x n F h [ a i t r t t r l ) n l s u w t c g s 0 c u m e i e s v i 5 d e p t e r c p h t a i i ⟩ l o e t m h t i s r o Ψ w o t n n c i o f i i e e u t 9 n ' y l b o , j . o i e h – m a n r s e - l e t o l f g r a t a i h i + r h e n h t t e o s p a 3 p R l n s e r o c i a c e i a a m 8 o w a e l h r o e n d . t v h v f t u y B o t m d t t r a n 1 † t [ t h d t s l n - n v t = ] i h e e ̸= c a e r y i m , a o o c o o Z e a e d p m h | i v [ y 1 o f 5 . u n l n o e e w o a e u o a + f T ↓ t - t e u i h l v i d r f 5 r r b r t t o u ¨ ) e r i p a o 2 n t c e P d − . y a t r s o d s c u a w ↑ , t c t e l e i b H . i r w s t r t y r r n v i i v i t s t s e p a e i t ] 1 n r r o e 6 n e a v i i h ⟩ e f s g d r n v ˆ y t o T i i t t o a h s e n c w t s b a o e a e ( a ( - n a x n a T c h o a b o p v - e r h c s # e d i l i P ] e x w G t i l e ( o n a r l ( h d h r a Ψ , ffi h a b l r c t n o s + s e l r x a d o a d , i i s h l h o c e $ l i e a n e c l o f n e f a , e b o s i e e s t j a s l n s p e s t r j b a o s v p i v d a ) a o i m ( b a o a e fi , o e f t o fi u m d d [ s u n i t e t p n S a 1 a s p g s ffi n f u 3 k o 1 c d i l ) k n c r e m x l s l n o x u h t o l n p e 1 w c a m n w c t h n a f s $ t i H e r e o x ) a r i ] e z i s v n o . u t - 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l e h j s a s o z a r t e Ψ s t o a t e u t c n w Ψ p h t e ( ( ( i t t p 1 o e h 1 1 1 a t t i r e ( ( i b o e h h n v 8 9 a e 1 2 0 c r = = T d n y e e e e e e ) ) s t ) ) ) On the Dynamical detection of Majorana fermions in current-biased nanowires

Fernando Dom´ınguez,1 Fabian Hassler,2 and Gloria Platero1 1Instituto de Ciencia de Materiales, CSIC, Cantoblanco, E-28049 Madrid, Spain 2Institute for Quantum Information, RWTH Aachen University, 52056 Aachen, Germany (Dated: October 29, 2012) We analyze the current-biased Shapiro experiment in a Josephson junction formed by two one- dimensional nanowires featuring Majorana fermions. Ideally, these junctions are predicted to have an unconventional 4π-periodic Josephson effect and thus only Shapiro steps at even multiples of the driving frequency. Taking additionally into account overlap between the Majorana fermions, due to the finite length of the wire, renders the Josephson junction conventional for any dc-experiments. We show that probing the current-phase relation in a current biased setup dynamically decouples the Majorana fermions. We find that besides the even integer Shapiro steps there are additional steps at odd and fractional values. However, different from the voltage biased case, the even steps dominate for a wide range of parameters even in the case of multiple modes thus giving a clear experimental signature of the presence of Majorana fermions. 2

PACS numbers: 73.23.-b, 05.60.Gg It is important to remark that in the experiment we are analyzing, the phase ϕ is biased by a noisy voltage coming from fixing an external current. These voltage fluctua- Majorana Fermions (MFs) have recently been pre- pair recombines and tthioensspareectiaralnpslarotpederttoiepshatsheafltutchtueatMioFns by the fact that dicted to occur in a multitude of different condensed- confer to the system ϕ˙are Vlo, satndimthmuse,ddiaepthealysin[g36en].terPs hinytsoi-play. We have es- ∝ matter systems [1–5, 5, 6, 8]. The interest in MFs stems cally, one can circumvtimenattetdhtishaptrtohbeledemphuasininggtima eJotDseisphm-uch shorter than from the non-Abelian quantum statistics which forms the son junction where ththee gtaimuegeneinedveadriatontchpanhgaesethise pthuansedby ϕ ϕ + 2π. → basis of topological quantum computation [2, 4, 9, 10]. non-adiabatically. InTthheisrefworaey, ,wteraanssuitmioensthbatetinwteeernfertehnece effects can be Majorana fermions naturally occur in half-vortices of chi- recombined fermions innedguleccetead,daynndamLaicndaal ud-eZceonuerplintragnsinitiotons (LZT) can be considered individually. Coherences between LZTs have ral p-wave superconductors. Although this type of su- Majorana fermions. been recently analyzed phenomenologically [35], and in perconductivity has not beenFIfGo.u1n:dJ,ositephwsoansjurnecatlizionewditrhea- nanowirIenonthtoisp.wRoerdkspwoetsanalymzoeretdheetoarilet[3ic4a] llyfor the ccausreroefnat bvoialtsaegde biased junction, cently that s-wave supercond(ucocltoorrontolingee)threeprrewseitnthMsatjroornanga fermions. Double arrows represent the overlap between the MajoSrhanaapirferomeixonpse. riment [3w2]hinereaafilsnoitaedd1itDioJnoasleApnhdsroenevjulenvcetlsio,nQP and inelastic spin-orbit and an applied magnetic field may emulate a p- where the MFs are retcroamnsbitinionesd h(asvee bFeigen. 1co).nsIidnertehde. pHreo-wever, we would wave superconductor [5, 5, 6, 11]. During the last months sented setup, the currlikenet tboiasstrtehssetghaaut gteheincvuarreiantntbiaphseadses,etup analyzed in three different experiments [1w2–h1er4e] EaJppisetahreeJdosinepthhsoenliteneerr-gy of inthde ujucninctgiodn,yϕnaismthiceal dtehcisouleptlintergfoorfthtehefirMst Ftims.e Mhaesatnwwo hailedva,ntages: contrary to the voltage biased case, it (1) shows a robust signal ature which may provide the gfiarusgteeinxvpaerriaimntepnhtasleedviidffeernenccee anditthine dopuecreastoarsvηoiltaargee difference that can be measured, pre- Hermitian η = η† and they fulfill the anticommutator of small odd integer Shapiro steps even if the case of a of MFs. i i senting the pattern of the periodicity of the junction. We relation ηiηj + ηj ηi = 2δi,j. Due to the presence of MF multimode wire and (2) the observation is not masked by Signatures of Majorana Fertmheiopnesrioapdpiceitayroinf tthheespeleecctrturi-m is 4hπa. vIen cfianlciteulasytsetdemtsh,e inindutecrefedrevnoclte aeffgectbsyasmtehaonses aorfetahbeseRnte-in our case. cal [15–17] and thermal conduinc-tpahnasceeM[1F8,m1a9y],reschoomtb-innoeisineto ususisaltfiveremlyioSnshuthnrtoeudghJunctioOnnc(eRwSeJh)amveoadneally[3ze1d].thIendaydndamitioicanl,transitions of the [19], Andreev-reflection [16, 2th0e] oavnedrlatphoef nthoenir-lowcaavle tfunnc-tionsw[1e, in36c].luIdneoerxdteratoAndjruenecvtiomno, dweesacrearreyadinygtoain2cπlupdeertiohediricdynamical effects neling [16, 21–23]. In this Laectctoeurntwweitwhillthisfopchuesnoomnenthone an ecxutraretnertmanshdouqlduabsipe artoicnlethpeocisuorrneinntg. (TQoPth).is aim we introduce the function added, so that the total Hamiltonian becomes I (ϕ) in the supercurrent measurement of the fractional Josephson effect, given In contrast to the infiMnite length case, where only even

when we put together two superHcon=diuEcJtocros (ϕfe/a2t)uηr1inη2g+ iδ (Sηh4ηa2p+iroη1ηs3t)e,ps a(p2)pear, the obtained results sho2w ∂small I(ϕ) = IM (ϕ) E(ϕ). (5) MFs [1, 4–6, 26–30]. Physically, this effect is produced contribution steps at odd and fractional multipEleJs∂oϕf the where we have introduced a parameter δ to account for by the fact that in the presence of a Majorana bound ac frequency, coming from the new features of the dynam- overlap between the in-phase MF which decreases expo- The function, IM (ϕ), can take the constant values IM , ± mode, the supercurrent carrienesntsiainllyglewitehleinctcrroenasining sdtisetaadnce beictwaelecnutrhreenMta(jsoereanFaig. 2w(hbe)rebeIloMwis). tNheevmeartxhimeleumss,valull ethoefsethe supercurrent, of the usual Cooper pairs. Tmhuosd,esth(siseefrFaigc.tio1)n. aCl oCnosidopereinr g thactonthtreibinu-ptiohansse aMrFesof thwehicorhdiseroofftthheeoordverrlaofpnbAe.twDeuerningint-he adiabatic pe- pairs affect the supercurrent bayretfuarnainwagyitcomfropamresdinw(itϕh)tthoose onpthhaesjeunMctFiosnawnedwnillegligribioled, tchoemfupnacrteiodntIoMt(hϕe) rheemigahints ocfontshteant and whenever sin(ϕ/2). use δ EJ . Diagonalizing the HaemviltenonSiahnapyirieoldsttehpes. RtehmeraerkisaablyL,ZwT,eIhMa(vϕe)fcohuanndgeas ritesgsimigne. To understand arXiv:1202.0642v3 [cond-mat.mes-hall] 26 Oct 2012 ≪ 2π-periodic energy spectrum (see Figw.h2e(rae))the effect of cotnhseidcehrainngge aof dthoemsinigannwteccoanntrcibomuptioarne in Fig. 2(b) the In Josephson junctions, Shapiro step experiments allow adiabatic and non-adiabatic passage through the anti- for the deduction of the periodicity of the current-ph2ase 2 o2f extra 2π periodic cAronsdsinregesv(smolidodaensd, ddaosehsednroetspmectoivdeiflyy). After each an- E(ϕ) = 4δ + EJ cos (ϕ/2). (3) relation of the junction [31, 32]. Very recently,±S!hapiro- the spectrum of eventicShroaspsinirgo tshteepcus,rvpercoovmidininggfroamroabLuZsTt acquires a neg- steps have been analyzed for Nvoltn-aagdeia-baiaticsecdhaMngaesjoorfatnheaphasme leeadsus rteomtreantsitoiof ntshe 4aπtivpe rsioigdnicritesyp.ecItntaodtdhite ioadnia, bwaetichapvaessage. Thus, we between the two eigenstates. Since sEeJen thδa, tthteytpricanasli-timedesscariblees t(hµes)dy[3na7m] icoaf lQeffPecptsroodnutchee acurrent produced wires [33–35]. However, the more experimentally real- ≫ istic current-biased experimentiot n[3p1r]orbeambilitainysisunnonex-vpalonisrhedin.g onlyneagt ligtheibalenticerffoescsint gosn thebydtyhneaLmZicTsbyofchthanegsinygsttehme .sign of IM . of the eigenspectrum, that is, for ϕ = (I2dne+ally1)π, ,awhgerneenric 1DWJeossetupdhysotnhejSuhnacptirioonexinpertimheenptrebsy-means of the re- In one-dimensional (1D) Misajaonrainntaegwerir(see,eMreFdsarweaills inapF-ig. 2(a)). Thus, as long as sistive shunted junction model (RSJ) in the overdamped pear at the end points [1]. nIonn-adniaidbaetaicl trsaitnusitatioionns ,octchuer, the eonvecrelaopfbMetawjeoernaMnaFfermlimioitns[3c1a,n39b].e dTehsecrinibduecdedbyvotlthaegeMoFnsthe junction can ends are infinitely apart fromis eeaffcehctivotehlyecraanvceoleiddin. Agstahecironsequpelanccee, dthaet4πthpeerjuiondc-tionb,eyciealclduinlagteda bHyasmolviltinogntiahne differential equation recombination. In turn, whenictithye inwirtheeiseigfiennitspee,ctthruemo,vaenrd- also in the supercurrent ! (I ∂ E ), is recovered. As we will see below the new lap, although very small, is differenϕt f±rom zero, thus MF H0 = iEJ cosI(0ϕ+/2I)1ηsin1η(2ω,act) = I(ϕ(t)) +(1) ϕ˙(t). (6) sha∝pe of the current does lead to the expected even steps 2eR and also to additional contributions of the order of δ at This equation is obtained from Kirchoff’s law where an odd and fractional multiples of the ac frequency [38]. external DC I0 and AC I1 sin(ωact) currents are applied In order to calculate the transition probability we con- to the junction. The outgoing current is modeled by sider the semiclassical approximation, and we make use a parallel circuit whose components are, I(ϕ(t)), given of the fact that the velocity at the anticrossings is linear, by Eq. (5), and a resistive current (!/2eR)ϕ˙ originating therefore, transitions between states can be obtained by from the existence of quasiparticles. The solution of the means of the Landau-Zener probability differential equation (6), allows to obtain the induced ! 2 voltage V = ϕ˙/2e. This equation is solved dynamically 4δ since I (ϕ) changes its sign depending on whether the PLZ = exp 2π ! . (4) M "− EJ ϕ˙ # LZT occurs or not. In order to include such a dynamical

Braiding Majorana Fermions R I I R R RI SI R I RS = RR I R IR SI R

Figure 30: TRSRhe Y a=ng -SRSBaxter equation

A solution to the Yang-Baxter equation, as described in the last paragraph is a matrix R, regarded as a mapping of a two-fold tensor product of a vector space V V to itself that satisﬁes the equation ⊗

(R I)(I R)(R I) = (I R)(R I)(I R). ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ From the point of view of topology, the matrix R is regarded as representing an elementary bit of braiding represented by one string crossing over another. In Figure 30 we have illustrated the braiding identity that corresponds to the Yang-Baxter equation. Each braiding picture with its three input lines (below) and output lines (above) corresponds to a mapping of the three fold tensor product of the vector space V to itself, as required by the algebraic equation quoted above. The pattern of placement of the crossings in the diagram corresponds to the factors R I and ⊗ I R. This crucial topological move has an algebraic expression in terms of such a matrix R. Ou⊗r approach in this section to relate topology, quantum computing, and quantum entanglement is through the use of the Yang-Baxter equation. In order to accomplish this aim, we need to study solutions of the Yang-Baxter equation that are unitary. Then the R matrix can be seen either as a braiding matrix or as a quantum gate in a quantum computer.

The problem of finding solutions to the Yang-Baxter equation that are unitary turns out to be surprisingly difficult. Dye [18] has classified all such matrices of size 4 4. A rough summary of her classification is that all 4 4 unitary solutions to the Yang-Baxter×equation are similar to × one of the following types of matrix: 1/√2 0 0 1/√2 0 1/√2 1/√2 0 R = ⎛ ⎞ √ − √ ⎜ 0 1/ 2 1/ 2 0 ⎟ ⎜ ⎟ ⎜ 1/√2 0 0 1/√2 ⎟ ⎜ ⎟ ⎝ − ⎠ a 0 0 0 0 0 b 0 R′ = ⎛ ⎞ 0 c 0 0 ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 0 d ⎟ ⎝ ⎠ 43

Ivanov’s Braiding Operators

Let a1, a2, a3, ..., a2n be Majorana Fermions. S(i) = (1 + a(i+1)a(i))/Sqrt(2) S^(i) = (1 - a(i+1)a(i))/Sqrt(2) The operators S(i) form a unitary representation of the Artin Braid Group: S(i)S(i+1)S(i) = S(i+1)S(i)S(i+1) S(i)S(j) = S(j)S(i) when |i=j|>2. Operators T(i) act on the space of MF’s via T(i)x = S(i) x S^(i). T(i)a(i) = a(i+1) T(i)a(i+1) = - a(i).

It is worth noting that a triple of Majorana fermions say a, b, c gives rise to a representation of the quaternion group. This is a generalization of the well-known association of Pauli matrices and quaternions. We have a2 = b2 = c2 = 1 and they anticommute. Let I = ba, J = cb, K = ac. Then I2 = J 2 = K2 = IJK = 1, − giving the quaternions. The operators

A = (1/√2)(1 + I)

B = (1/√2)(1 + J)

C = (1/√2)(1 + K)

braid one another: ABA = BAB, BCB = CBC, ACA = CAC. R I I R This is a special case of the braid group representation described above for an arbitrary list of Majorana fermionRs. TheAIse braiding operators are entanglingBaInd so Rcan be used for universal quantum computaItion, bRut they give only =partial topological qRuantumI computation due to the interaction with single quBbit operators not generated by them.A R IA BI R In Section 5 we show how the dynamics of the reentering mark leads to two (algebraic) Majorana fermions e and η that correspond to the spatial and temporal aspects of this recursive process. The corresponding standard fermion operators are then given by the formulas below. Figure 30: The Yang-Baxter equation ψ = (e + iη)/2 A solution to the Yang-Baxter equation, as described in the last paragraph is a matrix R, regardedaansda mapping of a two-fold tensor product of a vector space V V to itself that satisﬁes ψ† = (e iη)/2. ⊗ the equation − This gives a model of a fermion creation operator as a point in a non-commutative spacetime. This suggestive poin(tRof vieIw),(IbasedRo)n(Rknot lIo)gi=c a(nId LaRws)(oRf FormI),(wIill bRe e)x.plored in subsequent ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ From thepupbolicnattioofnsv.iew of topology, the matrix R is regarded as representing an elementary bit of braiding represented by one string crossing over another. In Figure 30 we have illustrated Topological quantum computing. This paper describes relationships between quantum topol- the braiding identity that corresponds to the Yang-Baxter equation. Each braiding picture with ogy and quantum computing as a modiﬁed version of Chapter 14 of the book [13] and an ex- its three pinanpduetdlivneerssio(bneolofw[6)7a]nadndouantpeuxtplaindeesd(avberosvioen) ocforarecshpaoptnedrsinto[8a1]m. aQpupaintgumofttohpeoltohgryeeisf,old tensor prrooduugchtlyosfptehaekivnegc,ttohratspaarcteofVlotwo-idtismelefn,saiosnraelqtoupiroeldogbyyththateinatlegreabctrsawicitehqsutattiisotincaqluaontdeqduaabno- ve. The patteturmn opfhypsliaccs.eMmaennyt ionfvatrhiaenctsrofsskinnogts, ilninkthseanddiathgrreaemdimcoernrseiospnaolnmdasntiofotldhsehfaavcetobresenRbornIofand ⊗ I R. Tthhiissinctreurcaciatilonto, panodlothgeicfaolrmmoofvtehehiansvaarnianatlsgiesbcrlaoiscelyexreplraetesdsitoonthienfotermrmosf othfescuocmhpuatamtiaotnrioxf R. Ou⊗r appraomapclhituindetshins qseucantitounmtmo ercehlanteictso. pCoolnosgeyq,uqenutalyn,tuitmis cfroumitfpuul ttionmg,oavnedbaqcukaanntdumforethntbaentgwleeemn ent is througqhutahnetuumsetoopfoltohgeicYaal nmge-tBhoadxstaenrdeqthueatteicohnn.iqInueosrodfeqrutaontaucmcoinmfoprlmisahtitohnisthaeiomry,. we need to study solutions of the Yang-Baxter equation that are unitary. Then the R matrix can be seen either as a 5 braiding matrix or as a quantum gate in a quantum computer.

Recall fermion algebra. One has fermion annihiliation operators ψ and their conjugate creation operators ψ†. One has ψ2 = 0 = (ψ†)2. There is a fundamental commutation relation

ψψ† + ψ†ψ = 1.

If you have more than one of them say ψ and φ, then they anti-commute:

ψφ = φψ. − The Majorana fermions c that satisfy c† = c so that they are their own anti-particles. There is a lot of interest in these as quasi-particles and they are related to braiding and to topological quantum computing. A group of researchers [?] claims, at this writing, to have found quasiparticle Majo- rana fermions in edge effects in nano-wires. (A line of fermions could have a Majorana fermion happen non-locally from one end of the line to the other.) The Fibonacci model that we discuss is also based on Majorana particles, possibly related to collecctive electronic excitations. If P is a Majorana fermion particle, then P can interact with itself to either produce itself or to annihilate itself. This is the simple “fusion algebra” for this particle. One can write P 2 = P + 1 to denote the two possible self-interactions the particle P. The patterns of interaction and braiding of such a particle P give rise to the Fibonacci model.

Majoranas are related to standard fermions as follows: The algebra for Majoranas is c = c† and cc′ = c′c if c and c′ are distinct Majorana fermions with c2 = 1 and c′2 = 1. One can make a standard−fermion from two Majoranas via

ψ = (c + ic′)/2,

ψ† = (c ic′)/2. − Similarly one can mathematically make two Majoranas from any single fermion. Now if you take a set of Majoranas c , c , c , , cn { 1 2 3 · · · } then there are natural braiding operators that act on the vector space with these ck as the basis. The operators are mediated by algebra elements

τk = (1 + ck+1ck)/√2,

−1 τ = (1 ck ck)/√2. k − +1 Then the braiding operators are

Tk : Span c , c , , , cn Span c , c , , , cn { 1 2 · · · } −→ { 1 2 · · · } via −1 Tk(x) = τkxτk .

30 Braiding Majorana Fermions -

x y x y +

x y x y

T(x) = y T(y) = - x

Figure 23: Braiding Action on a Pair of Fermions

1 1 + yx 1 yx T (p) = sps− = ( )p( − ), √2 √2 and verify that T (x) = y and T (y) = x. Now view Figure 23 where we have illustrated a − topological interpretation for the braiding of two fermions. In the topological interpretation the two fermions are connected by a flexible belt. On interchange, the belt becomes twisted by 2π. In the topological interpretation a twist of 2π corresponds to a phase change of 1. (For more − information on this topological interpretation of 2π rotation for fermions, see [45].) Without a further choice it is not evident which particle of the pair should receive the phase change. The topology alone tells us only the relative change of phase between the two particles. The Clifford algebra for Majorana fermions makes a specific choice in the matter and in this way fixes the representation of the braiding.

Finally, we remark that linear combinations of products in the Clifford algebra can be regarded as superpositions of the knot sets. Thus xy + xz is a superposition of the sets with members x, y and x, z . Superposition of sets suggests that we are creating a species of quantum set th{eory }and in{deed}Clifford algebra based quantum set theories have been suggested (see [19]) by David Finkelstein and others. It may come as a surprise to a quantum set theorist to ﬁnd that knot theoretic topology is directly related to this subject. It is also clear that this Clifford algebraic quantum set theory should be related to our previous constructions for quantum knots [60, 61, 62, 63, 64]. This requires more investigation, and it suggests that knot theory and the theory of braids occupy a fundamental place in the foundations of quantum mechanics.

26 root of minus one it is natural to introduce another one and call this one i, letting it commute with the other operators. Then we have the (ieη)2 = +1 and so we have a triple of Majorana fermions:

a = e, b = η, c = ieη and we can construct the quaternions

I = ba = ηe, J = cb = ie, K = ac = iη.

With the quaternions in place, we have the braiding operators 1 1 1 A = (1 + I), B = (1 + J), C = (1 + K), √2 √2 √2 and can continue as we did in Section 4.

There is one more comment that is appropriate for this section. Recall from Section 4 that a pair of Majorana fermions can be assembled to form a single standard fermion. In our case we have the two Marjorana fermions e and η and the corresponding standard fermion annihilation and creation operators are then given by the formulas below.

ψ = (e + iη)/2 and ψ† = (e iη)/2. Now we show how the− Majorana Fermion Since e represents a spatial view of the basic discrete oscillation and η is the time-shift operator for thalgebrais oscillation isit isatof theintere sbaset to not eoftha tthethe s tDiracandard fe requation,mion built by t handese tw o can be regarded as a quantum of spacetime, retrieved from the way that we decomposed the process into spachowe and ti mnilpotente. Since all th ioperatorss is initially built (representingin relation to extending Fermions)the Boolean logi c of the markariseto a no nnaturally-boolean recur sinive relationcontext, there tois f uplanerther ana lywavesis need esolutionsd of the relatio n of the physics and the logic. This towill bthee tak eDiracn up in a seequation.parate paper.

5.2 Relativity and the Dirac Equation Starting with the algebra structure of e and η and adding a commuting square root of 1, i, we have constructed fermion algebra and quaternion algebra. We can now go further and−construct the Dirac equation. This may sound circular, in that the fermions arise from solving the Dirac equation, but in fact the algebra underlying this equation has the same properties as the creation and annihilation algebra for fermions, so it is by way of this algebra that we will come to the Dirac equation. If the speed of light is equal to 1 (by convention), then energy E, momentum p and mass m are related by the (Einstein) equation

E2 = p2 + m2.

37 Dirac constructed his equation by looking for an algebraic square root of p2 +m2 so that he could have a linear operator for E that would take the same role as the Hamiltonian in the Schrodinger equation. We will get to this operator by ﬁrst taking the case where p is a scalar (we use one dimension of space and one dimension of time. Let E = αp + βm where α and β are elements of a a possibly non-commutative, associative algebra. Then

E2 = α2p2 + β2m2 + pm(αβ + βα).

Hence we will satisﬁy E2 = p2 + m2 if α2 = β2 = 1 and αβ + βα = 0. This is our familiar Clifford algebra pattern and we can use the iterant algebra generated by e and η if we wish. Then, because the quantum operator for momentum is i∂/∂x and the operator for energy is i∂/∂t, we have the Dirac equation − i∂ψ/∂t = iα∂ψ/∂x + βmψ. − Let = i∂/∂t + iα∂/∂x βm O − so that the Dirac equation takes the form

ψ(x, t) = 0. O Now note that i(px Et) i(px Et) e − = (E αp + βm)e − O − and that if U = (E αp + βm)βα = βαE + βp + αm, − then U 2 = E2 + p2 + m2 = 0, − from which it follows that i(px Et) ψ = Ue − is a (plane wave) solution to the Dirac equation.

In fact, this calculation suggests that we should multiply the operator by βα on the right, obtaining the operator O = βα = iβα∂/∂t + iβ∂/∂x + αm, D O and the equivalent Dirac equation ψ = 0. D i(px Et) 2 i(px Et) In fact for the speciﬁc ψ above we will now have (Ue − ) = U e − = 0. This way of reconﬁguring the Dirac equation in relation to Dnilpotent algebra elements U is due to Peter Rowlands [94]. We will explore this relationship with the Rowlands formulation in a separate paper.

38 so that the Dirac equation takes the form

ψ(x, t) = 0. O Now note that ei(px−Et) = (E αp βm)ei(px−Et). O − − We let ∆ = (E αp βm) − − and let U = ∆βα = (E αp βm)βα = βαE + βp + αm, − − then U 2 = E2 + p2 + m2 = 0. − This nilpotent element leads to a (plane wave) solution to the Dirac equation as follows: We have shown that ψ = ∆ψ O for ψ = ei(px−Et). It then follows that

(βα∆βαψ) = ∆βα∆βαψ = U 2ψ = 0, O from which it follows that ψ = βαUei(px−Et) is a (plane wave) solution to the Dirac equation.

In fact, this calculation suggests that we should multiply the operator by βα on the right, O obtaining the operator = βα = iβα∂/∂t + iβ∂/∂x + αm, D O and the equivalent Dirac equation ψ = 0. D In fact for the specific ψ above we will now have (Uei(px−Et)) = U 2ei(px−Et) = 0. This way of reconfiguring the Dirac equation in relation to Dnilpotent algebra elements U is due to Peter Rowlands [30]. Note that the solution to the Dirac equation that we have found is expressed in Clifford algebra or iterant algebra form. It can be articulated into specific vector solutions by using an iterant or matrix representation of the algebra.

We see that U = βαE + βp + αm with U 2 = 0 is really the essence of this plane wave solution to the Dirac equation. This means that a natural non-commutative algebra arises directly and can be regarded as the essential information in a Fermion. It is natural to compare this algebra structure with algebra of creation and annihilation operators that occur in quantum ﬁeld theory. to this end, let U † = αβE + αp βm. − 34 9.1 AnoRecapitulationther version of U an andd U † One More We start with ψ = ei(px−Et) and the operators

Eˆ = i∂/∂t and

pˆ = i∂/∂x − so that Eˆψ = Eψ and pˆψ = pψ.

The Dirac operator is = Eˆ αpˆ βm O − − and the modiﬁed Dirac operator is

= βα = βαEˆ + βpˆ αm, D O − so that ψ = (βαE + βp αm)ψ = Uψ. D − If we let ψ˜ = ei(px+Et)

(reversing time), then we have

ψ˜ = ( βαE + βp αm)ψ = U †ψ˜, D − − giving a deﬁnition of U † corresponding to the anti-particle for Uψ.

We have that U 2 = U †2 = 0 and UU † + U †U = 4E2.

Thus we have a direct appearance of the Fermion algebra corresponding to the Fermion plane wave solutions to the Dirac equation.

38 Dirac constructed his equation by looking for an algebraic square root of p2 +m2 so that he could have a linear operator for E that would take the same role as the Hamiltonian in the Schrodinger equation. We will get to this operator by ﬁrst taking the case where p is a scalar (we use one dimension of space and one dimension of time. Let E = αp + βm where α and β are elements of a a possibly non-commutative, associative algebra. Then

E2 = α2p2 + β2m2 + pm(αβ + βα).

In fact, this calculation suggests that we should multiply the operator by βα on the right, obtaining the operator O = βα = iβα∂/∂t + iβ∂/∂x + αm, D O and the equivalent Dirac equation ψ = 0. D i(px Et) 2 i(px Et) In fact for the speciﬁc ψ above we will now have (Ue − ) = U e − = 0. This way of reconﬁguring the Dirac equation in relation to Dnilpotent algebra elements U is due to Peter We have arrived at Rowlands [94]. We will explore this relationship with Peterthe Rowland’sRowla Nilpotentnds fo Solutionsrmula tion in a separate paper. to the Dirac Equation.

38 9.1 Another version of U and U † We start with ψ = ei(px−Et) and the operators

Eˆ = i∂/∂t and

pˆ = i∂/∂x − so that Eˆψ = Eψ and pˆψ = pψ.

The Dirac operator is = Eˆ αpˆ βm O − − and the modiﬁed Dirac operator is

= βα = βαEˆ + βpˆ αm, D O − so that ψ = (βαE + βp αm)ψ = Uψ. D − If we let ψ˜ = ei(px+Et)

(reversing time), then we have

ψ˜ = ( βαE + βp αm)ψ = U †ψ˜, D − − giving a deﬁnition of U † corresponding to the anti-particle for Uψ.

We have that U 2 = U †2 = 0 and UU † + U †U = 4E2.

Thus we have a direct appearance of the Fermion algebra corresponding to the Fermion plane wave solutions to the Dirac equation.

38 Note that we get different decompositions of the Fermion into Majorana operators according to what is reversed.

{E} and {p,m} for time reversal. {p} and {E,m} for spin reversal. {E,p} and {m} for spin and time reversal. In analogy to our previous discussion we let

ψ(x, t) = ei(p•x−Et) and construct solutions by first applying the Dirac operator to this ψ. The two Clifford algebras interact to generalize directly the nilpotent solutions and Fermion algebra that we have detailed for one spatial dimension to this three dimensional case. To this purpose the modified Dirac operator is = iβα∂/∂t + β σ αm. D ∇ • − And we have that ψ = Uψ D where U = βαE + βp σ αm. • − We have that U 2 = 0 and Uψ is a solution to the modified Dirac Equation, just as before. And just as before, we can articulate the structure of the Fermion operators and locate the corresponding Majorana Fermion operators. We leave these details to the reader.

10.3 Majorana Fermions at Last There is more to do. We will end with a brief discussion making Dirac algebra distinct from the one generated by α, β, σ1, σ2, σ3 to obtain an equation that can have real solutions. This was the strategy that Majorana [7] followed to construct his Majorana Fermions. A real equation can have solutions that are invariant under complex conjugation and so can correspond to particles that are their own anti-particles. We will describe this Majorana algebra in terms of the split quaternions ϵ and η. For convenience we use the matrix representation given below. The reader of this paper can substitute the corresponding iterants.

1 0 0 1 ϵ = − , η = . ! 0 1 " ! 1 0 " Let ϵˆ and ηˆ generate another, independent algebra of split quaternions, commuting with the ﬁrst algebra generated by ϵ and η. Then a totally real Majorana Dirac equation can be written as follows: (∂/∂t + ηˆη∂/∂x + ϵ∂/∂y + ϵˆη∂/∂z ϵˆηˆηm)ψ = 0. − To see that this is a correct Dirac equation, note that

Eˆ = αxpˆx + αypˆy + αzpˆz + βm

(Here the “hats” denote the quantum differential operators corresponding to the energy and momentum.) will satisfy 2 2 2 2 2 Eˆ = pˆx + pˆy + pˆz + m

51 In analogy to our previous discussion we let

10.3 MajoranaTheFerm ioMajorana-Diracns at Last Equation There is more to do. We will end with a brief discussion making Dirac algebra distinct from the one generated by α, β, σ1, σ2, σ3 to obtain an equation that can have real solutions. This was the strategy that Majorana [7] followed to construct his Majorana Fermions. A real equation can have solutions that are invariant under complex conjugation and so can correspond to particles that are their own anti-particles. We will describe this Majorana algebra in terms of the split quaternions ϵ and η. For convenience we use the matrix representation given below. The reader of this paper can substitute the corresponding iterants.

Eˆ = αxpˆx + αypˆy + αzpˆz + βm

(Here the “hats” denote the quantum differential operators corresponding to the energy and momentum.) will satisfy 2 2 2 2 2 Eˆ = pˆx + pˆy + pˆz + m

51 if the algebra generated by αx, αy, αz, β has each generator of square one and each distinct pair of generators anti-commuting. From there we obtain the general Dirac equation by replacing Eˆ by i∂/∂t, and pˆx with i∂/∂x (and same for y, z). −

(i∂/∂t + iαx∂/∂x + iαy∂/∂y + iαz∂/∂y βm)ψ = 0. − This is equivalent to

(∂/∂t + αx∂/∂x + αy∂/∂y + αz∂/∂y + iβm)ψ = 0.

Thus, here we take αx = ηˆη, αy = ϵ, αz = ϵˆη, β = iϵˆηˆη, and observe that these elements satisfy the requirements for the Dirac algebra. Note how we have a signiﬁcant interaction between the commuting square root of minus one (i) and the element ϵˆηˆ of square minus one in the split quaternions. This brings us back to our original considerations about the source of the square root of minus one. Both viewpoints combine in the element β = iϵˆηˆη that makes this Majorana algebra work. Since the algebra appearing in the Majorana Dirac operator is constructed entirely from two commuting copies of the split quaternions, there is no appearance of the complex numbers, and when written out in 2 2 matrices we obtain coupled real differential equations to be solved. Clearly this ending is a×ctually a beginning of a new study of Majorana Fermions. That will begin in a sequel to the present paper.

References

[1] G. Spencer–Brown, “Laws of Form,” George Allen and Unwin Ltd. London (1969).

[2] Kauffman, L. [1985], Sign and Space, In Religious Experience and Scientiﬁc Paradigms. Proceedings of the 1982 IASWR Conference, Stony Brook, New York: Institute of Ad- vanced Study of World Religions, (1985), 118-164.

[3] Kauffman, L. [1987], Self-reference and recursive forms, Journal of Social and Biological Structures (1987), 53-72.

[4] Kauffman, L. [1987], Special relativity and a calculus of distinctions. Proceedings of the 9th Annual Intl. Meeting of ANPA, Cambridge, England (1987). Pub. by ANPA West, pp. 290-311.

[5] Kauffman, L. [1987], Imaginary values in mathematical logic. Proceedings of the Seven- teenth International Conference on Multiple Valued Logic, May 26-28 (1987), Boston MA, IEEE Computer Society Press, 282-289.

[6] Kauffman, L. H., Knot Logic, In Knots and Applications ed. by L. Kauffman, World Scien- tiﬁc Pub. Co., (1994), pp. 1-110.

52 Core of our joint work with Peter Rowlands. Forming a nilpotent version of the Majorana-Dirac Equation. Using the Majorana Operators A and B with AB = -BA and A^2 = B^2 = - m^2. Thus we now see, via the nilpotent approach to the Dirac equation, how the Majorana Operators are directly related to real solutions to the Majorana-Dirac Equation. Thus ∂ψ 1 0 ∂ 0 i = + − ψ . (12) ∂t !! 0 1 " ∂x ! i 0 "" where Feynman− Checkerboard− z(z 1) (z n + 1) Hence Cz = − · · · − (30) n n! ∂ψ1 iψ2 ∂l denotes the choice coeﬃRII:cient. − = . (13) ! iψ " ! ∂ψ2 " − 1 ∂r We shall call (Eq. 13) the RII Diract eq(r,u )a = ti(7, 3o) n.

(r, ) = (7, 0) 3 Discrete Calculus and Solutions to the Dirac (r, ) = (0, 3)

Equation x (r, ) = (0, 0) r = (t+x)/2 3–96 = (t–x)/2 8131A10 Suppose that f = f(x) is a function of a variable x. Let ∆ be a ﬁxed non-zero constant. The discreteFigdureeri1: Rveactatingvulear laottifce fin Mwinkiothwski spreaces-tipmee.ct to ∆ is then deﬁned by the

We are thinking of r and ℓ as the light cone coordinates r = 1 (t + x), ℓ = equation 2 1 (t x). Hence, in a standard diagram for Minkowski space-time, a pair of values 2 − [r, ℓ] determines a rectangle with sides of lengfth(ℓxand+r o∆n th)e left afnd(rixgh)t pointing D∆f(x) = − . (14) light cones. (We take the speed of light c = 1.) This is sho∆wn in Figure. 1. Clearly, the simplest way to think about this combinatorics is to take ∆ = 1. If Consider thewefuwisnh ctotithoinnk about the usual continuum limit, then we shall ﬁx values of r and ℓ and choose ∆ small but such that r/∆ and ℓ/∆ are integers. The combinatorics of an r ℓ rectangle with integers r and ℓ is no diﬀerent in principle than the combinatorics × (n) of an (rx/∆) (ℓ=/∆)xrec(taxngle wi∆th i)(ntegxers r/∆2a∆nd ℓ)/∆. Acco(rdxingly, w(enshall tak1e)∆) . (15) × − − · · · − − ∆ = 1 for the rest of this discussion, and then make occasional comments to connect Lemma. this with the general case. (n) (n 1) D∆x8 = nx − . (16)

Proof. (x + ∆)(n) x(n) = − (x + ∆)(x)(x ∆) (x (n 2)∆) (x)(x ∆) (x (n 2)∆)(x (n 1)∆) = − · · · − − − − · · · − − − − (n 1) (n 1) [(x + ∆) (x (n 1)∆)]x − = n∆x − . − − − Thus (n 1) (n) n∆x − (n 1) D x = = nx − . (17) ∆ ∆ We are indebted to Eddie Grey for reminding us of this fact [8]. Note that as ∆ approaches zero x(n) approaches xn, the usual nth power of x. Note also that x(n) = Cx/∆ (18) ∆nn! n

5 Formalism, Metaphor and the Art of Mathematics

Louis H. Kauffman Department of Mathematics, Statistics and Computer Science (m/c 249) 851 South Morgan Street University of Illinois at Chicago Chicago, Illinois 60607-7045

Abstract Void where z(z 1) (z n + 1) Cz = − · · · − (30) n n! den1otes the cIntrhoice coeﬃcoductionient.The Feynman Checkerboard ⇥ = iC(P ) t P (r, ) = (7, 3) Dirac Amplitude

1/ (r, ) = (7, 0) ( 1) 1 = ( C(P) = ) P ⇥ ⇤ 1 P C(P ) number ψ = Σ i This(r, )is = (0, an3) essay about Art and Mathematics,of corners in written from the point of view of a path P mathematician. In that sense this isx an essay about the art of mathematics, not about art as a domain separate(r, ) = (0, 0from) mathematics. And yet ... there is, I believe, a subtle influence r = (t+x)/2 = – of the arts3–96 (music, literature, (t x)/2 poetry8131A10, painting, sculpture, theatre,...) on mathematics, and concomittantly, an influence of mathematics on these fields of aesthetic action. I can Fonlyigure 1: competentlyRectangular lattice in Mwriteinkowski saboutpace-time. the mathematical, but in musing on this theme

1 someWe are thiopinionsnking of r and ℓwillas the naturallylight cone coordicomenates r = forth.2 (t + x), ℓ I= had best say a few of them right at the 1 (t x). Hence, in a standard diagram for Minkowski space-time, a pair of values 2 outset.− I firmly believe that the creative source of good art and good mathematics is [r, ℓ] determines a rectangle with sides of length ℓ and r on the left and right pointing lighthet conessame.. (We take thI ebeliespeed ofvlieghtthatc = 1.)sourceThis is showton in beFigurethe. 1. human desire and need to go across apparent boundariesClearly, the simplestandway to findthink abcommonalityout this combinatorics isandto takecommunication∆ = 1. If between and among seemingly we wseparateish to think abdomains.out the usual conIntinuufmact,limit,thisthen wise shtheall fix enginevalues of r anofd ℓ metaphor. Metaphor declares the identity and choose ∆ small but such that r/∆ and ℓ/∆ are integers. The combinatorics of an r ℓofrectathatngle wiwhichth integers rcommonand ℓ is no differesensent in princdeclaresiple than the comdifbinaferent.torics “Juliet is the sun.” Only in the realm × of an (r/∆) (ℓ/∆) rectangle with integers r/∆ and ℓ/∆. Accordingly, we shall take of metaphor× can we make such an identification, and yet indeed Juliet and the sun ∆ = 1 for the rest of this discussion, and then make occasional comments to connect thisarewith thbounde general cainse. radience. It is the identification of Juliet and the sun that makes this a metaphor and not an analogy. The declaration of identity wipes away the superficial difference and directs8 us to the deep relation beneath the surface. 9.2 The Majorana VersioIn ntheo RII,f tMajoranahe D iFermionrac E casequ wea thaveion

∂ψ2/∂r = ψ2 ∂ψ /∂l = ψ 1 − 1

Return now to the originaThusl ve thersi oCheckerboardn of the D worksirac ewithqu aplus/minustion. cornering. This model gives a picture of how Majorana’s i∂equationsψ/∂t =can lookiα in∂ ψan/ explicit∂x + caseβm andψ . show how discrete− quantum physics may avoid complex numbers. We can rewrite this as ∂ψ/∂t = α∂ψ/∂x + iβmψ. We see that if iβ is real, then we can write a fully real version of the Dirac equation. For example, we can take the equation ∂ψ/∂t = e∂ψ/∂x + eηmψ. where we represent 1 0 e = − ! 0 1 " and 0 1 η = ! 1 0 " as matrix versions of the iterants associated with the reentering mark. For the case of one dimension of space and one dimension of time, this is the Majorana representation for the Dirac equation (compare [?]). Since the equation can have real solutions, these are their own complex conjugates and correspond to particles that are their own anti-particles. As the reader can check, the corresponding Rowland nilpotent U is given by the formula

U = iηE + ieηp + em. − For effective application to the topics in this paper, one needs to use two dimensions of space and one dimension of time.

References

[1] Dyson, F. J. [1990], Feynman’s proof of the Maxwell Equations, Am. J. Phys. 58 (3), March 1990, 209-211.

[2] Connes,Alain [1990], Non-commutative Geometry Academic Press.

[3] Deakin, A.M. [1999] Where does Schroedinger’s equation really come from?, in “Aspects II - Proceedings of ANPA 20”, edited by K.G. Bowden, published by ANPA.

In fact, this solution to the Feynmann Checkerboard model is different from the solutions from the path sum, suggesting new approaches to the original problem of path integrals for the Dirac equation.

In spacetime algebra terms the Dirac operator becomes

This provides a platform for deeper study of the Majorana Dirac Equation. But we should take a WIDER VIEW. The universal equation should be about the (state of) the Universe U. An operator D acts on U to produce Nothing. D U = 0. But the universe U is both operator and operand. So we take D = U and obtain the Universal Nilpotent Equation. UU = 0, of which the Dirac equation is one of the ﬁrst special cases. The Simplest example of the Universal Nilpotent Equation is given by the operator

Ux = x

Here the Universe U is that Universe (self) created by the Mark and taken to Nothing by the crossing from the marked state to the unmarked state.

UU = = .

UU = 0. In this formalism the mark is seen to make a distinction. 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 mark and the observer are seen, in the form, to be identical. The Calculus writes itself in terms of itself. Physical theory is seen to write itself in terms of the condition for observation to occur at all.