University of BRISTOL. Department of Physics. Final year project

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1 University of BRISTOL Deprtment of Physis Finl yer projet Angulr momentum oupling: spin networks nd their evlution y integrtion over SU(2) Author: Supervisor: N G Jones J H Hnny April 202 H H Wills Physis Lortory, University of Bristol, Tyndll Avenue, Bristol, BS8 TL

2 Astrt In this work, the use of group integrls in the evlution of spin networks is onsidered The introdution of integrtions over SU(2) is motivted y onsidering symmetry properties of spinors, nd how spinors nd their oupling properties re relted to spin networks Severl spin networks re evluted using integrl methods, greeing with prior lultions using different tehniques The ppliility of these methods is then disussed within the ontext of quntum mehnis nd spin network theory

3 Aknowledgements I wish to thnk John Hnny for suggesting suh n interesting nd rih topi to explore, s well s for the hours he hs spent helping nd dvising me with physis oth for the projet elow nd for nything else I trouled him with throughout the yer I lso wish to thnk two other Johns Firstly John Brrett for providing some lues when we were drowning in formlism Seondly I would hve struggled fr more if John Bez nd his seminrists from hd not pulished their disussions online Thnks go to them for providing suh wonderful resoure I thnk my friends Pete, Mike nd Ktie who hve gone through the ourse with me nd shred mny interesting disussions Finlly I thnk Mdeleine Fforde for her help nd support; she hs hd to tlk fr more out quntum mehnis thn ny lssiist should Nottion N G Jones April 202 The following nottion is used throughout: A generi element of group G will e denoted g, g i or R G = g G = dg is the size or volume of the group G The spin-j representtion spe of SU(2) is 2j + dimensionl vetor spe ted on y (2j + ) (2j + ) representtion mtries This spe my e represented y the symol V j, or just the symol j The (mn)th element of the spin-j representtion mtrix of g SU(2) is given y: [ ] j g = D (j) (g) m n mn = ρ j (g) mn The summtion onvention is enfored, tking the first index s ontrvrint nd the seond index s ovrint The hrter of the element g SU(2) in the spin-j representtion is written: [j; g] = χ j (g) ii

4 Contents Aknowledgements Nottion ii ii Introdution 2 Resoning with digrms 2 2 Digrmmti nottion 2 3 Spin nd groups 4 3 Symmetries of physil system nd representtions 4 32 SU(2) s symmetry of prtile with spin 4 33 Spinors 5 34 Spin wvefuntions s symmetri spinors 5 35 Orthogonlity reltions 6 36 The Wigner 3j nd 6j symols 6 37 The projetor 8 38 The 3j symol s the projetor 8 4 Spin networks 0 4 Penrose networks 0 42 Digrmmti spinor lger 43 Spin networks s opertors 2 44 The 3j nd 6j symols s networks 2 5 Evluting spin networks s integrls 5 5 Generl method 5 52 Computing integrls over SU(2) 5 53 The ylinder network 6 54 The prism network 8 6 Disussion 2 7 Conlusions 22 Biliogrphy 23 A Strightening out digrm kink 24 B The ylinder lultion in detil 25 iii

5 Introdution The im of this projet is to understnd how nd why integrls over SU(2) pper in the evlution of spin networks The nswer to this question requires n understnding of how to interpret spin networks s digrmmti spinor lulus, whih gives mening nd justifition to introduing integrls into the piture In this doument the min elements of the theory required to understnd how to go from spin network to n integrl re outlined A lind lgorithm is given in setion 5, nd this would suffie to solve ertin lss of prolems, however it is useful to try to see how it origintes The first three setions re thus n ttempt to summrise some very rod fields whih re relevnt to the prolem t hnd - the evlution of spin networks The first setion dels with digrmmti resoning: turning lger into digrms in order to mke things simpler This kind of resoning is ommon in tegory theory, nd spin networks n e interpreted s prtiulr type of digrm This helps to justify lter ssertions out spin networks The seond setion disusses the theory of ngulr momentum oupling nd why the group SU(2) is relevnt to the disussion t ll Representtions of SU(2) re disussed nd ojets suh s the Wigner 3j symol re defined, oth of whih re essentil for understnding spin networks Spin networks themselves re then introdued in the third setion Some disussion of the self ontined Penrose theory is given, efore useful interprettion is given in terms of spinors nd opertors From this understnding the lgorithm introdued in setion 5, n then e well justified Thus, with the kground in ple, some results re explined in detil First the method used to turn the network into n integrl is given This method is then illustrted in severl exmples Eh of these exmples lso hs some interest in the physis of spin networks, nd hene some disussions follow

6 2 Resoning with digrms 2 Digrmmti nottion Mnipulting multi index quntities is often diffiult nottionlly, nd mkes some rguments more opque thn neessry One wy of improving the sitution - populrised y Penrose - is to drw digrm to represent the mthemtil opertions in the mnner of flow hrt In ft, the whole of liner lger n e reformulted in the lnguge of these digrms A liner mp, f, from V to V 2, whih is given y mtrix of numers one ses for the vetor spes re hosen, is drwn s: V f V 2 The diretion of the digrm is top to ottom, so n element of the domin, V, is fed in t the top nd n element of the imge, V 2, is given out t the ottom (e wre, however, tht the opposite onvention is often used in the literture) The whole digrm orresponds to the mp f in the following wy: the symol f orresponds to the tul mp nd the edges in nd out represent the domin nd odomin The edges should thus e lelled y the vetor spes whih the irled funtions mp into nd from The numers of input nd output edges orrespond to how mny input nd output vetors there re - or more onretely tensor of vlene (p, q) will e represented y irle with p output lines nd q input lines We n now see how to represent n element of vetor spe in this piture - simply s funtion with single output, or s v : C V Funtion omposition is vertil, so given seond mp g : V 2 V 3, we n drw gf = g f : V V 3 s: V f V 2 = gf g V 3 V V 3 In mtrix lnguge, if the mtrix of f is Fn m nd the mtrix of g is G s m, then gf hs mtrix Mn s = G s mfn m where the strt index m is summed over Thus internl edges on the digrm re to e summed over The tensor produt is the horizontl diretion on the digrm For exmple the mp h = f g : V V 2 V 2 V 3 is represented y the digrm: V V 2 f g V 2 V 3 = h V V 2 V 2 V 3 See, for exmple, Penrose nd Rindler [2] 2

7 CHAPTER 2 REASONING WITH DIAGRAMS 3 There re lso some nturl funtions whih need not e lelled - they re relly prt of the piture 2 The identity mp, for one, ut there re lso the ounit nd unit, whih re nturl one we onsider the dul spe of ovetors (or liner funtionls) on V, denoted V The dul spe is represented on the digrms y n rrow lelled V ut direted up rther thn down the digrm These nturl mps hve the following definitions: the ounit tkes in ovetor nd vetor nd gives out numer: e V : V V C f v f(v) This is just the ovetor ting on the vetor Its representtion on the digrm is reminisent of up Then the unit tkes in numer nd gives out n element of V V : V i V : C V V V V Id V V This is just using tht ny vetor spe hs n identity mp Similrly one my sy it looks like p The power of digrmmti resoning n e seen y onsidering the following useful result whih involves these two mps nd lso the dul spe: V V ie tht this end in the digrm n e strightened out To prove this lgerilly, the left hnd side must e shown to lso represent the identity mp on V, whih is done in Appendix A Clerly the the digrmmti rule tht edges n e strightened out is muh simpler to use in prtie thn repeting the involved lgeri mnipultions This is one speil se of more generl result: tht ny two digrms whih n e ontinuously deformed into one nother represent identil mps Digrms tht re not equivlent in this wy n still represent the sme mp, ut re must e tken in unknotting them In ft when we onsider spin networks elow, the topologil invrine only holds up to sign 3 This rief disussion only gives the key notions in the theory of liner lger s digrms, s digrmmti resoning is used throughout in the following For longer exposition see for exmple Kuffmn [7] or Bez nd Muniin [3] 2 For exmple ny stright line n e interpreted s the identity mp, just s t ny stge in multiplition we n sy multiplition y hs ourred The word nturl n e mde preise, ut here we use it informlly 3 In the spinor se

8 3 Spin nd groups 3 Symmetries of physil system nd representtions A quntum system is speified y some Hmiltonin opertor, Ĥ If trnsformtion of the onfigurtion spe leves this Hmiltonin invrint it is symmetry of the system This sttement n e mde preise y sying tht if we trnsform wvefuntion ψ nd then operte with Ĥ we hve the sme result s if we hd pplied the trnsformtion fter operting with Ĥ Tht is, the opertor P ˆ R whih rries out the trnsformtion ommutes with Ĥ This disussion is ontinued in detil in Lndu nd Lifshitz [9] The set of trnsformtions R, whih leve the Hmiltonin invrint, will form group The trnsformtion will in generl tke the wvefuntion into liner omintion of equivlent wvefuntions The equivlent ψ will elong to the sme eigenvlue of the Hmiltonin (Ĥψ = Eψ implies Ĥ P ˆ R ψ = P ˆ R Ĥψ = EP ˆ R ψ) This mens tht in generl, if the eigenvlue is n-fold degenerte with eigenfuntions {ψ j }, then we n write P ˆ R ψ j = n i= D(R) ijψ i The D(R) will in this wy form representtion of the group A representtion should e onsidered s pir, vetor spe nd set of endomorphisms of this spe The endomorphisms re these D mtries, nd the vetor spe here is the portion of the Hilert spe of sttes of the system with prtiulr energy In ft we n think of the Hilert spe of the whole system s diret sum of these representtions The stte vetors will in generl e orthogonl to ll ut one of these spes if they hve prtiulr energy Then the tion of group on the stte vetor will e given y the pproprite representtion mtrix on eh prt of the spe To lrify, following Wigner [4], the stte vetor is written s: φ = (l)ν ψ ν (l) l where l indexes the different representtion spes Then fter pplying the trnsformtion: Pˆ R φ = [ ] (l)ν l R ψ µ ν µ (l) l nd we see tht the opertor hs not mixed the vetor omponents from the different surepresenttion spes 32 SU(2) s symmetry of prtile with spin From generl onsidertions, in prtiulr the isotropy of spe nd the equivlene of different oservers, it n e shown how spin-hlf wvefuntion of the form: ( ) u ψ(x, y, z, s) = u δ s, ψ(x, y, z) + u 2 δ s, ψ(x, y, z) = ψ(x, y, z) u 2 will trnsform fter rottion R in order to leve ll physil quntities invrint The possiility of spin-free experiment (eg the mesurement of the position of n eletron) mens tht ll wvefuntions in non-reltivisti quntum mehnis n e written in this form The sptil prt of the wvefuntion will trnsform s: Here s is the two vlued spin vrile ψ (x, y, z) = ˆ P R ψ(x, y, z) = ψ(r (x, y, z)) 4

9 CHAPTER 3 SPIN AND GROUPS 5 ( u ) The spin prt must trnsform unitrily to onserve proilities In ft the set of ll u 2 trnsformtion mtries D(R), together with the two wvefuntions (, 0) nd (0, ), will form representtion of the group SU(2) (this is disussed extensively in Wigner [3]) 2 Higher spins orrespond to higher dimensionl representtions of SU(2) Now, euse ny ojet will hve spin of some vlue (perhps zero), ny physis whih hnges the spin of the ojet(s) will e mpping etween representtions of SU(2) in wy omptile with this symmetry Tht is, it will e represented y n intertwining opertor or intertwiner Use of this term is ommon in the literture, nd simply mens mp etween representtions whih ommutes with the group tion This ommuttivity ould e motivted y onsidering tht redefining oordintes efore n event or fter n event should not mke differene to the physis As n exmple, Shur s lemm sttes tht there re no intertwining opertors etween the spin-j nd spin-k representtions for j k Physilly we interpret this s the onservtion of ngulr momentum of n isolted ody of spin-j 33 Spinors Spinors 3 re mthemtil ojets uilt up in the sme wy s tensors re uilt from vetors A rnk one spinor is n element ψ of V = C 2, with V hving uilt in sympleti struture This sympleti struture is n ntisymmetri inner produt, nd will e given s n ntisymmetri two-form ω ij In two dimensions this must e proportionl to ɛ ij, the Levi-Cevit symol This inner produt llows us to define dul spe V It ontins ojets ψ = ɛ ψ Then φ ψ = φ ɛ ψ = φ ψ From the ntisymmetry of the produt, ψ ψ = 0 The group SL(2, C) ts on spinors y mtrix multiplition in the usul wy: g ψ = g ψ The inner produt is invrint under this tion: g u ɛ gd vd = det gu ɛ d v d (fter doing the multiplition in omponents) nd det g = for this group Note tht s SU(2) is sugroup of SL(2, C), this inner produt is lso invrint under the smller group Rnk one spinors re rotted y n ngle φ out n xis n in spe y operting with Uˆ n = exp(i φ 2 n ˆσ) SU(2), where ˆσ is list of Puli mtries Higher rnk spinors re uilt y the outer produt: ψ = n χ (n) φ (n) Their trnsformtion properties under rottion re governed y those of the rnk one spinors 34 Spin wvefuntions s symmetri spinors Any wvefuntion of spin-j prtile n e represented s symmetri spinor: ψ(s) ψ αα2j This hs the dvntge tht the SU(2) symmetry under rottions is expliitly uilt in to the trnsformtion properties of the spinor Also, s the inner produt is invrint, proilities of finding prtile t point will e unhnged - s they must e The omponents of ψ will e given y: ψ(s) = (2j)! (j + s)!(j s)! ψ 22 j+s j s where the index ppers j + s times nd the index 2 ppers j s times 4 This uilds up the higher spins from spin-hlf omponents, whih is justified in Lndu nd Lifshitz [9] 2 In ft they will form the defining representtion of SU(2) 3 Preisely, 2-spinors There re different wys of defining spinors, this follows Bez nd Alvrez in [2] 4 We onsider the wvefuntion t the origin, whih is left invrint under rottions, so tht only the spin prt is relevnt

10 CHAPTER 3 SPIN AND GROUPS 6 A system of two prtiles will e produt of two symmetri spinors, whih is in generl not symmetri It n, however, e deomposed into symmetri spinor prts y ontrting indies pirwise (one from eh spinor - ontrtion with two indies from the sme spinor will yield zero s their ontrtion is ntisymmetri wheres the indies themselves re symmetri) nd t eh stge symmetrising 5 the result This proedure is desried in Chpter 3 of Penrose nd Rindler [2] In this wy, we n interpret the produt s superposition of systems with totl spin numer etween j j 2 nd j + j 2 The higher limit is evident s it omes from symmetrising the produt wvefuntion The lower limit omes from the lowest vlene symmetri spinor tht n e mde y ontrting ross the spinors - y summing over ll of the indies of the wvefuntion of the smller spin 35 Orthogonlity reltions The spin-j representtions re irreduile - tht is they ontin no proper surepresenttions This leds to numer of useful orthogonlity results, derived in Wigner [3], for oeffiients of the representtion mtries The most generl of whih is: [ ] [ ] j R j R lj lj m n r s G R = δ jj δ mr δ ns (3) where l j is the dimension of the spin-j representtion - in the se of SU(2) this is 2j + These reltions underlie mny results in group theory For ontinuous group, suh s SU(2), this sum is the invrint (Hr) integrl 36 The Wigner 3j nd 6j symols 36 In quntum mehnis Following Lndu nd Lifshitz, nd in similr wy to the proedure desried in setion 34, system omposed of three prtiles of spins j, j 2 nd j 3 n hve spin of zero if the produt of the three symmetri spinor wvefuntions n e redued to slr y ontrtion This is only possile if the dmissiility onditions re stisfied: tht the j i n form three sides of tringle nd tht their sum is n integer This produt wvefuntion is of the form ψ ()(αα2j ) ψ (2)(ββ2j 2 ) ψ (3)(γγ2j 3 ) To reh slr, ll indies must e ontrted ross pirs of wvefuntions Any ontrtion within one of the three ftor wvefuntions will yield zero, s eh is symmetri If the j i stisfy the tringle inequlity this n e done in nonil wy - tke j + j 2 j 3 upper indies from ψ () nd ontrt these with lowered ψ (2) indies, then j 2 + j 3 j upper indies from ψ (2) re ontrted with lowered ψ (3) indies nd finlly the remining j + j 3 j 2 upper indies of ψ (3) re ontrted with the lowered ψ () indies (We should ompre this proedure to digrm 44) When trnslted k to the wvefuntion piture, this gives the 3j symols s the oeffiients in the following sum: ψ system 0 = m,m 2,m 3 ( ) j j 2 j 3 ψ () m m 2 m j,m ψ (2) j 2,m 2 ψ (3) j 3,m 3 (32) 3 5 Ie dding omintions of the spinor with permuted indies suh tht the resulting omintion is symmetri For exmple ψ (ij) = 2 (ψij + ψ ji ), where the rkets round the indies on the left indite symmetristion

11 CHAPTER 3 SPIN AND GROUPS 7 By onsidering the pir ψ () nd ψ (2) s single oupled system, nd then oupling this oupled system to ψ (3), we n find two expressions for the slr ψ 0 This gives n eqution for the omined wvefuntion of system of two prtiles, expressed s superposition of single systems of ertin totl ngulr momentum, in terms of 3j oeffiients: ψ jm = m,m 2 ( j j 2 j m m 2 m ) ψ () j,m ψ (2) j 2,m 2 (33) Eqution 32 lso invites the ontrvrint nd ovrint notion for the spin indies m The left hnd side of this eqution is slr under rottion of the oordintes, so if the spinor omponents trnsform ontrvrintly under rottion, then the 3j symol will trnsform ovrintly in order for the sum to remin invrint This is slightly different to tensors in spetime, s in tht se eh set of omponents trnsforms in the sme wy Here, however, this trnsformtion depends on whih representtion eh spinor is in, nd hene whih j vlues pper in the 3j symol The 6j symol is slr uilt from the ontrtion of four 3j symols As eh 3j symol must hve j vlues whih form the sides of tringle, in order for this ontrtion to work the 6j symol must hve six j vlues whih n together form the sides of n irregulr tetrhedron; eh fe of whih orresponds to 3j symol The defining eqution is: { } j j 2 j 3 = ( ) ( ) i (ji mi) j j 2 j 3 j 4 j 5 j 6 m m 2 m 3 m k ( ) ( ) ( ) (34) j j 5 j 6 j4 j 2 j 6 j4 j 5 j 3 m m 5 m 6 m 4 m 2 m 6 m 4 m 5 m 3 where the sum is tken over ll m vlues 6 This symol hs multiple symmetries due to the symmetries of the 3j symol In ft the tetrhedron piture is gin useful - if the j i whih form n irregulr tetrhedron is deformed into regulr tetrhedron, then ny symmetry of this regulr tetrhedron is symmetry of the 6j symol on interhnge of the relevnt j i The symols rise in physis when we onsider the ddition of three ngulr moment If they re ll mutully wekly oupled, then one n proeed y oupling the first two with 3j symol nd then oupling this omposite system to the third Another eqully vlid method would e to ouple the seond two, nd then ouple this omposite system k to the first The 6j symol filittes the hnge from one method to the other 362 In group theory The 3j symol of simply reduile (SR) group 7, suh s SU(2), is defined y the unitry mtrix whih rings the tensor produt of two representtions into redued form s follows: ( ) U j3,m 3;m,m 2 j j = 2 j 3 (35) (2j 3 + ) 2 m m 2 m 3 6 The 3j symol is tken to e zero if its m vlues do not sum to zero 7 One in whih the tensor produt of two irreduile representtions ontins no irreduile representtion more thn one, see Wigner [4]

12 CHAPTER 3 SPIN AND GROUPS 8 where the 3j symol is the quntity in prentheses on the right hnd side U here is the unitry mtrix whih digonlises the produt representtion: [ j g ] [ j 2 g ] = D (j) D (j2) = U j +j 2 i= j j 2 D (i) U = U D (j+j2) D ( j j2 ) U (36) In ft, for generl S R group, the right hnd side should ontin the onjugte representtions D (j), ut for SU(2) these two representtions re essentilly the sme As the spinors form irreduile representtions of SU(2) this is just nother wy of looking t how to find prtiulr spin stte from the produt of two spinors From the group point of view the 3j selets the pproprite representtion of tht spin; from the spinor point of view the 3j symol gives the pproprite symmetri spinor 37 The projetor If we onsider representtion of G, ˆP = G dgd(g) is projetor This is euse if v = ˆP u = dgd(g)u, then: G ˆP 2 u = ˆP v = G G dgdg D(g)D(g )u = G G dgdg D(gg )u = G dgv = v = ˆP u The fourth equlity uses the trnsltion property of the normlised Hr mesure dg, whih is gurnteed to exist for ertin lss of groups (inluding SU(2)) s disussed in Kosmnn- Shwrzh [8] As u ws ritrry, we hve the projetor property ˆP 2 = ˆP Now onsider v = ˆP u Then: D(g )v = D(g )D(g)udg = D(g g)udg = D(h)udh = v G G As this is true for ll g G it must e tht D(g ) is the identity representtion nd tht v is in the spin-zero representtion spe Thus ˆP projets onto this representtion Note tht s the spin-j representtion is irreduile it will not ontin this representtion nd so this opertion will send vetor in the spin-j representtion to zero (not spin zero) 38 The 3j symol s the projetor Now onsider produt of representtions: A representtion mtrix in this spe (whih will e reduile) will e given y: [ ] [ ] [ ] g g g m n m 2 n 2 m 3 n 3 The projetor onto the spin-zero suspe will then e: [ ] [ ] [ ] g g g dg m n m 2 n 2 m 3 n 3 G G

13 CHAPTER 3 SPIN AND GROUPS 9 Using eqution 36, the orthogonlity reltions etween representtions nd lso the orthogonlity reltions expressing the unitrity of U (whih ppers in eqution 35) we n show: G [ ] [ ] [ ] ( g g g dg = m n m 2 n 2 m 3 n 3 = ) ( ) m m 2 m 3 n n 2 n 3 ( m m 2 m 3 ) ( ) (37) n n 2 n 3 This result is found in Wigner [4], long with disussion of why the omplex onjugted 3j symol is ontrvrint As the left-hnd side ts s projetor, the two 3j symols on the left do too

14 4 Spin networks Spin networks were introdued y Penrose in [] in n ttempt to estlish spetime s n emergent property of some underlying disrete physis The inspirtion seems to e the proilisti nture of quntum mehnis - perhps these fundmentl proilities of n event ourring re simply the rtio of the numer of wys irumstnes will led to the event nd the numer of ll possile irumstnes This prtiulr fundmentl reserh progrmme seems to no longer e pursued, however some onept of spin network hs survived nd is useful in modern theories This onept is somewht expnded nd generlised in the modern literture s disussed y Bez [] In wht follows, only SU(2) spin networks re onsidered They hve ler physil interprettion in stndrd quntum mehnis, s disussed elow 4 Penrose networks Closed network 2 2 Open network j j j 2 Figure 4: Spin networks In Penrose s originl oneption, spin network is trivlent grph with lel, j i, on eh edge Physilly eh edge would orrespond to n isolted system of totl ngulr momentum j i The only dynmil element of this piture is the intertion etween two systems, represented t vertex Thus eh vertex n e viewed s (ny) two of the systems oming together nd omining to give single system of the third given totl ngulr momentum These verties oey the tringle inequlity for ngulr momentum ddition (onservtion of ngulr momentum) nd lso j + j 2 + j 3 Z, whih n e interpreted s onservtion of fermion numer Assoited to eh losed spin network is numer, its norm This numer is kind of quntum mehnil proility mplitude ssoited to the results of ertin experiment: mesuring the ngle in spe etween two lrge spinning odies (represented y some lrge spin network) 2 This norm is derived from relted onept, the vlue The vlue is numer ssigned Atully j i 2 in the originl pper in order to hve grphs lelled y integers only - emphsising omintoris 2 This is n importnt experiment in this model, s the whole point of the reserh progrmme ws to see if tking some si ojet outside of spe nd seeing if spe emerges In ft, Penrose reovers Euliden 3-spe, ut it is unler how this is relted to tking the omintion rules from stndrd quntum mehnis whih itself depends on the geometry E 3 0

15 CHAPTER 4 SPIN NETWORKS j 3 Norm j j 2 j 3 Norm = j j 2 j 3 j j 2 Figure 42: The norm of n open network to losed network, whih is found omintorilly using formul desried y Penrose in [] The formul is evluted diretly from the digrm nd roughly involves ounting rossings t verties The norm of losed network is the modulus of this vlue, nd the norm of n open network is the modulus of the vlue of the squre of the open network We squre network y drwing seond opy of the sme network, nd onneting ll loose edges of the first opy to the orresponding loose edge on the seond opy (see figure 42) This is now onsistent losed network nd its vlue n e found 42 Digrmmti spinor lger As disussed ove, rnk one spinors ome with vetor spe struture - whih gives us the identity mp, unit nd ounit There is lso the sympleti struture, whih tkes in two spinors nd gives omplex numer, ntisymmetri in its two rguments This is drwn digrmmtilly s: (4) = V V 2 V 2 V where V,2 = C 2, the stte spe of spin-hlf prtile Comining this opertor with the ounit gives nturl isomorphism 3 : V (42) V going from V V This is just like index lowering defined in setion 33 Similrly there is n index rising opertor with the rrows inverted Beuse we n insert these opertors into the digrm nywhere, the rrows n e dispensed with ltogether All digrms with different distriutions of rrows (inditing the spin-hlf spe nd its dul) will e isomorphi The sympleti struture mkes unknotting digrms more diffiult - signs n hnge when pulling loop stright Wht look like thin strings in the digrm ould hene e etter represented y rions, with sign ssoited to twist Higher spins re onstruted s symmetrised produts of this spin-hlf spe The Hilert spe of spin-j prtile is V j = j = S 2j (V ), whih is the symmetri prt of V 2j We n now reformulte setion 36 s digrms, s ll lultions simply involved produts of spinors nd ontrtions The only other opertion required is symmetristion We indite this y hosen 3 There is sutlety depending on whih index of epsilon the ounit is oupled to Here the seond index is

16 CHAPTER 4 SPIN NETWORKS 2 wvey line over the indies (or spes) whih we re symmetrising over As spin-j is uilt from symmetrised spin-hlf spinors, the whole digrm only requires edges lelled y the spin-hlf spe The onvention is thus tht n unlelled line should e red s hving lel of 2 As S 2j (V ) V 2j, the following digrm projets into the symmetri prt of V 2j : j This digrm lso works upside down, whih is the mp inluding the spin-j spe into V 2j Now tht the higher representtions n e roken up into spin-hlf strnds, mps etween these representtions n e onstruted from the spin-hlf mps whih we hve lredy defined 4 - digrmmtilly the up shpe, the p shpe nd the identity 5 In ft it turns out tht ll intertwining mps n e onstruted in this wy 43 Spin networks s opertors Penrose networks were trivlent grphs with edges lelled y spin numers, whih hd quntum mehnis in the kground The previous setion on spinor digrms provides grphs with edges lelled y spin numers whih represent mps from produts of representtion spes to other produts of representtion spes These two pitures re tully lmost the sme, nd so Penrose networks n lso e interpreted s mps (or multiliner mps - tensors) The differene is explined in Kuffmn [7], nd is to do with sign miguity in the spinor digrms (tht only omes in when you identify V with V ) The solution is to hnge spinors to inors, nd this is wht Penrose did when he invented spin networks This introdues sign to eh rossing of spin-hlf strnds (nd so symmetrisers eome ntisymmetrisers, whih pper in Penrose s evlution of the network) This sign nels out the sign in eqution 4, so ll the sign informtion is ontined in the numer of rossings Penrose networks re fully topologilly invrint The spinor networks, when rrows re dispensed with, will hve sign miguities 44 The 3j nd 6j symols s networks 44 The 3j network The 3j network is essentil to oth pitures of spin networks The symol is the vertex in the Penrose network; this is ler if one onsiders Penrose s desription of the vertex s two odies slowly oupling their ngulr moment It is lso the digrmmti representtion of the intertwining mp from triple produt of irreduile representtions of SU(2) into C, s in eqution 32 The 3j symol is given, up to normlistion, y the digrm: = (43) This is tully the only mp uilt from epsilons from the produt spe into C whih is non-vnishing To void hving ny loose strnds t the ottom of the digrm, 4 Or intertwiners - the tehnil term for symmetry omptile mps disussed ove 5 Whether up, for exmple, is ounit or n epsilon doesn t mtter thnks to 42

17 CHAPTER 4 SPIN NETWORKS 3 every strnd t the top of the digrm must e onneted to nother If two strnds from the sme representtion re linked up, then n ntisymmetri produt of two symmetri indies is tken - using the whole digrm to vnish 6 In ft ll intertwining mps involving three representtions re proportionl to the 3j symol - this is the Wigner-Ekrt theorem of quntum mehnis 7 Now, using the invrine of the digrm under deformtion, the spe n e rotted round to the ottom of the digrm to give mp from to = (44) These digrms re simply different, perhps lerer, wy of writing the symmetri spinor oupling rules given in setion 34 The numers of indies mthed etween eh wvefuntion there re extly the sme s the numer of strnds in the pproprite undle ove These undle numers re lerly the only wy of linking up the digrm without joining representtion k to itself Similr deformtion possiilities give mp from one representtion into two ( spontneous dey) nd no representtions into three (orrelted prtile prodution) This is the sme s Wigner s o/ontrvrint formlism desried in Wigner [4] Digrm 43 is the fully ovrint 3j symol, wheres 44 is 3j symol ontrvrint in the index lelling the spe A more mthemtil wy of looking t it, s stted in Moussouris [0], is tht Hom(V V 2, C) is nturlly isomorphi to Hom(V, V2 ), nd tht V2 V 2 in the digrms 8 The thet network rises often in spin network lultions θ is just the 3j network onneted to itself, with edges lelled,, The right-hnd side of figure 42 is tully θ jj 2j The 6j network The Wigner 6j symol is represented y tetrhedrl network 9 It ppers nturlly onsidering mp from to d: f d This mp n e expnded in the following s indexed sis (where the verties re 3j networks): s 6 In Penrose s networks, this would e symmetri produt of two strnds emerging from n ntisymmetriser 7 This theorem nd Shur s lemm emerge nturlly in the spin network formlism In ll ses whih these theorems forid, you hve ontrtion of symmetri ojet with n ntisymmetri ojet, yielding zero This is disussed y Bez nd Alvrez in [2] 8 Hom(X, Y ) eing the set of homomorphisms etween X nd Y 9 In ft the vetor piture of the 3j symol is tringle, nd the network piture is the dul of this - three edges meeting t vertex The vetor piture of the 6j symol is tetrhedron, nd the dul of this is lso tetrhedron d

18 CHAPTER 4 SPIN NETWORKS 4 ut ould eqully well e expnded in the t indexed sis: d t The 6j symol is the oeffiient relting the two ses It ppers in the sum: s d = t { } t d s t d (45) found for exmple in Kuffmn [7] Agin, this eqution is relly just the sme s in the nonil ngulr momentum oupling theory In one sis is oupled to first, then ouples to In the seond sis the oupling is the other wy round Digrm mnipultions on oth sides of eqution 45 give the 6j symol s tetrhedrl network, modulo some simple ftors: { } t = ( )2s (2s + ) t d s d θ t θ td s (46) As motivtion for its onsidertion, the 6j network is very importnt in Ponzno-Regge model of (2+) dimensionl quntum grvity, s the spetime mnifold n lwys e uilt up out of tetrhedr This is explined in Brrett nd Nish-Guzmn [4]

19 5 Evluting spin networks s integrls 5 Generl method The key digrmmti identity for evluting spin network y group integrl is given y: = θ dg G g g g (5) This is relly just writing eqution 37 in digrmmti form Seeing the eqution s digrm gives n heuristi piture of wht the opertion relly is Eh side is mp from to itself On the left, the lower 3j symol projets out the spin-zero prt of the wvefuntion in the produt spe, nd then the upper 3j symol inludes this slr wvefuntion k into the produt spe On the right the wvefuntion in the produt spe is verged over the group, whih only leves the slr prt nonzero The θ ftor on the left-hnd side is there in order to normlise the projetion opertor written in terms of 3j symols The G ftor on the right-hnd side is there in order to emphsise tht the integrtion should lso e normlised If the normlised invrint mesure is used this ftor is simply equl to one Given network to evlute, we look for pirs of verties rrying the sme representtions The digrm should then e deformed until the pirs re rrnged s in the left-hnd side of the eqution ove - then this equlity llows us to introdue group integrls If ll the verties in the network n e pired off in this wy, then the network will e redued to set of loops, one in eh representtion spe, with eh loop hving possily multiple representtion mtries threded long it These n then ll e moved round next to eh other nd omposed - whih will here e mtrix multiplition - giving the representtion mtrix of one group element, whih is the group-produt of ll the others y the very nture of representtion This mtrix will e onneted to itself y line lelled y the vetor spe it ts on, nd this is simply the tre of the mtrix - in other words, the hrter of the group element 52 Computing integrls over SU(2) In order to tully evlute the group integrl, we first need prmeteristion of the group The preferred hoie here is the xis-ngle prmeteristion, with group element orresponding to diretion ˆn on the unit sphere, nd n ngle φ in the rnge [0, 2π] The vetor ˆn is then prmeterised using stndrd spheril oordintes This is good prmeteristion over the entire group (unlike, for exmple, the Euler ngles, whih re singulr t the poles) The invrint mesure dg whih filittes the extension of results in finite group theory to SU(2) ws found using n rgument due to Hnny The result is: SU(2) f(g)dg = S 2 2π 0 f sin 2 θ 2 dθdˆn = 2π 0 π 2π 0 0 f(θ, ρ, φ) sin 2 θ sin ρdθdρdφ 2 The derivtion of the sin 2 θ 2 weight ftor is explined in Jones [6] These irled g symols re relly group tions on the vetor spe, ut they will e expressile s mtries in some sis 5

20 CHAPTER 5 EVALUATING SPIN NETWORKS AS INTEGRALS 6 53 The ylinder network 53 4-Vertex Cylinder The ylinder network, whih is defined digrmmtilly y: C = f d is the simplest non-trivil spin network Using Penrose s methods its vlue ws found y Hnny to e: C = δ f θ θ def 2 +, (52) where δ is the Kroneker delt To evlute y group integrl, onsider first the se where = f (when the network is non-zero): e g g d = d = θ θ de dg dg 2 G 2 g d g 2 e e g 2 g 2 e All the loops ontining one group representtion mtrix re simply the tre of tht mtrix The other loop, in the spe, is: ([ ] [ ]) [ ] g g2 g g tr m n n m = 2 m m

21 CHAPTER 5 EVALUATING SPIN NETWORKS AS INTEGRALS 7 In the digrm, this orresponds to omposing the two representtion mtries Continuing the digrmmti rgument: g g C = θ θ de dg dg 2 G 2 g 2 d g g 2 g 2 e = θ θ de dg dg 2 G 2 [; g g 2 ][; g ][; g ][d; g 2 ][e; g 2 ] (53) After prmeterising SU(2), this integrl n e omputed numerilly Some low dimensionl ses were omputed nlytilly ut no generl integrtion method ws found The integrl n, however, e reognised s sum over 3j symols y using the lgeri identity (37) twie - one for g nd one for g 2 After severl summtions nd uses of the orthogonlity properties of representtions one rrives t: C = θ θ de i= (2 + ) 2 = θ θ de 2 + A detiled lultion is given in Appendix B This is lerly the sme s the strnd result for = f To fully reover the result (52), the squre of this network must e onsidered As losed network is just omplex numer, the squre of the ylinder is simply two opies pled side y side (C C C) C 2 = f d e d e f = f Inserting group integrls s efore nd tking the tre of the representtions round eh loop: C 2 = ( ) 4 dg i θ i [; g 2 g 3 ][; g g 2 ][; g g 2 ][d; g 3 g 4 ][e; g 3 g 4 ][f; g g 4 ] (54) G Verties, i i= Agin this integrl n e roken up into multiple simple sums over 3j symols, whih fter some mnipultions will eome proportionl to term f i= f δ f As this term will e zero for f, this utomtilly gives the result (52) k t this stge; however ontinuing the summtion does led to the squre of tht quntity, s it should d e d e f

22 CHAPTER 5 EVALUATING SPIN NETWORKS AS INTEGRALS n-vertex ylinder (ldder) This tehnique is redily extended to simple lrge network, defined digrmmtilly s: This digrm is deformed into: 2 n n n n+ 2 n 2 n n n n+, 2 n whih n now hve group integrls inserted etween similr, vertilly opposing, verties We n then evlute this integrl using the orthogonlity reltions to give: ( n 2 ) θ 0 i= 2 i + θ i i+ i+ 2 n + θ n n n+ (55) This expression holds for n > 2, otherwise the limits on the produt re n issue The se n = 2 is the 4-vertex ylinder ove nd the se n = is simply the thet network 54 The prism network m m P = l k = l k

23 CHAPTER 5 EVALUATING SPIN NETWORKS AS INTEGRALS 9 Then, using the group integrl introdution: g 2 g 2 g 2 m dg dg 2 dg 3 l k G 3 g g g g 3 g 3 g 3 Then, s ove, we ompose the representtion mtries nd then evlute the digrm y tking the tre giving: dg dg 2 dg 3 P G 3 [; g g 2 ][; g 3 g ][; g 2 g 3 ][l; g ][m; g 2 ][k; g 3 ] (56) The integrl form of the squre of the 6j symol is: { } 2 j j 2 j 3 = j 4 j 5 j 6 G 3 dg dg 2 dg 3 [j 6 ; g g2 ][j 5; g 3 g ][j 4; g 2 g3 ][j ; g ][j 2 ; g 2 ][j 3 ; g 3 ] (57) The similrity etween this integrl nd the prism network integrl is ler Using the yli property of the tre nd tht the hrter is lss funtion, eqution (56) n e rewritten s: dg dg 2 dg 3 P G 3 [; g g2 ][; g 3g ][; g 2g3 ][l; g ][m; g 2 ][k; g 3 ] (58) These onsidertions led to: { } 2 l m k P = up to some θ ftors The prism network s evlution is given y Moussouris in [0] (it should e noted tht Moussouris normlises ll θ networks to one) The result for generl prism is the produt of two different 6js: m = { } { } l m k l m k l k

24 CHAPTER 5 EVALUATING SPIN NETWORKS AS INTEGRALS 20 This grees with the result ove for the symmetri se =, = nd = The generl result tully follows quite simply from the Wigner-Ekrt theorem (stted in network form s in Bez nd Alvrez [2]) We just show tht the entre of the prism is proportionl to 3j network with 6j proportionlity ftor, nd then the outer edges of the prism lose up this 3j network into tetrhedrl 6j network The first step is illustrted here: m m = m k θ mlk l l k l k

25 6 Disussion In ll the onrete ses we looked t, the group integrl ws evluted using the orthogonlity reltions, nd not y doing the integrl A network ontining 3j symol pirs, s in the identity 5 n e diretly interpreted lgerilly in terms of sums over these symols, nd hene the group integrl step n e skipped to get the sme results Eqully, hving onverted the digrm to n lgeri sum, the group integrl n e introdued using eqution 37 The reltionship etween the methods of evlution of network of this form ould e summrised s follows: Penrose norm Closed Spin Network Alger Group Integrl Evlute Of ourse, the solute vlue of ny network n e found y squring the network - pling two opies of the network side y side Then there will e piring etween eh vertex nd its opy Also, the 3j symol ontins ertin ritrriness in its definition - this is fixed y mking its omplex phse zero A group integrl over SU(2) hs no suh phse informtion, or knowledge of this phse fixing, so ny omintion of 3j symols whih does ontin this ritrriness nnot e expressed s n integrl Hene not ll spin networks ould e expressed in integrl form We should lso note the one wy nture of the evlution A spin network is not reoverle from n eqution or integrl - for exmple two networks whih differ y some rossings of the edges n hve the sme vlue, s disussed in Brrett nd Nish-Gutzmn [4] The integrl form of the networks ove were not esy to work with in generl; this ws due to the need to tke hrters of group produts nd the reltive omplexity of finding the prmeters of this produt (in ft only the ngle ws required, ut this turned out to depend on the xes of the ftor elements, so ll the integrtion vriles were oupled together) The integrl form ws, however, suseptile to solution in low dimensionl ses, nd this gve the sme results s the sum over 3j symols Of ourse from the group theory this ws to e expeted, ut it showed tht there hd not een n error in the prmeteristion or in ny of the digrmmti mnipultions A similr onlusion regrding the use of the group integrl form of sums over 3j symols ws lso rehed y Wigner in [4]: In most ses, it will e esier to ompre the six-j-symols y the [reltions with three-j-symols] thn y [the group integrl reltions], nd use [the group integrl reltions] for the evlution of sums or integrls over the group The 6j symols (or omintions thereof) referred to y Wigner re representle s suitly symmetri spin networks The integrl form is still useful to e wre of nd hve ville Firstly it n e evluted numerilly; Monte Crlo integrtions were implemented suessfully for the ylinder network There re expliit formule for the 3j symols, for exmple in Lndu nd Lifshitz [9], ut these re over tehnilly infinite sums (lthough they redue to finite sums y noting tht for n N, ( n)! = ) nd so my use diffiulty in implementtion In ny se, hving two numeril possiilities is ertinly preferle to one if either is using omputtionl prolems Seondly, integrls re suseptile to ertin symptoti nlyses For exmple, the symptotis of the 6j symol re found using modified sttionry phse method y Friedel nd Loupre in [5] While there re symptoti methods ville for these sums, the integrl form is the most nturl Brrett nd Nish-Gutzmn [4] onvert digrms to integrls euse the integrls re esier to work with mthemtilly for some purposes thn digrms In their se this ws needed to show tht regulristion of the digrms is possile 2

26 It is lso interesting to onsider spin networks in two ontexts - one s digrmmti reformultion of the stndrd spinor oupling theory of quntum mehnis, nd one s strting point for fundmentl theory In the lultions in this projet the first ttitude ws tken, justifying the introdution of group integrls to the digrms As Penrose did not expliitly inlude these group integrls in his theory, perhps they should not e introdued into the digrms s freely The rules for his digrm evlutions re derived from quntum mehnis, so of ourse would gree with the integrl results ove, however if omintoril theory ws sought whih diverged from the kground quntum mehnis, this group integrl introdution would need further justifition (Penrose networks re nturlly SU(2) invrint, so perhps if ertin symmetry of the theory were postulted, then this ould ring in nturl group sum or integrl) The exmples we hose to evlute oth hve some importne The ldder network ws onsidered s nturl generlistion of the ylinder network; however, Hnny noted tht it is n importnt network in Penrose s theory, s it is the form of the network whih essentilly mesures the ngles etween two odies of lrge spin The ngle-mesuring experiment involves the exhnge of single spin units etween two odies emerging from one even lrger ody, nd must e repeted in order to find the ngle Hene from some lrge spin network ldder network emerges with i = 2 for ll i - representing multiple exhnges etween the two odies whih eh hve totl spin of i efore eh exhnge In generl, of ourse, the two spins my hve different totl spin, so less symmetri ldder thn the one evluted here represents the generl experiment The prism network deomposes into two 6j symols nd is the simplest network whih deomposes into tetrhedr In generl ny lrge network n e deomposed into tetrhedr, s disussed y Brrett nd Nish-Gutzmn [4], ut this is dedued from topologil rguments - here we sw it y evluting the network s n integrl 7 Conlusions Spin networks were seen s good wy to formulte quntum ngulr momentum oupling - visulising nd lrifying the spinor formlism The digrms illuminte ertin properties whih would e more opque in index form The onditions for spin network to e reduile to n integrl were found, nd how to reh this integrl ws explined The integrl form did not seem to e esier to work with nlytilly thn the spin network; however it is ertinly useful to hve ville We disussed its usge in numeril lultions, symptotis nd ringing some mthemtil methods of integrls to spin networks 22

27 Biliogrphy [] J C Bez, Spin networks in nonperturtive quntum grvity, The Interfe of Knots nd Physis (996), [2] J C Bez nd M C Alvrez, Quntum grvity, qg-fll2000/qgrvity/qgrvitypdf, 2005 [3] J C Bez nd J P Muniin, Guge fields, knots nd grvity, World Sientifi, 994 [4] J W Brrett nd I Nish-Guzmn, The Ponzno-Regge model, Clss Qunt Grv 26 (2009), no 5, 5504 [5] L Friedel nd D Loupre, Asymptotis of 6j nd 0j symols, Clss Qunt Grv 20 (2003), [6] N G Jones, Interim Projet Report: The Wigner 6j symol s group integrl formul, Unpulished (20) [7] L H Kuffmn, Knots nd physis, World Sientifi, 200 [8] Yvette Kosmnn-Shwrzh, Groups nd symmetries, Springer, 200 [9] L D Lndu nd E M Lifshitz, Quntum Mehnis (non-reltivisti theory), Heinemnn, 958 [0] J P Moussouris, Quntum models of spe-time sed on reoupling theory, PhD thesis, Oxford, 983 [] R Penrose, Angulr momentum: n pproh to omintoril spe-time, Quntum Theory nd Beyond (T Bstin, ed), Cmridge University Press, 97 [2] R Penrose nd W Rindler, Spinors nd spe-time, vol, Cmridge University Press, 984 [3] E Wigner, Group theory nd its pplition to the quntum mehnis of tomi spetr, Ademi Press, 959 [4], On the mtries whih redue the Kroneker produts of representtions of SR groups, Quntum Theory of Angulr Momentum (L C Biedenhrn nd H Vn Dm, eds), Ademi Press,

28 A Strightening out digrm kink p V V C V = V = I i V e V I π V V V C V V The digrm on the left is the omposition (vertil) of the four smller digrms in the entre Eh of these smller digrms is interpreted using the nturl mps nd the digrm rules (For exmple, n undistured line going from top to ottom is n identity mp The rrow on the line indites if it is n identity on V or V ) The mp p tkes vetor nd djoins C - this is like multiplying y one so doesn t hnge the vetor t ll The mp π is the inverse to p nd projets out the interesting prt of produt of vetor spe with C - ie the vetor spe Now, to show tht we n strighten out the digrm on the left, is to show tht it is simply the identity mp on V (s the rrows point upwrds in the digrm) Tke some element α V, long with ny sis {e (i) } for V Then pplying the digrm to α is simply evluting: Now expnd α in the sme sis to give: D(α) = (π (e V I) (I i V ) p)(α) = (π (e V I) (I i V ))(α ) = (π (e V I))(α e (i) e (i) ) D(α) = (π (e V I))(α j e (j) e (i) e (i) ) = (π)(α j e (j) (e (i) ) e (i) ), where e V is evluting e (j) (e (i) ) This gives δ j i y the definition of the dul sis Then, D(α) = (π)(α i e (i) ) = π( α i e (i) ) = α i e (i) = α Hene the digrm is indeed the identity mp on V 24

29 B The ylinder lultion in detil The following orthogonlity reltion omes from the 3j symols eing unitry: ( ) ( ) j j 2 j j j 2 j m m 2 m m m 2 m = δ jj δ mm 2j + m,m 2 (B) This is eqution (249) in Wigner s ook [3] Now, the integrl of interest is: C = θ θ de dg dg 2 G 2 [; g g 2 ][; g ][; g ][d; g 2 ][e; g 2 ] The following method to evlute C is slightly ounterprodutive s relly we re reversing the steps whih led from digrm to integrl These steps were: Turn 3j symol pirs into integrls over representtion mtries Compose these mtries Tke the tre round the loops To evlute the integrl using the orthogonlity reltion ove, we need to undo ll of these, ut lgerilly now rther thn digrmmtilly So first we write the hrters s the tre of mtries: C = θ θ de dg dg 2 G 2 m i [ ] [ ] [ ] [ ] [ ] g g 2 g g d g2 e g2 m m m 2 m 2 m 3 m 3 m 4 m 4 m 5 m 5 Then using the representtion property of these mtries, we n split up the spin- mtrix s follows: dg dg 2 [ ] [ ] [ ] [ ] [ ] [ ] g g2 g g d g2 e g2 C = θ θ de G 2 m n n m m 2 m 2 m 3 m 3 m 4 m 4 m 5 m 5 m i,n m i,n Now, we hve seprted the integrtion vriles in some sense, so n rewrite this s follows: [ ] [ ] [ ] [ ] [ ] [ ] dg g g g dg2 g2 d g2 e g2 C = θ θ de G m n m 2 m 2 m 3 m 3 G n m m 4 m 4 m 5 m 5 We n use eqution (37) to turn eh of those single integrls into pir of 3j symols: ( ) ( ) ( ) ( ) d e d e C = θ θ de m m 2 m 3 n m 2 m 3 n m 4 m 5 m m 4 m 5 m i,n To e thorough we relly need to use the symmetry properties of the 3j symol - tht is, tht they re unhnged under yli permuttions of the olumns Hene: ( ) ( ) ( ) ( ) d e d e C = θ θ de m 2 m 3 m m 2 m 3 n n m 4 m 5 m m 4 m 5 m,m 2,m 3,m 4,m 5,n Now we use eqution (2) nd evlute the sum over m 2 nd m 3 to give: δ δ mn C = θ θ de 2 + m,m 4,m 5,n ( ) ( ) d e d e n m 4 m 5 m m 4 m 5 25

30 APPENDIX B THE CYLINDER CALCULATION IN DETAIL 26 Tking the sum over n now just sets m = n, nd lerly δ =, so: ( ) ( ) d e d e C = θ θ de 2 + m m 4 m 5 m m 4 m 5 m,m 4,m 5 Permuting s efore nd summing now over m 4 nd m 5, with further use of eqution (2) gives: C = θ θ de δ δ mm m = θ θ de m = (2 + ) 2 = θ θ de (2 + ) This ws the required result Of ourse, in the finl step we need not hve permuted the olumns - the indies m, m 4 nd m 5 were symmetri in their pperne The outome of tht ended up eing nd this would hve hppened for ny (ie we ould eqully well hve piked out d or e s speil, we d still hve eventully multiplied y one) The reson ppers in the finl result is euse in the originl prolem ppered differently s it hd two indies ssoited to it (m nd n ) The ldder works in the sme wy - just with some more oupled indies, whih result in delts, whih re then summed over, just like m nd n