The SU (3) Algebra in a Cyclic Basis

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arXiv:1407.8360v2 [hep-ph] 30 Oct 2016 h he orsodn ekiopnguefil com- gauge-field weak-isospin ponents, corresponding three The na xlctyeulfoig ihn ekiopndi- weak-isospin no with appear footing, distinguished. will equal rection components explicitly field an weak on three all clearly that cyclic see part, [ pure-gauge Ref. the e.g. concerns as least (at Lagrangian et hc hnmxswt h ()hprhreboson hypercharge U(1) the with mixes compo- then non- weak-isospin which in 3rd the nent, direction the is define particular to it a space picks-out weak-isospin Model, which Standard field better Higgs the sym- zero In as spontaneous after seen breaking. states be metry physical might the context, with present aligned the in which, hvle ai,tkstecnncl(o-ylc form: (non-cyclic) canonical the (namely takes algebra basis, Lie Chevalley complex a as basis considering by expressed ˆ readily elements be may algebra invariance. gauge ensuring algebra, the struc- of the by constants constrained ture and thereby gluons bosons eight weak algebra, the three among the the each among of self-couplings representation in the adjoint bosons with the strong gauge in relevant and appear weak case The (purely-) respectively. of forces Model Standard the the under- generating groups lying symmetry algebras colour Lie and weak-isospin real unitary non-Abelian two the as loa h omtto eain ftenormalised the of relations ˆ ( commutation matrices or rule, the Pauli vector-product as 3-space ordinary also the as familiar uecntnsipii nEq. in implicit struc- constants the ture by determined as matrices, representation joint [ Introduction 1. 1 ˆ [ r eta ocneprr atcepyistheory, particle-physics contemporary to central are ] w nteohrhn,teagbaEq. algebra the hand, other the On eadn h eksco o xml [ example for sector weak the Regarding 1 , w ˆ om(q )frs()hsterasrn feature reassuring the has su(2) for 1) (Eq. form 2 ˆ [ ˆ = ] w W w 3 3 ′ )a offiinso h orsodn 3 corresponding the of coefficients as ]) i , 1 w w , aiyn h u2 ler nteform: the in algebra su(2) the satisying ler nQD n ol xetta hr hudeitaspe a exist should basis, there Cartan-Weyl a that in expect unlike would wherein, algebra one QCD, in algebra optrsac,adw niaeascae 3 associated indicate we and search, computer a te,epiil na qa otn.W eothr partic here report We footing. equal an on explicitly other, n htcci ae a loeitfrohrsu( other for a exist making also these may from bases obtained cyclic be may that su(3) and for bases cyclic all ˆ 3 1 W ′ ihteculnsbtenteegtgun osrie yt by constrained gluons eight the between couplings the With ˆ 2 = ] 2 w , ˆ [ i W ohtes()ads()algebras su(3) and su(2) the Both : w w ˆ [ − ↔ 2 3 w 1 , ′ a hnb noprtdi the in incorporated be then may , 1 w ′ ˆ , eateto hsc,Uiest fWrik oetyCV4 Coventry Warwick, of University Physics, of Department 3 w ˆ ˆ = ] iσ 2 ′ uhrodApeo aoaoy hlo,Ddo,O1 0QX OX11 Didcot, Chilton, Laboratory, Appleton Rutherford ˆ = ] i / w ,weeTr where 2, 1 1 w h above The . ˆ [ 3 ′ w h U3 ler naCci Basis Cyclic a in Algebra SU(3) The 3 ′ ˆ [ A w , w ˆ 3 1 , xrse nthe in expressed ) 2 ′ w = ] ˆ .F Harrison F. P. 1 1 hnviewed when , ˆ = ] σ − i .σ 2 ˆ 2 circulant h su(2) the ] w Dtd oebr1 2016) 1, November (Dated: w j 2 2 ′ , 2 = × .G Scott G. W. ad- 3 δ ij (2) (1) or ). ∗ all × n .Krishnan R. and n ersnain.W ojcueta essentially that conjecture We representations. 3 ), lositrc dnial ccial)wt each with (cyclically) identically interact gluons arxrpeetto ftes()agbai h form the in algebra su(3) the (ˆ of representation matrix Tr r eae yte(ope)transformation: (complex) the by related are ofr h htnadthe and photon the form to hr h i rcesqoe (Eq. quoted brackets Lie the where W anbss n esams meitl oee that however immediately almost su(3) sees One the Gell- basis. that the in Mann antisymmetric given totally are brackets, constants structure non-zero all determine to field (complex) (proportional defined are appropriately they components the that provided with utilised Lagrangian, coupled be the still constructing can in matrices representation sl(2,R) responding algebra Lie non-compact) but es opnnscmie a the (as combined components verse hl Eq. While gu otePuimtie o u2,i h aeo su(3) of case matrices, the Gell-Mann in the su(2), has for one matrices anal- Pauli Somewhat the way. to expressible ogous basis-independent readily sym- explicitly be averaged an hence or in and mea- summed colours any as over of metrically result prediction always we theoretical will and the surable unbroken, that be expect to may well-known is symmetry colour basis. of change undis- a remain such before by breaking, consequences, turbed symmetry physical spontaneous after all SU(2) or and remains invariant, theory the gauge Naturally, unchanged. mains odsrb h hre ekbosons, weak charged the describe to w om Eq. forms two [ˆ [ˆ g [ˆ > n g g g a ntecs ftesrn neato QD h SU(3) the (QCD) interaction strong the of case the In 2 − 4 1 λ , , − ↔ , a ‡ g ˆ g ( = ˆ g ˆ .λ 3. 4 8 2 ˆ = ] = ] lrsc ae,wihw aefudin found have we which bases, such ular ˆ = ] b W iλ 2 = poraecruattransformations, circulant ppropriate 1 g − g a 2 6 esrcuecntnso h su(3) the of constants structure he / 3 + √ ilbss(rsto ae)frthe for bases) of set (or basis cial / δ a evee sa xrsino h (real the of expression an as viewed be may 2): [ˆ 2 ab iW 3   † 1 / , w w w 2ˆ ˆ ˆ ˆ 2 ,b a, n Eq. and g g [ˆ 1 2 3 ) ′ ′ ′ 2 g 5 /   , 1 √ g ˆ , = 1 = 5 g ˆ and 2 ˆ = ]   4 A,UK 7AL, ˆ = ] i i UK , . . . [ˆ 2 g g − +1 7 6 g Z o h ope ler A algebra complex the (of ) hc osiuea3 a constitute which 8), W / , 7 1 g bsn ihtetotrans- two the with -boson, ˆ [ˆ 2 / 3 7 [ˆ 2 uhta h cinre- action the that such ) = ] 2 W i g λ   4 a − + ,   ± g ˆ [ g ˆ 4 ( = 4 5 w w w iesae of eigenstates 1 W g ≃ ˆ ˆ ˆ 3 ], ˆ = ] r ucetto sufficient are ) 1 / 1 2 3 , + + 2   a u11,tecor- the su(1,1), W g ˆ 5 g 1 = and . 1 = ] 3 √ / − + 2 3 . . . iW − W / 2ˆ g ˆ (with 8 − √ g 2 6 / 8 ) The . 3 / 2 / √ × I (3) 2ˆ 1 3 2, g 3 ) ) 8 (4) 2 the su(3) algebra in the form specified by Eq. 4 is far from cyclic (cf. Eq. 1), whereby the Pauli basis for su(2) 5 4 and Gell-Mann basis for su(3) cannot be considered fully analogous, at least in this respect. 2 6 3 Choice of basis for the algebra being of conceptual importance here, we find ourselves motivated to ask if 3 7 2 a cyclic basis, with all eight gluons in the same relative 3 1 relationship, analogous to Eq. 1 for su(2), actually exists 8 1 (or not) for the case of su(3). Our best literature search uncovered just one allusion [5] to a “cyclic basis” (Eq. 1) for su(2), but no mention of any such equivalent for FIG. 1. In a cyclic basis, the base elements of su(2) and su(3), or indeed for any su(n), n > 2 (or for any other su(3) are usefully visualised as being located at the vertices Lie algebra). Having thus had to rely on our own efforts of an equilateral triangle and regular octagon respectively. in attempting to answser this question, we are now able to report what we believe to be the complete set of such cyclic bases for su(3) (and for a few other very closely related Lie algebras). In the following sections to the cyclic form, Eq. 5, above: we briefly outline our methods and proceed to document the cyclic forms we obtained. As a by-product, we are able to give a set of 3 3 matrices which constitute T T (ˆga′ ) = C. (ˆga) (6) a matrix representation× of the su(3) algebra in cyclic form, very closely analogous to the Pauli matrices for su(2). These matrices (at least as regards their general where (ˆga)=(ˆg1,..., gˆ8) and (ˆga′ )=(ˆg1′ ,..., gˆ8′ ), we take form/symmetries etc.) in fact turn out to be familiar it as self-evident that, if the transformation takes the to us already in a somewhat different particle-physics form of a (+1)-circulant matrix [7]: context [6], as will be detailed later (see Section 4).

2. A ‘Theorem’ and a Computer Search: In a C = circ(c1,c2,c3,c4,c5,c6,c7,c8) (7) heuristic spirit, let us suppose that such cyclic forms exist for at least some Lie algebras beyond su(2). In particular, taking su(3) as an example, we would then expect (in that is, if: analogy to Eq. 1 for the su(2) case) to be able to write all brackets in the form: gˆ′ c c c c c c c c gˆ  1  1 2 3 4 5 6 7 8  1 gˆ2′ c8 c1 c2 c3 c4 c5 c6 c7 gˆ2 [ˆg , gˆ ]= f c gˆ with f c+x = f c , a b ab c a+x b+x ab gˆ′  c7 c8 c1 c2 c3 c4 c5 c6 gˆ3  3     where x =1 ... 8 indices mod 8, 1 (5) gˆ′  c6 c7 c8 c1 c2 c3 c4 c5 gˆ4  4 =     , (8) gˆ′  c5 c6 c7 c8 c1 c2 c3 c4 gˆ5  5     gˆ′  c4 c5 c6 c7 c8 c1 c2 c3 gˆ6 and where “indices mod 8,1” implies a cyclic interpreta-  6     gˆ′  c3 c4 c5 c6 c7 c8 c1 c2 gˆ7 tion of subscripts and superscipts, such that 7 + 1 = 8,  7     8 + 1 = 1 etc. The algebra being expressed in this form, gˆ8′  c2 c3 c4 c5 c6 c7 c8 c1 gˆ8 with all basis elements,g ˆ1 ... gˆ8, appearing cyclically on an explicitly equal footing, precisely generalises the su(2) case (Eq. 1). We add that it will prove useful to visu- then the transformed algebra (basisg â′ ) will also be cyclic. alise the base elements in such a basis as being located at This ‘theorem’ clearly generalises to complex transforma- the vertices of a regular polygon, i.e. at the corners of an tions, except of course that complex transformations may equilateral triangle in the su(2) case, or at the corners of result in a different real form of the same (complex) al- a regular octagon in the case of su(3), as illustrated (for gebra. This suggests that if we were just able to discover subsequent reference) in Figure 1. one cyclic form for su(3), we could immediately generate essentially all possible cyclic forms (for su(3) itself - or Now, given a real Lie algebra such as su(3), we can al- indeed for its complex extension A and hence also for ways transform to a new basis with any real non-singular 2 the sl(3,R) or su(2,1) real forms) by making appropriate linear transformation we choose. Along with the trans- (real or complex) circulant transformations. formation of the basis elements,g â, the structure con- c stants, fab, will transform, in a tensor-like fashion, with At this point therefore, we resorted to a computer- two covariant and one contravariant index. Applying based search aimed at finding at least one cyclic expres- such a transformation (with transformation matrix C) sion for the su(3) algebra. In a cyclic basis, in the 8- 3 dimensional case (Figure 1 right), the four brackets: algebra may be expressed in the cyclic form:

1 2 3 4 (1) (1) (1) (1) (1) (1) [ˆg1, gˆ2]= f12 g1 + f12 g2 + f12 g3 + f12 g4 hgâ+1, gâ+2i =g â+3 +ˆga+4 gâ+7 +ˆga+8 (15) 5 6 7 8 − +f12 g5 + f12 g6 + f12 g7 + f12 g8 (9) (1) (1) (1) (1) (1) (1) 1 2 3 4 hgâ+1, gâ+3i = gâ+2 gâ+4 gâ+6 +ˆga+8 (16) [ˆg1, gˆ3]= f13 g1 + f13 g2 + f13 g3 + f13 g4 − − − (1) (1) (1) (1) (1) (1) +f 5 g + f 6 g + f 7 g + f 8 g (10) gˆ , gˆ = gˆ +ˆg gˆ gˆ (17) 13 5 13 6 13 7 13 8 h a+1 a+4i − a+2 a+3 − a+6 − a+7 [ˆg , gˆ ]= f 1 g + f 2 g + f 3 g + f 4 g 1 4 14 1 14 2 14 3 14 4 gˆ(1) , gˆ(1) =0, a =1 ... 8 mod8, 1. (18) 5 6 7 8 h a+1 a+5i +f14 g5 + f14 g6 + f14 g7 + f14 g8 (11) [ˆg , gˆ ]= f 1 g + f 2 g + f 3 g + f 4 g (1) 1 5 15 1 15 2 15 3 15 4 The Killing form is diagonal in this basis (κab = 24δab) 5 6 7 8 − +f15 g5 + f15 g6 + f15 g7 + f15 g8, (12) and the structure constants are totally antisymmetric. As regards economy of expression, it turns out that all together with the cyclic constraint (Eq. 5) and the Lie an- structure constants Eqs. 15-18 could if necessary be read- tisymmetry condition, are clearly sufficient to readily de- ily inferred from Eq. 16 alone, exploiting the total anti- termine all brackets. Indeed, taking the cyclic constraint symmetry and the cyclic property (Eq. 5) together. and the Lie antisymmetry condition together, Eq. 12 is From Table 1: case 2, we have that the real algebra readily re-cast to involve just four parameters as follows: su(3) may also be expressed:

[ˆg , gˆ ]= f 1 g + f 2 g + f 3 g + f 4 g (2) (2) (2) (2) (2) 1 5 15 1 15 2 15 3 15 4 hgâ+1, gâ+2i =g â+4 gâ+7 +ˆga+8 (19) 1 2 3 4 − f15 g5 f15 g6 f15 g7 f15 g8. (13) − − − − gˆ(2) , gˆ(2) = gˆ(2) gˆ(2) gˆ(2) (20) h a+1 a+3i − a+2 − a+4 − a+6 For a manageable computer search, in place of Eq. 13 gˆ(2) , gˆ(2) = gˆ(2) +ˆg(2) gˆ(2) (21) (and thereby risking to miss valid instances), we in fact h a+1 a+4i − a+2 a+3 − a+6 implemented the simpler condition: gˆ(2) , gˆ(2) =0, a =1 8 mod8, 1. (22) h a+1 a+5i − [ˆg , gˆ ]=0. (14) 1 5 In this case the Killing form is circulant but not diagonal: Even so, we still had 24 structure constants f c , f c , f c , 12 13 14 κ(2) = circ 12, 0, 0, 0, 6, 0, 0, 0 . (23) c =1 ... 8 (Eqs. 9-11) to find, and our search was there- {− } fore restricted to trying only the values +1, 1 and 0 for − The transformation from su(3) (Eqs. 15-18) to su(3) each of these 24 parameters. After 100 hours running (Eqs. 19-22) is given by: on the RAL PPD linux farm [8], we∼ found that, out of 24 11 3 2.8 10 possibilities, a total of 972 choices fully C(21) = circ (1 + √3)/4, 0, 0, 0, (1 √3)/4, 0, 0, 0 (24) satisfied≃ the× Jacobi relation (excluding cases where any { − } c c c of f12, f13, f14=0, c =1 ... 8). Then, in a second pass, and the corresponding inverse transformation by: selecting only cases∀ with negative-definite Killing form (12) (corresponding to the real algebra su(3) itself), just two C = circ (1+1/√3), 0, 0, 0, (1 1/√3), 0, 0, 0 . (25) { − } cases remained as displayed in Table 1, excluding cases Cyclic forms satisfying Eq. 5 but related to case 1 and case 2 by trivial reversal of cyclic ordering and so c c c excluded from Table 1, may then be generated by a Case f12 f13 f14 metric No. 12345678 12345678 12345678 signature (-1)-circulant (or retro-circulant [7]) transformation:

(¯ii) 8 C = ( 1)circ (0, 0, 0, 0, 0, 0, 0, 1 (26) 1 001100 1¯ 1 0 1¯ 0 1¯ 0 1¯ 0 1 0 1100¯ 1¯ 1¯ 0 (−24) − { }

2 000100 1¯ 1 0 1¯ 0 1¯ 0 1¯ 0 0 0 1100¯ 1¯ 0 0 (−18)4, (−6)4 having non-zero (unit) entries only on the trailing dia- gional. Our claim to have found the complete set of cyclic TABLE I. The two sets of cyclic structure constants for bases for su(3) and hence for the complex algebra A2 and the Lie algebra su(3) found in our computer search (only any of its real forms, rests on the plausible conjecture that structure-constant values of +1, 0, −1 were tried, as indi- any such basis may be reached with a combination of the ¯ cated here by 1, 0, 1 respectively). The last column gives the transformation Eq. 26 and general circulant transforma- eigenvalues of the Killing metric with their multiplicities. tions Eqs. 6-8, with arbirary complex parameters. 4. Relation to the Gell-Mann Basis and the Gell- trivially related to these by overall sign change, or/and Mann Matrices: The Gell-Mann basis of su(3) is de- by simple reversal of the cyclic ordering - see below. fined by Eq. 4. If we define: 3. Transformations between Cyclic Forms: Di- rectly from Table 1: case 1 then, we have that the su(3) x =1/√3 1 x0 = 2/√3 x+ =1/√3+1, (27) − − − 4 then the (real) transformation from cyclic su(3) (taking su(3) algebra Eqs. 15-18 becomes instead: as an example the particular cyclic form Eqs. 15-18) to 1 the Gell-Mann basis (Eq. 4) is given by: gˆ(1) , gˆ(1) = (ˆg(1) +ˆg(1) gˆ(1) +ˆg(1) ) (32) h a+1 a+2i 2√2 a+3 a+4 − a+7 a+8 x+ x− x0 x− x+ x0 0 8 8 8 0 8 8 8 (1) (1) 1 (1) (1) (1) (1) x− x x x− gˆ , gˆ = ( gˆ gˆ gˆ +ˆg )(33)  − + − x0 − + −x0  h a+1 a+3i a+2 a+4 a+6 a+8 0 8 8 8 0 8 8 8 2√2 − − − √3 x− − − √3 x+ −  0 0 0 0 0 0  (1) (1) 1 (1) (1) (1) (1)  8 8  gˆ , gˆ = ( gˆ +ˆg gˆ gˆ )(34) − x0 x0 x0 x0 x0 x0  h a+1 a+4i a+2 a+3 a+6 a+7  0 8 8 8 0 8 8 8  2√2 − − −  − x0 x0 −x0 −x0 x0 − x0  .  0 8 8 8 0 8 8 8  (1) (1)  −x− x+ x0 x+ −x− − x0  hgâ+1, gâ+5i =0, a =1 ... 8 mod8, 1. (35)  0 8 8 8 0 8 8 8   −x+ x− − x0 − x− x+ −x0   0 8 8 8 0 8 8 8   √3 x+ − − √3 x− −  These cyclic structure constants could now be used di- 0 0 0 0 0 0  8 8  rectly in the QCD Lagrangian, in place of the Gell-Mann − − (28) structure constants in the pure-gauge part, with the ma- The corresponding inverse transformation from the Gell- trices Eq. 31 (with N =1/√2) replacing the Gell-Mann Mann basis (Eq. 4) back to the cyclic basis (Eqs. 15-18) matrices as concerns the interaction with the quarks. is given by the coresponding inverse matrix: Note that from Eq. 14, ‘opposite’ pairs of generators (i.e. diametrically ‘opposite’ in Figure 1) commute, 0 0 √3 x 0 0 0 0 √3 x+  − − −  whereby the two corresponding ‘opposite’ matrices (in x+ x 0 x0 x0 x x+ 0 − − − − − − Eqs. 31) have always very similar form. In particular  x x+ 0 x0 x0 x+ x 0   − − − −  λ1 and λ5 both turn out to be diagonal here. The  x0 x0 0 x0 x0 x0 x0 0  − − − − −  . remaining (off-diagonal) matrices, λ2, λ3, λ4, λ6, λ7, λ8,  0 0 √3 x+ 0 0 0 0 √3 x  o o  − − exhibit a characteristic 0 or 120 phase drop be-  x x+ 0 x0 x0 x+ x 0  ± − − − − − − −  tween cyclically-related off-diagonal elements equal  x+ x 0 x0 x0 x x+ 0   − −  in modulus, and thereby have the symmetry of the x x 0 x −x x x 0  0 0 0 0 0 0  circulant/circulative[9] forms introduced in Ref. [6]. − − − − (29) Indeed, simply adding a component proportional to Starting from the Gell-Mann matrices and exploiting the 3 3 identity to each generator λa, Eqs. 31, (or the above transformation, we may now readily construct × even just exponentiating the λa) produces precisely a 3 3 (i.e. the fundamental) representation of su(3) in × the matrix forms discussed in Ref. [6] which, taken to cyclic form (again, by way of example, in the particular act in the generation space as candidate fermion mass form Eqs. 15-18). Further defining: matrices, led us in 1994 to the prediction of ‘trimaximal’ mixing for quarks at very high energies [6]. (In fact x x0 x+ x x0 x+ b = − ω + + ω¯ ¯b = − ω¯ + + ω (30) matrices somewhat similar to Eq. 31 have been proposed 3 3 3 3 3 3 previously [10] in the su(3) context, but not yielding and normalising with normalisation constant N = 2, we explicit cyclic forms for the su(3) algebra as in the find that the 3 3 matrices: present work.) It may be readily verified (diagonalising × the individual λa, Eqs. 31), that the relative transfor- x 0 0 0 bω ¯b mation between any two ‘non-opposite’ λa (Figure 1) (1)  −  (1) ¯  λ1 = N 0 x0 0 λ2 = N bω¯ 0 bω¯ always takes the form of a trimaximal (i.e. 3 3 ‘flat’[11] − ¯ ×  0 0 x+  b bω 0  unitary) matrix, thereby to some degree confirming the 0 bω¯ ¯b 0 ¯b b underlying intuitive notion that our cyclic generators λ(1) = N ¯bω 0 bω λ(1) = N b 0 ¯b and their corresponding representation matrices are, 3 4 − in some meaningful sense, equally distributed (and  b ¯bω¯ 0  ¯b b 0 maximally separated) in the space in which they act. x+ 0 0 0 ¯bω b (1)   (1)   λ = N 0 x0 0 λ = N bω¯ 0 ¯bω¯ 5 6 − 5. Summary and a Final Conjecture: We have pre-  0 0 x   ¯b bω 0  − sented cyclic expressions for the real Lie algebra su(3), 0 ¯bω¯ b 0 b ¯b relevant in particle physics as the gauge group of QCD, (1)   (1)   λ = N bω 0 ¯bω λ = N ¯b 0 b (31) such that all gluons are seen to interact on an explic- 7 8 −  ¯b bω¯ 0  b ¯b 0 itly equal footing. We have plausibly conjectured that all cyclic forms of the complex algebra A2 (and its real do indeed constitute a matrix representation (ˆga forms) may be readily generated from the forms given ↔ iλa/2) of su(3) in the cyclic form Eqs. 15-18. Normal- here, using appropriate circulant transformations. We ising− Eq. 31 rather with N = 1/√2, we reproduce the have given 3 3 representations of these cyclic forms for × condition Tr λa.λb = 2δab, a,b = 1 ... 8, and the cyclic su(3) which faithfully generalise the 2 2 Pauli matrices. × 5

We close with a final (tentative) conjecture that with just one non-zero structure constant on the RHS in cyclic forms (analogous to Eq. 5) should exist for other each of Eqs. 36-38. We give here explicitly the (complex) Lie algebras, at the very least for su(n), n > 3. This transformation of Eqs. 36-39 into the Chevalley basis: supposition is based on the physical notion that in (3) a Grand Unified Theory (in which strong, weak and 10 0 0 10 0 0 gˆ1 gˆ′ − electromagnetic forces unify within a single gauge group, 1 ω 0 0 0 ω¯ 0 0 0   (3)   gˆ2 gˆ2′ i − i i e.g. SU(5) [12]), nothing should a priori distinguish the 0 0 0 − 0 − 0   (3) √3 √3 √3 gˆ3 gˆ3′   iω¯ iω i    various gauge bosons in the unbroken theory. Whether   0 − 00 0 0 −   (3) gˆ4′   √3 √3 √3  gˆ  cyclic bases have any computational advantages in   =  i iω¯ iω   4  , 0 0 0 − 0 − 0 (3) gˆ5′   √3 √3 √3  gˆ  practical calculations (e.g. in the search for classical    i i i   5  gˆ 0 − 00 0 0 − (3)  6′   √3 √3 √3  gˆ  solutions, in lattice calculations etc.) remains to be seen.    i iω iω¯   6  gˆ7′  0 0 0 − 0 − 0   (3)    √3 √3 √3  gˆ7  gˆ′   iω iω¯ i    Acknowledgement: PFH acknowledges support 8 0 − 00 0 0 − (3)  √3 √3 √3  gˆ8  from the UK Science and Technology Facilities Council (40) (STFC) on the STFC consolidated grant ST/H00369X/1. where ω = exp(2πi/3) andω ¯ = exp( 2πi/3) are the two PFH and RK also acknowledge the hospitality of the complex cube roots of unity. The resulting− (real) algebra, Centre for Fundamental Physics (CfFP) at the Ruther- having metric signature ( 3, +5), is sl(3,R): ford Appleton Laboratory (RAL). RK acknowledges − support from CfFP and the University of Warwick. WGS [ˆg′ , gˆ′ ]=0 [ˆg′ , gˆ′ ]=2ˆg′ [ˆg′ , gˆ′ ]=ˆg′ [ˆg′ , gˆ′ ]= gˆ′ 1 2 1 3 3 1 4 4 1 5 − 5 thanks Chan Hong Mo (RAL) for helpful discussions. [ˆg′ , gˆ′ ]= 2ˆg′ [ˆg′ , gˆ′ ]= gˆ′ [ˆg′ , gˆ′ ]=ˆg′ 1 6 − 6 1 7 − 7 1 8 8 [ˆg′ , gˆ′ ]= gˆ′ [ˆg′ , gˆ′ ]=ˆg′ [ˆg′ , gˆ′ ]=2ˆg′ Appendix: 2 3 − 3 2 4 4 2 5 5 [ˆg′ , gˆ′ ]=ˆg′ [ˆg′ , gˆ′ ]= gˆ′ [ˆg′ , gˆ′ ]= 2ˆg′ For completeness, it should be said that in our computer 2 6 6 2 7 − 7 2 8 − 8 search, if instead of requiring negative-definite Killing [ˆg′ , gˆ′ ]= gˆ′ [ˆg′ , gˆ′ ]=ˆg′ [ˆg′ , gˆ′ ]=ˆg′ 3 5 − 4 3 6 1 3 7 8 form in the second pass, we require only that the Killing [ˆg4′ , gˆ6′ ]=ˆg5′ [ˆg4′ , gˆ7′ ]=ˆg1′ +ˆg2′ [ˆg4′ , gˆ8′ ]= gˆ3′ form have non-zero determinant (i.e. if we require only − [ˆg′ , gˆ′ ]= gˆ′ [ˆg′ , gˆ′ ]=ˆg′ that the algebra be semi-simple, κ = 0) then, in ad- 5 7 − 6 5 8 2 | | 6 dition to case 1 and case 2 from the main text above [ˆg6′ , gˆ8′ ]=ˆg7′ (41) (reproduced once again in Table II below), we find also as is immediately evident switching to the more famil- the especially straightforward cyclic form Table II case 3. α β α α+β iar notation:g ˆ1′ = H ,g ˆ2′ = H ,g ˆ3′ = E ,g ˆ4′ = E , β α (α+β) β gˆ5′ = E ,g ˆ6′ = E− ,g ˆ7′ = E− ,g ˆ8′ = E− (see c c c e.g. Ref. [13]). The corresponding inverse transforma- Case f12 f13 f14 metric No. 12345678 12345678 12345678 signature tion from the Chevalley basis (Eq. 41) back to the cyclic basis for su(2,1) (Eqs. 36-39) is then:

8 1 001100 1¯ 1 0 1¯ 0 1¯ 0 1¯ 0 1 0 1100¯ 1¯ 1¯ 0 (−24) (3) iω¯ i − 00 000 0 gˆ1 √3 √3  (3)  iω i iω¯  gˆ1′ 2 000100 1¯ 1 0 1¯ 0 1¯ 0 1¯ 0 0 0 1100¯ 1¯ 0 0 (−18)4, (−6)4 gˆ 0 0 0 0 0   2 √3 √3 √3 gˆ′ (3) iω¯ i iω 2 gˆ   0 0 0 − 0 − 0 −  3 00100000 00000001 000000 1¯ 0 (−6)4, (+6)4  3   √3 √3 √3  gˆ3′   (3)  0 0 i 0 i 0 i 0    gˆ4   √3 √3 √3  gˆ4′   (3) =  iω i    . − 00 000 0 gˆ′ TABLE II. As for Table I, except that instaed of requiring gˆ5   √3 √3   5    i iω iω¯    negative definite Killing form, we require only that the Killing  (3)  0 0 0 0 0  gˆ6′  gˆ6   √3 √3 √3    form have non-zero determinant (i.e. we require only that the  (3)  i iω¯ iω  gˆ′  0 0 − 0 − 0 − 0 7 algebra be semi-simple, |κ| 6= 0). While case 1 and case 2 are gˆ7   √3 √3 √3       i i i  gˆ′  (3) 0 0 0 0 0 8 reproduced excatly as in Table I, the additional form case 3 gˆ8   √3 √3 √3  (c.f. Table I) corresponds to the non-compact algebra su(2,1). (42) The (complex) transformation C(13) from su(2,1) With metric signature ( 4, +4), Table II case 3 evidently (Eqs. 36-39) to su(3) (Eqs. 15-18) is given by: gives a cyclic form of the− non-compact algebra su(2,1): C(13) = circ (1 + i), 0, 0, 0, (1 i), 0, 0, 0 (43) (3) (3) (3) { − } hgâ+1, gâ+2i =ˆga+3 (36) and the corresponding inverse transformation is given by: (3) (3) (3) gâ+1, gâ+3 =ˆga+8 (37) h i (31) (21) 1 C = (C )− = circ (1 i)/4, 0, 0, 0, (1+i)/4, 0, 0, 0 . gˆ(3) , gˆ(3) = gˆ(3) (38) { − (44)} h a+1 a+4i − a+7 Finally, we note, with admittedly some benefit of hind- (3) (3) hgâ+1, gâ+5i =0, a =1 ... 8 mod8, 1. (39) sight, that our cyclic form Eqs. 15-18 for su(2,1) might 6 perhaps have been derived more analytically, possibly of 6 (mod 24), commute with each other: without the need to resort to a computer search, exploit- su(5) su(5) su(5) su(5) su(5) su(5) ing the ‘trigonometric structure constant’ (TSC) concept hgâ+1 , gâ+7 i = hgâ+1 , gâ+13 i = hgâ+1 , gâ+19 i =0, [10], by adopting suitable re-phasings, re-scalings and re- a =1 ... 24 mod24, 1 (46) orderings of the TSC operators in the A2 case. Con- sequently, recalling our (tentative) parting conjecture of su(5) while the Killing form is diagonal (κab = 40√5 ηδab). the main text above, an opportunity arises to try to ex- Possibly this ‘bi-cyclic’ form (Eqs. 45-46)− could be put ploit the A4 TSCs to work towards a cyclic basis (if such into a fully cyclic form, e.g. by applying suitable separate exists) for the real algebra su(5). Then, with all 24 grand- circulant transformations on even and odd generators. unified vector bosons of the (unbroken) SU(5) theory [12] appearing on an equal footing from the outset, we would finally have our ‘manifest bosonic democracy’. ∗ As a possible step in this direction, we present here [email protected] † [email protected] a ‘bi-cyclic’ form which we have obtained for the real ‡ [email protected] algebra su(5) itself, i.e. having metric signature ( 24). − [1] Following convention we have herein denoted the algebra We refer to this form as ‘bi-cyclic’ because coefficient su(n) when the group is SU(n). sets alternate as we step the indices by one, such that [2] S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967), ; S. L. the ‘odd’ brackets below are cyclic in themselves as are Glashow, Nucl. Phys. B 22, 579 (1961), ; A. Salam, the corresponding ‘even’ ones (which in fact differ from Proceedings of the Eigth Nobel Symposium on Elementary their ‘odd’ counterparts only by a factor of the golden Particle Theory (Almqvist & Wikell, Stockholm, 1968) . √ [3] M. Atiyah, Gauge Theories (OUP, 1988) ; J. Smit, Intro- ratio η = (1+ 5)/2 and in some cases by some re- duction to Quantum Fields on a Lattice (CUP, 2002) . arrangement of signs). Given this ‘bi-cyclicity’ and the [4] M. Gell-Mann, Phys. Rev. 125, 1067 (1962) ;M. Gell-Ma- total anti-symmetry inherent here, the following brackets nn,Y. Neeman, The Eightfold Way (Benjamin, 1964). are sufficient to fix all non-zero structure constants: [5] H. Rossi, Proc. Conf. Complex Analysis, Rice Univ., Houston, 1972 (Rice Univ. Studies, 59, 131-145 1973), su(5) su(5) su(5) su(5) su(5) su(5) http://www.dspace.rice.edu/handle/1911/63107 . hgâ+1 , gâ+3 i = ( gâ+4 +ˆga+6 gâ+16 gâ+18 ) − − − [6] P. F. Harrison and W. G. Scott, Phys. Lett. B 333, 471 su(5) su(5) su(5) su(5) su(5) su(5) (1994), arXiv:hep-ph/9406351 ; See also: W. G. Scott, gâ+2 , gâ+4 = η( gâ+5 +ˆga+7 gâ+17 gâ+19 ) h i − − − arXiv:hep-ph/9909431 . [7] P. J. Davis, Circulant Matrices (John Wiley, 1979) . su(5) su(5) su(5) su(5) su(5) su(5) [8] D. Kelsey, et al. Rutherford Appleton Laboratory Par- hgâ+1 , gâ+9 i = η( gâ+5 gâ+8 +ˆga+17 gâ+20 ) − − − ticle Physics Department (RAL PPD) Computing Group su(5) su(5) su(5) su(5) su(5) su(5) http://www.stfc.ac.uk/PPD/computing-group-people.html gâ+2 , gâ+10 = ( gâ+6 +ˆga+9 +ˆga+18 +ˆga+21 ) h i − . [9] H.-C. Chen, Siam J. Matrix Anal. Appl. 13, 1172 (1992) su su su su su su (5) (5) (5) (5) (5) (5) http://epubs.siam.org/doi/abs/10.1137/0613072 . hgâ+1 , gâ+5 i = η(+ˆga+9 gâ+12 +ˆga+21 +ˆga+24 ) − [10] J. Patera and H. Zassenhaus, J. Math. Phys. 29, 665 su(5) su(5) su(5) su(5) su(5) su(5) (1988) ; D. Fairlie, P. Fletcher, and C. K. Zachos, Phys. hgâ+2 , gâ+6 i = (+ˆga+10 +ˆga+13 +ˆga+22 gâ+1 ) − Lett. B 218, 203 (1989), ; D. Fairlie and C. K. Zachos, ibid. 620, 195 (2005), ; D. Fairlie, P. Fletcher, and C. K. su(5) su(5) su(5) su(5) su(5) su(5) 31 hgâ+1 , gâ+11 i = ( gâ+2 +ˆga+4 +ˆga+14 +ˆga+16 ) Zachos, J. Math. Phys. , 1088 (1990) . − [11] A. Belovs and J. Smotrovs, Mathematical Methods in su(5) su(5) su(5) su(5) su(5) su(5) Computer Science: Essays in Memory of Thomas Beth. hgâ+2 , gâ+12 i = η( gâ+3 +ˆga+5 +ˆga+15 +ˆga+17 ) − Springer-Verlag Berlin Heidelberg, (2008) p 57 . a =2, 4,... 24 mod24, 1 (45) [12] H. Giorgi and S. L. Glashow, Phys. Rev. Lett. 32, 438 (1974) . i.e. all structure constants are magnitude 1 or η in mod- [13] J.. Fuchs and C.. Schweigert, Symmetries, Lie Algebras ulus. Note that elements separated by one quarter of a and Representations (CUP, 1997) ; J.. Fuchs, Affine Lie full cycle on the associated 24-gon, i.e. by an index-count Algebras and Quantum Groups (CUP, 1992) .