OCTONIONIC TERNARY GAUGE FIELD THEORIES REVISITED

7/27/2019 OCTONIONIC TERNARY GAUGE FIELD THEORIES REVISITED

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OCTONIONIC TERNARY GAUGE

FIELD THEORIES REVISITED

CARLOS CASTRO

Center for Theoretical Studies of Physical SystemsClark Atlanta University, Atlanta, Georgia. 30314, USA

[email protected]

May 2013, Revised July 2013

Abstract

An octonionic ternary gauge field theory is explicitly constructed basedon a ternary-bracket defined earlier by Yamazaki. The ternary infinitesi-mal gauge transformations do obey the key closure relations [1, 2] = 3.An invariant action for the octonionic-valued gauge fields is displayed aftersolving the previous problems in formulating a non-associative octonionicternary gauge field theory. These octonionc ternary gauge field theoriesconstructed here deserve further investigation. In particular, to studytheir relation to Yang-Mills theories based on the G2 group which is theautomorphism group of the Octonions and their relevance to Noncommu-tative and Nonassociative Geometry.

Keywords : Octonions; Ternary Algebras; Gauge Fields.

1 Introduction

Exceptional, Jordan, Division, Clifford, noncommutative and nonassociative al-gebras are deeply related and are essential tools in many aspects in Physics, see[1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15] for references,among many others. A thorough discussion of the relevance of ternary andnonassociative structures in Physics has been provided in [16], [17], [18],[19],[20],[21]. The earliest example of nonassociative structures in Physics can be foundin Einsteins special theory of relativity. Only colinear velocities are commu-

tative and associative, but in general, the addition of non-colinear velocities isnon-associative and non-commutative.Great activity was launched by the seminal works of Bagger, Lambert and

Gustavsson (BLG) [22], [23], [24] who proposed a Chern-Simons type Lagrangiandescribing the world-volume theory of multiple M2-branes. The original BLGtheory requires the algebraic structures of generalized Lie 3-algebras and also

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of nonassociative algebras. Later developments by [25] provided a 3D Chern-Simons matter theory with N = 6 supersymmetry and with gauge groups

U(N) U(N), SU(N) SU(N). The original construction of [25] did notrequire generalized Lie 3-algebras, but it was later realized that it could beunderstood as a special class of models based on Hermitian 3-algebras [26], [27].

A Nonassociative Gauge theory based on the Moufang S7 loop product (not aLie algebra) has been constructed by [28], [29]. Taking the algebra of octonionswith a unit norm as the Moufang S7-loop, one reproduces a nonassociativeoctonionic gauge theory which is a generalization of the Maxwell and Yang-Mills gauge theories based on Lie algebras. BPST-like instantons solutionsin D = 8 were also found. These solutions represented the physical degreesof freedom of the transverse 8-dimensions of superstring solitons in D = 10preserving one and two of the 16 spacetime supersymmetries. Nonassociativedeformations of Yang-Mills Gauge theories involving the left and right bimodulesof the octonionic algebra were presented by [30]. Non-associative generalizationsof supersymmetry have been proposed by [31] which is very relevant to hiddenvariables theory and alternative Quantum Mechanics.

The ternary gauge theory developed in this work differs from the work by[22], [23], [24], [28], [29] in that our 3-Lie algebra-valued gauge field strengths Fare explicitly defined in terms of a 3-bracket [A, A, g] involving a 3-Lie algebra-valued coupling g = gata. Whereas the definition of F by [22], [23], [24] was

based on the standard commutator of the matrices (A)ac (A)

cb (A)

ac (A)

cb.

These matrices were defined as A = Aab f

cdab = (A)

cd and given in terms of

the structure constants f cdab of the 3-Lie algebra [ta, tb, tc] = f cdab td.

In the next section we shall analyze Nonassociative Octonionic TernaryGauge Field Theories based on a ternary octonionic product with the fundamen-tal difference, besides the nonassociativity, that the structure constants fabcd are

no longer totally antisymmetric in their indices. Thus the bracket in the octo-nion case [[A, B]] [A,B, g] is not effectively a Lie bracket (as it occurs in the3-Lie algebra case) because the bracket [[A, B]] in the octonion case does notobey the Jacobi identity since the structure constants fabcd are no longer totallyantisymmetric in their indices. This work is quite an improvement of our priorresults where we focused solely on the global rigid symmetries and homothecytransformations [32].

A gauge symmetry in the conventional sense has a spacetime dependentgauge parameter. Here the spacetime dependence of the gauge parameter entersthrough a scalar field via a nonlinear symmetry transformation. Hence we mustemphasize that one does not have a conventional gauge symmetry as such butit is inspired from gauge theories. 1 It is shown that the octonionic-valued fieldstrength F = F

aea transforms homogeneously (covariantly) under gauge

transformations and that the Yang-Mills-like action is indeed invariant underlocal gauge transformations involving ternary octonionic brackets and antisym-metric gauge parameters ab(x) = ba(x), a , b = 0, 1, 2, 3, ....7. Furthermore,there is closure of these transformations based on antisymmetric parameters

1We thank one referee for raising this important point

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ab = ba

2 Octonionic Ternary Gauge Field Theories

The nonassociative and noncommutative octonionic ternary gauge field theoryis based on a ternary-bracket structure involving the octonion algebra. Theternary bracket obeys the fundamental identity (generalized Jacobi identity)and was developed earlier by Yamazaki [33]. Given an octonion X it can beexpanded in a basis (eo, em) as

X = xo eo + xm em, m, n, p = 1, 2, 3, .....7. (2.1)

where eo is the identity element. The Noncommutative and Nonassociative

algebra of octonions is determined from the relations

e2o = eo, eoei = eieo = ei, eiej = ijeo + cijk ek, i ,j,k = 1, 2, 3, ....7. (2.2)

where the fully antisymmetric structure constants cijk are taken to be 1 for thecombinations (124), (235), (346), (457), (561), (672), (713). The octonion conju-gate is defined by eo = eo, ei = ei

X = xo eo xk ek. (2.3)

and the norm is

N(X) = | < X X > |1

2 = | Real (X X) |1

2 = | (xo xo + xk xk) |1

2 . (2.4)

The inverse

X1 =X

< X X >, X1X = XX1 = 1. (2.5)

The non-vanishing associator is defined by

(X, Y, Z) = (XY)ZX(YZ) (2.6)

In particular, the associator

(ei, ej , ek) = (eiej )ek ei(ej ek) = 2 dijkl el

dijkl =1

3!

ijklmnp cmnp, i,j,k.... = 1, 2, 3, .....7 (2.7)

Yamazaki [33] defined the three-bracket as

[ u, v, x ] Du,v x =1

2( u(vx) v(ux) + (xv)u (xu)v + u(xv) (ux)v ) .

(2.8)

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After a straightforward calculation when the indices span the imaginary ele-ments a,b,c,d = 1, 2, 3, ......, 7, and using the relationship [34]

cabd cdcm = dabcm + ac bm bc am (2.9a)

the ternary bracket becomes

[ ea, eb, ec ] = fabcd ed = [ dabcd + 2 ac bd 2 bc ad ] ed (2.9b)

whereas e0 has a vanishing ternary bracket

[ ea, eb, e0 ] = [ ea, e0, eb ] = [ e0, ea, eb ] = 0 (2.9c)

It is important to emphasize that fabcd = cabd cdcm otherwise one wouldhave been able to rewrite the ternary bracket in terms of ordinary 2-brackets

as follows [ea, eb, ec] 14 [ [ea, eb], ec ] and this would have defeated the wholepurpose of studying ternary structures.

The ternary bracket (2.8) obeys the fundamental identity

[ [x, u, v], y , z ] + [ x, [y, u, v], z ] + [ x, y, [z, u, v] ] = [ [x, y, z], u, v ](2.10)

A bilinear positive symmetric product < u, v >=< v, u > is required such thatthat the ternary bracket/derivation obeys what is called the metric compatibilitycondition

< [u, v , x], y > = < [u, v , y], x > = < x, [u, v , y] >

Du,v < x, y > = 0 (2.11)The symmetric product remains invariant under derivations. There is also theadditional symmetry condition required by [33]

< [u, v , x], y > = < [x, y, u], v > (2.12)

The ternary product provided by Yamazaki (2.8) obeys the key fundamentalidentity (2.10) and leads to the structure constants fabcd that are pairwiseantisymmetric but are not totally antisymmetric in all of their indices : fabcd =fbacd = fabdc = fcdab; however : fabcd = fcabd; and fabcd = fdbca. Theassociator ternary operation for octonions (x,y,z) = (xy)z x(yz) does notobey the fundamental identity (2.10) as emphasized by [33]. For this reason wecannot use the associator to construct the 3-bracket.

The physical motivation behind constructing an octonionic-valued field strengthin terms of ternary brackets is because the ordinary 2-bracket does not obeythe Jacobi identity

[ ei, [ ej , ek ] ] + [ ej , [ ek, ei ] ] + [ ek, [ ei, ej ] ] = 3 dijkl el = 0 ( 2.13)

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If one has the ordinary Yang-Mills expression for the field strength

F = A A + [ A, A ] (2.14)

because the 2-bracket does not obey the Jacobi identity, one has an extra (spu-rious) term in the expression for

[ D, D ] = [ F, ] + ( A, A, ) (2.15)

given by the crucial contribution of the non-vanishing associator (A, A, ) =(AA) A(A) = 0. For this reason, due to the non-vanishing condition(2.13), the ordinary Yang-Mills field strength does not transform homogeneouslyunder ordinary gauge transformations involving the parameters = aea

A = + [A, ] (2.16)

but it yields an extra contribution of the formF = [F, ] + ( , A, A) (2.17)

As a result of the additional contribution (, A, A) in eq-(2.17 ), the ordinaryYang-Mills action S =

< FF

> will no longer be gauge invariant. Underinfinitesimal variations (2.17), the variation of the action is no longer zero butreceives spurious contributions of the form S = 4Fl

iAjAk dijkl = 0 dueto the non-associativity of the octonion algebra. For these reasons we focus ourattention on ternary brackets.

We define the field strength in terms of the ternary bracket as

F = A A + [ A, A, g ] (2.18)

whereg

= g

a

ea is an octonionic-valued coupling function. One finds thatunder the naive infinitesimal ternary gauge transformations

(Aded) = (ded) + [

aea, Abeb, g

cec ] (Fd

ed) = [a(x)ea, F

beb, g

cec](2.19)

the ordinary quadratic action

S = 1

42

dDx < F F

> (2.20)

is not invariant under ternary infinitesimal gauge transformations as we shall seenext. is a suitable dimensionful constant introduced to render the action di-mensionless. The octonionic valued field strength is F = F

a ea, and has real

valued components F0

, F

i

; i = 1, 2, 3, ....., 7. The < > operation extractingthe e0 part is defined as < XY >= Real(XY) =< Y X >= Real(Y X). Underinfinitesimal ternary gauge transformations of the ordinary quadratic action onehas

S = 1

42

dDx < F (F

) + (F) F > =

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1

42 dDx < Fc ec [

aea, F b eb, g

nen] > +

1

42

dDx < [aea, F

b eb, g

nen] F c ec > =

1

42

dDx a Fc F

b ( < ec fabnk ek > + < fabnk ek ec > ) =

1

22

dDx a gn Fc F

b fabnc =

1

2

dDx

(aga) (F

bF

b ) (

aFa ) (gcFc

)= 0 (2.21)

Hence, because

fabnc = ( dabnc + 2 an bc 2 bn ac ) (2.22)

is not antisymmetric under the exchange of indices b c : fabnc = facnb, thevariation in eq-(3.15) is not zero. Had fabnc been fully antisymmetric then thevariation S would have been zero due to the fact that Fc F

b is symmetricunder b c.

Concluding, in the octonionic ternary algebra case, the naive transformations(2.19) do not leave the action (2.20) invariant : S= 0. One alternative wouldbe to find counter terms, if possible, to the action (2.20) S+ S so that (S+S) = 0. The authors [30] used counter terms of the form FAA+AAAAin the non-associative deformations of ordinary Yang-Mills theories based on theleft and right actions by octonions and ordinary brackets. Unfortunately it doesnot work in our case and for this reason we shall follow a different approach.

Another problem due to the fact that fabcd is not totally antisymmetric

in all of its indices is that there is no closure of the infinitesimal octonionicternary gauge transformations A = + [, A, g]. Furthermore, thebracket in the octonion case [[A, B]] [A,B, g] is not effectively a Lie bracket(as it is in the 3-Lie algebra case) since the bracket [[A, B]] in the octonion casedoes not obey the Jacobi identity because the structure constants fabcd are nolonger totally antisymmetric in their indices. As said previously, because theassociator ternary operation for octonions (x,y,z) = (xy)z x(yz) does notobey the fundamental identity (3.10) one cannot use the associator to constructthe 3-bracket and this rules out the use of the totally antisymmetric dabcd.

Nevertheless, as we shall show below, the quadratic Yang-Mills-like action(2.20) is invariant under the local octonionic ternary gauge transformationsdefined by

(Ad ed) = ab(x) [ ea, eb, Ac ec ] (2.23a)

and(gd ed) =

ab(x) [ ea, eb, gc ec] (2.23b)

where one introduces a local spacetime depedence on the antisymmetric gaugeparameters ab(x) = ba(x). One may notice now that the coupling gcec is

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not inert under the transformations (2.23b). Only the real part of the couplingg0 is inert.

After some straightforward algebra one can verify that the ternary fieldstrength F defined in terms of the 3-brackets transforms properly (homoge-neously) under the local transformations (2.23)

(Fm em) = ab [ ea, eb, F

c ec ] =

ab Fc fm

abc em Fm

= ab Fc f

mabc

(2.24)if the following conditions are satisfied

[ (ij ) Ak (

ij ) Ak ] fl

ijk = 0 (2.25)

Due to the key presence of the imaginary parts of the couplings gi we shallprove below that one can partially gauge the fields by setting Ai = g

i(x)(x)(in terms of an auxiliary scalar field (x)) and still leave room for residualsymmetries. One should note that even if the coupling functions gi are cho-sen to be constants gi = constant, it must be kept in mind that after gaugetransformations the new couplings gi will acquire a spacetime dependence viathe x-dependence of the ab(x) parameters. For this reason we should notset a priori the couplings to constants. Furthermore, the field strength will notbecome trivially zero. It can be rewritten, when the gauge fields are partiallygauged as Al = g

l(x)(x), in the following way

Fl = [Al] + A

i A

j g

k f lijk = (gl) () (g

l) () = 0 (2.26)

after using the conditions (gi)(gj )f

lijk = 0 due to the antisymmetry

of f lijk = fl

jik and the symmetry gigj = gj gi. Therefore, due to the x-

dependence of the imaginary parts of the couplings gi(x), the field strengthcomponents Fl are not zero and their contribution to the action (2.20) is nottrivially zero.

After this detour let us look for nontrivial solutions to (2.25). ij = constantare the trivial solutions leading to global rigid symmetries. A partial gauge fixingprovided by Ak = g

k, Ak = g

k yields in eq-(2.25), after factoring gk and

defining lk = fl

ijk ij , the following equation

(lk) () (

lk) () = 0 (2.27)

eq-(2.27) can also be rewritten as

( lk ) (

lk ) = 0 (2.28)

a solution to eq-(2.27) can be easily obtained when

lk =

lk ,

lk =

lk

lk(x) = C

lk e

(x), = 0 (2.29a)

is an arbitrary nonzero constant. Since lk = fl

ijk ij one can infer that

ij (x) = Cij e(x), Clk = fl

ijk Cij, = 0 (2.29b)

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where Clk and Cij are the constant entries of the arbitrary constant 7 7

matrices C, C when the range of indices is l, k = 1, 2, 3, ......, 7. Inserting thesolutions (2.29a) back into eq-(2.27) one immediately can verify that

lk ( ) = 0 (2.29c)

therefore, the expression lk(x) = Clk e

(x) is indeed a correct and nontrivialsolution to eq-(2.27). When = 0 one recovers in eqs-(2.29a, 2.29b) the trivialsolutions lk = constants,

ij = constants corresponding to the global (rigid)transformations.

To sum up, after partially gauging the field

Ak(x) = gk(x) (2.30)

the gauge parameters which solve eq-(2.25) are given by ij (x) = Cij e(x)

and, consequently, one has found nontrivial solutions for the gauge parameters.

From eq-(2.30) one can infer the explicit relation between and Ak given bythe following line integral

o =

x0

gk(x)Ak(x

)

g2(x)dx (2.31a)

The exponential of the constant o (x = 0) can be reabsorbed into thedefinition of the entries of the constant matrix so that the field dependence ofthe parameters becomes

ij (x) = Cij e(x) = Cij exp

x0

gk(x)Ak(x

)

g2(x)dx

(2.31b)

and the gauge transformations (2.23) are now nonlinear in the fields. Hence,we finally have arrived at eqs-(2.30, 2.31b) which are the nontrivial solutionsto eq-(2.25) when = 0. It is interesting that (2.31b) is given by an expressionwhich resembles the Wilson loop form with the main difference that one has aline integral instead of a loop. The value of the line integral is path-independentand solely depends on the initial point (the origin of coordinates) and the finalpoint x.

To conclude, when one partially gauges the fields as Ak = gk, there

is still a residual symmetry that remains such that the gauge transformations(2.23) become nonlinear in the fields and the nonvanishing field strength Fkgiven by eq-(2.26) transforms homogeneously (2.24) under the local gauge trans-formations (2.23) due to the vanishing conditions imposed by eq-(2.25). Thecomponent associated with the unit element of the octonion algebra F0 is inert

F0 = 0 under the gauge transformations.The reason the field strength Fk transforms homogeneously (2.24) when the

inhomogeneous terms (2.25) vanish is a direct consequence of the fundamentalidentity (2.10) because the 3-bracket (2.8) is defined as a derivation

[ [ea, eb, A], A, g ] + [ A, [ea, eb, A], g ] + [ A, A, [ea, eb, g] ] =

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[ ea, eb, [A, A, g] ] (2.32)

Another important condition due to the antisymmetry fijkl = fjikl = fijlk,and symmetry glgk = gkgl, is the invariance of g2 = glgl under gauge transfor-

mations

g2 = (glgl) = 2 gl g

l = 2 gl ij gk f lijk = 2 g

l gk ij fijkl = 0 ( 2.33)

Therefore, the full octonionic norm-squared (go)2+gigi of the octonionic-valued

coupling function g = goeo + giei is invariant under gauge transformations

(2.23b).An important remark is in order. There is a plausible caveat about the

conditions (2.25). One must ensure that such conditions, which do not appearto be explicitly gauge covariant, will not break the gauge covariance (invariance)of the theory one is trying to construct. In particular, after performing a gauge

variation of the conditions Cl

[] = 0 in (2.25) one would introduce the secondaryconditions Cl[] = 0. Performing yet another gauge variation of the secondary

conditions (Cl[]) = 0 .... , and so forth, one obtains a hierarchy of equations

to be satisfied by the gauge parameters ij (x). It is clear that the trivialsolutions ij = Cij = constants will satisfy automatically all the equationssuggesting, perhaps, that octonionic ternary field theories cannot be gauged.Nevertheless, as we shall show next, there is a very natural way to bypass thisproblem such that the gauge variation of the conditions (2.25) remains zerowithout introducing additional constraints on the parameters ij that mighthave forced them to be constants. From the gauge variations

Al = ij (x) Ak f

lijk =

ij (x) gk fl

ijk = (gl) =

(gl) + gl () =

ij (x) gk fl

ijk + gl () = 0 (2.34)

one learns that = 0, and in turn, we can infer from (2.31b) that ij = 0 sothat the variation of the condition (2.25) (when Ai = g

i, ij = = 0)

remains zero without introducing further constraints on the parameters. Avariation of (2.25) gives then

f lijk

(ij ) (gk) () (

ij) (gk) ()

= 0 (2.35)

therefore, after simple factorization of gk in (2.35) it leads to the exact sameequation (2.27) at the beginning obtained from a factorization of gk in (2.25)and which admits the solutions described above. Hence, the variations of (2.25)

do not impose additional constraints on the parameters ij which might haveforced us to have the trivial constant solutions for these gauge parameters.

Finally, given the octonionic valued field strength F = Fa

ea , with

real valued components F0, Fi

; i = 1, 2, 3, ....., 7, one can verify that thequadratic action (2.20) is indeed invariant under the ternary infinitesimal local

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gauge transformations (2.23) when the field strength transforms as provided byeq-(2.24)

S = 1

42

dDx < F (F

) + (F) F > =

1

42

dDx < Fc ec

ab [ea, eb, F n en] > +

1

42

dDx < ab [ea, eb, F

c ec] F

n en > =

1

42

dDx ab Fc F

n ( < ec fabnk ek > + < fabck ek en > ) = 0.

(2.36)this is a direct result of

< ec fabnk ek > + < fabck ek en > = (fabnk ck + fabck kn) = (fabnc + fabcn) = 0(2.37)

due to the property fabnc + fabcn = 0 which can be explicitly verified as follows

[ dabnc + 2 an bc 2 bn ac ] + [ dabcn + 2 ac bn 2 bc an ] = 0 ( 2.38)

because dabnc + dabcn = 0; dnabc + dcabn = 0, due to the total antisymme-try of the associator structure constant dnabc under the exchange of any pairof indices. A shortcut to prove the invariance S = 0 is simply (Fk)

2 =

2ij f lkij Fl F

k = 0 due to the antisymmetry flk

ij = fkl

ij and symmetryFl F

k = Fk F

l under the exchange of indices k l. The variation of

the components associated with the e0 generator is trivially zero (F0

)2 = 0

because [ei, ej , e0] = 0.The Bianchi identities are also satisfied. This work is not complete until weshow the closure of the infinitesimal transformations (2.23). To achieve thisone needs first to recast them as derivations

1A = 1(Ak ek) =

ab1 [ ea, eb, A

c ec ] =

ab1 Dea,eb A (2.39a)

2A = 2(Ak ek) =

cd2 [ ec, ed, A

l el ] =

cd2 Dec,ed A (2.39b)

by recurring to the fundamental identity (2.10) in order to evaluate the com-mutator of two derivations, and after relabeling indices, one arrives at

[1, 2] A = cd2

ab1 [ Dec,ed , Dea,eb ] A =

cd2 ab1

D[ec,ed,ea],eb + Dea,[ec,ed,eb]

A =

cd2 ab1 ( [ [ec, ed, ea], eb, A ] + [ ea, [ec, ed, eb], A ] ) =

cd2 ab1 fcdak [ ek, eb, A ] +

cd2

ab1 fcdbk [ ea, ek, A ] =

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( cd2 ak1 f

bcda

cd2

ab1 f

kcda ) [ ek, eb, A ] =

kb3 [ ek, eb, A ] = 3A

(2.40)

Therefore the antisymmetric parameter resulting from the closure of two trans-formations is given by

kb3 = ( cd2

ak1 f

bcda

cd2

ab1 f

kcda ) (2.41)

Inserting the solutions given by eq-(2.29b) into (2.41) give

kb3 = ( Ccd2 C

ak1 f

bcda C

cd2 C

ab1 f

kcda ) e

(1+2) = Ckb3 e3 (2.42)

where 3 = 1 + 2 and the constant entries of the antisymmetric C3 matrixare given in terms of the constant entries of the antisymmetric matrices C1, C2as follows

Ckb3 = ( Ccd2 C

ak1 f

bcda C

cd2 C

ab1 f

kcda ) = ( C

b2a C

ak1 C

k2a C

ab1 ) (2.43)

Therefore, because the solution for kb3 = Ckb3 e

3 given by (2.42) has the samefunctional form as that required by eq-(2.29b) in order to solve eq-(2.25), thereis closure of two gauge transformations as shown in the relations (2.42, 2.43)(3 = 1 + 2) among the parameters

ij1 ,

ij2 and

ij3 . One may write

such relation among the parameters symbolically as 3 = 1 2 such that[1, 2]A = 3A.

The finite ternary transformations can be obtained by exponentiation asfollows

F = F + F +1

2!(F) +

1

3!(((F))) + .... (2.44)

where (Fm em) = ab[ea, eb, F

cec]; (F) =

mn[ em, en, ab[ea, eb, F

cec] ];

...... To show that the action is invariant under finite ternary local transforma-tions requires to follow a few steps. Firstly, one defines

< x y > Real [ x y ] =1

2( x y + y x ) < x y > = < y x > (2.45)

Despite nonassociativity, the very special conditions

x(xu) = (xx)u; x(ux) = (xu)x; x(xu) = (xx)u; x(ux) = (xu)x (2.46)

are obeyed for octonions resulting from the Moufang identities. Despite that(xy)z = x(yz) one has that their real parts obey

Real [ (x y) z ] = Real [x (y z) ] (2.47)

Due to the nonassociativity of the algebra, in general one has that ( U F)U1 =U(F U1). However, if and only ifU1 = U U U = UU = 1, as a result of thethe very special conditions (2.46) one has that F = (U F)U1 = U(F U1) =

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U F U1 = U FU is unambiguously defined. One can equate the result of theexponentiation procedure in eq-(2.44) to the expression

F = U F U1 = U FU = ek(ab)ek (Fc tc) e

k(ab)ek ; k = 1, 2, 3, ...., 7.(2.48)

where k(ab)ek is a complicated function of ab. It yields the finite transforma-

tions which agree with the infinitesimal ternary ones when ab are infinitesimals.For instance, to lowest order in ab, one has that k satisfies 2kckcd =

abfabcdand which follows by comparing the transformations in (2.44) to those in (2.48),to lowest order.

Dropping the spacetime indices for convenience in the expressions for F, F,and by repeated use of eqs-(2.45, 2.46, 2.47), when U1 = U, the action den-sity is also invariant under (unambiguously defined) transformations of the formF = U F U1 = U FU,

< F F > = Re [F F] = Re [(UF U1) (U F U1)] = Re [(UF) ( U1 (U F U1) )] =

Re [(U F) (U1 U) (F U1)] = Re [(UF) (F U1)] = Re [(F U1) (UF)] =

Re [F ( U1 (U F) )] = Re [F (U1U) F] = Re [F F] = Re [F F] = < F F > .(2.49)

The real part of the coupling g0 is inert under the transformations (2.23b)and it decouples from the definition of the field strength F because e0 hasa vanishing 3-bracket with other elements of the octonion algebra. The cou-pling g0 = constant can be incorporated into the field strength in the samefashion as it occurs in ordinary Yang-Mills. One may rewrite the physical cou-pling g0 as a prefactor in front of the 3-bracket as F = A A +

g

0

[A, A, g], and reabsorb g0

into the definition of the A field as F =1g0

(g

0A) (g0A) + [g

0A, g0A, g]

. Thus F

1g0

F and the ac-

tion is rescaled as S 1(g0)2 S as it is customary in the Yang-Mills action.

To conclude this work, when Ai = gi we have an action

2

S = 1

4 2

dDx ( F0 F

0 F

i F

i ), i = 1, 2, 3, ....., 7 (2.50)

the (g0)2 coupling squared can be reabsorbed into the definition of 2. Thekinetic terms are explicitly given as

Fi F

i = [ [Al] + A

i A

j g

k f lijk ]2 = [ (g

i) () (gi) () ]

2 =

2

(gi)2 ()

2 (gi) () (gi) (

)= 0 (2.51)

F0 F0 = (A

0 A

0) (

A0 A

0 ) (2.52)

2One can choose< FF

> for the action so that the FiF

iterms also appear with

a positive sign inside the integrand.

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and which are invariant under the transformations (2.23a, 2.23b) due to eqs-(2.27) and = 0 (2.34). The solutions to eqs-(2.25, 2.27) are provided by

eqs-(2.30, 2.31b). It would have been desirable to avoid the conditions (2.25,2.27) which force the gauge parameters ij(x) to be field-dependent in the sensethat they are given by eq-(2.31b) and which has a Wilson-loop-like expression.There is closure of two gauge transformations as indicated by eqs-(2.40-2.43).After lengthy but straightforward algebra one can also verify that the Bianchiidentities D[F

i] = 0 are obeyed when A

i = g

i and DF = dF + [A ,F ,g].

The nontrivial role of the 7 + 1 scalars in the action (2.51) given by thecoupling functions gi(x) and (x) warrants to be studied further. We shouldemphasize that the kinetic terms FiF

i for the latter scalars do not appear

in the action in the usual canonical form (Di)2. One reason being that

Dl =

l + Aij gkf lijk does not transform homogeneously under gauge

transformations unless the gauge parameters are trivially constant. However,

when the scalars gi

(x), (x) appear in the action in the form displayed by eqs-(2.50, 2.51), the Fi terms transform homogeneously Fl

= ij Fkf

lijk as they

should, whereas F0 = 0.The terms (2.52) have the same functional form as the Maxwell action but

the transformations laws for A0 ( A0 = 0) differ from the U(1) gauge field (

A = ). The inclusion of potential terms for the scalar fields and Chern-Simons actions will be the subject of future investigation. These octonioncternary gauge field theories deserve further investigation. In particular, to studytheir relation to Yang-Mills theories based on the G2 group which is the auto-morphism group of the Octonions and their relevance to Noncommutative andNonassociative Geometry [35].

Acknowledgments

We thank M. Bowers for her assistance and to one of the referees for pointingan error and raising important points to improve this work.

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OCTONIONIC TERNARY GAUGE FIELD THEORIES REVISITED