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On the other hand (4+4)-space of the split octonions have Minkowskian subspaces, implying their structure contains light cone of special relativity. SO(8) group describing rotational symmetry of the Euclidean space is replaced by its non-compact analog for (4+4)-space, namely SO(4, 4), in which Lorentz group SO(1, 3) is contained as a subgroup. This makes the split octonions interesting to study in the context of geometry in physics [36–39]. Split octonions were used to provide possible explanation for the existence of three generations of fermionic elementary particles [40, 41]. In [42] generators of SO(8) and SO(7) groups were obtained and have been used to describe the rotational transformation in 7-dimensional space. In [43–45] real reducible 16 16-matrix representation of SO(4, 4) group utilizing the Clifford algebra × approach was constructed and it was shown that there are two inequivalent real 8 8 irreducible basic spinor representations, potential physical applications for 8-dimensional electrodynamics× [44] and gravity [45] was also considered. In [46] the basic features of Cartan’s triality of SO(8) and SO(4, 4) was analyzed in the Majorana-Weyl basis, it was shown that the three Majorana-Weyl spacetimes of signatures (4 + 4), (8 + 0), (0 + 8) are interrelated via the permutation group (signature-triality). Triality symmetry is also valid in (4+4)-space spanned by the split octonion algebra. Another concept associated with (4+4)-space is 4-ality. It’s similar to triality but deals with fourfold symmetry of modified Dynkin diagram D˜ 4 [47]. The main objective of this article is to recast results provided in [34] to (4+4)-space using the split octonions in place of regular octonions. The paper is organized as follows. In Sec. 2 we present (8 8) complex matrix representation of the Clifford algebra ℓ4 4. The Sec. 3 and Sec. 4 × C , are devoted to the vector and spinor representations in (4+4)-space, respectively. In the Sec. 5 the equivalence of SO(4, 4) vectors and spinors (triality) is explicitly demonstrated. In the Sec. 6 it is shown that the complete algebra of hyper-complex octonionic basis units can be recovered from the Moufang and Malcev relations. In Sec. 7 it is written the triality invariant trilinear form in terms of the split octonions. Finally, Sec. 8 presents our conclusions.

2 Matrix representation of ℓ , C 4 4 Geometric algebra of (4+4)-space is a Clifford algebra over the real number field with a diagonal metric gµν (Greek indices, e.g. µ, ν take on the values 0, 1,..., 7) having (4, 4) signature and is usually denoted as ℓ4,4(R). As all Clifford algebras, ℓ4,4 is associative and can be defined through anti-commutation relations:C C eµeν + eνeµ =2gµν , (2.1) where eµ are orthogonal basis units of grade-1 vectors. Let us denote the matrices representing the basis units eµ as Γµ = D (eµ). To obtain an exact form of the matrices for ℓ4,4, we can take the ℓ8,0 generating matrices Aµ described in [34] and multiply four of them byC complex imaginary unitC i,

Γµ = Aµ , (µ =0, 1, 2, 3) (2.2) Γν = iAν . (ν =4, 5, 6, 7)

This changes the Euclidean metric into the split metric of (4+4)-space. Here we use labeling and ordering of 16-dimensional Hermitian Aµ-matrices that differs from the one in [34],

0 αµ Aµ = † , (2.3)  αµ 0 

2 where the 8-dimensional αµ-matrices are:

1 i −  1   i  1 i      1   i  α0 =   , α1 =   ,  1   i   −     1   i   −     1   i   −     1   i     

1 i −  1   i  1 i  −     1   i  α2 =  −  , α3 =   ,  1   i   −   −   1   i   −   −   1   i     −   1   i     

1 i  1   i  1 i    −   1   i  α4 =  −  , α5 =  −  ,  1   i     −   1   i   −   −   1   i   −     1   i   −    1 i −  1   i  − 1 i    −   1   i  α6 =   , α7 =   .  1   i     −   1   i       1   i       1   i      We notice that four of the αµ-matrices, and thus four corresponding Aµ-matrices, are imaginary and four others are real. In general, ℓ4,4(R) is the algebra isomorphic to the ring of all 16 16 real matrices [48]. However, in the complexC representation defined above (2.2), some calculations× are easier and closer to those provided in [34] for Euclidean 8-space.

3 3 Vectors in (4+4)-space

Let us take x R8 to be a vector in (4+4)-space whose components are labeled as x . Object ∈ µ that transforms like a vector is represented by a matrix

7 = x Γ , (3.1) X β β Xβ=0 where Γβ are defined in (2.2). The vectors of (4+4)-space have the property that

2 2 2 2 2 2 2 2 2 = x0 + x1 + x2 + x3 x4 x5 x6 x7 , (3.2) X − − − − where we assume that the right-hand side is multiplied by the identity matrix. The similarity transformations

′ = L (ϑ) L−1 (ϑ) , (3.3) X µν X µν where 1 L (ϑ) = exp ϑΓ Γ , (3.4) µν −2 µ ν result in rotations of the vector xµ. This represents the SO(4, 4) group under which the quantity 2 is invariant. Transformations of x described by L can be divided into two types. One of X µ µν them comprise SO(4) compact rotations in maximal anisotropic subspaces and take place when either µ, ν = 0, 1, 2, 3 or µ, ν = 4, 5, 6, 7. Second type of transformations mix these subspaces in isotropic planes when µ = 0, 1, 2, 3 and ν = 4, 5, 6, 7, or vice versa. The later type is Lorentz-like non-compact boosts, i.e. hyperbolic transformations. To demonstrate these two different types of SO(4, 4)-transformations it is sufficient to study them in the tangential space. The space is spanned by Taylor expansion of transformation matrix (3.4) in the neighbourhood of the identity element up to the first order term 1 L (ϑ) 1 ϑΓ Γ . (3.5) µν ≃ − 2 µ ν

−1 Using the fact that Lµν = Lνµ, the formula (3.3) in the tangential space reduces to

1 x′ Γ = x Γ ϑx (Γ Γ Γ +Γ Γ Γ ) . (3.6) α α  β β − 2 β µ ν β β ν µ  Xα Xβ

Let us consider an example of rotations in the Γ4Γ5-plane. For β = 4, 5 the second term in ′ 6 (3.6) vanishes due to the algebraic relation (2.1) and we can write xβ = xβ. When β = 5 the ′ second term in (3.6) turns into ϑx5Γ4, which dictates that x4 = x4 + ϑx5. Similarly for β = 4 we ′ get x5 = x5 ϑx4. Since we have opposite sign in front of ϑ in these two infinitesimal coordinate − transformations, corresponding finite transformations would result in compact rotations:

′ x4 = x4 cos ϑ + x5 sin ϑ , ′ x5 = x5 cos ϑ x4 sin ϑ , (3.7) − x′ = x . (ρ =4, 5) ρ ρ 6 4 We have similar compact rotations in all anisotropic planes. Alternatively, the transformations that mix maximal anisotropic subspaces are non-compact. For example, if we apply calculations similar to the previous case to µ = 0 and ν = 4, we would get non-compact rotations of the form: ′ x0 = x0 cosh ϑ + x4 sinh ϑ , ′ x4 = x4 cosh ϑ + x0 sinh ϑ , (3.8) x′ = x . (µ =0, 4) ρ ρ 6 At the end of this section we want to introduce one of the 1680 possible grade-4 element of ℓ4 4, C , B = Γ1Γ3Γ5Γ7 , (3.9) − which due to the property T = B B (3.10) X X will become useful below.

4 Spinors in (4+4)-space

A spinor in the (4+4)-space can be represented as a 16-dimensional column vector η = φ + ψ , (4.1) where φ0 0  φ1   0  . .  .   .       φ7   0  φ =   and ψ =   (4.2)  0   ψ0       0   ψ1   .   .   .   .       0   ψ7      are spinors of different chirality. Here again φ, ψ R8, so η R16. The spinor transformations under Spin(4, 4)∈ (double cover∈ of SO(4, 4)) are described by the same matrix (3.4) that was used for vectors, but the transformation law is different ′ η = Lµν (ϑ) η . (4.3) Under this transformation the quantity ηT Bη = φT Bφ + ψT Bψ (4.4) is preserved. Let us prove this in the tangential space using the property of B-matrix (3.10), 1 1 η′T Bη′ = ηT 1+ ϑΓT ΓT B 1+ ϑΓ Γ η =  2 ν µ   2 µ ν (4.5) 1 1 = ηT B 1 ϑΓ Γ 1+ ϑΓ Γ η = ηT Bη .  − 2 µ µ  2 µ ν It can be noticed that two terms on the right hand side of the relation (4.4) are preserved inde- pendently, meaning that their terms do not mix.

5 5 Triality

It must be noted that the vector x considered in the Sec. 3 and two kind of spinors ψ and φ considered in the Sec. 4 are objects of same dimension in the underlying field. This kind of match between the dimensions of vector and chiral spinors only takes place in 8-dimensional space. In order to extract another peculiarity of (4+4)-space that relies on the previous one, we apply the following linear basis change to the spinor (4.1):

ϕ2 + iϕ3 −  ϕ0 iϕ1  − ϕ7 iϕ6  − −   ϕ5 + iϕ4   −   ϕ5 iϕ4   − −   ϕ7 iϕ6   −   ϕ0 iϕ1   − −  1  ϕ2 iϕ3  ξ =  − −  . (5.1) √2  ψ2 iψ3   −   ψ0 iψ1   − −   ψ7 iψ6   − −   ψ5 + iψ4   −   ψ5 + iψ4     ψ7 + iψ6   −   ψ0 + iψ1   −   ψ2 iψ3   − −  In this basis, the invariant quadratic form (4.4) for 8-spinors φ and ψ yields

T 2 2 2 2 2 2 2 2 φ Bφ = φ0 + φ1 + φ2 + φ3 φ4 φ5 φ6 φ7 , T 2 2 2 2− 2− 2− −2 2 (5.2) ψ Bψ = ψ0 + ψ1 + ψ2 + ψ3 ψ4 ψ5 ψ6 ψ7 . − − − − which are analogous to the invariant quadratic form for the vector (3.2). Then one can construct a trilinear form

: R8 R8 R8 R (5.3) F × × 7→ on x, φ and ψ (vector and spinors),

(φ, , ψ)= φT B ψ , (5.4) F X X which is preserved under simultaneously transforming and η = φ + ψ under the vector (3.3) and X spinor (4.3) transformation rules with the same Lµν . Proof is provided in the tangential space

φ′T B ′ψ′ = φT LT BL L L ψ = X µν µν X νµ µν 1 1 (5.5) = φT 1+ ϑΓT ΓT B 1+ ϑΓ Γ ψ = φT B ψ .  2 ν µ   2 µ ν X X

6 Let us look closely at these transformations. For example, the infinitesimal L01 (ϑ) rotations of vector and spinors are: ′ ′ 1 ′ 1 x0 = x0 ϑx1 φ0 = φ0 + 2 ϑφ1 ψ0 = ψ0 + 2 ϑψ1 ′ − ′ 1 ′ 1 x1 = x1 + ϑx0 φ1 = φ1 ϑφ0 ψ1 = ψ1 ϑψ0   − 2  − 2  ′  ′ 1  ′ 1 x2 = x2 φ2 = φ2 2 ϑφ3 ψ2 = ψ2 + 2 ϑψ3  ′  ′ − 1  ′ 1 x x3 φ φ3 ϑφ2 ψ ψ3 ϑψ2  3 =  3 = + 2  3 = 2  ′ ,  ′ 1 ,  ′ − 1 . (5.6) x4 = x4 φ4 = φ4 2 ϑφ5 ψ4 = ψ4 + 2 ϑψ5 ′ ′ − 1 ′ 1 x = x5 φ = φ5 + ϑφ4 ψ = ψ5 ϑψ4  5  5 2  5 2  ′  ′ 1  ′ − 1 x6 = x6 φ6 = φ6 + ϑφ7 ψ6 = ψ6 ϑψ7   2  − 2  ′  ′ 1  ′ 1 x7 = x7 φ7 = φ7 2 ϑφ6 ψ7 = ψ7 + 2 ϑψ6   −  As usual one full rotation for a vector x is only half a rotation for spinors φ and ψ. Here in all planes, except in the one where x is rotating, φ and ψ rotate in opposite directions to each other, which is the manifestation of their different chiralities. However, since Lµν -matrices form a group under matrix multiplication, we can construct transformations for x that exactly imitate transformations (5.6) of φ, ϑ ϑ ϑ ϑ 1 L10 L23 L54 L67 1 ϑ (Γ1Γ0 +Γ2Γ3 +Γ5Γ4 +Γ6Γ7) , (5.7)  2   2   2   2  ≃ − 4 which results in ′ 1 ′ 1 ′ x0 = x0 + 2 ϑx1 φ0 = φ0 + 2 ϑφ1 ψ0 = ψ0 ϑψ1 ′ 1 ′ 1 ′ − x1 = x1 ϑx0 φ1 = φ1 ϑφ0 ψ1 = ψ1 + ϑψ0  − 2  − 2   ′ 1  ′ 1  ′ x2 = x2 2 ϑx3 φ2 = φ2 + 2 ϑφ3 ψ2 = ψ2  ′ − 1  ′ 1  ′ x = x3 + ϑx2 φ = φ3 ϑφ2 ψ = ψ3  3 2  3 2  3  ′ 1 ,  ′ − 1 ,  ′ . (5.8) x4 = x4 2 ϑx5 φ4 = φ4 + 2 ϑφ5 ψ4 = ψ4 ′ − 1 ′ 1 ′ x = x5 + ϑx4 φ = φ5 ϑφ4 ψ = ψ5  5 2  5 2  5  ′ 1  ′ − 1  ′ x6 = x6 + ϑx7 φ6 = φ6 ϑφ7 ψ6 = ψ6  2  − 2   ′ 1  ′ 1  ′ x7 = x7 2 ϑx6 φ7 = φ7 + 2 ϑφ6 ψ7 = ψ7  −   Peculiar here is that roles of vector x and spinors φ and ψ have interchanged – x appears to behave like a spinor, since full rotation in ψ gives half a rotation in x and φ. This is the property of 8- dimensional space, which was named as triality, similar to the duality for dual vector spaces. Since these transformations preserve the trilinear form we’ll refer to them as triality transformations. Now for completeness we also write out boost-like non-compact transformations, which are always present in anisotropic spaces, let’s pick L04 (ϑ), ′ ′ 1 ′ 1 x0 = x0 + ϑx4 φ0 = φ0 2 ϑφ4 ψ0 = ψ0 2 ϑψ4 ′ ′ − 1 ′ − 1 x1 = x1 φ1 = φ1 φ5ϑ ψ1 = ψ1 + ϑψ5   − 2  2  ′  ′ 1  ′ 1 x2 = x2 φ2 = φ2 2 ϑφ6 ψ2 = ψ2 + 2 ϑψ6  ′  ′ − 1  ′ 1 x x3 φ φ3 ϑφ7 ψ ψ3 ϑψ7  3 =  3 = 2  3 = + 2  ′ ,  ′ − 1 ,  ′ 1 . (5.9) x4 = x4 + ϑx0 φ4 = φ4 2 φ0ϑ ψ4 = ψ4 2 ϑψ0 ′ ′ − 1 ′ − 1 x = x5 φ = φ5 ϑφ1 ψ = ψ5 + ϑψ1  5  5 2  5 2  ′  ′ − 1  ′ 1 x6 = x6 φ6 = φ6 φ2ϑ ψ6 = ψ6 + ϑψ2   − 2  2  ′  ′ 1  ′ 1 x7 = x7 φ7 = φ7 2 ϑφ3 ψ7 = ψ7 + 2 ϑψ3   −     7 We see that, similar to the compact case, the hyperbolic transformation of one of the three objects (vector and two kind of spinors) in the isotropic plane Γ0Γ4, generates spinorial transformations of other two objects in corresponding four isotropic planes Γ0Γ4, Γ1Γ5, Γ2Γ6 and Γ3Γ6. Again it is possible to replicate transformations of x in one of the spinors which would swap their behavior.

6 Split-octonions

It is known that spinors and vectors of (4+4)-space, considered in previous section, can also be represented using split octonions instead of matrices [43–46]. In our previous papers [36–39] it was suggested to describe geometry of the (4+4)-space by the elements of the split octonions:

n n s = ω + λ Jn + x jn + tI , (6.1)

n n m nm where Latin indices (e.g. n) take on the values 1, 2, 3 and x jn = δnmx j (δ is Kronecker’s delta). Four of the eight real parameters in (6.1), t and xn, denote the ordinary space-time coordinates, and ω and λn are interpreted as the phase (classical action) and the wavelengths associated with the octonionic signals. The eight octonionic basis units in (6.1) are represented by one scalar (denoted by 1), the three vector-like objects Jn, the three pseudo vector-like elements jn and one pseudo scalar-like unit I. Squares of the seven imaginary units of the split octonions give the identity element but with different signs, J 2 =1 , j2 = 1 , I2 =1 . (6.2) n n − Now we want to show that complete algebra of the seven hyper-complex basis units of the split oc- tonions follows from the Moufang and Malcev relations written for only three vector-like octonionic elements Jn. It is known that the anti-commuting basis units of octonions and split octonions, xy = yx, − are Moufang loops [49]. The algebra formed by them is not associative but instead is alternative, i.e. the associator 1 (x, y, z)= (xy)z x(yz) (6.3) A 2 −  is totally antisymmetric (x, y, z)= (y, x, z)= (x,z,y) . (6.4) A −A −A Consequently, any two units x and y generate an associative subalgebra and obey the following mild associative laws:

(xy)y = xy2 , x(xy)= x2y , (xy)x = x(yx) . (6.5)

The octonionic basis units also satisfy the flexible Moufang identities:

(xy)(zx)= x(yz)x , (zyz)x = z(y(zx)) , x(yzy)=((xy)z)y . (6.6)

In the algebra we have the following relationship

1 (x, y, z)= [x, [y, z]] + [y, [z, x]] + [z, [x, y]] (6.7) A 3 

8 between the associator and the commutator 1 [x, y]= (xy yx) . (6.8) 2 − Since the hyper-complex octonionic basis units anti-commute, their commutator can always be replaced by the simple product, [x, y]= xy. It is also known that basis units of octonions and split octonions form the Malcev algebra (see for example, [50,51]). Due to non-associativity, commutator algebra of octonionic units is non-Lie and instead of satisfying the Jacobi identity, they satisfy the Malcev relation:

(xy)(xz)=((xy)z)x + ((yz)x)x + ((zx)x)y , (6.9) or equivalently (x, y, (xz)) = (x, y, z)x , (6.10) J J where 1 (x, y, z)= (xy)z +(yz)x +(zx)y (6.11) J 3  is so-called Jacobiator of x, y and z. Indeed, using anti-commutativity of elements, we find:

3 (x, y, (xz))=(xy)(xz)+(y(xz))x + ((xz)x)y = J = ((xy)z)x + ((yz)x)x + ((zx)x)y +(y(xz))x + ((xz)x)y = (6.12) = ((xy)z +(yz)x +(zx)y)x =3 (x, y, z)x . J In Malcev algebra two types of products are defined: bilinear xy = yx and trilinear (x, y, z), − J which can be expressed using bilinear products as:

1 (x, y, z)= x(yz)+ y(zx)+ z(xy) = (y, x, z)= (x,z,y) . (6.13) J 3  −J J

We also have identities containing 4 and 5 elements of the algebra:

(xy,z,w)+ (yz,x,w)+ (zx,y,w)=0 , J J J (x,y,zw)= (x, y, z)w + z (x,y,w) , (6.14) J J J (x, y, (z,u,v)) = ( (x, y, z),u,v)+ (z, (x,y,u), v)+ (z, u, (x, y, v)) . J J J J J J J J One can generate a complete basis of the split octonions by the multiplication and distribution laws of only three vector-like elements Jn. Indeed, we can define pseudo-vector like basis units of the split octonions jn in (6.1) by the commutators (or simple binary products) of Jn, 1 j = ε J mJ k , (6.15) n 2 nmk where εnmk is the totally antisymmetric unit tensor. Also using Moufang identities for J1, J2 and J3 we can identify the seventh basis unit I with the Jacobiator,

1 2 3 J1j = J2j = J3j = (J1,J2,J3)= I. (6.16) −J

9 As a result we can recover the complete algebra of all seven hyper-complex octonionic basis units from the Moufang and Malcev relations:

J J = ε jk + δ , J I = IJ = j , m n mnk mn n − n n j j = ε jk δ , j I = Ij = J , (6.17) m n mnk − mn n − n n j J = ε J k δ I, I2 =1 . m n − mnk − mn The non-vanishing associators of the basis units of the split octonions in the above basis are:

(j , j ,J )= ε I δ J + δ J , (j , j ,I)= ε J k , A n m k − nmk − nk m mk n A n m nmk (j ,J ,J )= δ j δ j , (j ,J ,I)= ε jk , (6.18) A n m k nm k − nk m A n m − nmk (J ,J ,J )= ε I, (J ,J ,I)= ε J k . A n m k − nmk A n m nmk The conjugation of vector-like octonionic basis units,

J = J , (6.19) n − n can be understood as reflections. Conjugation reverses the order of Jn in products, i.e. 1 1 j = (ε J mJ k)= ε J kJ m = j , n 2 nmk 2 nmk − n I = (J1J2J3)= J3J2J1 = I. (6.20) − So the conjugation of the element (6.1) gives

s = ω λ J n x jn tI . (6.21) − n − n − The inner product of the split octonions s1 and s2 is defined as: 1 s1 s2 = (s1s2 + s2s1) R . (6.22) · 2 ∈ Then using (6.2), (6.17) and (6.21) one can find the norm of (6.1), the interval of the (4+4)-space,

2 s = ss = ss = ω2 λ2 + x2 t2 , (6.23) | | − − which is assumed to be non-negative in order to remain within the isotropic cone. A second condition is that for physical events the vector part of (6.1) should be time-like [38, 39],

2 n n t + λnλ > xnx . (6.24)

7 Split octonions and triality

Now let us define triality form in (4+4)-space in terms of the split octonions. We can write split octonionic representation of the 8-dimensional vector and chiral spinors in (4+4)-space as:

X= x0 + x1j1 + x2j2 + x3j3 + x4I + x5J1 + x6J2 + x7J3 , Φ= φ0 + φ1j1 + φ2j2 + φ3j3 + φ4I + φ5J1 + φ6J2 + φ7J3 , (7.1) Ψ= ψ0 + ψ1j1 + ψ2j2 + ψ3j3 + ψ4I + ψ5J1 + ψ6J2 + ψ7J3 .

10 Unlike the matrices, considered in Sec. 3 and Sec. 4, invariants constructed by the split octo- nionic vector and spinors (7.1), can be written identically to each other and we have the following correspondence between these two representations:

XX = 2 , X ΦΦ = φT Bφ , (7.2) ΨΨ = ψT Bψ .

These relations respects the fact that they evaluate to same quadratic forms (3.2) and (5.2) and are interchangeable as we have seen above. So trilinear form (5.4) represented with split octonions is

(Φ, X, Ψ) = Φ (XΨ) . (7.3) F − · 8 Conclusions

In this paper the known equivalence of 8-dimensional chiral spinors and vectors was discussed for (4+4)-space within the context of the algebra of the split octonions. It is shown that the complete algebra of hyper-complex octonionic basis units can be recovered from the Moufang and Malcev relations for the three vector-like elements of the split octonions. The trilinear form, which is invariant under triality transformations of (4+4)-space, is explicitly written using the matrix and split octonionic representations. This trilinear relation is exactly of the form used in supersymmetry theories (see, for example [32,33]), so it is only natural that the overall symmetry of such models is given by triality algebras.

Acknowledgments:

This work was supported by Shota Rustaveli National Science Foundation of Georgia (SRNSFG) through the grant DI-18-335 and the joint grant of Volkswagen Foundation and SRNSF (Ref. 93 562 & #04/48).

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