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Split Octonions and Triality in (4+4)-Space

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Split Octonions and Triality in (4+4)-Space Split Octonions and Triality in (4+4)-Space Merab Gogberashvili1,2 and Alexandre Gurchumelia1,3 1 Iv. Javakhishvili Tbilisi State University, 3 Chavchavadze Avenue, Tbilisi 0179, Georgia 2 E. Andronikashvili Institute of Physics, 6 Tamarashvili Street, Tbilisi 0177, Georgia 3 E. Kharadze Georgian National Astrophysical Observatory, Abastumani 0301, Georgia December 15, 2020 Abstract The known equivalence of 8-dimensional chiral spinors and vectors is 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 Mal- cev relations for the three vector-like elements of the split octonions. Trilinear form, which is invariant under SO(4,4) transformations for vectors and corresponding Spin(4,4) trans- formations for spinors, is explicitly written using both purely matrix and purely octonionic representations. PACS numbers: 02.10.Ud; 02.20.-a; 11.25.Hf Keywords: Split octonions; Triality; (4+4)-Space 1 Introduction Non-associative algebras have never been systematically utilized in physics, only some attempts have been made toward this goal. Nevertheless, there are some intriguing hints that these kinds of algebras may play essential role in the ultimate theory, yet to be discovered. arXiv:2012.02255v2 [math-ph] 13 Dec 2020 Octonions, as an example of such non-associative algebra, form the largest normed algebra after the algebras of real, complex and quaternionic numbers [1–3]. Since their discovery in 1844-1845 there have been various attempts to find appropriate uses for octonions in physics (see reviews [4–7]). One can point to the possible impact of octonions on: Color symmetry [8,9]; GUTs [10–12]; Representation of Clifford algebras [13, 14]; Quantum mechanics [15, 16]; Space-time symmetries [17, 18]; Formulations of wave equations [19–21]; Quantum Hall effect [22]; Kaluza-Klein program without extra dimensions [23, 24]; Strings and M-theory [25–29]; SUSY [30–33]; etc. Eight-dimensional Euclidean space, in which ordinary octonions reside, possesses certain pecu- liarities, namely both vector and two chiral spinors are eight dimensional and there exists rotation invariant trilinear form in which vectors and chiral spinors act indistinguishably from one another. This property called triality [34,35] is usually formulated in terms of spin group automormphisms and symmetry of D4 Dynkin diagram [13]. Properties of spinors and vectors have been also discussed within the context of split octonions. Unlike ordinary octonions, the split algebra lacks the advantage of being a division algebra because 1 it contains zero divisors. 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.
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