- Split-octonion
In
mathematics , the split-octonions are anonassociative extension of thequaternion s (or thesplit-quaternion s). They differ from theoctonion s in the signature ofquadratic form : the split-octonions have a split-signature (4,4) whereas the octonions have a positive-definite signature (8,0).The split-octonions form the unique
split octonion algebra over the real numbers. There are corresponding algebras over any field "F".Definition
Cayley-Dickson construction
The octonions and the split-octonions can be obtained from the
Cayley-Dickson construction by defining a multiplication on pairs of quaternions. We introduce a new imaginary unit ℓ and write a pair of quaterions ("a", "b") in the form "a" + ℓ"b". The product is defined by the rule::a + ell b)(c + ell d) = (ac + lambda dar b) + ell(ar a d + c b)where:lambda = ell^2.If λ is chosen to be −1, we get the octonions. If, instead, it is taken to be +1 we get the split-octonions. One can also obtain the split-octonions via a Cayley-Dickson doubling of thesplit-quaternion s. Here either choice of λ (±1) gives the split-octonions. See alsosplit-complex numbers in general.Multiplication table
A basis for the split-octonions is given by the set {1, "i", "j", "k", ℓ, ℓ"i", ℓ"j", ℓ"k"}. Every split-octonion "x" can be written as a
linear combination of the basis elements,:x = x_0 + x_1,i + x_2,j + x_3,k + x_4,ell + x_5,ell i + x_6,ell j + x_7,ell k,with real coefficients "x""a". By linearity, multiplication of split-octonions is completely determined by the followingmultiplication table :Conjugate, norm and inverse
The "conjugate" of a split-octonion "x" is given by:ar x = x_0 - x_1,i - x_2,j - x_3,k - x_4,ell - x_5,ell i - x_6,ell j - x_7,ell kjust as for the octonions. The
quadratic form (or "square norm") on "x" is given by:N(x) = ar x x = (x_0^2 + x_1^2 + x_2^2 + x_3^2) - (x_4^2 + x_5^2 + x_6^2 + x_7^2)This norm is the standard pseudo-Euclidean norm on R4,4. Due to the split signature the norm "N" is isotropic, meaning there are nonzero "x" for which "N"("x") = 0. An element "x" has an (two-sided) inverse "x"−1 if and only if "N"("x") ≠ 0. In this case the inverse is given by:x^{-1} = frac{ar x}{N(x)}.Properties
The split-octonions, like the octonions, are noncommutative and nonassociative. Also like the octonions, they form a
composition algebra since the quadratic form "N" is multiplicative. That is,:N(xy) = N(x)N(y).,The split-octonions satisfy theMoufang identities and so form analternative algebra . Therefore, byArtin's theorem , the subalgebra generated by any two elements is associative. The set of all invertible elements (i.e. those elements for which "N"("x") ≠ 0) form aMoufang loop .Zorn's vector-matrix algebra
Since the split-octonions are nonassociative they cannot be represented by ordinary matrices (matrix multiplication is always associative). Zorn found a way to represent them as "matrices" containing both scalars and vectors using a modified version of matrix multiplication. Specifically, define a "vector-matrix" to be a 2×2 matrix of the form:egin{bmatrix}a & mathbf v\ mathbf w & bend{bmatrix}where "a" and "b" are real numbers and v and w and vectors in R3. Define multiplication of these matrices by the rule:egin{bmatrix}a & mathbf v\ mathbf w & bend{bmatrix} egin{bmatrix}a' & mathbf v'\ mathbf w' & b'end{bmatrix} = egin{bmatrix}aa' + mathbf vcdotmathbf w' & amathbf v' + b'mathbf v + mathbf w imes mathbf w'\ a'mathbf w + bmathbf w' - mathbf v imesmathbf v' & bb' + mathbf v'cdotmathbf w end{bmatrix}where · and × are the ordinary
dot product andcross product of 3-vectors. With addition and scalar multiplication defined as usual the set of all such matrices forms a nonassociative unital 8-dimensional algebra over the reals, called Zorn's vector-matrix algebra.Define the "
determinant " of a vector-matrix by the rule:detegin{bmatrix}a & mathbf v\ mathbf w & bend{bmatrix} = ab - mathbf vcdotmathbf w.This determinant is a quadratic form on the Zorn's algebra which satisfies the composition rule::det(AB) = det(A)det(B).,Zorn's vector-matrix algebra is, in fact, isomorphic to the algebra of split-octonions. Write an octonion "x" in the form:x = (a + mathbf a) + ell(b + mathbf b)where "a" and "b" are real numbers and a and b are pure quaternions regarded as vectors in R3. The isomorphism from the split-octonions to the Zorn's algebra is given by:xmapsto phi(x) = egin{bmatrix}a + b & mathbf a + mathbf b \ -mathbf a + mathbf b & a - bend{bmatrix}.This isomorphism preserves the norm since N(x) = det(phi(x)).
Applications
Split-octonions are used in the description of physical law. For example, (a) the
Dirac equation in physics (the equation of motion of a free spin 1/2 particle, like e.g. an electron or a proton) can be expressed on native split-octonion arithmetic, (b) the supersymmetric quantum mechanics has an octonionic extension (see references below; split-octonions are isomorphic to hyperbolic octonions fromMusean hypernumber s).References
*cite book
first = F. Reese
last = Harvey
year = 1990
title = Spinors and Calibrations
publisher = Academic Press
location = San Diego
id = ISBN 0-12-329650-1
*cite book
first = T. A.
last = Springer
coauthors = F. D. Veldkamp
year = 2000
title = Octonions, Jordan Algebras and Exceptional Groups
publisher = Springer-Verlag
id = ISBN 3-540-66337-1For physics on native split-octonion arithmetic see e.g.
* M. Gogberashvili, Octonionic Electrodynamics, "J. Phys. A: Math. Gen." 39 (2006) 7099-7104. [http://dx.doi.org/10.1088/0305-4470/39/22/020 doi:10.1088/0305-4470/39/22/020]
* J. Köplinger, Dirac equation on hyperbolic octonions. "Appl. Math. Computation" (2006) [http://dx.doi.org/10.1016/j.amc.2006.04.005 doi:10.1016/j.amc.2006.04.005]
* V. Dzhunushaliev, Non-associativity, supersymmetry and hidden variables, "J. Math. Phys." 49, 042108 (2008); doi:10.1063/1.2907868; arXiv:0712.1647 [quant-ph] .
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