Eight-dimensional Octonion-like but Associative Normed Division Algebra
aa r X i v : . [ m a t h . G M ] M a y Eight-dimensional Octonion-like but Associative Normed Division Algebra
Joy Christian ∗ Einstein Centre for Local-Realistic Physics, 15 Thackley End, Oxford OX2 6LB, United Kingdom
ABSTRACT:We present an eight-dimensional even sub-algebra of the 2 = 16-dimensional associative Cliffordalgebra Cl , and show that its eight-dimensional elements denoted as X and Y respect the normrelation || XY || = || X || || Y || , thus forming an octonion-like but associative normed division algebra,where the norms are calculated using the fundamental geometric product instead of the usual scalarproduct. The corresponding 7-sphere has a topology that differs from that of octonionic 7-sphere.Consider the following eight-dimensional vector space with graded Clifford-algebraic basis and orientation λ = ± Cl λ , = span { , λ e x , λ e y , λ e z , λ e x e y , λ e z e x , λ e y e z , λ e x e y e z } . (1)As we shall see, the choice of orientation, λ = +1 or λ = − { e x , e y , e z } is a set of anti-commuting orthonormal vectors in IR such that e j e i = − e i e j for any i = j = x, y , or z .In general the vectors e i satisfy the following geometric product in this associative but non-commutative algebra [1][2]: e i e j = e i · e j + e i ∧ e j , (2)with e i · e j := 12 { e i e j + e j e i } (3)being the symmetric inner product and e i ∧ e j := 12 { e i e j − e j e i } (4)being the anti-symmetric outer product, giving ( e i ∧ e j ) = −
1. There are thus basis elements of four different gradesin this algebra: An identity element e i = 1 of grade-0, three orthonormal vectors e i of grade-1, three orthonormalbivectors e j e k of grade-2, and a trivector I = e i e j e k of grade-3 representing a volume element in IR . Since in IR there are 2 = 8 ways to combine the vectors e i using the geometric product (2) such that no two products are linearlydependent, the resulting algebra, Cl λ , , is a linear vector space of eight dimensions, spanned by these graded bases.In this paper we are interested in a certain conformal completion of this algebra, originally presented in Ref. [3].This is accomplished by using an additional vector, e ∞ , to close the lines and volumes of the Euclidean space, giving K λ = span { , λ e x e y , λ e z e x , λ e y e z , λ e x e ∞ , λ e y e ∞ , λ e z e ∞ , λI e ∞ } . (5) ∗ Electronic address: [email protected] The conformal space we are considering is an in -homogeneous version of the space usually studied in Conformal Geometric Algebra [2].It can be viewed as an 8-dimensional subspace of the 32-dimensional representation space postulated in Conformal Geometric Algebra.The larger representation space results from a homogeneous freedom of the origin within E , which does not concern us in this paper. ∗ λ e x e y λ e z e x λ e y e z λ e x e ∞ λ e y e ∞ λ e z e ∞ λ I e ∞ λ e x e y λ e z e x λ e y e z λ e x e ∞ λ e y e ∞ λ e z e ∞ λ I e ∞ λ e x e y λ e x e y − e y e z − e z e x − e y e ∞ e x e ∞ I e ∞ − e z e ∞ λ e z e x λ e z e x − e y e z − e x e y e z e ∞ I e ∞ − e x e ∞ − e y e ∞ λ e y e z λ e y e z e z e x − e x e y − I e ∞ − e z e ∞ e y e ∞ − e x e ∞ λ e x e ∞ λ e x e ∞ e y e ∞ − e z e ∞ I e ∞ − − e x e y e z e x − e y e z λ e y e ∞ λ e y e ∞ − e x e ∞ I e ∞ e z e ∞ e x e y − − e y e z − e z e x λ e z e ∞ λ e z e ∞ I e ∞ e x e ∞ − e y e ∞ − e z e x e y e z − − e x e y λ I e ∞ λ I e ∞ − e z e ∞ − e y e ∞ − e x e ∞ − e y e z − e z e x − e x e y ” of E . Here I = e x e y e z , e ∞ = +1, and λ = ± With unit vector e ∞ , this is an eight-dimensional even sub-algebra of the 2 = 16-dimensional Clifford algebra Cl , .Unlike the seven imaginaries of octonions, there are only six basis elements of K λ that are imaginary. The seventh, λI e ∞ , squares to +1. This is evident from the multiplication table I. We therefore call it an “octonian-like” algebra.As an eight-dimensional linear vector space, K λ has some remarkable properties [3]. To begin with, it is closed undermultiplication. Suppose X and Y are two vectors in K λ . Then X and Y can be expanded in the graded basis of K λ : X = X + X λ e x e y + X λ e z e x + X λ e y e z + X λ e x e ∞ + X λ e y e ∞ + X λ e z e ∞ + X λI e ∞ (6)and Y = Y + Y λ e x e y + Y λ e z e x + Y λ e y e z + Y λ e x e ∞ + Y λ e y e ∞ + Y λ e z e ∞ + Y λI e ∞ . (7)And using the definition || X || := X · X † for the quadratic form Q ( X ) (where † represents the reverse operation [1]and X · X † represents the scalar part of the geometric product XX † ) the multivectors X and Y can be normalized as || X || = X µ = 0 X µ = 1 and || Y || = X ν = 0 Y ν = 1 . (8)Now it is evident from the multiplication table above (Table I) that if X , Y ∈ K λ , then so is their product Z = XY : Z = Z + Z λ e x e y + Z λ e z e x + Z λ e y e z + Z λ e x e ∞ + Z λ e y e ∞ + Z λ e z e ∞ + Z λI e ∞ = XY . (9)Thus K λ remains closed under arbitrary number of multiplications of its elements. This is a powerful property. Moreimportantly, we shall soon see that for vectors X and Y in K λ (not necessarily unit) the following norm relation holds: || XY || = || X || || Y || , (10) S S K λ Q z = q r + q d ε q d ε q r IR IR FIG. 1: An illustration of the 8D plane of K λ , which may be interpreted as an Argand diagram for a pair of quaternions. provided the norms are calculated employing the fundamental geometric product instead of the usual scalar product.In particular, this means that for any two unit vectors X and Y in K λ with the geometric product Z = XY we have || Z || = X ρ = 0 Z ρ = 1 . (11)Now, in order to prove the norm relation (10), it is convenient to express the elements of K λ as “dual” quaternions.The idea of dual numbers, z , analogous to complex numbers, was introduced by Clifford in his seminal work as follows: z = r + d ε, where ε = 0 but ε = 0 . (12)Here ε is the dual operator, r is the real part, and d is the dual part [2][4]. Similar to how the “imaginary” operator i is introduced in the complex number theory to distinguish the “real” and “imaginary” parts of a complex number,Clifford introduced the dual operator ε to distinguish the “real” and “dual” parts of a dual number. The dual numbertheory can be extended to numbers of higher grades, including to numbers of composite grades, such as quaternions.In analogy with dual numbers, but with ε = +1, it is convenient for our purposes to write the elements of K λ as Q z = q r + q d ε , (13)where q r and q d are quaternions and Q z may now be viewed as a “dual”-quaternion (or in Clifford’s terminology, asa bi-quaternion). Next, recall that the set of unit quaternions is a 3-sphere, which can be normalized to a radius ̺ r and written as the set S = (cid:26) q r := q + q λ e x e y + q λ e z e x + q λ e y e z (cid:12)(cid:12)(cid:12) || q r || = q q r q † r = ̺ r (cid:27) . (14)Consider now a second, dual copy of the set of quaternions within K λ , corresponding to the fixed orientation λ = +1: S = (cid:26) q d := − q + q e x e y + q e z e x + q e y e z (cid:12)(cid:12)(cid:12) || q d || = q q d q † d = ̺ d (cid:27) . (15)If we now identify λ I e ∞ appearing in (5) as the duality operator − ε , then (in the reverse additive order) we obtain ε ≡ − λ I e ∞ with ε † = ε and ε = +1 (since e ∞ is a unit vector in K λ ) (16)and q d ε ≡ − q d λ I e ∞ = q λ e x e ∞ + q λ e y e ∞ + q λ e z e ∞ + q λ I e ∞ , (17)which is a multi-vector “dual” to the quaternion q d . Note that we write ε as if it were a scalar because it commuteswith all element of K λ in (5). Comparing (14) and (17) with (5) we now wish to write K λ as a set of paired quaternions, K λ = (cid:26) Q z := q r + q d ε (cid:12)(cid:12)(cid:12) || Q z || = q Q z Q † z = q ̺ r + ̺ d , < ̺ r < ∞ , < ̺ d < ∞ (cid:27) , (18)in analogy with (14) or (15), with Q z Q † z being the geometric product between Q z and Q † z (instead of the inner product Q z · Q † z used in (8) to calculate the value of || Q z || ). But this definition || Q z || = q Q z Q † z = p ̺ r + ̺ d for the norm ispossible only if we require q r q † d + q d q † r = 0, rendering every q r orthogonal to its dual q d (cf. Fig. 1). In other words, || Q z || = q Q z Q † z = q ̺ r + ̺ d ⇐⇒ q r q † d + q d q † r = 0 , (19)or equivalently, ( q r q † d ) s = 0; i.e. , q r q † d must be a pure quaternion (for a pedagogical discussion of (19) see section7.1 of Ref. [4]). We can see this by working out the geometric product of Q z with Q † z while using ε = +1, which gives Q z Q † z = (cid:16) q r q † r + q d q † d (cid:17) + (cid:16) q r q † d + q d q † r (cid:17) ε . (20)Now, using definitions (14) and (15), it is easy to see that q r q † r = ̺ r and q d q † d = ̺ d , reducing the above product to Q z Q † z = ̺ r + ̺ d + (cid:16) q r q † d + q d q † r (cid:17) ε . (21)It is thus clear that for Q z Q † z to be a scalar q r q † d + q d q † r must vanish, or equivalently q r must be orthogonal to q d .But there is more to the normalization condition q r q † d + q d q † r = 0 than meets the eye. It also leads to the crucialnorm relation (10), which is at the heart of the only known four normed division algebras R , C , H and O associatedwith the four parallelizable spheres S , S , S and S , with octonions forming a non-associative algebra in addition toforming a non-commutative algebra [5][6]. However, before we prove the norm relation (10), let us take a closer lookat definition (19) within Geometric Algebra. In (8) we used the following definition of norm for a general multivector: || X || = √ X · X † . (22)The left-hand side of this equation — by definition — is a scalar number; namely, || X || . But what is important torecognize for our purpose of proving (10) is that there are two equivalent ways of working out this scalar number:(a) || X || = square root of the scalar part X · X † of the geometric product XX † between X and X † ,or(b) || X || = square root of the geometric product XX † with the non-scalar part of XX † set to zero.The above two definitions of the norm || X || are entirely equivalent . They give one and the same scalar value for thenorm. Moreover, in general, given a product denoted by ∗ , the quantity X † is said to be the conjugate of X if X ∗ X † happens to be equal to unity, X ∗ X † = 1, as in the case of quaternionic products in (14) and (15). On the other hand,in Geometric Algebra the fundamental product between any two multivectors X and Y is the geometric product, XY , not the scalar product X · Y (or the wedge product X ∧ Y for that matter). Therefore the product that mustbe used in computing the norm || X || that preserves the Clifford algebraic structure of K λ is the geometric product XX † , not the scalar product X · X † . To be sure, in practice, if one is interested only in working out the value of thenorm || X || , then it is often convenient to use the definition (a) above. However, our primary purpose here in workingout the norms of X and Y is to preserve the algebraic structure of the space K λ in the fundamental relation (10), andtherefore the definition of the norm we must use is necessarily the second definition stated above; i.e. , definition (b).With the above comments in mind, we are now ready to prove the norm relation (10). To this end, suppose themultivectors X and Y belonging to K λ as spelled out in (6) and (7) are normalized using the definition (b) as follows: || X || = √ XX † = ̺ X (23)and || Y || = √ YY † = ̺ Y , (24)where ̺ X and ̺ Y are fixed scalars. Then, according to (9), their product XY in K λ is another multivector, giving || XY || = q ( XY )( XY ) † (25)= √ XYY † X † (26)= q X ̺ Y X † (27)= (cid:16) √ XX † (cid:17) ̺ Y (28)= ̺ X ̺ Y (29)= || X || || Y || . (30)Thus, at first sight, the norm relation (10) appears to be trivially true. However, the above simple proof is not quitesatisfactory because we have assumed that X and Y are normalized using the definition (b), which requires us to setthe non-scalar parts of the geometric products XX † and YY † equal to zero. That is not difficult to do for both X and Y , but what is involved in the above proof is the geometric product XY and its conjugate ( XY ) † , which makesthe proof less convincing. It is therefore important to spell out the proof in full detail without resorting to shortcuts.To that end, we first work out the right-hand side of the norm relation (10) in the notations of the condition (19): || Q z || || Q z || = (cid:18)q ̺ r + ̺ d (cid:19) (cid:18)q ̺ r + ̺ d (cid:19) = q ̺ r ̺ r + ̺ r ̺ d + ̺ d ̺ r + ̺ d ̺ d . (31)Now, to verify the left-hand side of the norm relation (10), consider a product of two distinct members of the set K λ , Q z Q z = ( q r q r + q d q d ) + ( q r q d + q d q r ) ε , (32)together with their individual definitions Q z = q r + q d ε and Q z = q r + q d ε . (33)If we now use the fact that ε , along with ε † = ε and ε = 1, commutes with every element of K λ defined in (5) andconsequently with all q r , q † r , q d and q † d , and work out Q † z , Q † z and the products Q z Q † z , Q z Q † z and ( Q z Q z ) † as Q † z = q † r + q † d ε , (34) Q † z = q † r + q † d ε , (35) Q z Q † z = (cid:16) q r q † r + q d q † d (cid:17) + (cid:16) q r q † d + q d q † r (cid:17) ε , (36) Q z Q † z = (cid:16) q r q † r + q d q † d (cid:17) + (cid:16) q r q † d + q d q † r (cid:17) ε , (37)and ( Q z Q z ) † = (cid:16) q † r q † r + q † d q † d (cid:17) + (cid:16) q † d q † r + q † r q † d (cid:17) ε , (38)then, using the same normalization condition q r q † d + q d q † r = 0 of (19), the norm relation (10) is not difficult to verify.To that end, we first work out the geometric product ( Q z Q z )( Q z Q z ) † using expressions (32) and (38), which gives( Q z Q z )( Q z Q z ) † = n ( q r q r + q d q d ) (cid:16) q † r q † r + q † d q † d (cid:17) + ( q r q d + q d q r ) (cid:16) q † d q † r + q † r q † d (cid:17)o + n ( q r q d + q d q r ) (cid:16) q † r q † r + q † d q † d (cid:17) + ( q r q r + q d q d ) (cid:16) q † d q † r + q † r q † d (cid:17)o ε . (39)Now the “real” part of the above product simplifies to (42) as follows: (cid:8) ( Q z Q z )( Q z Q z ) † (cid:9) real = q r q r q † r q † r + q d q d q † r q † r + q r q r q † d q † d + q d q d q † d q † d + q r q d q † d q † r + q d q r q † d q † r + q r q d q † r q † d + q d q r q † r q † d (40)= q r q r q † r q † r + q d q d q † d q † d + q r q d q † d q † r + q d q r q † r q † d (41)= ̺ r ̺ r + ̺ d ̺ d + ̺ r ̺ d + ̺ d ̺ r . (42)Here (41) follows from (40) upon inserting the normalization condition (19) in the form q r q † d = − q d q † r into thesecond and third terms of (40), which then cancel out with the sixth and seventh terms of (40), respectively; and(42) follows from (41) upon inserting the normalization conditions || q || = qq † = ̺ for the real and dual quaternionsspecified in (14) and (15), for each of the four terms of (41). Similarly, the “dual” part of the product (39) simplifies to (cid:8) ( Q z Q z )( Q z Q z ) † (cid:9) dual = n q r q d q † r q † r + q d q r q † r q † r + q r q d q † d q † d + q d q r q † d q † d + q r q r q † d q † r + q d q d q † d q † r + q r q r q † r q † d + q d q d q † r q † d o ε (43)= 0 . (44)We can see this again by inserting into (43) the normalization condition (19) in the form q r q † d = − q d q † r and thenormalization conditions || q || = qq † = ̺ for the quaternions in (14) and (15), which cancels out the first four termsof (43) with the last four. Consequently, combining the results of (42) and (44), for the left-hand side of (10) we have || Q z Q z || = q ̺ r ̺ r + ̺ r ̺ d + ̺ d ̺ r + ̺ d ̺ d . (45)Thus, comparing the results in (45) and (31), we finally arrive at the relation || Q z Q z || = || Q z || || Q z || , (46)which is evidently the same as the norm relation (10) in every respect apart from the appropriate change in notation.This result is facilitated by the definition (b) of the norm [or of the quadratic form Q ( X )] explained below Eq. (22).We have thus proved that the finite-dimensional algebra K λ over the reals can be equipped with a positive definitequadratic form Q (the square of the norm) such that Q ( XY ) = Q ( X ) Q ( Y ) for all X and Y in K λ . Consequently, aproduct XY would vanish if and only if X or Y vanishes. In other words, K λ , equipped with Q , is a division algebra.Without loss of generality we can now restrict K λ in (18) to a unit 7-sphere by setting the radii ̺ r and ̺ d to √ : K λ ⊃ S := n Q z := q r + q d ε (cid:12)(cid:12)(cid:12) || Q z || = 1 and q r q † d + q d q † r = 0 o , (47)where ε = − λ I e ∞ , ε † = ε , ε = e ∞ = +1 , q r = q + q λ e x e y + q λ e z e x + q λ e y e z , and q d = − q + q e x e y + q e z e x + q e y e z , (48)so that Q z = q + q λ e x e y + q λ e z e x + q λ e y e z + q λ e x e ∞ + q λ e y e ∞ + q λ e z e ∞ + q λI e ∞ . (49)Needless to say, since all Clifford algebras are associative algebras by definition, unlike the non-associative octonionicalgebra the 7-sphere we have constructed here corresponds to an associative (but non-commutative) division algebra.Note that in terms of the components of q r and q d the condition q r q † d + q d q † r = 0 is equivalent to the constraint f K = q q + λ q q + λ q q + λ q q = 0 . (50)This constraint reduces the space K λ to the sphere S , thereby reducing the 8 dimensions of K λ to the 7 dimensionsof S defined in (47). But the 7-sphere thus constructed has a topology [7] that is different from that of the octonionic7-sphere, and the difference between the two is captured by the difference in the corresponding normalizing constraints f ( q , q , q , q , q , q , q , q ) = 0 . (51)More precisely, the two normalizing constraints giving rise to the two topologically distinct 7-spheres of radius ρ are: f O = q + q + q + q + q + q + q + q − ρ = 0 , (52)which reduces the set O of unit octonions to the sphere S made up of eight-dimensional vectors of fixed length ρ , and f K = q q + λ q q + λ q q + λ q q = 0 , (50)which reduces the set K λ to the sphere S made up of a different collection of eight-dimensional vectors of fixed length ρ = p ̺ r + ̺ d . Both constraints, (50) and (52), involve the same eight variables of the embedding space IR , namely, q , q , q , q , q , q , q , and q , giving the same dimensions for the sphere S of radius ρ , albeit respecting differenttopologies [7]. This difference arises because we have used the geometric product XX † rather than the scalar product X · X † to derive the constraint f K = 0. But both definitions of the norm || X || give identical results, as explained above.Given the quadratic form Q ( X ) and the norm relation (46), we may now view the four associative normed divisionalgebras in the only possible dimensions 1, 2, 4 and 8, respectively [5], as even sub-algebras of the Clifford algebras Cl λ , = span { , λ e x } , (53) Cl λ , = span { , λ e x , λ e y , λ e x e y } , (54) Cl λ , = span { , λ e x , λ e y , λ e z , λ e x e y , λ e z e x , λ e y e z , λ e x e y e z } , (55)and Cl λ , = span (cid:8) , λ e x , λ e y , λ e z , λ e ∞ , λ e x e y , λ e z e x , λ e y e z , λ e x e ∞ , λ e y e ∞ , λ e z e ∞ ,λ e x e y e z , λ e x e y e ∞ , λ e z e x e ∞ , λ e y e z e ∞ , λ e x e y e z e ∞ (cid:9) . (56)It is easy to verify that the even subalgebras of Cl λ , , Cl λ , and Cl λ , are indeed isomorphic to R , C and H , respectively.In practice, the above eight-dimensional algebra sometimes appears in the guise of a ‘1d up’ approach to ConformalGeometric Algebra in the engineering and computer vision applications [8][9]. Such physical applications would benefitfrom explicitly using the quadratic form Q ( X ) and the corresponding division algebra we have presented in this paper.For instance, it may help in removing the “singularities” or non-zero zero divisors from occurring in such applications.An illustration of how that may work can be found in Ref. [3] where we have applied the quadratic form Q ( X ) andthe corresponding division algebra to understand the geometrical origins of quantum correlations within the 7-sphereconstructed in this paper. In the broader context of relativistic quantum theory, it is well known that between1932 and 1952 Jordan attempted to use an alternative ring of octonions with non-associative multiplication rulesto transfer the probabilistic interpretation of quantum theory to what is now known as exceptional Jordan algebra[10]. But as Dirac has noted [11], Jordan’s attempt to obtain a generalized quantum theory in this manner was notsuccessful, because the non-associative multiplication rules are not compatible with any physically meaningful groupof transformations such as the Poincar´e group. However, the octonion-like algebra K λ with six rather than sevenimaginaries we have presented in this paper is associative by construction, and therefore it will be amenable to Jordantype application to quantum theory. Apart from these applications, in Section 5 of Ref. [6] Baez has discussed moremathematically oriented applications of the norm division algebras in four dimensions. These application can now beextended to eight dimensions, thanks to the associativity of K λ . The Clifford-algebraic investigations by Lounesto innormed division algebras, octonions, and triality may also benefit from the associativity of K λ [12]. To facilitate theseapplications, in the appendix below we illustrate how zero divisors are precluded from the K λ equipped with Q ( X ). Appendix: Illustration of How the Definition (b) of the Norm Precludes Zero Divisors
According to Frobenius theorem [13], a finite-dimensional associative division algebra over the reals is necessarilyisomorphic to either R , C , or H in the 1, 2, and 4 dimensions, respectively. Since Clifford algebras are finite-dimensionalassociative algebras, Frobenius theorem suggests that those Clifford algebras that are not isomorphic to R , C , or H will contain non-zero zero divisors or idempotent elements. It is therefore important to understand how the definition(b) of the norm leading to the quadratic form Q ( X ) prevents non-zero zero divisors from occurring in the algebra K λ .To that end, recall that the elements of K λ are of the following general form in terms of quaternions q r and q d : Q z = q r + q d ε , (A.1)where ε † = ε and ε = +1 as in (16) and the normalization of Q z requires that q r and q d must satisfy the condition q r q † d + q d q † r = 0 ⇐⇒ q r q † d = − q d q † r . (A.2)This condition follows from the definition (b) of the norm || Q z || discussed below Eq. (22). It respects the fundamentalgeometric product Q z Q † z and gives the same scalar value for the norm || Q z || as that calculated using definition (a).Now, for the sake of argument, consider the following idempotent quantities, which are also non-zero zero divisors: Z ± = 12 (1 ± ε ) . (A.3)We call the quantities Z ± idempotent quantities because they square to themselves, which can be easily verified: Z = Z + Z † + = Z + and Z − = Z − Z †− = Z − . (A.4)But Z + and Z − are also orthogonal to each other because their products vanish, which can also be easily verified: Z + Z − = Z − Z + = 0 . (A.5)Now consider two multivectors, X and Y , confined to a two-dimensional subspace of K λ , by setting X = Y = √ , Y = − X = √ , and the remaining twelve coefficients equal to zero in the Eqs. (6) and (7), along with ε ≡ − λ I e ∞ : X = 1 √ ε ) = √ Z + and Y = 1 √ − ε ) = √ Z − . (A.6)Then (A.5) implies that for the LHS of Eq. (10) we have || XY || = 0. On the other hand, if we neglect the normalizationcondition (A.2) and insist on using definition (22) for calculating the norm, then using (A.4) and (A.3) we obtain || X || = 1 and || Y || = 1, and then their product || X || || Y || for the RHS of Eq. (10) gives || X || || Y || = 1, implying that || XY || 6 = || X || || Y || . (A.7)This contradiction between Eqs. (10) and (A.7) stems from the fact that the multivectors X and Y defined in (A.6)are incompatible with the definition (b) of the norm we have discussed just below Eq. (22). That definition impliesthat X and Y defined in (A.6) amount to assuming +1 = −
1, as a consequence of the condition (A.2). To appreciatethis, consider first the multivector X defined in (A.6). According to the general element (A.1), this particular X isobtained by setting q r = √ and q d = √ . Substituting these values, together with q † r = √ and q † d = √ (because q r = √ and q d = √ are scalars), into the normalization condition q r q † d = − q d q † r leads to the said contradiction:1 √ × √ − √ × √ ⇒ +1 = − . (A.8)Similarly, consider the multivector Y defined in (A.6). According to the general element (A.1), this Y is obtained bysetting q r = √ and q d = − √ . Substituting these values, together with q † r = √ and q † d = − √ (because q r = √ and q d = − √ are scalars), into the normalization condition q r q † d = − q d q † r again leads to the same contradiction: − √ × √ √ × √ ⇒ − . (A.9)This proves that the ad hoc coefficients chosen in (A.6) to define X and Y confined to a two-dimensional subspace of K λ are not compatible with the normalization condition (A.2), and thus they are not compatible with the definition(b) of the norm used to construct the 7-sphere defined in (47). Consequently, these multivectors are not members ofthe set (47) that constitutes that 7-sphere. On the other hand, rejecting the definition (b) to calculate the norms || X || and || Y || is not an option because it is entirely equivalent to the definition (a), and, moreover, uses the fundamentalgeometric product Q z Q † z necessary to preserve the Clifford algebraic structure of K λ instead of the scalar product Q z · Q † z . This completes the illustration of how using the definition (b) for the norm precludes zero divisors from K λ . [1] C. Doran and A. Lasenby, Geometric Algebra for Physicists (Cambridge University Press, Cambridge, 2003).[2] L. Dorst, D. Fontijne, and S. Mann,
Geometric Algebra for Computer Science (Elsevier, Amsterdam, 2007).[3] J. Christian,
Quantum correlations are weaved by the spinors of the Euclidean primitives , R. Soc. Open Sci., , 180526(2018); https://doi.org/10.1098/rsos.180526; See also https://arxiv.org/abs/1806.02392 (2018).[4] B. Kenwright, A beginners guide to dual-quaternions: what they are, how they work, and how to use them for 3D characterhierarchies , in Proceedings of the 20th International Conferences on Computer Graphics, Visualization and ComputerVision, 1–10 (2012).[5] A. Hurwitz, ¨Uber die Composition der quadratischen Formen von beliebig vielen Variabeln , Nachr. Ges. Wiss. G¨ottingen, , 309–316 (1898).[6] J. C. Baez,
The octonions , Bull. Am. Math. Soc., , 145–205 (2002).[7] J. W. Milnor, Topology from the Differentiable Viewpoint (Princeton University Press, Princeton, New Jersey, 1997).[8] A. Lasenby,
Recent applications of conformal geometric algebra , in Computer Algebra and Geometric Algebra with Appli-cations, 298–328 (Springer, New York, 2004).[9] A. Lasenby,
Rigid body dynamics in a constant curvature space and the 1D up approach to conformal geometric algebra , inGuide to Geometric Algebra in Practice, 371–389 (Springer, New York, 2011).[10] P. Jordan, ¨Uber die Multiplikation quantenmechanischer Gr¨oßen , Zeitschrift f¨ur Physik, , 285–291 (1933).[11] P. A. M. Dirac, The relation between mathematics and physics , Proceedings of the Royal Society (Edinburgh), , Part II,122–129 (1939).[12] P. Lounesto, Octonions and triality , Advances in Applied Clifford Algebras, , 191 (2001).[13] F. G. Frobenius, ¨ber lineare Substitutionen und bilineare Formen , Journal fr die reine und angewandte Mathematik,84