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Multi-dimensional vector product

Z. K. Silagadze

arXiv:math/0204357v1 [math.RA] 30 Apr 2002

Budker Institute of Nuclear Physics, 630 090, Novosibirsk, Russia

Abstract It is shown that multi-dimensional generalization of the vector product is only possible in seven dimensional space.

The three-dimensional vector product proved to be useful in various physical problems. A natural question is whether multi-dimensional generalization of the vector product is possible. This apparently simple question has somewhat unexpected answer, not widely known in physics community, that generalization is only possible in seven dimensional space. In mathematics this fact was known since forties [1], but only recently quite simple proof (in comparison to previous ones) was given by Markus Rost [2]. Below I present a version of this proof to make it more accessible to physicists. For contemporary physics seven-dimensional vector product represents not only an academic interest. It turned out that the corresponding construction is useful in considering self-dual Yang-Mills ?elds depending only upon time (Nahm equations) which by themselves originate in the context of M-theory [3, 4]. Other possible applications include Kaluza-Klein compacti?cations of d = 11, N = 1 Supergravity [5]. That is why I think that this beautiful mathematical result should be known by a general audience of physicists. Let us consider n-dimensional vector space Rn over the real numbers with the standard Euclidean scalar product. Which properties we want the multi-dimensional bilinear vector product in Rn to satisfy? It is natural to choose as de?ning axioms the following (intuitively most evident) properties of the usual three-dimensional vector product: A × A = 0, (A × B ) · A = (A × B ) · B = 0, |A × B | = |A| |B |, if A · B = 0. Here |A|2 = A · A is the norm of the vector A. Then 0 = (A + B ) × (A + B ) = A × B + B × A shows that the vector product is anti-commutative. By the same trick one can prove that (A × B ) · C is alternating in A, B, C . For example 0 = ((A + C ) × B ) · (A + C ) = (C × B ) · A + (A × B ) · C 1 (1) (2) (3)

shows that (C × B ) · A = ?(A × B ) · C . For any two vectors A and B the norm |A × B |2 equals to

2 2

A?

A·B |B |2

B

×B

= A?

A·B |B |2

B |B |2 = |A|2 |B |2 ? (A · B )2 .

Therefore for any two vectors we should have (A × B ) · (A × B ) = (A · A)(B · B ) ? (A · B )2 . Now consider |A × (B × A) ? (A · A)B + (A · B )A|2 = = |A × (B × A)|2 + |A|4 |B |2 ? (A · B )2 |A|2 ? 2|A|2 (A × (B × A)) · B. But this is zero because |A × (B × A)|2 = |A|2 |B × A|2 = |A|4 |B |2 ? (A · B )2 |A|2 and (A × (B × A)) · B = (B × A) · (B × A) = |A|2 |B |2 ? (A · B )2 . Therefore we have proved the identity A × (B × A) = (A · A)B ? (A · B )A . (5) (4)

Note that arrangement of the brackets in the l.f.s. is in fact irrelevant because the vector product is anti-commutative. However, familiar identity A × (B × C ) = B (A · C ) ? C (A · B ) (6)

does not follow in general from the intuitively evident properties (1-3) of the vector product [2]. To show this, let us introduce the ternary product [6] (which is zero if (6) is valid) {A, B, C } = A × (B × C ) ? B (A · C ) + C (A · B ). Equation (5) implies that this ternary product is alternating in its arguments. For example 0 = {A + B, A + B, C } = {A, B, C } + {B, A, C }. If ei , i = 1, . . . , n is an orthonormal basis in the vector space, then (ei × A) · (ei × B ) = ((ei × B ) × ei ) · A = [B ? (B · ei )ei ] · A and, therefore,

n

(ei × A) · (ei × B ) = (n ? 1)A · B.

i=1

(7)

2

Using this identity we obtain

n

{ei , A, B } · {ei , C, D } =

i=1

= (n ? 5)(A × B ) · (C × D ) + 2(A · C )(B · D) ? 2(A · D )(B · C ). Hence

n

(8)

{ei , ej , A} · {ei , ej , B } = (n ? 1)(n ? 3)A · B

i,j =1

(9)

and [6]

n

{ei , ej , ek } · {ei , ej , ek } = n(n ? 1)(n ? 3).

i,j,k =1

(10)

The last equation shows that there exists some {ei , ej , ek } that is not zero if n > 3. So equation (6) is valid only for the usual three-dimensional vector product (n = 1 case is, of course, not interesting because it corresponds to identically vanishing vector product). Surprisingly, we do not have much choice for n even in case when validity of (6) is not required. In fact the space dimension n should satisfy [2] (see also [7]) n(n ? 1)(n ? 3)(n ? 7) = 0. To prove this statement, let us note that using A × (B × C ) + (A × B ) × C = (A + C ) × B × (A + C ) ? A × B × A ? C × B × C = = 2A · C B ? A · B C ? B · C A and A × (B × (C × D)) = 1 A × (B × (C × D)) + (A × B ) × (C × D)? 2 (11)

?(A × B ) × (C × D) ? ((A × B ) × C ) × D + ((A × B ) × C ) × D + +(A × (B × C )) × D ? (A × (B × C )) × D ? A × ((B × C ) × D)+ +A × ((B × C ) × D ) + A × (B × (C × D )) we can check the equation A × {B, C, D } = ?{A, B, C × D } + A × (B × (C × D )) ? {A, C, D × B }+ +A × (C × (D × B )) ? {A, D, B × C } + A × (D × (B × C )) = = ?{A, B, C × D } ? {A, C, D × B } ? {A, D, B × C } + 3A × {B, C, D}. The last step follows from 3{B, C, D } = {B, C, D } + {C, D, B } + {D, B, C } = 3

= B × (C × D ) + C × (D × B ) + D × (B × C ). Therefore the ternary product satis?es an interesting identity 2 A × {B, C, D} = {A, B, C × D} + {A, C, D × B } + {A, D, B × C } Hence we should have 4

i,j,k,l=1 n n

(12)

|ei × {ej , ek , el }|2 =

=

i,j,k,l=1

|{ei , ej , ek × el } + {ei , ek , el × ej } + {ei , el , ej × ek }|2 .

L.h.s. is easily calculated by means of (7) and (10):

n

4

i,j,k,l=1

|ei × {ej , ek , el }|2 = 4n(n ? 1)2 (n ? 3).

To calculate the r.h.s. the following identity is useful

n

{ei , ej , A} · {ei , ej × B, C } = ?(n ? 3)(n ? 6)A · (B × C )

i,j =1

(13)

which follows from (8) and from the identity

n

(ei × A) · ((ei × B ) × C ) =

i=1 n

=

i=1

(ei × A) · [2ei · C B ? B · C ei ? ei · B C ? ei × (B × C )] = = ?(n ? 4)A · (B × C ).

Now, with (9) and (13) at hands, it becomes an easy task to calculate

n

|{ei , ej , ek × el } + {ei , ek , el × ej } + {ei , el , ej × ek }|2 =

i,j,k,l=1

= 3n(n ? 1)2 (n ? 3) + 6n(n ? 1)(n ? 3)(n ? 6) = 3n(n ? 1)(n ? 3)(3n ? 13). Therefore we should have 4n(n ? 1)2 (n ? 3) = 3n(n ? 1)(n ? 3)(3n ? 13). But 3n(n ? 1)(n ? 3)(3n ? 13) ? 4n(n ? 1)2 (n ? 3) = 5n(n ? 1)(n ? 3)(n ? 7) and hence (11) follows. As we see, the space dimension must equal to the magic number seven if unique generalization of the ordinary three-dimensional vector product is possible. 4

So far we only have shown that seven-dimensional vector product can exist in principle. What about its detailed realization? To answer this question, it is useful to realize that the vector products are closely related to composition algebras [1] (in fact these two notions are equivalent [2]). Namely, for any composition algebra with unit element e we can de?ne the vector product in the subspace orthogonal to e by x × y = 1 (xy ? yx). Therefore from 2 a viewpoint of composition algebra, the vector product is just the commutator divided by two. According to Hurwitz theorem [8] the only composition algebras are real numbers, complex numbers, quaternions and octonions. The ?rst two of them give identically zero vector products. Quaternions produce the usual three-dimensional vector product. The seven-dimensional vector product is generated by octonions [9]. It is interesting to note that this seven-dimensional vector product is covariant with respect to smallest exceptional Lie group G2 [10] which is the automorphism group of octonions. Using the octonion multiplication table [9] one can realize the seven-dimensional vector product as follows

7

ei × ej =

k =1

fijk ek ,

i, j = 1, 2, . . . , 7,

(14)

where fijk is totally antisymmetric G2 -invariant tensor and the only nonzero independent components are f123 = f246 = f435 = f651 = f572 = f714 = f367 = 1. Note that in contrast to the three-dimensional case fijk fkmn = δim δjn ? δin δjm . Instead we have fijk fkmn = gijmn + δim δjn ? δin δjm where gijmn = ei · {ej , em , en }. In fact gijmn is totally antisymmetric G2 -invariant tensor. For example gijmn = ei · {ej , em , en } = ?ei · {em , ej , en } = = ?ei · (em × (ej × en )) + (ei · ej )(em · en ) ? (ei · en )(em · ej ) = = ?em · (ei × (en × ej )) + (ei · ej )(em · en ) ? (ei · en )(em · ej ) = = ?em · {ei , en , ej } = ?em · {ej , ei , en } = ?gmjin . The only nonzero independent components are g1254 = g1267 = g1364 = g1375 = g2347 = g2365 = g4576 = 1. In conclusion, generalization of the vector product we have considered is only possible in seven-dimensional space and is closely related to octonions – the largest composition algebra which ties up together many exceptional structures in mathematics [10]. In a general case of p-fold vector products other options arise [1, 5, 11]. We recommend that interested reader consults references to explore these possibilities and possible physical applications. (15)

5

References

[1] B. Eckmann, ”Stetige L¨ osungen linearer Gleichungssysteme,” Comm. Math. Helv. 15, 318-339 (1943). [2] M. Rost, ”On the Dimension of a Composition Algebra,” Doc. Math. J. DMV 1, 209-214 (1996). [3] D. B. Fairlie and T. Ueno, “Higher-dimensional generalizations of the Euler top equations,” hep-th/9710079. [4] T. Ueno, “General solution of 7D octonionic top equation,” Phys. Lett. A 245, 373-381 (1998). [5] R. Dundarer, F. Gursey and C. Tze, “Generalized Vector Products, Duality And Octonionic Identities In D = 8 Geometry,” J. Math. Phys. 25, 1496-1506 (1984). [6] S. Maurer, Vektorproduktalgebren, Diplomarbeit, Universit¨ at Regensburg, 1998. http://www.math.ohio-state.edu/?rost/tensors.html#maurer [7] J. A. Nieto and L. N. Alejo-Armenta, “Hurwitz theorem and parallelizable spheres from tensor analysis,” Int. J. Mod. Phys. A 16, 4207-4222 (2001). [8] For rigorous formulation see, for example M. Kocher and R. Remmert, Composition algebras, in Numbers, Graduate Texts in Mathematics, vol. 123 (Springer, 1990). Edited by J. H. Ewing, pp. 265-280. [9] For applications of octonions in physics see M. G¨ unaydin and F. G¨ ursey, “Quark Structure And Octonions,” J. Math. Phys. 14, 1651-1667 (1973); V. de Alfaro, S. Fubini and G. Furlan, “Why We Like Octonions,” Prog. Theor. Phys. Suppl. 86, 274-286 (1986); F. G¨ ursey and C. H. Tze, On the role of division, Jordan and related algebras in particle physics (World Scienti?c, Singapore, 1996); G. M. Dixon, Division algebras: octonions, quaternions, complex numbers and the algebraic design of physics (Kluwer Academic Publishers Group, Dordrecht, 1994); S. Okubo, Introduction to octonion and other nonassociative algebras in physics (Cambridge University Press, Cambridge, 1995). [10] J. C. Baez, “The Octonions,” math.ra/0105155. [11] R. L. Brown and A. Gray, “Vector cross products,” Comm. Math. Helv. 42, 222-236 (1967); A. Gray, “Vector cross products on manifolds,” Trans. Am. Math. Soc. 141, 465504 (1969); “Errata to ’Vector cross products on manifolds’,” ibid. 148, 625 (1970) (erratum).

6

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