A pr 2 00 2 Multi - dimensional vector product

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 fields depending only upon time (Nahm equations) which by themselves originate in the context of M-theory [3, 4]. Other possible applications include Kaluza-Klein compactifications 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 R n over the real numbers with the standard Euclidean scalar product. Which properties we want the multi-dimensional bilinear vector product in R n to satisfy? It is natural to choose as defining axioms the following (intuitively most evident) properties of the usual three-dimensional vector product: A × A = 0, (1) (A × B) · A = (A × B) · B = 0, (2)