Alternative multiplications and non-associativity in physics

Abstract: Some physical systems like the quantum mechanics with magnetic charges or field theoretical models appearing in the context of string theory are formulated in terms of non-associative algebras. Hence, demand non-associative star products for its description. However, unlike its associative counterpart the situation with non-associative star products is quite unclear. At the moment it is not evident which condition should be used instead of the associativity in the definition of the non-associative star multiplication. In mathematics, the natural generalization of the associativity is the requirement that the multiplication should be alternative, the associator of three elements should vanish if each two of them are equal. We show that for the alternative closed star product the integrated associator vanishes. In particular, it means that on-shell there will be no violation of associativity in string scattering amplitudes. We discuss the examples of the alternative and closed star products. In particular, it is constructed the star product corresponding to the algebra of octonions.