Observable Algebra

A physical applicability of normed split-algebras, such as hyperbolic numbers, split-quaternions and split-octonions is considered. We argue that the observable geometry can be described by the algebra of split-octonions, which is naturally equipped by zero divisors (the elements of split- algebras corresponding to zero norm). In such a picture physical phenomena are described by the ordinary elements of chosen algebra, while the zero divisors give raise the coordinatization and two fundamental constants, namely velocity of light and Planck constant. It turns to be possible that uncertainty principle appears from the condition of positively defined norm, and has the same geometrical meaning as the existence of the maximal value of speed. The property of non-associativity of octonions could correspond to the appearance of fundamental probabilities in physics. Grassmann elements and a non-commutativity of space coordinates, which are widely used in various physical theories, appear naturally in our approach