Why Vectors?

Abstract

Quantum kinematics represents initial and final actions on any physical system by vectors, internal operations by operators on these vectors, and logical operations by the linear algebraic operations of + and x. Why is the complex vector space so basic for kinematics in nature? What physical combination of actions does the + stand for? Whence the imaginary i? The sum of two vectors depends on their phases, which we ignore when we use the vectors to represent external actions. The phase of an initial vector is meaningless. Is it not possible to formulate a quantum theory entirely in terms of experimentally meaningful entities? For this we must give these phases empirical meanings where possible and eliminate them where not.