Position operators in relativistic quantum theory: Analysis of algebraic operator relationships

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Abstract
This work is the first of a series of three papers examining different aspects of position operators in relativistic quantum theory. In this paper the properties of the position operator X and the velocity operator V are derived for single particle matrix elements in the context of the Poincaré generator algebra. Both the physical meaning and the mathematical implications of each property are discussed. The algebraic structure of the extended set of relationships including the Poincaré generators, X and V is examined. It is found that this set defines an infinite algebra which is intractable mathematically. The Casimir operators of the Poincaré algebra are required to be Casimir operators for X and V, a new condition on V is formulated, and a simple solution for K is constructed. These conditions, together with familiar position operator properties, give the constraints and solutions for the extended algebra.

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