The Conceptual Basis and Use of the Geometric Invariance Principles

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Abstract
Invariance principles are used, in physics, in two distinct manners. First, they are used as superlaws of nature in that, once their validity has been suggested by their consistency with the known laws of nature, they serve as guides in our search for as yet unknown laws of nature. Second, they can serve as tools for obtaining properties of the solutions of the equations provided by the laws of nature. It is desirable for the first use to give a formulation of invariances directly in terms of the primitive concepts of physical theory, i.e., in terms of observations, or measurements, and their results. Invariances which can be so formulated are called geometric invariances. The present paper contains an attempt at such a formulation of geometric invariances. This formulation is then applied, in detail, to the classical mechanics of point particles, to a relativistic mechanics of interacting point particles, and to quantum theory. With the exception of the relativistic mechanics of point particles, these applications form a review, from a single point of view, of earlier work on this subject. The last part of the paper contains a review of the second use of invariances.

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