ORIGIN, PROBLEMATIC ASPECTS AND INVARIANT FORMULATION OF CLASSICAL AND OPERATOR DEFORMATIONS

Abstract

In this paper we study three aspects of generalized classical and operator theories, herein generically called deformations, which do not appear to have propagated in the rather vast literature in the field: (1) the first known studies on classical and operator deformations; (2) their rather serious physical and mathematical shortcomings due to lack of invariance when conventionally formulated; and (3) the ongoing efforts for the achievement of invariant formulations preserving the axiomatic consistency of the original theories. We begin by recalling the mathematical beauty, axiomatic consistency and experimental verifications of the special relativity at both classical and quantum levels, and its main axiomatic properties: universal invariance of the fundamental units of space and time; preservation of hermiticity-observability at all times; uniqueness and invariance of numerical predictions; and other known properties. We then review the first known, generally ignored, classical and operator deformations. We then study the generally ignored problematic aspects of classical and operator deformations in their current formulation which include: lack of invariance of the fundamental units of space and times with consequential inapplicability to real measurements; loss of observability in time; lack of uniqueness and invariance of numerical predictions; violation of causality and probability laws; and, above all, violation of Einstein's special relativity. We finally outline the generally ignored ongoing efforts for the resolutions of the above shortcomings, and show that they require the necessary use of new mathematics specifically constructed for the task. We finally present a systematic study for the identical reformulation of existing classical and operator deformations in an invariant form.