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4. Observability in Classical Physics, Commutative Algebra and Differential Calculus

Mathematical correspondence principle—affirming the existence of the Quantum Differential Calculus and showing the departing point to look for it—obviously is still too generic and is unable to define its own boundaries. More precise hints about the direction that must be followed can be obtained by trying to clarify the mathematical contents of the base paradigm of contemporary Physics stating that it exists only that which can be observed. Fundamental difference between Classical and Quantum Physics—as anyone knows—is that observations in Classical Physics, unlike those in Quantum Physics, are independent one from another and do not change the state of the observed object. Thus it is natural, before moving on and looking for Quantum Differential Calculus, to formalize mathematically the observation procedure in Classical Physics.
From a mathematical viewpoint, the measuring devices gathered into a classical lab can be interpreted as the generators of a commutative algebra on the real numbers which can be called "Algebra of Observables". Any concrete observation of the physical system under inspection is nothing but an homomorphism from the algebra of observables onto the algebra of real numbers, and thus a concrete state of the system is identified with such an homomorphism. In other words, the variety of all the states of the physical system under consideration coincides with the R-spectrum of the algebra of observables. This fact leads to think that all the information about the dynamical structure, etc., of the system might be expressed exclusively in terms of the algebra of observables. On the other hand we know de facto that such an information can be lent by Differential Calculus. Ergo, by putting together these two considerations we are lead to the conclusion that Differential Calculus is an aspect of Commutative Algebra, and as such it can be built only on the basis of arithmetic operations without adding ingredients of topological or other nature.
Actually we found out that all the aspects of Differential Calculus—and consequently the theory of differential equations, Differential Geometry, Differential Topology, and so on—can be constructed over an arbitrary commutative algebra. The logical scheme of such a construction is furnished by the so-called functors of Differential Calculus, ad it might be called "Primary" Differential Calculus. Moreover, the fact that Differential Calculus is an aspect of Commutative Algebra can be interpreted as a confirmation of the validity of the principles we have been formulating so far.