An overlooked but nevertheless very important problem concerns the role of the differentiable and smooth categories in mathematical physics, that is, the category of maps having continuous up to a certain order or all order. Our question is: why use such a class of maps rather than (real) analytic ones (or semi-analytic) ? The equations of physics have analytic coeficients. Known solutions are analytic (for simplicity we do not distinguish analytic from meromorphic). In fact known solution are analytic having power series representations with computable coeficients (for a standard notion of computability for sequences of real numbers). And in fact all numerical methods for mathematical physics depend on working in the domain of computable analytic functions.
The class of computable analytic functions is related to the problem of integration of elementary functions. It is also very elegant and simple in itself as it reduces the problems of the foundation of analysis (infinitesimals, non-standard analysis) to the algebra of (convergent) power series.
Deep results in the theory of smooth maps depend on the theory of several complex variables.
The existence and domain of analytic solutions to analytic equations is an interesting and difficult area of mathematics. Several results rule out associating analytic equations with any kind of global determinism (in the terminology of Poincaré, solutions in power series diverge). That is if we wish to equate determinism with computability and thus with computable analytic functions in physics. Thus that computable determinism is essentially local is not philosophy but a hard result in mathematics.
An interesting mathematical questions: are there analytic equations which (locally) admit smooth but analytic solutions ?
Another vexing question: why are there not abundantly more applications of the theory of functions of several complex variables (and complex analytic geometry) to mathematical physics ?
There are objections against the analytic class. For instance it rules out the test function used in distribution theory or more generally functions with compact support. Thus we cannot represent a completely localized field or soliton wave (but notice how Newton's law of gravitation posits that a single mass will influence the totality of space). And yet the most general functions constructed (like the test function) are often simply the result of gluing together analytic functions along a certain boundary. Most concrete examples of smooth function but not analytic functions are precisely of this sort. We could call these piecewise analytic maps. Thus additional arguments are required to justify why we have to go beyond piecewise analytic maps. An obvious objection would be: distributions and weak solutions. But here again we invoke the theory of hyperfunctions. It seems plausible that there could be a piecewise analytic version of distribution theory (using sheaf cohomology) - even a computable piecewise analytic version.
In another note we investigate other incarnations of computability in mathematical physics (and their possible role in interpreting quantum theory). Can we consider measurable but not continuous maps which are yet computable ? Together with obvious examples of locally or almost-everyhwere (except on a computable analytic set) computable analytic maps we can seek for examples of nowhere continuous measurable maps which are yet computable (in some adequate sense). The philosophy behind this is the computable determinism may go beyond the differential and analytic category, the equations of physics in this case however only expressing (in a non-exhaustively determining way) measure-theoretic properties of the solutions.
We end with a discussion of what constitutes exactly a computable real analytic function and how we can define the most interesting and natural classes of such functions. Obvious examples are so-called ’elementary functions’ which have very simple coeficient series in their Taylor expansions. Also it is clearly interesting to study real analytic functions whose coeficient series are computable in terms of n. And can we decide mechanically when one of these functions is elementary ?
Consider the class of real elementary functions defined on a real interval I. These are real analytic functions. How can we characterise their power series ? That is, what can we say about the series of their coeficients ? For instance there are coeficients an given by rational functions in n , or given by combinations of rational functions and factorials functions, primitive recursive coeficients, coeficients given by recurrence relations, etc. It is easy to give an example of a real analytic function which is not elementary. Just solve the equation x′′ − tx = 0 using power series. This equation is known not to have any non-trivial elementary solution, in fact it has no Liouville solution (indefinite integrals of elementary functions).
Let ELEM be the problem: given a convergent Taylor series, does it represent an elementary function ? Let INT be the problem: given an elementary function does it have a primitive/indefinite integral which is an elementary function ? An observation that can be made is that if ELEM is decidable then so too is INT. Given an elementary function write down its Taylor series and integrate each term. Then apply the decision procedure for ELEM (of course we must be more precise here, this is just the general idea). Thus to show that ELEM is undecidable it suffices to show that INT is.
In the literature there is defined the class holomonic functions which can be characterised
either as:
1) Being solutions of a homogenous linear differential equation with polynomial coeficients.
2) Having Taylor series coeficients given by polynomial recurrence relations.
There is an algorithm to pass between these two presentations. The holonomic class includes elementary functions, the hypergeometric functions, the Bessel function, etc. The question naturally arises: given a sequence of real numbers, is it decidable if they obey a polynomial recurrence relation ?
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