Abstract
Arithmetic operations can be defined in various ways, even if one assumes commutativity and associativity of addition and multiplication, and distributivity of multiplication with respect to addition. In consequence, whenever one encounters `plus' or `times' one has certain freedom of interpreting this operation. This leads to some freedom in definitions of derivatives, integrals and, thus, practically all equations occurring in natural sciences. A change of realization of arithmetic, without altering the remaining structures of a given equation, plays the same role as a symmetry transformation. An appropriate construction of arithmetic turns out to be particularly important for dynamical systems in fractal space-times. Simple examples from classical and quantum, relativistic and nonrelativistic physics are discussed, including the eigenvalue problem for a quantum harmonic oscillator. It is explained why the change of arithmetic is not equivalent to the usual change of variables, and why it may have implications for the Bell theorem.
Highlights
Symmetries of physical systems can be rather obvious or very abstract
Lorentz transformations, discovered as a formal symmetry of Maxwell’s equations, seemed abstract until their physical meaning was understood by Einstein
It has taught us that differences in mathematical realizations of a symmetry may directly reflect physical differences
Summary
Symmetries of physical systems can be rather obvious or very abstract. Lorentz transformations, discovered as a formal symmetry of Maxwell’s equations, seemed abstract until their physical meaning was understood by Einstein. The approach developed in this paper is not in any sense close to those generalizations of physics that involve padic mathematics [1,2], the context of fractals and the Cantor set might create such an impression. Another formalism that can be confused with what we do is based on ‘arithmetic dynamics’ [3] which involves the so-called linear conjugates F f = f −1 ◦ F ◦ f (which will play a role here), but where the calculus is defined in the standard way. Neither Burgin nor Benioff made the essential step of formulating a non-Diophantine calculus
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