1 Introduction

[page 207, §1]   
[207.1.1] A fractional dynamical system has been defined [1] as a dynamical system involving fractional (i.e. noninteger order) time derivatives instead of integer order time derivatives. [207.1.2] Despite the long history of fractional calculus in mathematics (see [2, 3, 4, 5, 6] for reviews), despite numerous publications on fractional powers of infinitesimal generators [7, 8, 9, 10, 11, 12, 13, 14, 15, 16], and despite a rapidly growing literature on possible applications of fractional dynamical systems to physical phenomena (see [17, 18, 19, 20] and the present volume for reviews), there seem to exist only few publications discussing the physical foundations of fractional dynamics.^a (This is a footnote:) ^a The term “fractional dynamics” is used synonymously with “fractional dynamical system”

[207.2.1] My objective in this chapter is to call attention to the foundations of fractional dynamics and fractional time evolution by reformulating the problem stated originally in [21, 1] and briefly summarizing some known results. [207.2.2] As everyone knows, fractional time derivatives do not appear in any established fundamental theory of physics such as classical mechanics, electrodynamics, or quantum mechanics. [207.2.3] Instead, integer (first and second) order time derivatives occur in all fundamental theories of physics. [page 208, §0]    [208.0.1] Obviously, time is a primordial and fundamental concept from the foundations of physics. [208.0.2] Replacing integer order with fractional order time derivatives therefore changes the fundamental concept of time and with it the concept of evolution in the foundations of physics. [208.0.3] Evolution equations in physics do not contain fractional time derivatives, because it would contradict the deep and fundamental principle, that time evolution is time translation. [208.0.4] Most publications on fractional dynamics proceed directly to applied problems, but do not justify, discuss or even mention, that they remove the fundamental concept of time evolution (=time translation) from the foundations of physics.

[208.1.1] Difficulties with fractional dynamics arise also, because fractional derivative operators can be defined in numerous ways [4, 5, 6]. [208.1.2] Embedding a conventional dynamical system into a family of fractional dynamical systems is not unique. [208.1.3] In fact, an infinite number of choices are possible and many publications fail to justify or discuss their particular choice.

[208.2.1] Given the need for a fundamental justification of fractional dynamics, the article is structured as follows. [208.2.2] Let me first recall some basic ideas about time. [208.2.3] Observables, states and their time evolution are discussed next. [208.2.4] Restricting attention to conservative dynamical systems raises a fundamental problem for the time evolution of macroscopic states. [208.2.5] Induced measure preserving transformations are then introduced to solve this problem. [208.2.6] Averaging them shows, that macroscopic states evolve in time by convolution rather than translation. [208.2.7] My short account of the foundations of fractional dynamics concludes with remarks about irreversibility, experimental evidence, and dissipative systems.