General introduction to Scale Relativity

The essential question that is addressed here is the problem of scales in Nature. This is not a recent question. From Plato, Euclid, and Aristotle, to Leibniz, Laplace, and Poincaré, many philosophers, mathematicians and physicists have thought over scales and their transformations, dilations and contractions. What determines the universal scales in Nature? What is the origin of the elementary particles scales, of the unification and symmetry breaking scales, of the large scale structures in the Universe? Not only are fundamental or characteristic scales observed to occur in the world, but physical laws may in some situations depend themselves on scale: this leads us to the concepts of scaling and of scale invariance.

As reminded later, this scale dependence may in some cases be very fundamental: hence in quantum mechanics the results of measurements explicitly depend on the resolution of the measurement apparatus, as described by the Heisenberg relations; in cosmology, it is the whole set of interdistances between the objects of the Universe that depends on a time varying universal scale factor (this is the expansion of the Universe). Moreover, scale laws and scaling behaviours are encountered in many situations, at small scales (microphysics), large scales (extragalactic astrophysics and cosmology) and intermediate scales (complex self-organized systems), but most of the time such laws are found in an empirical way, since we still lack a fundamental theory allowing us to understand them from fundamental principles.

Our proposal is that such a fundamental principle upon which a theory of scale laws may be founded is the principle of relativity itself. But, by `principle of relativity' we mean something more general than its application to particular laws: we actually mean a universal method of thought. Following Einstein, we shall express it by postulating that the laws of Nature must be such that they apply to reference systems whatever their state. The present theory of relativity, after the work of Galileo, Poincaré and Einstein, results from the application of this principle to space and time coordinate systems and to their state of position (origin and axis orientation) and of motion (which may be eventually included into axis orientation in space-time).

We have suggested that the principle of relativity also applies to laws of scale. Taking advantage of the relative character of every length and time scales in Nature, we define the resolution of measurements (more generally, the characteristic scale of a given phenomenon) as the state of scale of the reference system. This allows us to set a principle of scale relativity, according to which the laws of physics must be such that they apply to coordinate systems whatever their state of scale, whose mathematical translation is the requirement of scale covariance of the equations of physics. While the classical domain is apparently unchanged by such an analysis, its fundamental laws being scale independent (but situations where dynamical chaos occurs may call for a reopening of the question), there are two fundamental scale-dependent domains on which this extension of the principle of relativity sheds new light, namely quantum physics and cosmology.

In order to describe physical laws complying to this principle, one needs some mathematical tools capable of achieving such a fundamental and explicit dependence of physics on scale in their very definition. There is one geometrical concept that immediately comes to mind in this respect, Mandelbrot's so-called fractals, a word that names objects, sets and functions whose forms are extremely irregular and fragmented on all scales. But on the other hand, physical laws that are explicitly scale dependent have already been introduced in physics from algebraic methods, mainly by the development of the renormalization group in Wilson's many scales of length approach. The connection of both tools is revealed by the remark that the standard measures on fractals (based on their topological dimension, such as length, area, volume...) are solutions of renormalization group-like equations.
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Quoted from: Nottale L., 1993, "Fractal Space-Time and Microphysics: Toward of Theory of Scale Relativity", Chap. 1 (© World Scientific, 1993).