Relativity of Scales

We have considered, in the previous Section, the various transformations of the coordinate systems corresponding to changing the origin and axes. The subsequent state of coordinate systems which may be submitted to a transformation are units. Some attempts at including such a transformation in physical laws have been made, in particular in the framework of the conformal group.^42,43 Conformal transformations include, in addition to the Poincaré ones, dilatations and special conformal transformations; both of which may be interpreted as related to changes of units. However, while electromagnetic waves are subjected to the conformal symmetry, this is not the case for matter, so that the conformal symmetry cannot be an exact symmetry of nature. Moreover the choice of the unit is in most situations a purely arbitrary one, which does not describe the conditions of measurement, but only their translation into a number.

Let us nevertheless analyse further the physical meaning of units. Their introduction for measuring lengths and times is made necessary by the relativity of every scales in Nature. When we say that we measure a length, what we actually do is to measure the ratio of the lengths of two bodies. In the same way as the absolute velocity of a body has no physical meaning, but only the relative velocity of one body with respect to another, as demonstrated by Galileo, the length of a body or the periods of a clock has no physical meaning, but only the ratio of the lengths of two bodies and the ratio of the periods of two clocks.

When we say that a body has a length of 132 cm, we mean that a second body, to which we have arbitrarily attributed a length of 1 cm and which we call the unit, must be dilated 132 times in order to obtain the first body's length. Measurements of length and time intervals always amount, in the end, to dilatations. The tendency for physics to define a unique system of units was certainly a good thing, since this was necessary for a rational comparison of measurement results from different laboratories and countries. However, this means that the length of all bodies are referred to a same unique body and the period of all clocks to a same clock, this giving a false impression of absoluteness: such a method masks the actual relation between lengths of all bodies in Nature, which is a two-by-two relation.

The fact which allows us to use a unique unit is the simple law of composition of dilatation, r"=rr'. This law is certainly extremely well verified in the classical domain: there is no doubt that a body having a length of 21 m also measures 2100 cm. We however claim that we know nothing about the actual law of dilatation in the two domains of quantum microphysics and cosmology, in which explicit measurements of length and time become impossible. In these two domains length and time intervals are deduced from observation of other variables (energy-momentum at small scale, apparent luminosity and diameter at large scale) and from underlying accepted theories (quantum mechanics and general relativity) which have been constructed assuming implicitly the hereabove standard law of dilatation. (Compare with the status of velocities before the coming of special relativity, when it also seemed self-evident that their law of composition was w = u + v, see [57] Chapter 6 and in Sec. 7.1 for cosmology).

The status of resolutions is related to that of units (in particular hey are subject to the relativity of scales), but is actually different ad of more far reaching physical importance. Changing the resolution of measurement corresponds to an explicit change of the experimental conditions. Measuring a length with a resolution of 1/10^th mm implies the use of a magnifying glass; with 10 mm, we need a microscope; with 0.1 mm, an electron microscope; with 1Å, a tunnel microscope. For even smaller resolutions, the measurements of length become indirect, since the atom sizes are reached and exceeded. When we enter the quantum domain, i.e., for resolutions smaller than the de Broglie length and time of a system (as will be specified afterwards), the physical status of resolutions radically changes. While classically it may be interpreted as precision of measurements (measuring with two different resolutions yields the same result with different precisions), resolution plays a completely different role in microphysics: the results of measurements explicitly depend on the resolution of the apparatus, as indicated by Heisenberg's relations. This is the reason why we think that the introduction of resolution into the description of coordinate systems (as a state of scale) is not trivial, but will instead lead to a genuine theory of scale relativity and the emergence of new physical laws (see [57] Chapter 6).

In present quantum mechanics, the scale dependence is already implicitly present. However it is explicitly present neither in the axioms nor in the basic equations. It comes from the interpretation of these equations thanks to a theory of measurements. Specifically, one writes Schrödinger's equation, then solve it. This yields a probability amplitude from which one deduces the probability density; then one may compute the dispersion of the variable considered, and Born's statistical interpretation of quantum mechanics ensures that this will give us the standard error of a statistical ensemble of values resulting from several measurements of this variable. By extension, this dispersion may also be interpreted as the resolution of the measurements: e.g., if one makes position measurements with a resolution Dx, one expects a subsequent dispersion in the values of the momentum given by Heisenberg's relation s_p ~ h/Dx.

However one may require that a complete physical theory includes in its equations the whole set of physical information yielded by experiment. In other words, it may be demanded that a theory of measurement, instead of being externally added to a given theory, becomes an integral part of it.

Such a requirement of explicit expression of the measurement resolutions in the equations of physics begins to be fulfilled in present physics, even though the interpretation is different from ours. Let us quote two approaches where scale-dependent equations are actually written.

One of these domains is that of the theory of measurement in quantum mechanics, concerning in particular the problem of the so-called reduction of the wave-packet (i.e., sudden collapse of the state vector caused by a measurement). This problem, which underlies that of the quantum-classical dualism (where is the transition from quantum to classical; are classical laws approximations of the quantum ones...?), has recently known a resurgence of interest (see Refs. 44 and 45 and references therein). The basic idea of these works is that reduction of the wave packets originates from an interaction of a quantum system with the environment. This interaction is described by a master equation which is explicitly resolution-dependent, so that the transition from quantum to classical behaviour is found to depend directly on resolution in terms of a decoherence time scale⁴⁴

t_D ~ (l_T/Dx)²,

where l_T is the thermal de Broglie length of the system (more on this approach in [57] Sec. 5.7).

A second domain where explicitly scale-dependent equations have been introduced is the renormalization group approach.^46-48 First introduced in quantum electrodynamics as the group of transformation between the various ways to renormalize the theoretical divergences, it became under Wilson's influence a general method of description of problems involving multiple scales of length.^48-50 In the renormalization group approach, one writes differential equations describing the infinitesimal variation of physical quantities (fields, couplings) under an infinitesimal variation of scale. The renormalization group will play a leading part in our approach. We shall indeed demonstrate that the renormalization group equations (i) can be interpreted as the simplest lowest order differential equations describing the measure on fractal geometry; (ii) are for scale laws the equivalent of Galileo's group for motion laws. As a consequence we shall propose (in [57] Chapter 6) a generalization of its structure aimed at making it consistent with the principle of scale relativity, which is stated above.

Our first proposal for implementing the idea of scale relativity was to extend the notion of reference system by defining "supersystems" of coordinates which contain not only the usual coordinates but also spatio-temporal resolutions, i.e., (t,x,y,z;Dt,Dx,Dy,Dz).³³ The axes of such a reference supersystem would be endowed with a thickness: this corresponds, indeed, to actual measurements. Then we proposed an extension of the principle of relativity, according to which the laws of nature should apply to any coordinate supersystem. In other words, not only general (motion) covariance is needed, but also scale covariance.

Consider now the fundamental behaviour of the quantum world in the light of these ideas. Recall our assumption that when the physicist finds universal properties for physical objects, these properties may be attributed to the nature of space-time itself. This analysis applies particularly well to some of the quantum properties, accounting for their universal character: de Broglie's⁵¹ and Heisenberg's⁵² relations.

Let us recall indeed that the wave-particle duality is postulated to apply to any physical system, and that the Heisenberg relations are consequences of the basic formalism of quantum mechanics (see On the Present State of Fundamental Physics). The existence of a minimal value for the product Dx.Dp is a universal law of nature. Such a law, in spite of its universality, is considered in the current quantum theory as a property of the quantum objects themselves (it becomes a property of the measurement process because measurement apparatus are in part quantum and precisely because it is universal). But it is remarkable that it may be established without any hint to any particular effective measurement (recall that it arises from the requirement that the momentum and position wave functions are reciprocal Fourier transforms). So we shall assume that the dependence of physics on resolution pre-exists any measurement and that actual measurements do nothing but reveal to us this universal property of nature: then a natural achievement of the principle of scale relativity is to attribute this universal property of scale dependence to space-time itself.

By such a route, we finally arrive at the same conclusion as that at the end of the previous Section, but this conclusion is now reached by basing ourselves on the principle of scale relativity rather than on the extension of the principle of motion relativity to non-differentiable motion. Namely, the quantum space-time is scale-divergent, according to Heisenberg's relations, i.e., by our adopted definition (see The Need for a New Extension of Principle of Relativity and [57] Chapter 3), fractal. This idea are more fully developed in [57] Chapters 4 and 5.

However this first formulation of the principle of scale relativity³³ is not fully satisfactory. It does not incorporate the complete analysis of the relativity of scales (see above), and treats resolutions on equal footing with space-time variables. However, we have shown that resolutions are more accurately described as a relative state of scale of the coordinate system, in the same way as velocity describes its state of motion. So, in parallel with Einstein's formulation of the principle of motion relativity, we shall finally set the principle of scale relativity in the form⁵³

"The laws of physics must apply to coordinate systems whatever their state of scale."

The full principle of relativity will then require validity of the laws of physics in any coordinate system, whatever its state of motion and of scale. This is completed by a principle of scale covariance (in addition to motion covariance):

"The equations of physics keep the same form (are covariant) under any transformation of scale (i.e. contractions and dilatations)."

We can see in [57] Chapter 6 that in this form the principles of scale relativity and scale covariance imply a profound modification of the structure of space-time at very small scales: we find that there appears a universal, unpassable, limiting scale in Nature, which is invariant under dilatation, and plays for scale laws a role quite similar to that played by the velocity of light for motion laws (i.e., the limiting scale is neither a cut-off, nor a quantization, nor a discontinuity of space-time). Such a modification has observable consequences at presently accessible energies, which may be expressed in terms of 'scale-relativistic' corrections.
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Quoted from: Nottale L., 1993, "Fractal Space-Time and Microphysics: Toward of Theory of Scale Relativity", Chap. 2 (© World Scientific, 1993).