On the Role of Self-similarity in the Dynamics of Effective Field Theory

The relevance of continuous symmetries in theoretical physics can hardly be overstated. Conservation laws form the backbone of both quantum and relativistic phenomenology and are organically tied to Lorentz symmetry, gauge groups and the diffeomorphism invariance of General Relativity. A set of observables 1 2 { , ,..., } N q q q q  associated with a system may be naturally represented as coordinates of an N dimensional vector q in q  space. The general principle of invariance demands that the laws of physics must be independent of any origin or orientation of q . Moreover, the general principle of covariance demands form Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 1 December 2020 doi:10.20944/preprints202012.0012.v1


Introduction
The relevance of continuous symmetries in theoretical physics can hardly be overstated.
Conservation laws form the backbone of both quantum and relativistic phenomenology and are organically tied to Lorentz symmetry, gauge groups and the diffeomorphism invariance of General Relativity.
A set of observables 12 { , ,..., } N q q q q  associated with a system may be naturally represented as coordinates of an N  dimensional vector q in q  space. The general principle of invariance demands that the laws of physics must be independent of any origin or orientation of q . Moreover, the general principle of covariance demands form independence of the laws of physics from any particular choice of coordinates in q  space.
The goal of this brief report is to show that fundamental symmetries of effective field theories emerge from self-similarity, the latter being inspired by the geometry of fractal and multifractal sets. As we shall see below, in the context of theoretical physics, selfsimilarity reflects invariance of the action under the most general scale transformation related to dilations, rotations, and translations [Appendix A, 1, 14].

Noether theorem from scale transformations
Consider a classical field theory with fields i  and action An infinitesimal transformation of coordinates and fields can be formulated in terms of a set of 1, 2,3,..., lN  infinitesimal parameters l  as in [2] ' ( ) In what follows, () x  and () x  are referred to as scaling operators of magnitude substantially close to one In keeping with the standard terminology, the scaling operator is deemed local if it is coordinate-dependent and global otherwise. On account of (4), the infinitesimal Noether transformations are equivalent to local scaling operations that have no effect on the action. This insight suggests that invariance under properly defined scale transformations, and in particular self-similarity, must lie behind the fundamental symmetries of effective field theories.

Self-similarity as underlying source of invariance
Consider again the set of observables 12 { , ,..., } N q q q q  . Self-similarity in field theory and statistical physics enters through the generalized homogeneous function [3] 12 Some textbooks display (6) in an alternative form, using the substitution which turns (6) into the condition As detailed in [4][5], self-similarity has several applications in theoretical physics.
Noteworthy examples include the behavior of classical oscillators, Newtonian gravitation, the virial theorem, Renormalization Group theory, random walk models, correlations and partition functions of Statistical Physics, and local scale invariance in field theory [6].
Equally important is the link between self-similarity and self-organized criticality (SOC), the latter being a universal paradigm for the onset of complex dynamics in systems outside equilibrium. On the same note, we mention here that surprising connections between SOC and several unsettled topics of contemporary theoretical physics have been brought up in [7][8][9][10][11]16].
To bridge the gap between self-similarity and SOC, one starts from the generalized homogeneous function of two variables and show that (10) satisfies the power-law scaling [3] A generic paradigm for SOC is the sandpile model, whose "avalanche"-size probability distribution is given by in which s is the avalanche size, L the system size and , s D  denote the avalanche-size exponent and its dimension, respectively. Term by term identification of (11) and (13)- The key observation of this report is that the scaling operator  of (5) can be naturally interpreted as a phase angle. Appealing to the concept of analytic continuation in Euclidean field theory [12][13], leads to in which E  is the Euclidean phase. Since any phase angle is the product of an angular

Self-similarity and the Lorentz group
The Lorentz group represents the group of linear coordinate transformations that leave invariant the relativistic interval Here,   denotes the six independent elements of the antisymmetric matrix entering (21) and J  are the group generators. A similar line of reasoning is outlined in [14], where Lorentz transformations are regarded as similitudes.

Self-similarity and gauge theory
The QED Lagrangian written in the Feynman gauge is given by [2] Demanding that the QED Lagrangian conforms to local (1) U gauge symmetry amounts to requiring that (24) stays unchanged under It is known that imposing local gauge invariance leads to the introduction of the covariant derivative operator 8 | P a g e

Self-similarity and General Relativity
The gauge field concept brought up in the last section may be built from a straightforward geometric interpretation [13]. Consider the parallel transport of a complex vector  It is seen that the effect of parallel transport is to induce a non-vanishing rotation of  in internal space proportional to the strength of the gauge field. Likewise, the curvature tensor of General Relativity (GR) may be motivated through similar arguments. Taking a vector V  on a round trip by parallel transport, the difference between the initial and final components of the vector amounts to 9 | P a g e It is apparent that a self-similar transformation is a composition of a dilation, a rotation and a translation of the types encoded in the Lorentz and gauge groups of effective field theory.

APPENDIX B
The