Mapping Effective Field Theory to Multifractal Geometry

Fractals and multifractals are well-known trademarks of nonlinear dynamics and classical chaos. The goal of this work is to tentatively uncover the unforeseen path from multifractals and selfsimilarity to the framework of effective field theory (EFT). An intriguing finding is that the partition function of multifractal geometry includes a signature analogous to that of gravitational interaction. Our results also suggest that multifractal geometry may offer insights into the non-renormalizable interactions presumed to develop beyond the Standard Model scale.


Introduction
There is a vast panorama of objects and processes in Nature that exhibit self-similarity, either as "shape invariance" under scaling operations or invariance under scaling of variables defining a system. Examples include sets of fractional dimensions (fractals and multifractals), Levy flights and random walks, fluid turbulence, the geometry of quantum mechanical paths, anomalous diffusion, non-differentiable functions and fractional operators, percolation and crystal growth, self-organized criticality and so on. As defining property of nonlinear dynamics, self-similarity has emerged as common denominator of 2 | P a g e many theoretical frameworks, from the mathematics of chaos and complexity to critical behavior and the Renormalization Group approach to Quantum Field Theory (QFT) [8].
The concepts of continuous dimension and entropy play a pivotal role in the analysis of selfsimilarity. Unlike simple fractals, multifractals are selfsimilar structures endowed with multiple dimensions and are naturally fit to describe the long-run evolution of chaotic phenomena. In particular,  Nonlinear dynamical systems and iterated maps can generate multifractals by fragmentation of the phase space. The crux of this observation is that both statistical behavior and fragmentation of the phase-space follow from strictly deterministic equations of motion, with no apriori assumptions about randomness and probability distributions [4].
 The long-term chaotic orbits of many dynamical systems are confined to invariant sets called strange attractors, whose characterization requires the language of multifractal geometry.
 Fully developed chaos is, in fact, a faithful replica of equilibrium statistical mechanics. It can be shown that dynamics on a strange attractor displays thermodynamic-like behavior consistent with ergodicity, fluctuation-dissipation theorem, and invariant probability distributions [1-2, 4-5, 8].
The goal of this work is to tentatively uncover the step-by-step route from multifractals and selfsimilarity to effective field theory (EFT). The reader is cautioned upfront that our approach is far from being either fully rigorous and/or formally complete and that many relevant details are left out for concision and clarity. In a nutshell, our sole motivation is 3 | P a g e to initiate a new research avenue and to lay the groundwork for subsequent modeling efforts.
The paper is organized in the following way: next couple of sections present the working assumptions of the paper and a brief pedagogical introduction to nonlinear maps and Lyapunov stability. Section 4 elaborates on the correspondence between the partition function of multifractal geometry and its counterpart of classical Thermodynamics.
Drawing from the link between Lyapunov exponents and the Gaussian curvature of geodesic trajectories, section 5 argues that the partition function of multifractal geometry includes a signature analogous to that of gravitational interaction. Section 6 suggests that multifractal geometry may offer insights into the hypothetical non-renormalizable interactions beyond the Standard Model (SM) scale. Concluding remarks are outlined in the last section.
As pointed out earlier, further analysis and independent evaluation are needed to refute, confirm, or develop these lines of reasoning and determine their long-term viability.

A1)
We confine the discussion to low-dimensional nonlinear systems exhibiting dissipative behavior. Typical examples of such systems include non-invertible onedimensional maps and non-conservative two-dimensional maps.
The rationale for choosing low-dimensional systems as starting point echoes the center manifold theory, where a multivariable system of differential equations is shown to reduce in the long run to a lower dimensional system of universal equations dependent on a single emerging variable [19][20]. On the same note, we recall that the 4 | P a g e dissipative behavior of many nonlinear systems falls in line with non-equilibrium dynamics of high energy observation scales [4][5]17].

A2)
Section 3) focuses exclusively on Lyapunov stability applied to trajectories having a limited extent in spacetime or phase space. A3) Section 5) focuses exclusively on weak and slowly varying gravitational fields, as typically described in introductory textbook on General Relativity.

Nonlinear maps and Lyapunov exponents
A hallmark feature of chaotic dynamics is sensitivity to initial conditions, which leads to the exponential instability of nearby phase-space trajectories. The separation between adjacent trajectories grows exponentially in time according to for 0   . To fix ideas, consider a one-dimensional nonlinear system whose time evolution is described by the iterated map where the iterates On the other hand, by (5), one can write Comparative inspection of (6)-(8) in the limit 1 n  yields a consolidated expression of the maximal Lyapunov exponent in the form Similar arguments apply to the Lyapunov exponents of higher dimensional maps.
Consider, for example, a two-dimensional map given by It can be shown that an area element n a  of the phase space defined by

Multifractal geometry as analog of classical Thermodynamics
Although the thermodynamic formalism of multifractal structures is not new, we briefly introduce the topic here to make the paper self-contained and accessible to a large audience.
A remarkable property of non-invertible maps of the type (2) is that, when 1 n  , consecutive iterations [ , ] zz [1,4]. In geometric terms, it is customary to refer to this partition as a multifractal set. In the context of multifractal sets, a key concept is the generating function defined as [4,6] () subject to the normalization condition where q D labels the so-called Rényi entropy (or generalized dimension) of multifractal geometry generated by (12).
For an even distribution of iterates ( i pp  ) and 1 n  , the generating function (12) translates to the condition (16) below, namely [4] A closer look at (16) suggests that the even distribution of iterates in the long-time limit The so-called backward map iteration starts from the whole phase space min max [ , ] zzand partitions it in a distribution of disjoint intervals, a process that is formally equivalent to 9 | P a g e coarse graining 0f the phase space [4]. It is intuitively clear that coarse graining by backward iteration mirrors the error propagation equation (18), which describes the progressive growth of separation between nearby trajectories. As a result, setting the maximal propagation error to unity, and performing the identification [4] () n ii zr   (20) leads to Accordingly, (23) turns into It is apparent that (26) generalizes the canonical partition function of Thermodynamics, by including in its expression the Lyapunov exponents  and the number of map iterations n . A glance at (1) and (7) shows that a vanishing  signals conservative dynamics, whereby the trajectory error

Non-Euclidean metric and multifractal geometry
The goal of next two sections is to bridge the gap between multifractal geometry and the framework of effective field theory (EFT), with emphasis on General Relativity and the Standard Model of particle physics (SM).
Let us first recall that the inherent sensitivity of geodesics to initial conditions connects their Lyapunov exponents ( i  ) to the local Gaussian curvature ( K ) via [3,6]  where the integral is taken over the phase space  , whose differential measure is d  .
Taken together, (29), (31a) and (31b) hint that there is an intriguing relationship between the partition function of multifractal geometry and the non-Euclidean metric of General Relativity. In particular, the Lyapunov exponents entering (29) describe effects analogous to those induced by gravitation, from which they decouple in the corresponding limit of flat spacetime, where 0 K  together with 0 An important point is now in order. It is known that all the points on a trajectory are described by the same Lyapunov exponent. This may raise a challenge to the validity of (31a), since K is a local and not a global attribute. Using assumptions A2) and A3), one way out of this challenge is to replace K in (31a) with the average curvature across the trajectory span as in in which L  denotes the arclength of the trajectory.
By (6) and (7), the only setting consistent with 0 which means a stationary deviation from initial conditions. It follows from these considerations that conditions akin to Euclidean spacetime and equilibrium Thermodynamics are recovered in the asymptotic regime defined by 0 These results can be symbolically summarized as follows Appealing to (11), one finds that (33) matches the concept of non-conservative maps   echoes the settings of both Thermodynamics and field theory in Euclidean spacetime.

Given the known analogy between Thermodynamics and Euclidean Quantum Field
Theory (QFT) [11], we believe that these findings point to a hypothetical connection between QFT and General Relativity and a possible path towards unification based upon multifractal geometry.

Multifractals and physics beyond the Standard Model
Moving onto the EFT, we recall the expression of the effective field Lagrangian in d spacetime dimensions [12]