Analysis of Vacuum Energy as a Simple Fractal

**Abstract**

In this letter, we analyze the vacuum energy as a simple fractal. With elementary mathematics and a novel approach, we study its properties.

Keywords: Vacuum energy, compact dimensions, relative fractal dimension, transition of dimensions, hypothetical quantum generalization

Importante

Una línea, aunque tiene dimensión topológica 1, es capaz de cubrir un espacio de 3 dimensiones: su dimensión fractal será de 3. De la misma forma, considerando la energía del vacío (dimensión topológica 3) como un fractal obtenemos un valor de 9 para su dimensión fractal. Según este resultado la energía del vacío sería capaz de cubrir un espacio de 9 dimensiones.

Otro resultado importante  es que la propia naturaleza del cuanto de acción queda definida por la especial geometría entre dimensiones ordinarias y compactadas.

Important

A line, although it has a topological dimension of 1, is capable of covering a 3-dimensional space: its fractal dimension will be 3. Similarly, considering vacuum energy (topological dimension 3) as a fractal, we obtain a value of 9 for its fractal dimension. According to this result, vacuum energy would be capable of covering a 9-dimensional space.

Another important result is that the very nature of the quantum of action is defined by the special geometry between ordinary and compactified dimensions.

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**1 Introduction**

The existence of Planck's quantum of action turns Newton's classical and deterministic universe into a quantum universe, with Heisenberg's uncertainty principle. The vacuum is filled with zero-point energy (ZPE), which increases as the distance considered decreases. The minimum length considered, called Planck's length (lp), is associated with a maximum energy called Planck's energy (Ep). For a distance n (lp), the associated energy is (Ep)/n, where "n" is a natural number. This property, conserved at all known scales, will help us analyze this fractal.

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**2 Fractal dimension, study of Brownian motion, and the Koch snowflake**

The fractal dimension is made up of two components: the topological dimension and a dimensional coefficient (topol_dim + dimens_coef.). The more irregular the fractal, the higher the dimensional coefficient. For our study, it is interesting to analyze simple fractals such as the fractal path of Brownian motion, which has a topological dimension of 1.

Brownian motion (britannica.com, December 23, 2021), also called Brownian movement, is any of various physical phenomena in which some quantity constantly undergoes small, random fluctuations. It was named after the Scottish botanist Robert Brown, the first to study such fluctuations (1827).

For a particle moving with Brownian motion to move away N effective steps, it must take N² total steps. The N effective steps are considered in a straight line, in one dimension. The N² steps occur in a space of two or more dimensions. The relation log(N²) / log(N) = 2 gives us the value of its fractal dimension (a basic property of fractal lines) [1]. The topological dimension is 1 and the dimensional coefficient is also 1. The value 2 of the fractal dimension indicates that a linear movement, of topological dimension 1, can fill a plane, of topological dimension 2.

In Brownian motion, and in general, fractal value = N² = distance^fractal_dimension.

This can also be observed in the Koch curve, as shown in Figure 1. In the first iteration, the side that measures 3 segments becomes 4 segments. The fractal dimension is log 4 / log 3 = 1.26186. In one dimension, 3 segments become 4 segments in two dimensions (the plane): 4 = 3^1.26186 (Mandelbrot, 1987).

FIG.1

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**3 Fractal dimension of vacuum energy**

We know the dependence of vacuum energy on distance: En = Ep / n = (Ep)(distance-1). We assume that we live in hyperspace (string theory), and we know the dependence of vacuum energy on distance. Let En-1 be the value of the energy in hyperspace, then:

Log (En-1) / log (En) = -1. This implies that vacuum energy is proportional to distance in hyperspace. Although the energy has no topological dimension of 1, the quotient of the two logarithms behaves the same as in the case of Brownian motion. When comparing two energies, the topological dimension no longer matters because the result is a relative fractal dimension:

Relative fractal dimension = (topol_dim. + dimens_coef.)/(topol_dim.). To simplify, we will write:

Relat_fr_dim. = (δ+ε)/δ (1).

So, we have: Relat_fr_dim. = Log (En-1) / log (En) = -1 = (δ+ε)/δ.

The value -1 reminds us of the compacted dimensions of string theory, since while a positive dimensional coefficient indicates that the fractal occupies a space greater than its topological dimension, a negative dimensional coefficient indicates dimension compaction (Ruiz-Fargueta, 2004). The situation indicates a transition of dimensions such that: T: δ → δ-ε.

The expression (1), with this transition, becomes: δ/(δ-ε) (2).

If the dimensional coefficient is the same as the number of compact dimensions.

Expression (2) is consistent with the value -1, since for d = 3 it gives us the value -6 for the number of compact dimensions, which coincides with the value predicted by string theory. Applying these values to expression (1):

(δ+ε)/δ = (3+6)/3 = 3. 3 is the relative fractal dimension of the vacuum energy, 9 is its true fractal dimension.

The same result is found in the following equivalent transformations:

T11: 1/n → n } log(n)/log(1/n) = -1. Apparent result in relative fractal dimension.

T12: n → n³ } log(n³)/log(n) = 3. True result in relative fractal dimension.

The T11 transformation gives us the apparent result -1. But the transformation T12 gives us the true result 3.

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**4 Generalization and possible transition of dimensions**

The value -1 is the result of En, as a function of distance, in the expression (En)(n) < Constant, where we have replaced the time (energy-time uncertainty principle) with the space (n) traveled by the light in that time. If in this expression we add a fictitious coefficient f, we will have:

• (En)(nf) < Constant (3) (Hypothetical quantum generalization)

Now the transformations T11 and T12 will be:

T11: 1/nf → n }

T12: n → n²+f }

The true generalized result of the relative fractal dimension is log(n²+f)/log(n) = 2+f, with the expression (1): (δ+ε)/δ = 2+f (4).

During the transition of dimensions, the value of the fictitious coefficient f, associated with the very nature of the quantum (hypothetically), was defined. We will analyze the transition of dimensions by combining expressions (3) and (4), for ε=9-δ.

(En)(n^(ε-δ)/δ) < Constant, multiplying and dividing by nδ which is the generalized volume to ordinary dimensions δ:

(Energy_density)(nφ) < Constant. The value of φ = (δ²-2δ+9)/δ, and is represented in Figure 2.

FIG.2 

For δ = 3, there is a minimum that corresponds to a maximum in energy density.

For δ = 0, the value is infinite and corresponds to a minimum density equal to zero. The transition of dimensions from δ = 0, ordinary dimensions, to δ = 3, ordinary dimensions, takes us from a vacuum energy equal to zero to a maximum value. "In particular, our laws of physics arise from the geometry of the extra dimensions. Understanding this geometry ties string theory to some of the most interesting questions in modern mathematics, and has shed new light on them, such as mirror symmetry" (Polchinski, 2015).

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**5 Conclusion**

Possibly, there was a transition of dimensions that maximized the energy density of the vacuum for δ=3 and ε=6 (δ= ordinary dimensions, ε= compact dimensions). The nature of the quantum of action is tied to these values of δ and ε.

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**Acknowledgments**

My thanks to the great popularizers of science: George Gamow (Biography of Physics), Richard Feynman (The Feynman Lectures on Physics), Benoit Mandelbrot (Fractal Objects), Isaac Asimov (The Universe), Ken Kilber, David Bohm (The Holographic Paradigm), Roger Penrose (The Emperor's New Mind), Kip S. Thorne (Black Holes and Time Warps), Stephen Hawking (Black Holes), Leonard Susskind (The Cosmic Landscape), Brian Greene (The Elegant Universe), Steven Weinberg (The First Three Minutes), Ilya Prigogine (Only an Illusion), Michio Kaku (Hyperspace), Joseph Polchinski (String Theory)...

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**References**

Mandelbrot, B. (1987), Los objetos fractales, Barcelona, Tusquets Editores.

Polchinski, J. (2015), String theory to the rescue. ArXiv: 1512.02477 v5 [hep-th]

Ruiz-Fargueta, J.S. (2004) El sorprendente vacío cuántico. Revista Elementos, Universidad de Puebla BUAP.MX, 53, pp.52-53. (16/01/2022) https://elementos.buap.mx/directus/storage/uploads/00000002608.pdf

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