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2.5.12 Presumption of the Whole Inclusive Theory of Everything 1

2.5.12.1 Overview

The above mentioned are major theories of present-day physics. However, major theories don't seem reached the final solution.
Then own theories begin from here to the partial fractal structure, the final solution. The introduction here is Causal Dynamical Triangulation (CDT). Since CDT wouldn't be a major theory yet, CDT is placed here as an own (minor) theory.

2.5.12.2 Details

2.5.12.2.1 Loll, Ambjorn, and Jurkiewicz's Causal Dynamical Triangulation in 2005 CE

2.5.12.2.1.1 Background

Loop Quantum Gravity Theory was presented keeping strict Symmetry. However, it seemed stuck because of its strict Symmetry.

2.5.12.2.1.2 Causal Dynamical Triangulation

Loll, Ambjorn, and Jurkiewicz presented Causal Dyanamical Triangulation Theory loosening Symmetry of Loop Quantum Gravity.
Causal Dynamical Triangulation (CDT) is like a play building LEGO blocks complying with already-known physical laws. Physicists (possibly tentatively be called CDTists) virtually build blocks complying with already-known physical laws to create models of universes in Minkowski spacetime as depicted before ("Expansion of the Observable Universe") (or models of a part of universes, a part of spacetime) employing computers (including simple desktop computers).
(The depiction of "Expansion of the Observable Universe" consists of spatial 1-dimension and temporal 1-dimension. On the other hand, CDTists would naturally aim at models of Whole Universe including unobservable universe in spatial 3-dimensions and temporary 1-dimension, unobservable universe would be unclear though.)
CDTists' blocks are analogues of triangles of various dimensions, Simplexes. There are various types of Simplexes depending on numbers of dimensions.
For example, triangles are 2-simplexes (2-dimensional simplexes). Tetrahedrons are 3-simplexes (3-dimensional simplexes). 4-simplexes (4-dimensional simplexes) are called "Pentachorons," "5-Cells," and so on.

CDTists build these simplex blocks in computers to create (simulate) spacetime models of universes (or a part of universes). CDTists set conditions of these simplex blocks and relevant programs in computers and wait for the created models as results of computer calculation.
CDTists mostly use 4-simplexes (pentachorons) because the actual universe or spacetime and already-known physical laws consists of 3 spatial dimensions and 1 temporary dimension, consequently 4-dimensions.
However, as a simple explanation, the way of building blocks is explained employing triangles for now.
* "Causal Dynamical Triangulation in Wikipedia" http://en.wikipedia.org/wiki/Causal_dynamical_triangulation
* "Simplex in Wikipedia" http://en.wikipedia.org/wiki/Simplex
* "Scientific American Self Organizing Quantum Universe" http://www.scientificamerican.com/article.cfm?id=the-self-organizing-quantum-universe
* "The Emergence of Spacetime or Quantum Gravity on Your Desktop arvix" http://arxiv.org/abs/0711.0273
* "Reconstructing the Universe arvix" http://arxiv.org/abs/hep-th/0505154

2.5.12.2.1.3 Modified Triangles

CDTists presume modified Triangles like below.
First, Minkowski Spacetime is presumed. Then horizontal direction corresponds to spatial location and the upward direction corresponds to time passage.
Then secondly, horizontal edge lines of the triangles (here in blue) are designated as "Spatial-Like Edge (Line)." Thirdly, slanting edge lines of the triangles are designated as "Time-Like Edge (Line)."
Fourthly, an upward arrow (here in green) is designated for each triangle. The arrow represents Causality. Since things (including physical states) in the world change in accordance with causality along with time passage, Causality associated with time passage was introduced.
Fifth, upside-down triangles with upward causality arrows are presumed as well.
Sixth, the length of a "Spatial-Like Edge (Line)" would be like the Planck length (1.616*10-35 m). Seventh, the height of a triangle would be like the Planck time (5.39*10-44 sec). Eighth, triangles should naturally be connected at the corners (vertexes) each other.
*"Spatial-Like Edge (Line)" of twice Planck length (2*1.616*10-35 m) might be preferable, since in this case the slant of the triangle correspond to the speed of light. (Speed of light = Planck length / Planck time) However, the presumption here is just "like the Planck length" for now in accordance with traditional CDT.
*The triangles would be originally equilateral triangles or isosceles triangles. However, if Gravity presents (if Gravitons work), Triangles and Causality Arrows would be warped.
* "Planck Length in Wikipedia" http://en.wikipedia.org/wiki/Planck_length
* "Planck Time in Wikipedia" http://en.wikipedia.org/wiki/Planck_time

Such conditions and relevant programs were set in computers. Then computers started to combine the triangles aiming at models of Spacetime.
An image of a result would be as follows. A layered area between blue lines is called "time slice."

*Just for reference, the figure above is a modification of the following figure shown again, formerly shown in reference to the explanation of the General Theory of Relativity. The color is replaced with pink representing particles (or strings or waves) associated with the Uncertainty Principle, since issues here are microscopic questions.


2.5.12.2.1.4 Pentachorons

Pentachorons should be employed in actual 4-dimensional simulations.
Pentachorons show various features in 3-dimension. "Flat 4 Simplex Type (4, 1)" and "Flat 4 Simplex Type (3, 2)" were examplified.

Then an image of connected pentachorons between horizons might be as follows.


2.5.12.2.1.5 Results of Simulations and Dimension

2.5.12.2.1.5.1 Introduction to Dimension

CDTists claim computers dynamically simulated proper spacetime models like actual spacetime of the universe by introducing conditions (rules) of arrows representing Causality (or Time Passage).
CDTists analized the models and calculated dimension of the models.
There would be 2 kinds of definitions relating to mathematical dimension, Hausdorff Dimension and Spectral Dimension.

Hausdorff Dimension
Hausdorff Dimension is defined as "log multiplying factor of edges multuplying factor of contents." For example, supposing a cube with an edge length of "1", the quantity of the content (volume) would be "1". Then supposing a cube with an edge length of "2", the quantity of the content (volume) would be "8". In this case, log 2 8 = 3 (log multiplying factor of the edges multiplying factor of the contents) (23=8), then the Hausdorff Dimension of cubes is 3.

For example, supposing a square with an edge length of "1", the quantity of the content (area) would be "1". Then supposing a square with an edge length of "2", the quantity of the content (area) would be "4". In this case, log 2 4 = 2 (22=4), then the Hausdorff Dimension of squares is 2.

Other than that, the concept of Hausdorff Dimension could be applied in other ways.
For example, supposing triangles below and extending the 2 edge length from the extreme lower left triangle, triangles could be counted as the content depending on the numbers corresponding to the extended edge length. If the edge length is 3, 5 triangles are counted. If the edge length is 6, 14 triangles are counted. In this case, the Hausdorff Dimension would be "log 6/3 14/5."


Spectral Dimension
On the other hand, Spectral Dimension is defined as "log multiplying factor of diffusion times multiplying factor of the contents." For example, starting from the extreme lower left No.1 triangle in the following figure, when the diffusion time is 1, the content is one No.1 triangle. When the diffusion time is 2, neighbor two No.2 triangles are added and the content is three triangles. When the diffusion time is 3, neighbor two No.3 triangles are added and the content is five triangles. When the diffusion time is 4, neighbor two No.4 triangles are added and the content is seven triangles. When the diffusion time is 6, the content is 13 triangles. Thus adding neighbor elements one by one, the number of times added the neighbor things is "diffusion time." Since it is like diffusing something, it is called diffusion times. In this case, the Spectral Dimension is for example " log 6/3 13/5 ."
Thus the Spectral Dimension is quite similar to the Hausdorff Dimension associated with the traditional concept of Dimension.


2.5.12.2.1.5.2 Dimension Reduction

CDTists analized Dimension (Spectral Dimension) of the simulated spacetime. The dimension was mostly some 4.0.
*Since 4-dimensional blocks (4-Simplexes) were mostly employed, the result (mostly 4-dimensional) makes sense.
However, a strange result was seen in low diffusion times. The dimension reduced in low diffusion times. It was illustrated as follows. (Ds means Spectral Dimension)


*The dimension reduction on small scales would suggest derivation from the pointy corners of Exotic Differentiable Structure of 4-Dimensional Euclidean (conceptual) space mentioned before.

CDTists interpreted it that structure of spacetime changes into fractal structure on small scales.
Fractal structure is for example like below. (A small triangle is presumed consisting of 3 smaller triangles.)


*Dimension Reduction due to Fractal structure on small scales would be accounted for as follows.
The definition of "fractal" is controversial, while the definition of "Fractal" here is "having the same pattern regardless of integral multiple enlargement or reduction on scales."
* "Fracral in Wikipedia" http://en.wikipedia.org/wiki/Fractal
The following is a typical Fractal similarly to the above example. It shows the same pattern regardless of multiple enlargement or reduction on scales as far as the multiple is an integer (integral multiple).
(In the following cases, the length is mostly doubled.)
(The small green triangles are presumed consisting of further smaller triangles.)

In contrast, the following is an example of continuous homogeneous state. Even if the small green triangles are presumed consisting of further smaller triangles, it wouldn't show the same pattern through integral multiple enlargement or reduction. Then it is not Fractal.
*Superstring Theory was based on continuous homogeneous Spacetime assuming the absence of minimum units. In contrast, continuous homogeneous state (Spacetime) here assumes the presence of a minimum unit.

Calculating the Spectral Dimension, Diffusion Time (DT), Number of green triangles (N), and Spectral Dimension (Ds) are summarized below. Continuous Homogeneous Structure shows higher dimension and Fractal Structure shows lower dimension.

The next is an example of "Partial Fractal Structure."

Then the Spectral Dimensions of the Partial Fracral Structure depending on scales are sumnarized as follows.
From a macroscopic view, the "Partial Fractal Structure" is Continuous Homogeneous. However, from a microscopic view, it is Fractal.


CDTists' Dimension Reduction implies such Partial Fractal Structure on small scales.






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