Intro To Causal Dynamical Triangulations

[JMJones0424]

First, an obligatory caveat. This thread is intended as a request for information, not an assertion of fact or a claim.

 

Likely everyone knows of the conflicting descriptions of reality in the realms of the very great and the very small. For some time, a workable theory of quantum gravity that incorporates explanations of observations that led to both general relativity and quantum mechanics has been elusive. It seems (to me), from the LHC's apparent null result of supersymmetry, that string theory, brane theory, and the whole lot may be discarded as insufficient.

 

I've recently come across an excellent, layman friendly, introduction to a promising avenue of investigation for an explanation of quantum gravity called (unfortunately) Causal Dynamical Triangulations. Only an economist could dream up a more confuscating name. The idea will suffer from a lack of popular support over string theory for twenty years on its name alone.

 

However, I'm not interested in names; I'm interested in testable descriptions of observations. I'm also, like most people, limited in both education and time required to develop the skills necessary to adequately evaluate the claims of CDT. I know there are contributors here that do have those skills, and I selfishly ask for your input on the subject.

Edited October 14, 2013 by JMJones0424

[Aethelwulf]

It's a very promising approach. If I had to choose one theory, I'd say I agreed with this theory. Do I think this is how reality is built up... I don't quite think Triangulation has everything right, but I think the premise is correct. The idea that space and time, is in fact a geometric property of a configuration space involves a relativistic concept that particles are not independent as such and they do form a geometric reality at the world of the small. This idea of geometry can be linked with geometrogenesis: The emergence of geometry is when the universe sufficiently cooled down enough to create matter. There are geometric interpretations of other quantum effects, such as the Uncertainty Principle. It's geometric form is the Cauchy Schwarz inequality - not only does triangulation determine effects on spacetime, but the lengths of that triangulation may also hold inherent secrets about the very nature of the universe - including the effects of quantum mechanics. Another interesting thing I noticed over the years of lightly studying the subject, is that there is what, I called the ''Fotini energy'' - basically you can only assign an energy in a graphical tensor notation involving the vertices of other locations of particles forming the geometry. If you don't have an energy in this model, you can't be talking about triangulation.

[Aethelwulf]

Since there is no such thing as ''free space'' or ''empty space'' we must assume that all vertices making the geometric triangulation are in fact locations of particles, therefore that in all cases we are dealing with ''three neighbouring points'' on what I have come to call a Fotini graph. Really, the graph has a different name and is usually denoted with something like [math]E(G)[/math] and is sometimes called the graphical tensor notation. In our phase space, we will be dealing with a finite amount of particles [math]i[/math] and [math]j[/math] but asked to keep in mind that the neighbouring particles are usually seen at a minimum three and that each particle should be seen as a configuration of spins - this configuration space is called the spin network. I should perhaps say, that to any point, there are two neighbours.

 

Of course, as I said, we have two particles in this model [math](i,j)[/math], probably defined by a set of interactions [math]k \equiv (i,j)[/math] (an approach Fotini has made in the form of on-off nodes). In my approach we simply define it with an interaction term:

 

[math]V = \sum^{N-1}_{i=1} \sum^{N}_{i+1} g(r_{ij})[/math]

 

I have found it customary to place a coupling constant here [math]g[/math] for any constant forces which may be experienced between the two distances made in a semi-metric which mathematicians often denote as [math]r_{ij}[/math].

 

If [math]A(G)[/math] are adjacent vertices and [math]E(G)[/math] is the set of edges in our phase space, (to get some idea of this space, look up casual triangulation and how particles would be laid out in such a configuration space), then

 

[math](i,j) \in E(G)[/math]

 

It so happens, that Fotini's approach will in fact treat [math]E(G)[/math] as assigning energy to a graph

 

[math]E(G) = <\psi_G|H|\psi_G>[/math]

 

which most will recognize as an expection value. The Fotini total state spin space is

 

[math]H = \otimes \frac{N(N-1)}{2} H_{ab}[/math]

 

Going back to my interaction term, the potential energy between particles [math](i,j)[/math] or all [math]N[/math]-particles due to pairwise interctions involves a minimum of [math]\frac{N(N-1)}{2}[/math] contributions and you will see this term in Fotini's previous yet remarkably simple equation.

[Aethelwulf]

I think this is cool.

 

Going back to our set of particles defined as [math]k = (i,j)[/math] both these particles are associated with a spin

 

[math]\sqrt{j_i(j_i + 1)}[/math]

 

[math]\sqrt{j_j(j_j + 1)}[/math]

 

Then according to loop quantum gravity and their respective spin network, you can actually quantize their area's

 

[math]A_{\Sigma} = 8 \pi \ell^{2}_{pl} \gamma \Sigma_i \sqrt{j_i(j_i + 1)}[/math]

 

 

[math]A_{\Sigma}' = 8 \pi \ell^{2}_{pl} \gamma \Sigma_j \sqrt{j_j(j_j + 1)}[/math]

Edited November 7, 2013 by Aethelwulf

[Rade]

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Edited November 13, 2013 by Rade