Fractal dynamics(quantum geometry)

[Finbar]

Fractal dynamics is the name I give to a research program I hope one day to embark on. My idea is that nature is actually a dynamical fractal in the following sense: there are many scales on which we can describe nature if we stick to a single scale then we can describe the physics using some form of field theory that has a classical geometrical interpetation i.e. classical gravity is intepretated as riemannian geometry, QCD can be understood as fibre bundles. In this sense we have a "geometrodynamics" associated with each scale so when we probe different scales we see a new geometry. So in this sense actually have a fractal (picture in your mind some geometical shape then if you look at a small bit with a microscope you see it is made up of a new geometry, you could then zoom further in ad see yet another geometry).

Remember though that these geomtries don't describe a static physical state but a dymanical process(spacetime not just space). This is important as I believe it to be an explanation of the uncertainty principle. Consider for a secound classical physics such that there are no quantum effects. In this instance we could describe the classical dynamics by a single geometry. This would be the same as looking at just one scale(call it A) on this single scale we could have a well defined dynamics but dues to the fact that we are only looking at the one scale we will always have an uncertainty in the momentum and position ;we don't care about actally where the particle is because we're only concered with a given length scale say a metre so if two particles are 1.0001 meters apart we could just call that a meter. But when we look at the same phyical system at a new maller scale(call it B) we see a new set of dymanics associated with a new geometry and the concepts of lengths and momentums on the old geometry no longer apply. From this we see that the uncertainties associated with scale A were real because geomery A was only valid at scale A. Its importantant to note that this makes sense in terms of the unceratity princple. The equation states:

delta x delta p > 2 pi

working in units such that Plancks constant and the spped of light are one. x has mass dimension [x]=-1 and p has mass dimension [p]=1. This implies that the uncertainty pricplle is scale independent i.e one can choose the mass unit we measure p and x in and the constant just cancels.


Now curreently all this is in some way all ready understood in terms of QFT and the renormalisation group. What I would like to explore is a new frame work. The current frame work is that we take a classical action and "quantize" it such that we then have a QFT. If we then look at that QFT at different energy scales we see that we can define an effective action which describes the physics at that level. The renormalisation group is the set of transformations that take us from one scale to another. My idea is that prehaps we can circumvent the quatization procedure and instead of starting with a classical theory given by a geometrodynamics we start with a quantum theory given by a fractal dynamics. This will involve understanding how fractal could ultimatly describe the universe in an analogy to how cosmogolgists currently understand the universe as a geometry.

I think though that there maybe a consequence of what I am saying...If my explanation of the unceratinty principle holds and we assume that the unertainty principle applies on all scales then there must alawys be a scale smaller i.e. there is no smallest scale! This may seem to be at odds with the idea of a smallest area in loop quantum gravity. But as loop gravity does not present itself as a final theory there is no real contdriction. The concept of a smallest area only makes sense when you look at a single geometry when you go to a smaller scale you can no longer define that area...as such there should be a smallest are in QG as gravity is the theory that defines areas.


Something I like about my idea is that it explains quantum mechanics but only in the light of relativity inparticular one needs the notion of locality lacking in newtoinan physics which essentially lead Einsten to a geometrical interpretation of gravity. One then extends this idea of locality such that it has some scale dependence.

A small note: when i say geomery I really mean in the most universal way possible not just geomeries defined by metrics (probably). Also I'm not really sure what I mean by fractal more than something that can be well approximated by a geometry at a single scale. Again what I mean by "scale" may also come into question sense geometry could be seen as defing scale so one should understand it as an energy scale i.e associated with the relative dynamics of geomery(where a lower energy scale would desribe the local area of a point in the fractal as smooth the higher energy scale would describe the local area as more dynamic ie more energtic.

 

[Finbar]

I've been thinking over the ideas again. I think the simplest way to express the idea of a fractal corresdonding directly to the full dynamics of a system is the following:

If we have a classical theory we can express the theory in terms of an action S (for example the einstein hilbert action). From the action one can find the classical equations of motion(by minimising the S, dS=0). Further more if one specifies intial conditions and boundary conditions one has all the information to find the complete dynamics of the classical system. In the case of general relativity knowing the Einstein hilbert action and the intial/boundary conditions gives you all the information to find the geometry of spacetime. Now its well known that GR is a gauge theory. Now a by gauge theory one generally means that we take a global symmetry(in the GR case lorentz invariance) and make it local. It is true then that all four forces can be understood as gauge theories. What is maybe less appreciated is that these gauge theries can all be understood as geometries. So for any claasical gauge theory if we have the action S and some intial/boundary conditions B we have all the information we need to find the dynamics which are expressed as a geometry.

(S,B)--> G


This is the classical picture. To get a quantum theory from a classical theory we define the path integral Z such that we sum over all possible paths from A to B but from this geometrical point of view we are actually summing over all geometries as each geometry represents a different dynmaical trasition from A to B.

Z = Sum(integral) over all geometries( exp[i S])

Now this some over geometries can be reexpressed as a sum over momentum modes. Now if one preforms the sum over only modes with frequency greater than some infrared cut-off k one can then(with a bit of algebra and a few transformations) define a few action S_k.

S_k is now a scale dependent action. If one takes a fixed value of k then one gets an action which we could then treat as if it was a classical action and hence define a geometry for a set of boundary conditions. If we then set a new value for k we can get a new geometry. Now quite how the boundary conditions should change for different values of k I am not sure none the less we could now say


(S_k,B_k)--> G_k

so we have a geometry G_k for every value of k then what I call a fractal is the Set of geometries {G_k} plus a continuous transormation that takes G_p--->G_q where p is greater than q.

Now S_k is called the effective average action you can read about it and the transformation law(the Renormalisation group equation) in papers by Wettrich and Reuter on ArXiv.org. It is the basis of current RG approaches to quantum gravity.



My essential idea is to covert this physical picture expressed in terms of an action into a purely geometrical approach which was a dream of John Archibald Wheeler known as Geometrodynamics:

" mass without mass,
charge without charge,
field without field."

I guess my idea is to add renormalisation without renormalisation to this set of ideas; that is express the renormalisation flow purely in terms of geometry.

 

[Finbar]

As nobody is replying I'm going to post this work and it continuation as a blog.

Fin.

 

[apeiron]

In this sense we have a "geometrodynamics" associated with each scale so when we probe different scales we see a new geometry. So in this sense actually have a fractal (picture in your mind some geometical shape then if you look at a small bit with a microscope you see it is made up of a new geometry, you could then zoom further in ad see yet another geometry).
.

Perhaps there is a basic problem here in your conception of fractals? Self-similarity would argue that the expectation is you should see the same dynamical geometrical set-up over all scales.

I agree that actually the bigger story is that geometry will change abruptly as you reach the limits of scale. Which is what we find in practice, in observation, when we either approach the lower scale (the transition from classical to QM regimes), or the upper scale (the transition from classical to GR).

So something is going on. And it can be modeled in terms of some notion of fractal maths. But it is something other than what you are getting at I would argue.

The fractal part of the universe is the flat middleground, the renormalised region, where the two opposed tendencies - QM and GR - are smoothly equilibrated over all available scales.

Crossing the boundary into these other realms of scale is then the breaking down of this smooth mixing. Cross into pure QM territory and you have only local point-like raw actions - the foam of possibility in loop-speak. Cross into pure GR territory and you only have the global continuity of constraints, the frozen block universe type view where foaming possibility has been dissipated. The dynamically organised region is the classical flat (ie: expanding/cooling) region of scale that lies inbetween these two extremes.

Many people are trying to apply fractal insights to cosmological problems. But the basis of this kind of maths - especially in terms of real life dynamical processes - does not yet seem widely understood.

It is a mistake to think it is fractals "all the way down". But probably an even bigger mistake to think that it is a succession of different fractals all the way down. The story, in my view, is that it fractals all the way down to the lower limit (and all the way up to the global limit).

We can thus account for the middle ground of worlds in terms of fractal geometrodynamics, but we need to account for boundaries as boundaries. The place where fractal balances effectively become undone in their complementary directions.

Also I'm not really sure what I mean by fractal more than something that can be well approximated by a geometry at a single scale. .

To take a step back for a minute, think again about the generation of fractals.

You have some sort of "seed" shape or operation that is then freely expressed over all scales. In familiar fractal examples like a koch curve or a branching structure, you have a kink in a line or fork in a path as the seed.

So this is indeed a geometrical design of "a single scale". But note that it embodies two opposing extremes of scale. You have local and global limits in interaction.

With bifurcations, for example, you have the global geometry that is a continuous path, then the local geometry which is the discrete or point-like act of bifurcation. Fractal dimensionality arises because the resulting geometrical process or system then exists in the space between the continuous 1D line and the discrete 0D point.

This what I mean about a fractal middle zone that is bounded to either side. The world of the fractal is anchored by its 0D and 1D extremes. It emerges poised exactly and flatly between them as an axis of scale symmetry.

Now put the microscope on the fractal and at the lower limit you find the "new geometry" of pure 0D points - a dust of possibility. At the global limit, using your telescope, you also find a different "new geometry" - the 1D realm of continuous static linearity. And neither of these pure states are fractal anymore. You have hit the limit and there is no beyond.

So I think it is absolutely correct to take a dynamical systems approach to cosmological questions. But the expectation then would be to find the system breaking up into three kinds of thing under close examination. (A basic triadic process as it has been dubbed by Stan Salthe, a semiotic relation as it was earlier described by CS Peirce).

You will have a local limit, a global limit, and then the middleground of actual stuff when two opposed possibilities are freely mixed over all possible scales of being. This middleground will have the powerlaw equilibrium structure we call fractal.

 

[MTd2]

Finbar, you write the lagrangian of those fractals, otherwise you won't get anything.

 

[Finbar]

Finbar, you write the lagrangian of those fractals, otherwise you won't get anything.

Indeed. Well that's already possible its just the lagrangian of the effective field theory with running couplings such that its scale dependent. So now the lagragion depends on both x the coordinates and k the momentum scale. This is just QFT. The tricky part is to go from the this lagrangian to a fractal. What I think will be needed is some kind of "coordinate transformation" Such that x and k are converted into some new "coordinates" which now span the fractal space. So really my starting point is to assume I have a lagragian and a set of beta functionds for the couplings. As for any given value of k I have a classical theory I can go from the lagrangian to the Euler-Lagrange equations of motion. But now when I have these equations. The field phi=phi(x,k) it now depends on the scale. I don't want to go into any more detail for now.