What does like a fractal mean, talking smallscale spacetime

[marcus]

... so what I can imagine is that for short distances (what in the paper is the same as for short times) spacetime is connected in some way that in average you can move only in two directions (2D) from some point, but if you do that for a long time (or long distances) the average behaviour is as if you could move in more...

this helped me get a clearer picture of how it might work. thanks.
discussing with others can help.

 

[mccrone]

The spectral dimension measured comes from the probability of return to a point so what I can imagine is that for short distances (what in the paper is the same as for short times) spacetime is connected in some way that in average you can move only in two directions (2D) from some point, but if you do that for a long time (or long distances) the average behaviour is as if you could move in more. ..

But isn't this just a fact written into the model? The triangles are flat space with all the kinkiness pushed to the edges. So as you shrink them, the ratio of edge to surface grows. For a point to wander off, it helps to be traveling in flatness. As the triangles get shrunk, there is more probability of getting bent back. Put a box around the space to measure the behaviour of wandering points and it seems to have less degrees of freedom as it finds it more difficult to escape the locality.

To me, this seems in some way more a rise in dimensionality. A wandering point in expanded space is more like a 1D trajectory. A wandering point in shrunk space is acting more like a volume filling entity. There is an increase in symmetry at least? And thus a loss of asymmetry that gives you more crisp directions.

The whole thing is not very convincing at all. A Planckian cut-off is being generated by the sharp division of flat surfaces and wonky edges when the simplices are large. The cut-off is not emerging as an output but being specified as one of the inputs.

Correct me if I'm wrong of course.

 

[selfAdjoint]

The whole thing is not very convincing at all. A Planckian cut-off is being generated by the sharp division of flat surfaces and wonky edges when the simplices are large. The cut-off is not emerging as an output but being specified as one of the inputs.

But note that Lauscher and Reiter in
http://www.arxiv.org/abs/hep-th/0508202

Find the same thing, fractal structure and dimension running to 2 at small scales, and that's coming from studying the quantized continuum theory side with renormalization group methods.

 

[Alamino]

To me, this seems in some way more a rise in dimensionality. A wandering point in expanded space is more like a 1D trajectory. A wandering point in shrunk space is acting more like a volume filling entity. There is an increase in symmetry at least? And thus a loss of asymmetry that gives you more crisp directions.

Filling all the space in a random walk is akin to saying that the probability of return to a point is 1, what happens (in flat space) for dimension $d \leq 2$. The path per se is always one-dimensional, but the capacity of filling all the space decreases with the increasing of the dimension. Probably that is the meaning of this dimensional increasing. The fractality is related, it seems, to the fact that as time goes by the probability of returning decreases in a way that the calculated spectral dimension goes from 2 to 4 in a continuous way. That´s because the probability of return is given by

$P(t) \sim \frac1{t^{d_s/2}}$

where $t$ is the time elapsed (or number of steps) and $d_s$ is the spectral dimension. I even guess that this IS the definition of $d_s$.

 

[marcus]

That´s because the probability of return is given by

$P(t) \sim \frac1{t^{d_s/2}}$

where $t$ is the time elapsed (or number of steps) and $d_s$ is the spectral dimension. I even guess that this IS the definition of $d_s$.

the result for regular random walks in a FLAT space, and on the other hand the definition for more general spaces, seem to be mutually supportive.

If one just studies a diffusion process or a random walk in a d dimensional ordinary flat euclidean space, then the classical result (if I am not mistaken) is just what you wrote

$P(T) \sim \frac1{T^{d_s/2}}$

where T is the time (think of it as the number of steps taken.) the probability that the particle has returned after the first T steps is proportional to T^-d/2.

Therefore in a standard 4D euclidean space the probability that it has returned after T steps falls off as the square of T.

there is actually a precise formula given on page 22 of "The Universe from Scratch"


$P(T) = \frac{1}{4\pi T^{d_s/2}}$


BUT THIS IS NOT EXACTLY TRUE ON A CURVED MANIFOLD. that's obvious enough, running a diffusion on a rounded surface will not give exactly the same result so the exact return-probability formula doesn't work

However one can still use that formula IN REVERSE to DEFINE THE DIMENSION! What a good idea! And that is where we get the concept of "spectral dimension".

the good thing about that is that it can be applied to structures that don't resemble euclidean space at all!

You can define a spectral dimension on ANYTHING YOU CAN RUN DIFFUSION ON or in other words anything you can run a random walk on.

So the thing does not have to have coordinate charts or to look even remotely like a piece of euclidean space, it can be a hairball, or a branchy tangle.

ooops, have to go, we have company. back later

(probably it isn't quite that simple, but I think that is the idea of "spectral dimension")

 

[Alamino]

However one can still use that formula IN REVERSE to DEFINE THE DIMENSION! What a good idea! And that is where we get the concept of "spectral dimension".

I like the idea of defining dimension w.r.t. a physical phenomenon (a random walk). It has a feeling of Einstein´s definition of time interval given in relativity w.r.t. the propagation of light signals. Just a thought.

 

[mccrone]

Filling all the space in a random walk is akin to saying that the probability of return to a point is 1, what happens (in flat space) for dimension $d \leq 2$. The path per se is always one-dimensional, but the capacity of filling all the space decreases with the increasing of the dimension.

I think you are saying here that there are two ways to impose a cut-off. The CDT approach is to push QM-kinky space to the edges of the simplices which hides the kinkiness at large scales and gets revealed at small scales.

What you describe sounds more like a coarse-graining - a cut-off imposed from above. So you take a point (with zero dimensions) and say wait for it to get back to "exactly" the same place. In reality, this would be infinitely unlikely. But you are deciding that close is near enough after a certain stage. So this is putting the location in a coarse-grain box and saying if the point re-enters the box - at anyone of the box's infinity of locations - then you have your return and the clock can be stopped.

Again a reduction in dimension would be imposed by the model rather than generated by the model as you are effectively saying a 3D solid (a box of space) is a single zero-D point for the sake of your measurement needs.

What I would find more convincing would be models in which the Planckian realm was treated as a hyperbolic roil - ye olde spacefoam - and a Feynman topological averaging to flatness emerges with the context of scale.

In effect, an isolated Planck scrap of spacetime would fluctuate with any curvature. But surrounded by other scraps, it knows how to line up. Context has a smoothing effect - as in any SO story such as a spin glass.

So in this view, the hyperbolic fluctuations on the QM scale are a bit of a fiction. They don't actually occur because spacetime has sufficient size - a relativistic ambience - to iron out such fluctuations. It would only be an isolated Planck-sized scrap disconnected from an actual Universe that could behave in a hyperbolic fashion.

This is why attempts to merge QM and relativity generally seem to get things backwards. The QM wildness is a behaviour that emerges as there is a loss of relativistic context. So a quantum gravity theory would be a model of gravity (a contextual feature) in a realm too small to support a stabilising context.

In this view, it would be a good thing that the two can't be completely merged, only asymptotically reconciled. If QM and relativity are boundaries or limits that lie in opposite directions, then the nonsense of UV infinities is what we should expect if we try to imagine a realm so lacking in scale that it has no idea which way to orientate itself and so apparently (according to the calculations) is curving in all directions at once.

Other ontologies would suggest that really an isolated Planckian scrap is just as much curving in no particular direction at all. It's behaviour would be vague and meaningless rather than powerful and directed.

 

[marcus]

I like the idea of defining dimension w.r.t. a physical phenomenon (a random walk)...

I too. It uses an entirely artificial notion of time, I notice.
they run the Monte Carlo randomization and come of with one sample triangulated spacetime. that geometry is "frozen" then, so that they can study it and discover dimensionality etc., volumes etc.

then using an artificial time, they run a diffusion in this frozen 4D geometry.

it is just a minor observation but i thought I would remark it anyway

 

[Alamino]

What you describe sounds more like a coarse-graining - a cut-off imposed from above. So you take a point (with zero dimensions) and say wait for it to get back to "exactly" the same place. In reality, this would be infinitely unlikely. But you are deciding that close is near enough after a certain stage. So this is putting the location in a coarse-grain box and saying if the point re-enters the box - at anyone of the box's infinity of locations - then you have your return and the clock can be stopped.

I see. I´m not sure, I need to get back to the books to confirm, but I guess that return probability is indeed a probability density and the continuum limit is already done, so there is no coarse graining in the equations. Of course the simulations are coarse-grained, but I suppose that AJL are trying to check their results with larger lattices each time and seeing if they have some limit.
I don´t know if they are already in a stage to say for sure that the effect of dimension reduction does not disappear in the continuum. Does anyone know?

 

[marcus]

...
I don´t know if they are already in a stage to say for sure that the effect of dimension reduction does not disappear in the continuum. Does anyone know?

when large numbers of 4-simplices are used the effect is independent of the exact number----it does not go away as one increases the number of simplices used in the experiment.

this is because it arises locally, unaffected by the total number (except for very long times that probe the boundaries) so one would expect it to persist effectively unchanged in the limit

there is a delicate question that you raise here! this is the PHYSICAL INTERPRETATION of the scale at which this lower-dimension fractal-like structure predominates.

the "size" of the simplexes in the Monte Carlo computer simulations is arbitrary (just some number not associated with any physical unit of distance)------so far the question of appropriate units of measurment has been left open!

one sees the effect, but one can only CONJECTURE about the scale at which it occurs. Is it, for example, the scale of Planck length? there is no definite answer, so far.

the passage to read about this is at the bottom of page 8 and top of 9 in
http://arxiv.org/hep-th/0505113 "The Spectral Dimension..." Here the important word is "TEMPTING".

---quote---
Translating our lattice results to a continuum notation requires a “dimensional transmutation” to dimensionful quantities, in accordance with the renormalization of the lattice theory. Because of the perturbative nonrenormalizability of gravity, this is expected to be quite subtle. CDT provides a concrete framework for addressing this issue and we will return to it elsewhere. However, since $\sigma$ from (1) can be assigned the length dimension two, and since we expect the
short-distance behaviour of the theory to be governed by the continuum gravitational coupling G_N, it is tempting to write the continuum version of (10) as


$$P_V (\sigma) \sim \frac1 {\sigma^2} \frac1 {1 + const\cdot G_N/\sigma}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (16)$$

where const. is a constant of order one. The relation (16) describes a universe whose spectral dimension is four on scales large compared to the Planck scale. Below this scale, the quantum-gravitational excitations of geometry lead to a nonperturbative dynamical dimensional reduction to two, a dimensionality where gravity is known to be renormalizable.

---end quote---

it might be helpful to glance at equation (4) of this paper, where one seens that the ficticious time of the diffusion process MUST HAVE DIMENSION SQUARED LENGTH, in order for the exponential to be meaningful.

as the paper says, attaching units to these results and saying at what scale to expect such and such effects has so far NOT been done and doing it will raise very interesting questions, including some questions (as I suspect) related to DSR---the modified lorentz invariance that keeps not only the speed of light but also a certain length scale the same for all observers

 

[marcus]

As long as we have a little technical detail, like equation (16) in preceding post, and what Alamino put earlier, I could also say why the Newton constant G has dimension of a square length.

this is true in the units everybody likes which have
$$\hbar = c = 1$$

one either must memorize that the usual Planck area = G in these units, or one must have some way to rediscover this fact. here is one way to rediscover it:

1. everybody familiar with Planck $\hbar$ knows that its dimension is "energy x time" and also knows that the dimension of
$\hbar c$ is "energy x length" or equivalently "force x area"

2. people familiar with the Einstein equation (main equation of Gen Rel) know that the coefficient is reciprocal force

$$8 \pi G/c^4$$

so $G/c^4$ is reciprocal force and $\hbar c$ is force x area. So what happens if you multiply them? you get AREA!

$$G/c^4 \times \hbar c = \hbar G/c^3$$

and that is what is usually called Planck area.

So if one is using lazy units which relativists like hbar = c= 1, then G is just the same as Planck area. and it has dimension of a squared length.

that is why in equation (16) previous post you have G/sigma.
both G and sigma have dimensions of squared length so the ratio is a plain number---dimensionless. that way it does not screw up the fraction.

========EDIT: afterthought=======
This bit of explanation is meant to apply to the preceding post where it says:

...the passage to read about this is at the bottom of page 8 and top of 9 in
http://arxiv.org/hep-th/0505113 "The Spectral Dimension..."

---quote---
Translating our lattice results to a continuum notation requires a “dimensional transmutation” to dimensionful quantities, in accordance with the renormalization of the lattice theory. Because of the perturbative nonrenormalizability of gravity, this is expected to be quite subtle. CDT provides a concrete framework for addressing this issue and we will return to it elsewhere. However, since $\sigma$ from (1) can be assigned the length dimension two, and since we expect the
short-distance behaviour of the theory to be governed by the continuum gravitational coupling G_N, it is tempting to write the continuum version of (10) as


$$P_V (\sigma) \sim \frac1 {\sigma^2} \ \frac1 {1 + const\cdot G_N/\sigma}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (16)$$

where const. is a constant of order one. The relation (16) describes a universe whose spectral dimension is four on scales large compared to the Planck scale. Below this scale, the quantum-gravitational excitations of geometry lead to a nonperturbative dynamical dimensional reduction to two, a dimensionality where gravity is known to be renormalizable.

---end quote---

 

[Spin_Network]

Things may be even clearer if you continue on to physical models like Benard cells and Bak sandpiles. Or scale-free networks.

A fractal system that expands or dissipates at a steady rate has special edge of criticality features. I think this is where connections with spacetime models are waiting to be made.

I think you hit the nail on the head!

The close packing of emerging fractals, and Bak Sandpiles, have a functional quantity for close particle packing.

The scale free networks, or even internal spin-networks, can be incorperated by using Smolins and Loll, Dynamical Triangulation principles.

In this bad image
:http://groups.msn.com/RelativityandtheMind/shoebox.msnw?action=ShowPhoto&PhotoID=25

there is a relation to a Smolin paper here:http://arxiv.org/abs/hep-th/0409057

and thus here:http://arxiv.org/abs/hep-th/0412307

the inner products being 2-D, self-organize the outer 3-D volumes.

In the second smolin paper above, I believe it shows (page 3 ) the frozen spin-network, and its dual, the triangulated triangle?

The spin-network (inner 2-D area), can influence the outer 3-D (volumes), that would obviously have a corresponding limiting effect, that may have its releveance here:https://www.physicsforums.com/showthread.php?t=90044

P.S there is a lot of distracting "doodling" in my linked pic, but..I want to continue the fractals (emerging from the righthand side) soon, so ignore.

 

[mccrone]

Thanks for the pointer to the Smolin paper and the other links.

If spacetime is an SO system, then we may well imagine it to be a sandpile. Let's start off with a bout of inflation for our sandpile. We dump a load of sand so fast that it sides are too steep and there is a massive general collapse towards some kind of "flat" balance of its slopes. We approach the criticality described by Smolin which is just a (Plankian?) hair's breadth on the side of not completely flat. The sand grains form local dips and bumps - enough to give the system a dynamic tension, but too flat for the system to observe and thus erase with a slippery avalanche.

In a real sandpile, the friction between grains stores a bit of configuration energy so that the slope may approach the zero point energy value, but cannot actually reach such perfect continuous flatness. Something analogous must be the case for the vacuum state of a universe - Smolin of course just plugs the necessary variety into the model.

Anyway, we have a flat spacetime void that is at the edge of criticality. And being there, it will respond in a scale-free fractal manner to (Planckian) perturbations. An event triggered by a grain of sand can occur avalanches over all scales. It is the flatness of the vacuum (the angle of the slope) that guarantees a behaviour with a new axis of symmetry - a freedom to make avalanches over any physical scale.

A vacuum with less than this critical slope would not respond to (Planckian) events. Like a sandpile that is too flat, nothing propagates. A vacuum too energetic would also fail to show this powerlaw respose to the tiniest events - it would be a sandhill so steep that it would be slumping spontaneously in global fashion, not a sandhill sprinkled with a powerlaw range of events.

So the fracticity here is not in the fine-grain structure but in the wider systems-level behaviour. It is an emergent property of "flatness" - or rather the attempts of a Universe to ease itself as near to flatness as its self-observing organisation will allow.

BTW what is the structure of the triangulated space in your and Smolin's diagrams? You have a sequence of 1, 4, 16, 64...

As in...
http://members.cox.net/tessellations/1 4 16 64.html

 

[Spin_Network]

Thanks for the pointer to the Smolin paper and the other links.

If spacetime is an SO system, then we may well imagine it to be a sandpile. Let's start off with a bout of inflation for our sandpile. We dump a load of sand so fast that it sides are too steep and there is a massive general collapse towards some kind of "flat" balance of its slopes. We approach the criticality described by Smolin which is just a (Plankian?) hair's breadth on the side of not completely flat. The sand grains form local dips and bumps - enough to give the system a dynamic tension, but too flat for the system to observe and thus erase with a slippery avalanche.

In a real sandpile, the friction between grains stores a bit of configuration energy so that the slope may approach the zero point energy value, but cannot actually reach such perfect continuous flatness. Something analogous must be the case for the vacuum state of a universe - Smolin of course just plugs the necessary variety into the model.

Anyway, we have a flat spacetime void that is at the edge of criticality. And being there, it will respond in a scale-free fractal manner to (Planckian) perturbations. An event triggered by a grain of sand can occur avalanches over all scales. It is the flatness of the vacuum (the angle of the slope) that guarantees a behaviour with a new axis of symmetry - a freedom to make avalanches over any physical scale.

A vacuum with less than this critical slope would not respond to (Planckian) events. Like a sandpile that is too flat, nothing propagates. A vacuum too energetic would also fail to show this powerlaw respose to the tiniest events - it would be a sandhill so steep that it would be slumping spontaneously in global fashion, not a sandhill sprinkled with a powerlaw range of events.

So the fracticity here is not in the fine-grain structure but in the wider systems-level behaviour. It is an emergent property of "flatness" - or rather the attempts of a Universe to ease itself as near to flatness as its self-observing organisation will allow.

BTW what is the structure of the triangulated space in your and Smolin's diagrams? You have a sequence of 1, 4, 16, 64...

As in...
http://members.cox.net/tessellations/1 4 16 64.html

I see a great link you provided!..but forgive me for a while if I do not reply directly,(reletive to the numbers! ) that is, except to say if you could see a "spacetime vacuum field", then your image is pretty damn close! I need to collate a few things relevant to what I am doing, hopefully I am going to post soon, about 2 days before the loop conference, thanks again for a really interesting post, and great link.

 

[rtharbaugh1]

Hi all

I finally finished reading through this thread, although I have not visited all the links. I certainly won't pretend to understand all of it. But I have been reading and working through Barnsley's Fractals in Nature[\B] book offline, so I am beginning to get the idea of Hausdorf dimensions. I also have been reading about random walks and following the "ball of wire" model of dimensionality. The recent turn of this discussion to the sandpile analogy has me concerned.

If we are going to try modeling spacetime as a sandpile, what part of the model replaces the force of gravity in the sandpile? It seems to me that gravity, Planck scale graininess, fractals, foams, strong-weak-EM fields, standard model of particles and standard model of cosmology all have to emerge from any acceptable fundamental model. I don't think we can make progress by starting out assuming gravity is already in effect. We need to start from something like Machian space, or the zero point. Probably the sandpile stuff will come in at larger scales on accelerated surfaces.

In my meditations I have been trying to imagine an empty space of infinite dimension in which probability alone demands the emergence of form. Probability also demands that form emerges in more than one spacetime location, and that the simplest forms are the most common ones to emerge. Probability further demands that some forms have duration, and that some forms will emerge in common spaces where, having duration, there will be time for them to interact.

Perhaps all kinds of interactions are possible, but some kinds of interactions will have a longer duration than others. We need to look for a fundamental model that results in long term durations, such as we see in the standard models. Probability suggests that some forms emergent in common spaces will be expanding and will thereby come to interact for long duration. These forms are the ones most likely to have sufficient duration and interaction to become observable.

Simple expanding forms in common space are therefore a likely model to test for properties that result in emergent phenomena like gravity and the other forces.

Thanks for your comments,

Richard, the ex-Nightcleaner
(currently seeking employment)

 

[selfAdjoint]

These comments are well-said, Richard. I don't think the sandpile was intended to be taken literally, just a visual example of an emergent critical point. I have been trying to learn about anomalous dimension in Renormalization group theory, but it's very hard, and I believe over my head in terms of what I know and can do now. I'm reading the book http://www.pupress.princeton.edu/titles/5772.html by Giuseppe Benfatto and Giovanni Gallavotti, and they state in their introduction that the minumum level for reading the book with profit is ability to do simple first order calculations in QFT, and I'm not quite there yet. The problem is deducing the pattern of thought from the blizzard of integral and distributional developments.

 

[marcus]

Hi all

Hello Richard, pleased to see you back, glad you could make it.

M

 

[mccrone]

If we are going to try modeling spacetime as a sandpile, what part of the model replaces the force of gravity in the sandpile?

In Smolin, it would be the links in the spin network. They are spacetime paths and their curvature would be gravity.

In my meditations I have been trying to imagine an empty space of infinite dimension in which probability alone demands the emergence of form. Probability also demands that form emerges in more than one spacetime location, and that the simplest forms are the most common ones to emerge. Probability further demands that some forms have duration, and that some forms will emerge in common spaces where, having duration, there will be time for them to interact.

Invoking the process of natural selection of a space of possibility is definitely the way to go. But why start with nothingness? How could a something develop from a nothing? Why not instead consider a prior state of everythingness, a plenitude, and see the emergence of universes with only a very few dimensions as an act of self-organised constraint, a phase transition, of this everythingness?

You can chip away at a large block of stone to reveal the forms inside. But of course a block of stone is too substantial for our purposes. Instead the prior everythingness must be maximally vague.

Now imagine something more like the crackle of white noise on an untuned TV set. Are you watching nothing or everything - does potentially every TV show you've ever seen, or could see, exist in that crackle?

Why do you think string theorists talk about landscapes, Smolin about multiverses, Linde about eternal fractal inflation? Even CDT is based on Feynman averaging.

The only problem is that this is all probability theory based on discrete entities, crisp variety. Which IMHO is not the view of probability that you need for fundamental theories.

 

[selfAdjoint]

Jacques Distler has now done what I was unable to, critiqued Reuter's work on the asymptotic safeness of Quantum General Relativity. Distler's main point is that Reuter's exact theory is not exact in the usual sense of the word, and it is not non-perturbative. Furthermore the theory is unable to deal with the infinity of equations it generates and works with a finite truncated set. But as Distler says, the RG fixed point in the full QGR will surely be affected by non-perturbative and non-truncated contributions.Therefore Reuter's stated results on fixed points do not have any persuasive surety of carrying over to the full non-perturbative, untruncated case. This is a point I intuited and wanted to make but lacked the computational guns to do it. Distler has done the work and showed the result.

Reuter's work is interesting and valid as far as it goes, but it does not go far enough to be convincing about the asymptotic safety of QGR.

 

[rtharbaugh1]

Mccrone said:
"Invoking the process of natural selection of a space of possibility is definitely the way to go. But why start with nothingness? How could a something develop from a nothing? Why not instead consider a prior state of everythingness, a plenitude, and see the emergence of universes with only a very few dimensions as an act of self-organised constraint, a phase transition, of this everythingness?"

nothing is everything and everything is nothing. A blank sheet of paper is still blank no matter if the blank-ness is black or white or some other color. Barnsley takes a set theory approach to fractals. He says there are two sets that are simultaneously open and closed. One is the universal set, or as he calls it, the complete metric set, and the other is the set containing only one point.

I am reading about the metric that is natural to fractals. Barnsley denotes it by putting a fancy, squiggley capital H in front of the notation for a metric set...

H {X,d}

I suppose, although he has not said, that this is the Hausdorf metric set. He says that fractals have to live in such a metric. What does this mean in terms of background independence, I wonder?

SelfAdjoint, I have Zee's book on Quantum Field theory but have made little headway. I don't know the authors you mention. Barnesly is a difficult read for me. He likes to pose puzzles and leave you on your own for the answers. Still, I have been able to read some of it.


Hi again Marcus. The maples have turned red and are dropping their leaves, in embarrasment I suppose. It was a dry summer but recent rains have returned a hope for a colorful autumn. My winter nest isn't quite finished and I have a serious shortage of aged firewood at hand. Time passes, or we pass. Thanks for keeping a light in the library.

Richard

 

[marcus]

...
Hi again Marcus. The maples have turned red and are dropping their leaves, in embarrasment I suppose. It was a dry summer but recent rains have returned a hope for a colorful autumn. My winter nest isn't quite finished and I have a serious shortage of aged firewood at hand. Time passes, or we pass. Thanks for keeping a light in the library.
...

we do this for each other, Richard----keeping the library occupied so the chairs don't get dusty. Yeats mentioned that light. It was in an uncomplimentary poem about political leaders:

THE LEADERS OF THE CROWD

They must to keep their certainty accuse
All that are different of a base intent;
Pull down established honour; hawk for news
Whatever their loose fantasy invent
And murmur it with bated breath, as though
The abounding gutter had been Helicon
Or calumny a song. How can they know
Truth flourishes where the student's lamp has shone,
And there alone, that have no Solitude?...

 

[selfAdjoint]

Richard, don't worry about Zee, stick with your fractals book. I was just venting because I got the Benfatto & Gallavotti book to help me understand anomalous dimension and it's such a slog.

Yes the curly H means Hausdorf, and Hausdorf dimension is roughly the same as fractal dimension. It can come out non-integer. All the nice spaces are Hausdorf from the git-go; it means given any two distinct points you can find two non-intersecting neighborhoods in the space such that each point is in one of the neighborhoods (the neighborhoods are specified by the TOPOLOGY on the space, and the Hausdorf property is a property of that topology). The real interest here is in spaces that are Hausdorf but not any nicer than that. Fractal spaces for example.

The drive currently is to derive spacetime with its nice metric and curvature properties from something much simpler; "bare causality" maybe. Or tangles of colliding hyper-triangles. A constraint on all such efforts is that spacetime when you get to it has to be an interacting player in the physics, i.e. "background free".

 

[Spin_Network]

I see a great link you provided!..but forgive me for a while if I do not reply directly,(reletive to the numbers! ) that is, except to say if you could see a "spacetime vacuum field", then your image is pretty damn close! I need to collate a few things relevant to what I am doing, hopefully I am going to post soon, about 2 days before the loop conference, thanks again for a really interesting post, and great link.

John Baez is seeking some "handwaving" here:https://www.physicsforums.com/showthread.php?t=91813

one of his linked papers here:http://arxiv.org/abs/http://www.arxiv.org/abs/hep-th/9401137
gives an early Ambjorn interesting paper.

Having just read this paper today for the first time, I see the light!

Its just as I thought, the Loll et al Dynamical Triangulations, do not transform from 4-D to 2-D, there is a fundamental flaw "running" through the proposed model, along the lines of linearlity "dimensional renormalization".

Stringtheory, has world-lines that are not continueous from one dimension to another, example if one has a string source starting in 5-D , the only route to bring the string-path in a linear transformation from 5-D to 3-D,is to compact from a 5-D boundary, inversely. So 5-D volume space extends around a 3-D space, (5...(4...(3-+-3)...)4...)...5).

Extending dimensions exponentially form a compact lower volume source, starting from a zero-mode, stringtheory uses the interconnecting spaces at above ie (... as the linear world-line. If one starts at a low-energy source, and join the total volumes together, one gets a specific sheet of space forming a "background" .A 2-D area has only non-directional transforms, due to the fact that a 2-D world-sheet has no 3-D directional routes contained within, you only get rotational values.

http://groups.msn.com/RelativityandtheMind/shoebox.msnw?action=ShowPhoto&PhotoID=25

 

[selfAdjoint]

Stringtheory, has world-lines that are not continueous from one dimension to another, example if one has a string source starting in 5-D , the only route to bring the string-path in a linear transformation from 5-D to 3-D,is to compact from a 5-D boundary, inversely. So 5-D volume space extends around a 3-D space, (5...(4...(3-+-3)...)4...)...5).

Extending dimensions exponentially form a compact lower volume source, starting from a zero-mode, stringtheory uses the interconnecting spaces at above ie (... as the linear world-line. If one starts at a low-energy source, and join the total volumes together, one gets a specific sheet of space forming a "background" .A 2-D area has only non-directional transforms, due to the fact that a 2-D world-sheet has no 3-D directional routes contained within, you only get rotational values.

This evidently means something to you, but it sure doesn't convey anything to me. In string theory the compact 6D manifolds are orthogonal at every point of 4D spacetime, and a normal thing would be for the string to move smoothly through them. Of course they can get hung up on the non-trivial topology of the manifolds, wrapped around a handle, for example, but that is supposed to correspond to real and interesting physics. Your idea that there's some stupid error that nobody but you has spotted in either string theory or dynamic triangulations is just mistaken. You are not going to figure this stuff out with just the ininformed, untutored imagination.