View Full Version : Two World-theories (neither one especially stringy)
The two that look most promising to me are Lorentzian DT and Loop.
To look at the raw numbers---sheer quantity of research papers written per year---you'd say LQG was growing rapidly and DT was flat.
Lorentzian DT was first proposed in 1998 (a paper by Ambjorn and Loll), here are some preprint numbers:
http://arxiv.org/find/grp_physics/1/OR+OR+abs:+AND+triangulations+AND+Lorentzian+dynam ical+abs:+AND+triangulations+AND+causal+dynamical+ ti:+AND+gravity+AND+Lorentzian+quantum/0/1/0/1998/0/1
http://arxiv.org/find/grp_physics/1/OR+OR+abs:+AND+triangulations+AND+Lorentzian+dynam ical+abs:+AND+triangulations+AND+causal+dynamical+ ti:+AND+gravity+AND+Lorentzian+quantum/0/1/0/1999/0/1
http://arxiv.org/find/grp_physics/1/OR+OR+abs:+AND+triangulations+AND+Lorentzian+dynam ical+abs:+AND+triangulations+AND+causal+dynamical+ ti:+AND+gravity+AND+Lorentzian+quantum/0/1/0/2000/0/1
http://arxiv.org/find/grp_physics/1/OR+OR+abs:+AND+triangulations+AND+Lorentzian+dynam ical+abs:+AND+triangulations+AND+causal+dynamical+ ti:+AND+gravity+AND+Lorentzian+quantum/0/1/0/2001/0/1
http://arxiv.org/find/grp_physics/1/OR+OR+abs:+AND+triangulations+AND+Lorentzian+dynam ical+abs:+AND+triangulations+AND+causal+dynamical+ ti:+AND+gravity+AND+Lorentzian+quantum/0/1/0/2002/0/1
http://arxiv.org/find/grp_physics/1/OR+OR+abs:+AND+triangulations+AND+Lorentzian+dynam ical+abs:+AND+triangulations+AND+causal+dynamical+ ti:+AND+gravity+AND+Lorentzian+quantum/0/1/0/2003/0/1
http://arxiv.org/find/grp_physics/1/OR+OR+abs:+AND+triangulations+AND+Lorentzian+dynam ical+abs:+AND+triangulations+AND+causal+dynamical+ ti:+AND+gravity+AND+Lorentzian+quantum/0/1/0/2004/0/1
LORENTZIAN DT (etc.) PREPRINTS
1998 3
1999 3
2000 5
2001 4
2002 6
2003 4
2004 4
Numberwise, DT doesn't look like much is happening.
Loop has been going longer, at least since the early 1990s. Here are output numbers for Loop and allied QG approaches.
Year 1994:
http://arXiv.org/find/nucl-ex,astro-ph,nucl-th,math-ph,hep-ex,physics,cond-mat,hep-lat,quant-ph,gr-qc,hep-ph,hep-th/1/OR+OR+abs:+AND+AND+loop+quantum+OR+cosmology+gravi ty+abs:+AND+AND+quantum+gravity+OR+OR+discrete+phe nomenology+OR+canonical+nonperturbative+abs:+OR+OR +spinfoam+AND+spin+foam+AND+OR+triply+doubly+speci al/0/1/0/1994/0/1
Year 1995:
http://arXiv.org/find/nucl-ex,astro-ph,nucl-th,math-ph,hep-ex,physics,cond-mat,hep-lat,quant-ph,gr-qc,hep-ph,hep-th/1/OR+OR+abs:+AND+AND+loop+quantum+OR+cosmology+gravi ty+abs:+AND+AND+quantum+gravity+OR+OR+discrete+phe nomenology+OR+canonical+nonperturbative+abs:+OR+OR +spinfoam+AND+spin+foam+AND+OR+triply+doubly+speci al/0/1/0/1995/0/1
Year 1996:
http://arXiv.org/find/nucl-ex,astro-ph,nucl-th,math-ph,hep-ex,physics,cond-mat,hep-lat,quant-ph,gr-qc,hep-ph,hep-th/1/OR+OR+abs:+AND+AND+loop+quantum+OR+cosmology+gravi ty+abs:+AND+AND+quantum+gravity+OR+OR+discrete+phe nomenology+OR+canonical+nonperturbative+abs:+OR+OR +spinfoam+AND+spin+foam+AND+OR+triply+doubly+speci al/0/1/0/1996/0/1
Year 1997:
http://arXiv.org/find/nucl-ex,astro-ph,nucl-th,math-ph,hep-ex,physics,cond-mat,hep-lat,quant-ph,gr-qc,hep-ph,hep-th/1/OR+OR+abs:+AND+AND+loop+quantum+OR+cosmology+gravi ty+abs:+AND+AND+quantum+gravity+OR+OR+discrete+phe nomenology+OR+canonical+nonperturbative+abs:+OR+OR +spinfoam+AND+spin+foam+AND+OR+triply+doubly+speci al/0/1/0/1997/0/1
Year 1998:
http://arXiv.org/find/nucl-ex,astro-ph,nucl-th,math-ph,hep-ex,physics,cond-mat,hep-lat,quant-ph,gr-qc,hep-ph,hep-th/1/OR+OR+abs:+AND+AND+loop+quantum+OR+cosmology+gravi ty+abs:+AND+AND+quantum+gravity+OR+OR+discrete+phe nomenology+OR+canonical+nonperturbative+abs:+OR+OR +spinfoam+AND+spin+foam+AND+OR+triply+doubly+speci al/0/1/0/1998/0/1
Year 1999:
http://arXiv.org/find/nucl-ex,astro-ph,nucl-th,math-ph,hep-ex,physics,cond-mat,hep-lat,quant-ph,gr-qc,hep-ph,hep-th/1/OR+OR+abs:+AND+AND+loop+quantum+OR+cosmology+gravi ty+abs:+AND+AND+quantum+gravity+OR+OR+discrete+phe nomenology+OR+canonical+nonperturbative+abs:+OR+OR +spinfoam+AND+spin+foam+AND+OR+triply+doubly+speci al/0/1/0/1999/0/1
Year 2000:
http://arXiv.org/find/nucl-ex,astro-ph,nucl-th,math-ph,hep-ex,physics,cond-mat,hep-lat,quant-ph,gr-qc,hep-ph,hep-th/1/OR+OR+abs:+AND+AND+loop+quantum+OR+cosmology+gravi ty+abs:+AND+AND+quantum+gravity+OR+OR+discrete+phe nomenology+OR+canonical+nonperturbative+abs:+OR+OR +spinfoam+AND+spin+foam+AND+OR+triply+doubly+speci al/0/1/0/2000/0/1
Year 2001:
http://arXiv.org/find/nucl-ex,astro-ph,nucl-th,math-ph,hep-ex,physics,cond-mat,hep-lat,quant-ph,gr-qc,hep-ph,hep-th/1/OR+OR+abs:+AND+AND+loop+quantum+OR+cosmology+gravi ty+abs:+AND+AND+quantum+gravity+OR+OR+discrete+phe nomenology+OR+canonical+nonperturbative+abs:+OR+OR +spinfoam+AND+spin+foam+AND+OR+triply+doubly+speci al/0/1/0/2001/0/1
Year 2002:
http://arXiv.org/find/nucl-ex,astro-ph,nucl-th,math-ph,hep-ex,physics,cond-mat,hep-lat,quant-ph,gr-qc,hep-ph,hep-th/1/OR+OR+abs:+AND+AND+loop+quantum+OR+cosmology+gravi ty+abs:+AND+AND+quantum+gravity+OR+OR+discrete+phe nomenology+OR+canonical+nonperturbative+abs:+OR+OR +spinfoam+AND+spin+foam+AND+OR+triply+doubly+speci al/0/1/0/2002/0/1
Year 2003:
http://arXiv.org/find/nucl-ex,astro-ph,nucl-th,math-ph,hep-ex,physics,cond-mat,hep-lat,quant-ph,gr-qc,hep-ph,hep-th/1/OR+OR+abs:+AND+AND+loop+quantum+OR+cosmology+gravi ty+abs:+AND+AND+quantum+gravity+OR+OR+discrete+phe nomenology+OR+canonical+nonperturbative+abs:+OR+OR +spinfoam+AND+spin+foam+AND+OR+triply+doubly+speci al/0/1/0/2003/0/1
Year 2004:
http://arXiv.org/find/nucl-ex,astro-ph,nucl-th,math-ph,hep-ex,physics,cond-mat,hep-lat,quant-ph,gr-qc,hep-ph,hep-th/1/OR+OR+abs:+AND+AND+loop+quantum+OR+cosmology+gravi ty+abs:+AND+AND+quantum+gravity+OR+OR+discrete+phe nomenology+OR+canonical+nonperturbative+abs:+OR+OR +spinfoam+AND+spin+foam+AND+OR+triply+doubly+speci al/0/1/0/2004/0/1
LOOP (etc.) PREPRINTS
1994 61
1995 83
1996 72
1997 70
1998 67
1999 76
2000 89
2001 98
2002 121
2003 139
2004 178
The 2004 figures are up through 19 December, which is close enough to yearend so one gets an idea.
I have been reading nothing but DT papers this morning. the approach has some unique and impressive advantages working in its favor. I would like to be able to compare these two quantum spacetime theories on an equal footing.
Their most noticeable disagreement is apt to concern the area and volume operators. As yet no indication that in DT these will have discrete spectra.
I would like to know why the research output in DT is essentially flat. Given its apparent promise and the recent (2004) success, why arent more people getting into DT?
I will quote some about Lorentzian path integral from
the most pedagogical paper I know----Renate Loll
http://arxiv.org/hep-th/0212340
----quote from "A Discrete History"----
The desire to understand the quantum physics of the gravitational interactions lies at the root of many recent developments in theoretical high-energy physics. By quantum gravity I will mean a consistent fundamental quantum description of space-time geometry (with or without matter) whose classical limit is general relativity. Among the possible ramifications of such a theory are a model for the structure of space-time near the Planck scale, a consistent calculational scheme to compute gravitational effects at all energies, a description of (quantum) geometry near space-time singularities and a non-perturbative quantum description of four-dimensional black holes. It might also help us in understanding cosmological issues about the beginning (and end?) of our universe, although it should be said that some questions (for example, that of the “initial conditions”) are likely to remain outside the scope of any physical theory.
---end quote---
I guess anyone interested in this thread already has realized this: one of the unusual things about this approach is there are no coordinates.
That was the headline on Regge's 1961 paper that set things up for Renate Loll and friends------"General Relativity Without Coordinates".
they can consider the space Geom(M) of all spacetime geometries on some manifold-----each geometry is described by listing interconnections between uniformsized simplexes, some kind of computer data structure.
that is a point in Geom(M), it is real elementary barebones
there is no "gauge" or chaff of arbitrary choice (as when things are presented using coordinates)
and that barebones reality is what the quantum mechanics is about
I have always appreciated the spareness of LQG----it doesnt seem to have anything in it that isn't needed to describe a quantum theory of 4D spacetime. But to get started, LQG does employ a differentiable manifold and connections thereon. That takes in a batch of arbitrary mathematical equipage (physically meaningless "gauge" accessory) which then has to be factored out later. But I thought that LQG kept gauge to a bare miniumum. After all, how could one ever get started without an underlying smooth manifold?
Systems of coordinates are an arbitrary physically meaningless choice but how do you get started without them?
Well the framework for Lorentzian path integral, or DT, is even more stripped down nitty. No coordinate system. It seems right. have to go, will try to get back to this later.
nightcleaner
Dec20-04, 07:47 PM
0 1
1
1 1 1 1
0 1
2 2 2 1 2 1
0 1 2
3 3 3 3 1 3 3 1
0 1 2 3
4 4 4 4 4 1 4 6 4 1
0 1 2 3 4
n pick k pascal
bravo Cleaner
I will go over to the other thread and do an example
============
some side comments
since the essential thing about space is relations (next-to, between, around) so that space is in some sense a compendium of all those spatial relations then it is intuitive to me that a basic piece of space would be a simplex.
and it seems reasonable that global geometry would consist of saying HOW THEY ARE GLUED
but the minimal element of space, I could see, might be a tetrahedron---basically just 4 points
and for a path integral describing the evolution of space one would want to build it of the fivepointer analog (the simplex with 5 points and 5 tetrahedral walls)
OR, if that one would get a quantum model of spacetime by TAKING THE LIMIT with smaller and smaller simplices.
I mean that space is not actually to be imagined as diced up into little simplices, because maybe there IS no minimal distance. Maybe we just THINK that planck length indicates some fundamental minimal length and it really doesnt! But even so it could be that the right approach to quantizing is to divvy up into simplexes and then make the simplexes smaller and smaller. Because the simplex approximation is a good approximation to how space behaves.
suppose we go along with Loll and Ambjorn and we say OK
simplexes are basic
and we are going to have a "path integral"
then we are going to get to recognize two kinds of fivepoint simplexes
or, if you'd rather, two types of orientation.
that is because CAUSALITY layers spacetime
a simplex can stand like a pyramid with 4 points in one spatial layer and the remaining 5th point upstairs in the next layer
or it can be upside down with the 5th point in the prior layer
(that is really the same kind)
in this case there are 6 spacelike edges and 4 timelike edges (connecting the 5th point to the other 4)
but there is another kind of fivepointer you maybe did not expect that may be thought of as dual to this one----it has 4 spacelike edges and 6 timelike!
this kind has 3 points on the ground and 2 points upstairs in the next layer (or turning it over) downstairs in the prior layer.
In Ambjorn and Loll's path integral approach each 4simplex has a piece of MINKOWSKI space in it. What could be a nicer material for them to be made of? All the simplexes are chunks cut out of the familiar 4D flat space of 1905 special relativity.
The two types of 4simplexes----call them (4,1) and (3,2) and remember there are flipped versions (1,4) and (3,2) that are so similar to the first two that we dont make a point of distinguishing----are two ways that Minkowski space can be oriented so as to sit in the simplex.
when these little lego-bricks are glued together to make a spacetime PATH (from some initial to some final geometry of space)
then the GLUING HAS TO RESPECT the lightcones in each block. The fitting of face to face has to respect the way Minkowski space sits in each simplex.
that is why Renate Loll tells us about the two types of 4simplex. So we wont forget and try to stick two faces together in a way that disrespects the Minkowski causality, or lightcones, in the two neighbor pieces.
the two types are shown in her picture Figure 5 on page 11
of
http://arxiv.org/hep-th/0212340
this paper I esteem more and more because of its
occasional kindergartenness
I just wish it were that way all the time
the simpler the better. amen.
On page 10:
"... no local curvature degrees of freedom are suppressed by fixing the edge lengths; deficit angles in all directions are still present, although they take on only a discretized set of values. In this sense, in dynamical triangulations all geometry is in the gluing of the fundamental building blocks. This is dual to how quantum Regge calculus is set up, where one usually fixes a triangulation T and then “scans” the space of geometries by letting the li's run continuously over all values compatible with the triangular inequalities..."
This is just a reminder that after all spacetime is nothing but a PATH between two geometries of space---the way it is now and the way it will be later (or was earlier)
In the Feynman path integral spirit, one says that the path a particle follows in getting from here to there DOES NOT EXIST. the path does not exist and there is no unique path that it follows!
It just somehow gets from here to there, and to calculate a quantum mechanics amplitude of it doing so, we make a weighted sum over all the paths. An integral that mooshes together all the paths from here to there, even crazy ones.
OK now SPACETIME DOES NOT EXIST EITHER
there is just the way space was shaped before
and the way it is shaped now
and there are LOTS OF PATHS of geometry to connect from then to now.
And we have to be prepared to average---to take a weighted sum including all the paths even ones that seem quite unlikely
this is the Feynman path integral philosophy (which has a pretty good track record so probably isnt totally out of step with nature)
Now what Ambjorn and Loll need is a machine that will generate a random geometry path from spatial shape A to spatial shape B.
And all geometry is in the gluing this means the machine
has to be able to output a random 4D TRIANGULATION that gets from shape A to shape B
which means it has to find ways of gluing uniformsize fivepointer simplexes
of those two orientation types together, so as to connect from A to B (3D conditions of space before and after)----and do it in an orderly layered way.
the more I read of this explanation by Loll the more I think that this is actually what a quantum theory of gravity ought to look like.
I mean that it ought to provide a path integral in the space of geometries.
Or a measure on the space Geom(M) of 4D geometries.
I DONT CARE IF IT IS SIMPLICIAL or not. Simplexes and gluing is just one way of describing a point in Geom(M)-----just one way of specifying a 4D geometry-----i.e. a path from A to B.
If someone can find a general way of describing a 4D geometry that is less messy than with simplexes that would be great! however my experience with coordinates is that the minute you try to do it with coordinates and metrix and ten-sores and coneckshuns, in that moment you have opened the closet of the nineteenth century and it is very difficult to close the door back up.
Another nice thing is that it doesnt matter if the path is jagged and zigzag because its QUANTUM so it gets blurred with other paths.
this is a great thing, and it is reminiscent of the original Feynman path integrals with were zigzag piecewise linear jagged and thus completely unrealistic paths---- the real particle wouldnt behave like that but it DOESNT MATTER you still calculate good results because the jagged things are blurred together in the weighted average. Well the same thing happens here: the Ambjorn Loll approach is intrinsically quantum because when you glue simplexes together, especially these uniformsized ones, you almost never get anything FLAT you get something which a PF poster has called a "broken glass" look. But all that averages out and the overall effect can be smooth.
Renate mentions that somewhere. i will try to find the page.
nightcleaner
Dec21-04, 09:21 PM
I mean that space is not actually to be imagined as diced up into little simplices, because maybe there IS no minimal distance. Maybe we just THINK that planck length indicates some fundamental minimal length and it really doesnt!.
Maybe. But consider the Compton wavelength, where for example the Compton wavelenght of an electron is calculated from the energy of the electron. If a universe has a measurable amount of energy, then it should have a minimum Compton wavelength. Since more energy means smaller wavelength, the Compton wavelength of a universe should be the smallest length possible in that universe.
You could substitute "measurment" for "universe" in the case of any real world observational system.
If there is a minimal length, then there is a minimal time, given the maximum velocity c where c= minimum length/minimum velocity. All the other units, such as energy, power, voltage, current, resistance, etc can be calculated from these base unitis, see Wikipedia, Natural units.
http://en.wikipedia.org/wiki/Natural_units
I would like to challenge the assertion found in Kaku (Hyperspace p. 10) and elsewhere that humans cannot visualize in four dimensions. I base my proposition on my personal experience of discovering binocular vision at a delayed age, so that I can remember vividly the experience of "seeing" the world in 3d for the first time, when before I had "seen" only in the flat, monocular view. I can tell you it was a very exciting experience, given to me originally by my opthamologist through the careful selection of lenses and mirrors. I was sixteen years old.
Since I have had the distinct pleasure of "popping up" into the 3d vision from the 2d world, I see no reason why a further progression should not be possible from our usual and common 3d vision into a 4d vision. Or five or six or any required number of dimensions. It just takes careful thought and practice. I see regularly in 3d now, using both eyes at once, because once I had seen how beautiful it is, I practiced it until I could do it without mirrors and lenses.
Vision in four dimensions is not much more difficult. Anyone who can catch a baseball should have no trouble with it. A catcher has to know where the object is, where it was a moment ago, and where it will be by the time of interception. Moving across a field to catch a ball on the run is clearly a four dimensional activity. Why should we have difficulty seeing what we are already able to do?
nightcleaner
Dec21-04, 10:02 PM
but there is another kind of fivepointer you maybe did not expect that may be thought of as dual to this one----it has 4 spacelike edges and 6 timelike!
this kind has 3 points on the ground and 2 points upstairs in the next layer (or turning it over) downstairs in the prior layer.
Ok so by upstairs and downstairs here you are meaning in the instant next in the future or in the past? Please confirm this so I know I have gotten your point.
nightcleaner
Dec21-04, 10:22 PM
This is just a reminder that after all spacetime is nothing but a PATH between two geometries of space---the way it is now and the way it will be later (or was earlier)
In the Feynman path integral spirit, one says that the path a particle follows in getting from here to there DOES NOT EXIST. the path does not exist and there is no unique path that it follows!
It just somehow gets from here to there, and to calculate a quantum mechanics amplitude of it doing so, we make a weighted sum over all the paths. An integral that mooshes together all the paths from here to there, even crazy ones.
OK now SPACETIME DOES NOT EXIST EITHER
there is just the way space was shaped before
and the way it is shaped now
I have to challenge this view of the Feynman path integral. This interpretation depends on the idea that "really" the particle can only follow one path, no matter what the math says, no matter what the two slit experiment says. One particle, one path.
However, there is another interpretation, one which is fully consistant with the Feynman path intergral, as described in Quantum Electrodynamics. That is the many worlds interpretation of Everette, Deutsch and others, or in my idiosyncratic formulation, the many times interpretation. In this paradigm, the particle does indeed follow every path available to it, just as a single photon is shown to reflect off of every part of a mirror in QED, which is an unavoidable result. This is what the photon or the particle actually does. What we see in our 3d 1t vision is only one edge of the simplex which the object occupies.
This view makes all the path integrals real, as opposed to the method you have chosen, which says only one can be real, while all the others are some sort of statistical trick, a mathematical illusion. Why not use Occam? You already have huge evidence that the other dimensions are present, and that they have a definable geometry, and that our usual vision is limited to 3d in space and one in time.
Four pick three, Marcus. There are four dimensions in spacetime, of which we pick three to hold our view of space, leaving one of time. How many ways are there to do this? Four ways, Marcus. Four possible paths of time from any instant. You choose.
When we move through four dimensional spacetime, at any instant there will be three dimensions of choice (space) and immediately beyond them but still in our quantum view a fourth dimension, which is time. The fourth in this case is not unique, but one of four possible time dimensions. The fifth point of the simpex? You are standing upon it. You the observer, three space, one time, five points.
Ok so by upstairs and downstairs here you are meaning in the instant next in the future or in the past? Please confirm this so I know I have gotten your point.
yes.
I am picturing spacetime as (they often say) folliated
that is to say "leaved", or layered
like philo dough
this is still at an intuitive stage for me and I cant
answer any very rigorous questions
but we do seem to be seeing it similarly
so on a visual level things are ok.
this folliation is a kind of representation of causality or
temporal ordering, as I see it, deeper down is into the past
...
Four pick three, Marcus. There are four dimensions in spacetime, of which we pick three to hold our view of space, leaving one of time. How many ways are there to do this? Four ways, Marcus. Four possible paths of time from any instant. You choose.
When we move through four dimensional spacetime, at any instant there will be three dimensions of choice (space) and immediately beyond them but still in our quantum view a fourth dimension, which is time. The fourth in this case is not unique, but one of four possible time dimensions. The fifth point of the simpex? You are standing upon it. You the observer, three space, one time, five points.
this is witty and entertaining but...
well and provocative too, but...
I probably am not going to respond because of a deeply engrained
intellectual laziness.
besides my wife is playing an Elvis Presley's greatest hits collection which she does whenever she sews (it helps regress back to the 1950s when women DID sew and it all seems to fit) and how can I think philosophy under the circumstances
I liked your reasoning about the Compton wavelength
It is a little like the Zen experiment of dropping a drop of ink into
a glass of water and watching it until......
As I recall, the planck mass is 22 micrograms
so if the mass of the universe is a billion planck masses (of course it is really much more, since that is only 22 kilograms!)
then the compton of the universe is one billionth of the planck length.
nightcleaner
Dec21-04, 10:54 PM
Now what Ambjorn and Loll need is a machine that will generate a random geometry path from spatial shape A to spatial shape B.
And all geometry is in the gluing this means the machine
has to be able to output a random 4D TRIANGULATION that gets from shape A to shape B
which means it has to find ways of gluing uniformsize fivepointer simplexes
of those two orientation types together, so as to connect from A to B (3D conditions of space before and after)----and do it in an orderly layered way.
This is what I have been trying to give you. Consider the Compton wavelength of the universe. Consider a sphere of radius one universal Compton wavelength (call it a Planck, it is shorter to spell and afaik it is the same thing). Consider a dense stack of these spheres. That is what 4d spacetime looks like.
Now consider the observer as if the observer could occupy a single sphere. I know we are too big to fit into a single sphere of that size, but suspend your disbelief on this point for a moment and I will try to remember to come back to it. For now, just accept that there are larger spheres which we do fit into, and they behave the same way as the Planck size spheres I am describing.
If the observer occupies one sphere, then there are twelve spheres around the observer. Each of these spheres is a next instant. In a sense, they make a layer around the observer, a layer of events that are infalling at the speed of light. The observer in the one sphere must wait until the next instant to know what is happening in the next layer. In a sense, the universe of the observer is growing one layer per instant.
But the observer is always moving. I can tell you why the observer has to move in a moment, but stay with me here. Because the observer is moving, there are some spheres which are left behind. No information from those spheres can catch up to the moving observer. Altho all twelve spheres are anext to any spacetime instant, the observer only "sees" the ones directly in the path of the movement. The others are left behind.
Hence, the observer seems to occupy a four dimensional spacetime universe in which there are three visable spatial dimensions and one time dimension. The observer follows a path, and sees all other objects following paths. The real higher dimensional structure is not seen, but only the path edges of the simplices. But it exists and we know it exists because 1. The math requires it (eg string theory); and 2. Observations in the laboratory confirm it (eg 2 slit wave particle duality experiments); and 3. Cosmological observations confirm it (eg, GR and dark energy/ dark matter).
Now for gluing simplices. Yes, you can build spaces by gluing simplices. Not all spaces that are possible in geometry are possible in our universal conditions. AJL seem to want to use the octet space formed by building entirely with tetrahedrons. I have explained elsewhere why this is not the optimal space for modeling our univerese. One must ask where the tetrahedrons come from? I have given a derivation from Planck lengths. It does not involve only tetrahedrons, but also includes cubes. It can be seen to hold both triangular simplices and tetrahedral ones, as well a cubic forms.
The isomatrix. The cubeoctahedron. Face Centered Cubic. Please look at the link.
nc
nightcleaner
Dec21-04, 10:59 PM
Hi Marcus. I just now saw that you were online and replying. By all means be with your family. We have to keep our priorities straight, and this stuff here is all starlight on a distant sea. I trust you will return again, refreshed and ready to trade points with me? Be well, on this longest of nights.
nc
nightcleaner
Dec21-04, 11:05 PM
I liked your reasoning about the Compton wavelength
It is a little like the Zen experiment of dropping a drop of ink into
a glass of water and watching it until......
As I recall, the planck mass is 22 micrograms
so if the mass of the universe is a billion planck masses (of course it is really much more, since that is only 22 kilograms!)
then the compton of the universe is one billionth of the planck length.
The Planck mass is derived in a different way from the Planck length, but I value your attempt to quantify this, since I work easier with words and images than with numbers. I hope we can work through this so I can see if the ideas match up with the observations. At first glance you have provided a challenge. Thanks!
I have been studying this for a while now and lets see if I can get it right in one go.
The Planck length and Planck time were given in pretty much their current form by Max Planck about a hundred years ago. No one seems to know how he came on the right numbers, but the Planck length is about 10^-34 cm, and the Planck time is about 10^-43 seconds, which when divided should give the speed of light in cm per second at about c=10^9cm/s. I'll have to check and see if I got those numbers right.
The Planck mass is derived by considering how much mass can be crammed into a small space before it collapses into a Schwartzchilde singularity. I think if memory serves that the small space is a proton diameter. The mass that can be crammed into a proton diameter is about the mass of a small flea.
So, your estimate may not be based on first principles.
I am going to go look up the numbers and derivations. Maybe I'll even find out how to calculate the Compton Wavelength while I am at it.
Thanks,
nc
nightcleaner
Dec21-04, 11:47 PM
yes.
I am picturing spacetime as (they often say) folliated
that is to say "leaved", or layered
like philo dough
this is still at an intuitive stage for me and I cant
answer any very rigorous questions
but we do seem to be seeing it similarly
so on a visual level things are ok.
this folliation is a kind of representation of causality or
temporal ordering, as I see it, deeper down is into the past
just have to answer this first. Jumping to the end of my line of reasoning without going over the middle ground, when we get to our current spacetime, the universe we find ourselves in is very nearly flat, and we are very nearly large. The past is in, the future is out, and we exist in a thin layer. The layer is very nearly regular, but has some flaws in it which come from the difference between the close pack face centered cubic form and the curvature of spacetime at our distance, quite a large distance, from the origin. This is a spacetime distance and is reflected in our observations of the cosmos, eg CMBE, but is not to be thought of as distant from us in space. The origin now and always is within. The flaws are matter and energy, the very regular areas are "vacuum" or empty space, and the vacuum fluctuations come about because of the uncertainty of position of the flaws. You could think of the Planck spheres as being very nearly perfectly densely packed, but not quite perfectly. The little bit of slop in the fit accounts for all the phenomena we observe. We never observe the spacetime directly.
nc
nightcleaner
Dec21-04, 11:54 PM
From this source:
http://en.wikipedia.org/wiki/Natural_units
I get this derivation:
<tr><td>'''[[Planck mass]]'''</td>
<td>[[Mass]] (M)</td>
<td><math>m_P = \sqrt{\frac{\hbar c}{G}}</math></td>
<td>[[1 E-8 kg|2.17645 × 10<sup>-8</sup>]] [[kilogram|kg]]</td>
</tr>
but I see it does not show the formula in this forum. I'll have to review my latex skills.
m_p=\sqrt{\frac{\hbar c}{G}
That was easy. Just replace math with tex and <> with []
so
m_p=\sqrt{\frac{\hbar c}{G} = 2.17645 x 10^-8 kilogram
I guess that's about 2.2x10^-5 grams, or 2.2x10^-2 micrograms, or .02 micrograms? Anyway within a couple orders of magnitude.
Now to find out how to calculate Compton wavelength.
I find the Compton wavelength of the electron to be listed at :
http://en.wikipedia.org/wiki/Physical_constant
as
\lambda_e=h/m_e c
This is not helpful. I see that the Compton wavelength already assumes the value of the Planck length, h, is known. If we write
\lambda_U=h/m_U c
then lambda_U=h when m_U c =1. Since c is 1 in base units, then m_U is just 1 also. But what if we go back to CGS units? When is the mass of the universe times the speed of light equal one? When the mass of the universe is the inverse of the speed of light. c = 3x10^8 meters/second so 1/c = 3x10^-9 seconds per meter. What kind of a mass is that?
Lets try a different tack. I interpret Lambda_e as the radius of the region in which an electron is most probably found. (Since an electron is a point particle, this radius is really the radius of the area in which the electron is most likely to interact with photons via quantum fluctuations and virtual particles.) So Lambda_U would be the likely radius of the universe. We can get that from cosmological data. But should we use the inflation value of 78 billion light years, or the age of the universe data of 13 billion light years? Well they are only an order of magnitude or so apart.
Anyway the radius of the universe is equal to the smallest possible length in the universe divided by the mass of the universe times the speed of light. Sounds easy enough. Solving for h, h equals the radius of the unverse times the mass of the universe times the speed of light. Ooops. That looks like a big number.
But the extremely small is the inverse of the extremely large. Can we be justified to say then that the formula should be inverted on one side? Then we would have radius universe = mass universe times speed of light, divided by smallest possible length. So smallest possible length equals mass of universe times speed of light, divided by radius of universe.
\lambda_U=1/\lambda_p
\lambda_U=m_U c/h
h=m_U c/\lambda_U
nightcleaner
Dec22-04, 02:59 AM
ok I think I see it better now. A Compton wavelength is not a length at all, but a frequency, T^-1. So to recover a length from the Compton wavelength, it is necessary to divide a velocity by that frequency.
working on it. Have to sleep. nc
m_p=\sqrt{\frac{\hbar c}{G}
= 2.17645 x 10^-8 kilogram
I guess that's about 2.2x10^-5 grams,..
that is the same as 22 x 10^-6 grams
that is the same as 22 micrograms
because a microgram is a millionth of a gram----that is, 10^-6 gram
I am glad to see you using Latex
in many discussions you will find h standing for a version of planck's constant (not planck length) and hbar standing for a reduced version of planck's constant, namely h divided by 2 pi.
the formulas defining the planck quantities customarily use hbar, as you did when you wrote:
m_p=\sqrt{\frac{\hbar c}{G}
but the formula for the Compton sometimes uses h and sometimes hbar, so what people call the compton can vary by a factor of 2 pi.
humans, imperfect as they are, sometimes waffle a bit in their notation
requiring tolerance and goodwill on everyone's part
in the path integral approach to quantum spacetime
one has an integral which is an average over all geometries
(a weighted sum of 4D geometries that get you from one 3D condition of space to another one later on, but we have quantum uncertainty about what went on in between)
now Geom(M) the vast warehouse of all possible 4D geometries is a huge place to wander around in
and actually summing or integrating over all those possibilities (as a very dutiful consciencious person would do when asked to find the average) is next to impossible
(there are more degrees of freedom in the geometry of a whole spacetime than, for instance, in the mere path of a single particle going from here to there----so the vastness of the possibilities is vaster)
nevertheless that is the idea of quantizing, you have a bigspace of all possibilities and you define wavefunctions on the bigspace and you define quantum states and you integrate and so on----you have to be able to describe a blur of possibilities and an indefiniteness about how you got from one situation to another
so what to do? the Monty approach says to define a small set of MOVES which allow you to do a RANDOM WALK in the vast warehouse of geometries------and go for a walk, and get lost, and wander about AVERAGING AS YOU GO
this is very zen because it uses the vastness in order to overcome the vastness----because it is very huge you can wander randomly and be sure of not coming back or getting caught in a loop---and therefore you can make a RANDOM SAMPLE and the average of a random sample is a good estimate of the real average.
So Renate "the Fox" Loll defines what she calls the Monte Carlo moves, which are rearrangements of the simplex-gluings which get you from one geometry to another "nearby" geometry
these moves are socalled "ergodic" which means that if you do them enough you will eventually pass thru every configuration in the warehouse.
ergodic is an idea about mixing which means THOROUGH
the moves are very little but but they thoroughly stir the geometry
so if you do enough of these little moves you will completely stir things up.
Jan Ambjorn should be praised for this too. And it goes back to 1980s and 1990s when people applied it to "euclidean" (not lorentzian) path integrals and dynamical triangulations. but even though the Monty method did not originate with Ambjorn and Loll I find it admirable and they are the ones that finally applied this approach to the Lorentzian setup and finally, in 2004, made it work right.
nightcleaner
Dec22-04, 09:32 AM
Thanks Marcus. I see I need to brush up on my metric prefixes. I really am being dragged away from here by commitments. Be well, I'll be back tomorrow or later today if I can manage. Richard.
Thanks Marcus. I see I need to brush up on my metric prefixes. I really am being dragged away from here by commitments. Be well, I'll be back tomorrow or later today if I can manage. Richard.
no rush
take care of the commitments
the green arrow in the title of my previous post was an accident of the fingers
Bianca Dittrich did her Diploma thesis on dynamical triangulations approach to quantum gravity, back in 2001.
then she moved over to Loop and working with Thiemann.
she could move back.
Renate Loll has also migrated between Loop and DT.
they are kindred approaches, even though they do not agree about the
discrete spectrum of area and volume (which I think makes it likely that only one can be successful in the long run)
Bianca's 2001 (undergrad? masters?) thesis was in German and is probably on file in Univ. Potsdam, and is called
Dynamische Triangulierung von Schwarzloch-Geometrien
...
so what to do? the Monty approach says to define a small set of MOVES which allow you to do a RANDOM WALK in the vast warehouse of geometries------and go for a walk, and get lost, and wander about AVERAGING AS YOU GO
...
these moves are socalled "ergodic" which means that if you do them enough you will eventually pass thru every configuration in the warehouse.
ergodic is an idea about mixing which means THOROUGH
the moves are very little but but they thoroughly stir the geometry
so if you do enough of these little moves you will completely stir things up.
...
in lower dimension versions of the theory the moves are few and comparatively easy to visualize. I want to get more familiarity with the Monty moves in 4D (where there are more moves and harder to visualize)
In 4D the authors (hep-th/0105267) give us a set of 10 moves
but some are just the REVERSE of others so, depending how you count there are really only 5 (or maybe 6) different moves
these can be given names which are just a couple of numbers like for example these:
(2,8)
(4,6)
(2,4)
(3,3)
The move called (4,6) takes a cluster of 4 chunks and redivides it so it becomes 6 chunks
the reverse of that could be named (6,4) and simply does the reverse.
the authors have a picture of how this redividing is done---Figure 7 on page 25---and they also spell it out by listing the vertices of the chunks---on page 24. It is actually pretty simple.
This redividing up of clusters of chunks RESPECTS the layering, or causality, or foliation, or Lorentzianity-----whatever you want to call it. If some tetrahedron starts out purely spatial and it gets divided up then the resulting things are purely spatial and so on.
I guess I would like to run down the list of these moves and try to say in words what they do.
Remember that Geom(M) is the bunch of all the 4D geometries of the universe from bang to crunch (so far in their simulations they have a finite life universe with a crunch)
We want to see how the geometry of the universe works---quantum style---and so we are exploring this huge Geom(M) set of possibilities by doing a random walk in it. We make little bitty steps from one 4D geometry to another by just changing some detail of some cluster of simplexes.
the marvel is if we make enough of these moves, take enough of these little bitty steps, we get a representative random sample of how the geometry of the universe works
check out some of their graphics, like Figure 5 of hep-th/0411152
On page 2 of hep-th/0105267 the authors refer to Geom(M) as
"the mother of all spaces" and also as the "space of geometries" which
seems reasonable since it contains all the possible spacetimes as its elements-----a spacetime geometry is a POINT in Geom(M)
this means that wandering around in Geom(M), the space of geometries, means visiting many different spacetimes geometries in turn.
these Monty moves may seem very weak and insignificant, each one changes just one cluster somewhere in the universe, and not by very much either. but the effect of the steps is cumulative
Anyway, I want to review these 5 or so Monty moves
...
In 4D the authors (hep-th/0105267) give us a set of 10 moves
but some are just the REVERSE of others so, depending how you count there are really only 5 (or maybe 6) different moves
these can be given names which are just a couple of numbers like for example these:
(2,8)
(4,6)
(2,4)
(3,3)
The move called (4,6) takes a cluster of 4 chunks and redivides it so it becomes 6 chunks
...
Instead of saying 4D simplex I will say "chunk"
A chunk has 5 points and 5 tetrahedron for its sides.
the easiest way to picture a chunk is to put down a tetrahedron as a base
(like the base of a pyramid, sort of) and then put a new apex point "up in the air" and imagine drawing lines from each point of the tetrahedron base up to the the new point.
here is something you can visualize because it is in ordinary 3D:
take a tetrahedron and put a point in the middle of it and connect the 4 orig points to the centerpoint. Presto you have divided the orig tet
into FOUR tets.
each of the four orig. faces becomes like the base of one of the four new tets and the centerpoint becomes like the common apex
Now to business:
(2,8)
now we are in 4D and we have a spatial tetrahedron in the present and two apexes one up in the future and one down in the past. So we have a CLUSTER OF TWO CHUNKS meeting at a shared tet.
put a centerpoint into that shared tet, dividing it into FOUR tets, and connect each of them to the two (future and past) vertices.
pretty clearly we now have a CLUSTER OF EIGHT CHUNKS.
so that is what is called the move (2,8)
and there is an obvious reverse move (8,2)
(4,6)
there is a 3D thing that is easy to visulize where you start with TWO tets butting together at a shared triangle, like two pyramids base-to-base with one's apex out East and the other's apex out West. And you erase that triangle and draw a line between the east and west apex points and suddenly you see that you have THREE tetrahedrons.
well now in 4D suppose you have two tets in the present, butted together like that, and each connected to an apex up in the future and to an apex down in the past-----so you have a cluster of FOUR chunks
if you do that 3D redivision of the spatial pair of tetrahedrons so you now have 3 tetrahedrons in the present----and then connect each of them up and down to the future and past apexes as before, then you have a
cluster of SIX chunks. that is the (4,6) move and the reverse move is obvious.
The (3,3) and the (2,4) moves involve cluster of fewer chunks. I guess I will take a break here and describe them later.
These are spelled out and depicted around page 23 of hep-th/0105267
selfAdjoint
Dec23-04, 10:20 AM
Marcus, I just found this guy (http://yolanda3.dynalias.org/wb/whoiswb.html); check him out.
Marcus, I just found
http://yolanda3.dynalias.org/wb/whoiswb.html
check him out.
-----------
thanks, I will do so at once
strange URL
-----------
with Fahrenheit near zero by Lake Winnebago
you are well situated to appreciate the photo album
of life in the Bahamas. Wolfgang Beirl has a nice
offshore lifestyle.
his real business appears to be modeling financial markets
like the US stock market
his most recent lattice gravity posting (that he gives a link to)
is mid 2003
http://arxiv.org/find/hep-lat/1/au:+Beirl_W/0/1/0/all/0/1
19 simplicial/lattice quantum gravity papers
3 in 1996 but only two since then
Here is Beirl's record of some communications with Lubos Motl
http://yolanda3.dynalias.org/tsm/tsm.html
that began with Lubos comment on the recent AJL paper
nightcleaner
Dec23-04, 01:11 PM
Marcus, in regard to LQG and DT, you said:
"they are kindred approaches, even though they do not agree about the
discrete spectrum of area and volume (which I think makes it likely that only one can be successful in the long run)"
Marcus could you take a moment to expand on this? I would like to understand more about what is meant by discrete spectrum of area and volume. I think I know what area and volume are. I have a notion of discrete spectrum but need to verify. And then, in what respects do not agree?
Thanks
nc
nightcleaner
Dec23-04, 02:46 PM
Instead of saying 4D simplex I will say "chunk"
A chunk has 5 points and 5 tetrahedron for its sides.
the easiest way to picture a chunk is to put down a tetrahedron as a base
(like the base of a pyramid, sort of) and then put a new apex point "up in the air" and imagine drawing lines from each point of the tetrahedron base up to the the new point.
here is something you can visualize because it is in ordinary 3D:
take a tetrahedron and put a point in the middle of it and connect the 4 orig points to the centerpoint. Presto you have divided the orig tet
into FOUR tets.
each of the four orig. faces becomes like the base of one of the four new tets and the centerpoint becomes like the common apex
Now to business:
(2,8)
now we are in 4D and we have a spatial tetrahedron in the present and two apexes one up in the future and one down in the past. So we have a CLUSTER OF TWO CHUNKS meeting at a shared tet.
put a centerpoint into that shared tet, dividing it into FOUR tets, and connect each of them to the two (future and past) vertices.
pretty clearly we now have a CLUSTER OF EIGHT CHUNKS.
so that is what is called the move (2,8)
and there is an obvious reverse move (8,2)
(4,6)
there is a 3D thing that is easy to visulize where you start with TWO tets butting together at a shared triangle, like two pyramids base-to-base with one's apex out East and the other's apex out West. And you erase that triangle and draw a line between the east and west apex points and suddenly you see that you have THREE tetrahedrons.
well now in 4D suppose you have two tets in the present, butted together like that, and each connected to an apex up in the future and to an apex down in the past-----so you have a cluster of FOUR chunks
if you do that 3D redivision of the spatial pair of tetrahedrons so you now have 3 tetrahedrons in the present----and then connect each of them up and down to the future and past apexes as before, then you have a
cluster of SIX chunks. that is the (4,6) move and the reverse move is obvious.
The (3,3) and the (2,4) moves involve cluster of fewer chunks. I guess I will take a break here and describe them later.
These are spelled out and depicted around page 23 of hep-th/0105267
Marcus,
You said
"here is something you can visualize because it is in ordinary 3D:
take a tetrahedron and put a point in the middle of it and connect the 4 orig points to the centerpoint. Presto you have divided the orig tet
into FOUR tets."
Are you accounting here for the condition that the new point has to be non-co-spatial with the original 4 of the tetrahedron?
You said
"each of the four orig. faces becomes like the base of one of the four new tets and the centerpoint becomes like the common apex"
The four new tets each have a 2-simpex base, and these four bases share a common 3-space. The apex however is in 4d, and is not really in the common 3-space of the bases. The apex is best thought of as being infinitely removed from the 3-space of the bases, which is to say it is not in the common 3-space of the bases at all. So the lines that join the points of the bases to the apex are infinitely long parallel time-like lines, and the apex itself is not a point, but another 3-space tetrahedron offset an infinite distance, so that it could be represented as a point anywhere in the 3-space of the bases. You chose to represent it as if it were in the center of the original tetrahedron, which is a good choice since it represents each of the infinitly long timelike lines as equal in length in the 3-space, but of course infinity does not equal infinity, so the apparent equality does not hold, but is best remembered as an artifact of the visualization process, much as in a two dimensional drawing of a three dimensional object two points that seem to be close together in the drawing can actually represent points that are far apart in the 3-space. Think of a drawing of a cube. The near corner and the far corner can seem to be in the same place on the drawing, but we mentally recall that they are really in separate planes.
I am trying to follow your analysis of the chunks and moves, but have worked myself into some kind of a cross-eyed cross-legged 4-d stance and need to take time out for sugar and protien to balance the caffeine fix. As the Arnold says, with a steely gleam in his android eye, "I'll be back."
nc
281 views of this thread as of this posting
in ordinary 2D, on a sheet of paper, draw an equilateral triangle.
then put a point in the middle and connect it to each of the orig. vertices.
this divides the triangle into 3 triangles (still on the orig. 2D piece of paper)
everything is "co-spatial"
do the same with a tetrahedron,
Are you accounting here for the condition that the new point has to be non-co-spatial with the original 4 of the tetrahedron?
...
no I am counting on the new point being actually inside the tetrahedron
and so part of the same 3D space that the tetrahedron occupies
[clarification: in this case we are not making a 4simplex, we are
DIVIDING UP an existing tet to make 4 smaller tets, by placing a point in the center]
Marcus, in regard to LQG and DT, you said:
"they are kindred approaches, even though they do not agree about the
discrete spectrum of area and volume (which I think makes it likely that only one can be successful in the long run)"
Marcus could you take a moment to expand on this? I would like to understand more about what is meant by discrete spectrum of area and volume. ..., in what respects do not agree?
...
Here is a brief response to this question. I would be glad if anyone wants to explain it in more detail.
this issue points to a crucial difference between LQG and DT.
DT is fairly new and i do not know of the area and volume operators being studied yet. I suspect that when they are defined they could turn out NOT to have discrete spectrum.
But in LQG associated with any material surface, like a desktop, there is an operator on the hilbert space (the quantum states of geometry) which corresponds to measuring the area of that surface----it is an operator called an "observable" and it has a discrete set of possible values, including a smallest positive value.
the language is a bit technical, even a bit awkward, and gives the impression of more difficulty than there really is.
the essential is that from LQG one has learned to expect that measuring the area will give discrete levels of area, like the energy levels of an atom.
as an atom can be excited and made to go up into a higher energy level (or an electron of the atom can, however you think about it)
so also the gravitational field can be excited and go up into a stage where things have more area, and also more volume.
but the areas and volumes only go up in little "jumps"
(too small to measure with today instruments)
and this is a characteristic of LQG-------indeed in LQG COSMOLOGY even the size of the universe goes up in miniscule "jumps" and at the time of the bigbang this turns out to be significant (even tho the steps are very very small it still matters)
But I have not seen any hints that when they get DT more developed and get area and volume operators, that they will have a discrete set of possible values---a discrete spectrum---and that areas and volumes will increase and decrease in jumps (as the gravitational field changes).
So this could be a serious disagreement between DT and LQG
nightcleaner
Dec24-04, 12:49 AM
Hi all
Some more on 4d visuals.
We have seen how the 3d simplex is a tetrahedron. Now we wish to consider the shape of things in the 4th dimension. We have discussed how logical development by geometric principles lead us to think that the 4d simplex will have five points. Moreover, these five points consist of a 3d simplex tetrahedron, and one point which is not in the same 3d space as the tetrahedron. This follows from the condition placed on upper dimensional points that they not be part of the lower dimensional simplex.
One way to model the 4space simplex, which will have five points, is to place the fifth point somewhere in the same 3space as the tetrahedron and then try to remember that it is not really a part of the same 3space. We then draw lines from each point of the tetrahedron through 3space to the new point. One convenient way to do this is to place the point at the center of the tetrahedron, thus dividing the tetrahedron into four 3spaces which are interior to the tetrahedron.
This model may prove useful, but we need to remember that the central point is not allowed to be in the same space as the original tetrahedron. I have suggested that we think of the fifth point as being offset in time, or alternately, offset to another 3space which is at a sufficient distance so that there can be no contact with the original 3 space. This offset distance is, for any practical purpose, infinite. This means that the interior lines to the central point in the tetrahedron are infinite. We could represent this infinity either by a gauge variance or by a space-like curvature.
Infinities are not welcome in physics problems because they lead to divergent conclusions. In the math sense, infinity is not equal to infinity so calculations are nearly impossible, making any theory which relies on infinities non-physical. Most serious physics reasearchers rule out any such theory on the grounds that it cannot describe the physical processes we find around us in this universe. I would like to suggest that we hold on to the tetrahedron with its infinitely removed center for a moment, if for no other reason than it gives us the most symmetric possible 3space model of this 4space system.
Meanwhile, let us return to the idea that the offset is not in space, but is in time. This has the advantage of allowing us to place the fifth point in the same space as the tetrahedron but offset one unit of time, thereby removing the infinities. We can still use our model of the tetrahedron with a central point, with a few modifications.
First, we must keep in mind that this central point is a representative point, and our choice of placement at the center is merely a convenience. The point could equally well be placed anywhere in the 3space of the model, because it is not really in the 3space of the model at all, but is offset by one unit of time. The only limitation to the placement comes from the speed of light, which causes the set of possible placements in the next unit of time to be limited to a three dimensional sphere. Any placement outside the limits of the sphere results in a discontinuity between time units, and perhaps it would be best for now to regard time as continuous.
Now we can take our tetrahedron simplex in 3space, and displace it one unit of time, and regard them side by side, as if they were two tetrahedrons side by side in 4space. We can say that the tetrahedron has not actually moved at all in 3space, so all of its points in the offset space are in the same relationship to each other as they were in the original 3space tetrahedron.
But what relationship do the two tetrahedrons have, in the 3space model, to each other? That is, if we represent the original tetrahedron, call it tet1, and the offset tetrahedron, call it tet2, in the same 3space, do we have any justification for saying that the lines in tet1 are parallel to the lines in tet2? It would be convenient for visualization purposes if we could say that tet2 is not rotated compared to tet1 in the 3space model, but is it justified?
We have to remember that tet2 can be placed anywhere in the 3space model. It could be placed offset to the right of the viewer and then up, and then forward or back, with no preferred position. A sequence of these moves can result in any desired rotation, at any desired location, so we cannot justify the convenient proposition that the two tetrahedrons should have parallel lines in the 3space model
Moreover, tet2 can have any size compared to tet1, within the 3space limits set by the continuity provision as determined by the spacetime ratio, c. Tet2 could be represented as entirely within or entirely outside of tet1, and any size from a single point to the full extension of 3space surrounding tet1. Within the limits of the tet1 3space set by the discrete unit of time, every point has to be considered as equal in terms of representation for the offset.
The model now seems rather blurry, and of little use in visualization. However, we can make some improvements. We can justify some preferred conditions. For example, no matter where we place tet2, and no matter what rotation, there is always a one to one correlation between the four points in tet1 and the four points in tet2. From any apex in tet1, there are four lines leading to tet2. Likewise from any apex in tet2, there are four lines leading to tet1. We can count these lines. There are sixteen lines leading from tet1 to tet2, and sixteen lines leading from tet2 to tet1. Can we say that the sixteen lines 2=>1 are the same sixteen lines 1=>2? No.
We have to remember that there is no preferred orientation. All rotations must be considered. The lines from 2 to 1 are therefore not simple one dimensional lines. They are cones. They start as a single point in the tet of origin, but by the time they arrive at the tet of destination, they are no longer zero dimensional in cross section. Because of this fact, we cannot say that a line from tet1 to tet2 is matched by any line from tet2 to tet1. We are left with the unavoidable conclusion that there are thirty two cones.
These cones are not without structure. They expand from the origin to the offset, and moreover, they are not of consistant internal density. Rather, there is a spectrum of preferred densities within the cone. This results from the fact that the offset tetrahedron can take any rotated position. To draw the cone, we must consider all possible rotations. A moment of consideration will convince you that not all points in three space will be equally represented in these rotations.
The tet2 can be represented, in its own 3space, as unchanged in any way from tet1, except for the offset in time. This identity can be asserted as long as we keep the two 3spaces apart in our mind. The blurring of the model only comes about when we try to represent the two tets as if they were in one 3space. But it is allowed, for example, to indicate the two tets on one sheet of paper, so long as we keep in mind that they are seperated by one unit of time. We can do this by drawing a circle around one of the tets to remind is that it is not in the same space as the other tet. In fact it might be a good idea to draw circles representing their spaces around each of the two tets. Then we could label one circle 3space1 and it contains tet1, and the other circle is 3space2 containing tet2. Now when we draw the lines, we have to draw not from point one tet1 to point one tet2, but from point one tet1 to the entire circle containing tet2. Each resulting cone then represents all four lines from point one to the four apexes of tet2.
Now lets consider all the possible rotations of one tet in 3space.
First we have to choose a point of origin for the rotation. The simplest choice would be the center point of the tetrahedron. This choice gives us a sphere when the tet is rotated in every possible way around it. The sphere is equal density everywhere on its surface, but the interior of the sphere has a density structure.
This density structure is definable from the existance of edge lines and surfaces of the tetrahedron. Consider for example the density of the region close to the center. This density is defined only by lines radial from the center point to the apexes of the tet. Then consider a region near to the interior surface of the sphere formed by the rotated apex points. The density of this region is also defined by the rotation of the radial lines, but in addition has definition provided by the existance of lines between the apexes. These lines are the edges of the tet. When rotated, these lines define a region limited to the difference between the radial distance to the center of the cord defined by two apexes and the radial distance to the apex itself. The center of the cord is closer to the center of the tet (origin of rotation) than is the apex.
This results in a density of definition which is not merely a spherical surface of points equidistant from the center, but is a sphere with a surface that has thickness, an inner and an outer surface with an incremental space defined between them.
But we also have to consider rotating the tet around the apex points. This gives another equal surface density sphere, but one which is larger than the rotation from the center point. It is larger because the distance from the center of the tet in the first rotation is less than the distance of each apex from the other three apexes. The sphere also has its own internal density structure formed in a similar manner to that described above. Then we have to rotate the tet around each of the other apex points.
Now we begin to see the 4d stucture in some detail. There are spheres within spheres, and spheres intersecting spheres. There are unique definable points within this 4d structure, and it has a spectrum of densities in different regions.
In this discussion, I have tried to show the limitations and conditions which can be placed on a three dimensional model of a four dimensional structure. The simplest model in three dimensions, that of a tetrahedron with a central point, is certainly useful, as long as we keep the conditions in mind.
I have also shown that a four dimensional structure is not merely an undifferentiated space, a blur of 3space, but can be shown to have definable geometric points, lines, planes, surfaces, densities, and spectra.
I have shown that mappings from points in one 3space to their corresponding points in another 3space, even when there is a one to one correspondance between the points, can not be taken as one dimensional lines but have to be considered as three dimensional objects (cones) with internal structure.
In conclusion, this discussion has application to the Regge calculus in that it shows that it is not sufficient to model 4space objects in 3space by a dynamic triangulation with variable but discrete side lengths. I suggest that a better fit to physical measurements can be achieved by spectral analysis of rotated 3space objects.
Richard
343 views as of posting
I am still working on the small project of reviewing the 4D Monte Carlo moves. BTW other sets of moves would probably be OK, the moves just have to be simple and easy to program in a computer and ergodic in the sense that if you do them enough it thoroughly explores the space of geometries Geom(M).
Part of this is a re-edit of post #25 to make it clearer. In that post I got halfway thru the list of moves and then got distracted, so i will review that and then try to proceed further with it.
To have fewer syllables to say: Instead of saying 4D simplex I will say "chunk" and instead of tetrahedron I will sometimes say "tet"
A chunk has 5 vertices and 5 sides, which are tets.
the easiest way to picture a chunk is to put down a tetrahedron as a base
(like the base of a pyramid, sort of) and then put a new apex point "up in the air" and imagine drawing lines from each point of the tetrahedron base up to the the new point.
As preliminaries, here is some things you can visualize in ordinary 3D and THEY DONT REQUIRE 4D:
take a tetrahedron and put a point in the middle of it and connect the 4 orig points to the centerpoint. Presto you have divided the orig tet
into FOUR smaller tets.
(each of the four orig. faces becomes like the base of one of the four new tets and the centerpoint becomes like the common apex)
here is another 3D thing where 4D IS NOT REQUIRED. You start with TWO tets butting together at a shared triangle, like two pyramids base-to-base with one's apex out East and the other's apex out West. And you erase that triangle and draw a line between the east and west apex points and suddenly you see that you have THREE tetrahedrons. they all share that eastwest line.
we will use these two maneuvers----one is a purely 3D way to divide ONE tet into FOUR (by adding a point) and the other is a purely 3D way to divide TWO tets into THREE (by adding a line)
Now let's consider some 4D moves:
(2,8)
Suppose we have a CLUSTER OF TWO CHUNKS meeting at a shared tet. Imagine it is a spatial tet in the present plus two apexes one up in the future and one down in the past.
Put a centerpoint into that shared tet, dividing it into FOUR tets.
(as described earlier)
Then connect each of them to the two (future and past) vertices.
We now have a CLUSTER OF EIGHT CHUNKS.
That is what is called the move (2,8)
and there is an obvious reverse move (8,2)
(4,6)
Now suppose you have two spatial tets in the present, butted together at a common triangle. Suppose each is connected to an apex up in the future and to an apex down in the past-----so you have a cluster of FOUR chunks
Purely in 3D, we can redivide the spatial pair of tets to make 3 tets (as described earlier). So we now have 3 tetrahedrons in the present, and we connect each of them up and down to the future and past apexes as before, so that we have have a cluster of SIX chunks. That is the (4,6) move and the reverse move is obvious.
-----------that finishes that, the rest is just a comment------
I still have to review the (3,3) and the (2,4) moves.
All these are spelled out and depicted around page 23 of hep-th/0105267
Maybe it is worth mentioning that the moves discussed already both require 7 points. In the case of the move (2,8) the initial cluster needed only 6 points to define but one had to add a point, and in the other move (4,6) the initial cluster already needed 7 points to define-----5 for the pair of abutting tets in the present, plus two apexes in future and past.
But the moves I still have to describe, namely (2,4) and (3,3),
are simpler in the sense that they take place in the context of just 6 points.
Also these moves are interesting because THEY DONT CHANGE THE PRESENT AT ALL. More is true: they dont change spatial layer triangulation either in the present or in the future. They only change HOW YOU GET FROM ONE LAYER TO THE NEXT.
they just change the timework links that are sandwiched between spacework layers.
so these two remaining types of moves are in a way more simple, but actually I had more struggle visualizing them from the pictures in
http://arxiv.org/hep-th/0105267
maybe this is their fault (the moves really arent simpler) or else my fault, or maybe the picture's fault (maybe better pictures could be made, say by coloring)
Imagine 6 points, 3 in the present and 3 in the future------from here on we only are dealing with 2 layers at a time so I will choose them to be present and future and not refer to past.
So there are two triangles, call the downstairs triangle 134 and the upstairs triangle 256.
Now in this little venue marked by these 6 points, we can imagine a tet named 3456------this is not a spatial tet, it is timelike! It goes slanting up between layers,
and by joining this tet to apexes 1 and 2, we can make two chunks!
HERE IS WHAT MOVE (2,4) DOES.
It erases tet 3456, and shoots a line from 1 to 2, and the same bulk (which used to be TWO chunks butted together at a shared tet) is now divided up into FOUR chunks all meeting at the common line 12.
The authors write this move like this, where the underline shows what the chunks have in common
13456 + 23456 goes to 12345 +12346 + 12356 + 12456
-----analogy in 3D----
this is analogous to a move we described earlier that you can do entirely in 3D where you have two tets meeting at a triangle and you erase the triangle and shoot a line from the east vertex to the west vertex and then you have three tets meeting at that line.
---------
Now let's see what is left to do. We have dealt with at least one instance of the move (2,4)
-----comment-----
keep remembering the analogy of shuffling a deck of cards.
if you do enough shuffles you will explore all the possible orderings of the deck.
these moves are very simple modifications of a 4D triangulated geometry ("all the geometry is in the gluing") and they are only dealing with one small local cluster of chunks chosen randomly from perhaps zillions.
these moves are like a shuffle so simple that it only swaps two cards or permutes 3 or 4 cards.
but if you do enough of these very simple shuffles then in the end you make a kind of random walk thru the whole space of possibilities.
this is at the core of the MonteCarlo method which the authors have programmed, which explores the 4D geometries of their small universes
(so far at most a third of a million chunks) evolving under the Einstein rules of dynamic geometry------which rules Tullio Regge translated into rules about simplexes.
you might like to check out the animations at Jan Ambjorn website.
they give some of the flavor.
there is a minor variation on the move (2,4) just described
I mentioned that the set up was 3 points in the present and 3 in the future and it can also be set up with 4 and 2-----for example 4 points in the present and 2 in the future.
the move is written exactly the same way
13456 + 23456 goes to 12345 +12346 + 12356 + 12456
and the same thing happens as described in the previous post.
godzilla7
Dec25-04, 02:18 AM
hi guys not sure if this is appropriate to this discussion, but seeing as you mentioned the big bang; read an article about the IMAP data being flawed because of polorizations in our solar system, if this is true does it have implications on the amount of matter we have calculated, and if it does would dark matter be irrelevant, kind of a naive question but the article was a little vague can someone clarify the errors and make a guess at the implications :confused:
... read an article about the IMAP data being flawed because of polorizations in our solar system, if this is true does it have implications ...
I read an article about that too. I think you mean WMAP.
the first few bumps in the CMB seem "too" alligned with the solar system for that to have been an accident. It could be coincidence, or it could indicate that something we dont know about in the the solar system is acting as a source or sink of microwave and affecting the low-order poles.
My take on it (just my personal reaction) is that it is overblown----the estimates of dark energy and other good cosmological stuff depend on the higher order poles----the smaller bumps that make the CMB skymap all speckly. Cosmologists estimates about the universe dont depend on those few low-order huge bumps. they might be affected by some dust or crud or other unexpected local effect and one would then just factor that out and throw that away and one would still have the essential WMAP picture of the microwave background and all that can be inferred from it.
so if there is something in that coincidence then it just tells us some minor new detail about the solar system that we didnt know, and doesnt affect the picture of the universe at large (guess).
we should try to maintain our focus on quantum gravity in this thread.
quantum gravity means quantum models of the universe
specifically of the GEOMETRY of the universe
the 1915 Gen Rel insight was that gravity equals geometry
ultimately you cannot have a quantum theory of gravity unless you
have a model of the evolving geometry of the universe
oddly rather few approaches to quantum gravity
actually model the universe
or even offer a quantum version of Einstein's 1915 Gen Rel equation
(which describes how the geometry of space evolves, along various possible paths called spacetimes)
some of the most visible lines of theory do not bother to quantize the main equation and do not come up with a model of evolving geometry.
so given this anomalous situation and I want to focus on an approach that DOES do the requisite stuff, namely Lorentzian DT.
In DT ("dynamical triangulations") you have a spacetime manifold M and you have a bigspace of all the possible geometries on M, called Geom(M).
And for these researchers Geom(M) is not just an abstract idea but they are able to get a handle on the possible geometries and express them
and code them as data structures into a computer
and RUN the little mothers
and do various kinds of counts and extract statistics on them and get specs.
so this is a hard-edge hands-on approach to quantizing the evolving geometry of the universe
and the bizarre thing is even tho it looks like an obvious thing to do the main bulk of theorists work on stuff with no connection or relevance to it
I want to recall a quote from the most recent DT paper
"The idea to construct a quantum theory of gravity by using Causal Dynamical Triangulations was motivated by the desire to formulate a quantum gravity theory with the correct Lorentzian signature and causal properties [14], and to have a path integral formulation which may be closely related to attempts to quantize the theory canonically...."
this is page 3 of http://arxiv.org/hep-th/0411152
Semiclassical Universe from First Principles
by Ambjorn, Jurkiewicz and Loll
it begins a section called "Observing the bounce"
this points up some unresolved issues for me.
one of the two main aim of developing Lorentzian DT, the authors say, is to get close to Loop Quantum Gravity (the canonical approach to quantizing Gen Rel)
and yet there are conspicuous differences
OF COURSE THE BIG BANG SINGULARITY GOES AWAY IN EITHER CASE
but with Loop it is replaced by a real bounce, where a (possibly very small) contracting phase turns inside out to become an expanding universe.
In DT there is also no singularity but in this case the "bounce" seems to get started from nothing-----you dont see a prior contraction.
the cosmological constant appears naturally in Lorentzian DT and must be positive every computer run they do they choose a value for Lambda . I do not understand this. for them it seems related to volume of their model universe.
another conspicuous difference is that in DT I do not see area and volume observables, and I dont see any indication that they will turn out to have discrete values (when and if constructed)
with DT I see a real direct relation to matter and field theory
because DT already has a kind of lattice that QFT is often defined on.
the only real novelty is that it is not a fixed, pre-arranged, lattice, but instead an evolving one.
Anyway, what is puzzling me right now is that there are important differences from LQG, even though the aim is to get a workable path integral approach that connects with LQG
nightcleaner
Dec26-04, 03:04 PM
this is page 3 of http://arxiv.org/hep-th/0411152
Semiclassical Universe from First Principles
by Ambjorn, Jurkiewicz and Loll
it begins a section called "Observing the bounce"
Hi Marcus
I have been thinking about the bounce and was wondering if my insight could be correct, or if it is contradicted by the math. My idea is that the bounce is not really a bounce at all, in the same sense that the event horizon is not really a membrane. As I recall an observer falling through an event horizon doesn't really see any "there" there. No observable thermal barrior or gravitational tidal effect is measurable by the observer because of distortions in spacetime. Basically, the horizon effect is not observed by a free falling observer because all the gauges are distorted along with the spacetime distortion.
Anyway the horizon which appears to be present in embedding diagrams and in the universe as seen by an outside observer is never actually reached by the free-falling observer, just as you can see the earth's horizon quite easily but if you set out to find the horizon you discover that you never acturally get there....because there isn't any "there" there, or maybe more precisely, the there that is there is everywhere, so that even when you go there, you find the there that is there is the same as the there that was there before you went there. Ha!
So if the changeover from the approach to infinity to the approach to unity is a horizon, then there will be no locally observable effect on crossover, no bounce. Bounce, after all, implies a change in acceleration, or delta L over T^3. The observer doesn't feel the third factor in the inverse of time because that dimension of time goes to zero as the observer passes through the limit. I am not sure what to call the limit in LQG. It seems to me that it is quite symmetrical, so we could speak of a concave limit and a convex limit, we could speak of a limit approaching infinity (never gets there because of restrictions on the maxumum value of c, which is L/T) and a limit approaching unity (never gets there because of discrete quantum intervals). These limits are different only to the observer watching the fall, and cannot be determined or felt by the falling observer.
Correct me if I am missing something.
Meanwhile, you seem to have a better grasp of the math than I have. Would it be possible for you to translate into words the terms in equation one from your reference, the partition function for Quantum gravity? I mean a literal translation of the formula into English. If it would not be too much trouble for you to do so, I am sure it would help me understand, and perhaps help others also.
I sat in one of Brian Greene's topology classes and was very excited to find that I could understand the formulas he was writing on the board, because as he wrote them he also spoke the meaning of the symbols. For some reason this made an incredible difference to me, and gave me renewed confidence that i could understand the math if only someone would walk me through it a few times. I have been longing desperately for a return to that fluidity of understanding.
When you speak of dynamics in the spacetime structure, as if spacetime itself changes over time in a manner of a history, or of a path integral, how many dimensions are you counting? I guess i could ask, what order equation? For example, velocity is length divided by time, L/T, a first order equation, but acceleration is lenght over time squared, L/T^2, a second order equation. Spacetime in the Minkowski-Einstein sense is four dimensions of spacetime equivalence, in which case L^3/T, a third order equation? So if we take that as changing over time, are we talking of a mathematical expression involving something like L^3/T^2?
I am feeling confused by this and hope you can help me get clear.
Thanks,
nc
484
Would it be possible for you to translate into words the terms in equation one from your reference, the partition function for Quantum gravity? I mean a literal translation of the formula into English. If it would not be too much trouble for you to do so, I am sure it would help me understand, and perhaps help others also.
When you speak of dynamics in the spacetime structure, as if spacetime itself changes over time in a manner of a history, or of a path integral, how many dimensions are you counting?
Richard
Your physical intuition is much better than many people I know with
the mathematical expertise.
The first equation in
http://arxiv.org/abs/hep-th/0411152
is Z ( \Lambda , G ) = \int \mathcal{D} [g] e^{i S [g]}
\hspace{4mm} S[g] = \frac{1}{G} \int \textrm{dx}^{4}
\sqrt{| \textrm{det} g |} (R - 2 \Lambda)
\Lambda is the cosmological constant. G
is Newton's constant. The fancy \mathcal{D} [g] means
that the integral is over some ridiculously huge space of fields; in
this case the metrics g .
The e^{i S [g]} is the 'weight' for the contribution
from any given field. S is the action functional, an
integral over a four dimensional manifold. The dimension could be
altered.
Under the square root
is the determinant of g , which recall at a given point
is like a matrix. R is the Ricci scalar, defined by
R = g^{\mu \nu} R_{\mu \nu} where one uses the Einstein
convention that one sums over matching indices, which range from
1 to 4. Here R_{\mu \nu} is the Ricci tensor, defined by
R_{\mu \nu} = R^{\lambda}_{\mu \lambda \nu} in terms
of the Riemann tensor
R^{\rho}_{\sigma \mu \nu} = \partial_{\mu}
\Gamma^{\rho}_{\nu \sigma} - \partial_{\nu}
\Gamma^{\rho}_{\mu \sigma} + \Gamma^{\rho}_{\mu \lambda}
\Gamma^{\lambda}_{\nu \sigma} - \Gamma^{\rho}_{\nu \lambda}
\Gamma^{\lambda}_{\mu \sigma}
using the Christoffel symbol
\Gamma^{\lambda}_{\mu \nu} = \frac{1}{2}
g^{\lambda \sigma} ( \partial_{\mu} g_{\nu \sigma} +
\partial_{\nu} g_{\sigma \mu} - \partial_{\sigma} g_{\mu \nu})
Hope this is a little helpful. As you will see, this mathematics falls far short
from being a satisfactory description of 'cosmic bounces', which is
a term combining the 'bounce' of CT (not a great idea as you say)
and the 'cosmic duality' associated to Strings or Cosmic Galois groups
and other fun things.
Regards
Kea
:smile:
nightcleaner
Dec26-04, 05:57 PM
Hi Kea. Thank you for the kind words.
So we might say something like "The partition function for quantum gravity in a universe with a given cosmological constant and gravitational constant \zeta_(\lamda G) is the integral over the metrics (g) multiplied by the weight factor expressed as the natural log of the action functional for any given field, where the action functional is the inverse of the gravitational constant times the integral in four dimensions of the square root of the absolute value of the determinants of (g) times the ......ok I'm lost.
Is that gamma the inverse of the hyperbolic tangent function? I ran into it once before and came out with the right table of numbers after some coaching.
Anyway I am grateful for your exposition and am still working on it. Just to let you know I am here paying attention.
thanks,
nc
573
nc
573
What are the numbers?
\Gamma hasn't anything to do with such functions. It
is the Christoffel symbol associated to a connection, which we need
to understand how to move about the manifold, and to see the
'curvature'.
Try this webite:
Carroll's lecture notes on General Relativity
http://pancake.uchicago.edu/~carroll/notes/
Kea
nightcleaner
Dec27-04, 06:30 AM
The numbers are pure vanity. Silly, really, but that is how many views there were when i posted. So now there are 608, and i know that there are maybe as many as thirty people checking in.
I will check out the Christoffel symbol. Thanks.
Later..... I did check out the link, but my browser didn't want to read that file type. Anyway I looked it up in Wikipedia and amazing coincidence, it is the same topic Brian Greene lectured on in his topology class the day I was there. And it might even be the same gamma I remembered from my work with DW on the relitivity board. I noticed on Wiki, although I have not had time to study it, that gamma deals with tangential vectors. Tanh would just be the same vector under relitivity, I guess.
nc
You're quite right about tanh, of course. Are you sure it wasn't a
course in Differential Geometry?
Are you sure it wasn't a
course in Differential Geometry?
No it was an undergrad topology class.
No it was an undergrad topology class.
You're lucky. I never had a proper topology course as an
undergrad.
So - do you know Nightcleaner?
You're lucky. I never had a proper topology course as an
undergrad.
So - do you know Nightcleaner?
Correct. It is difficult to find a topology course at many Universities, but Columbia University offers undergrad topology. My first topology professor at Columbia University was Michael Thaddeus. Amazing speaker!
Yes I am of good fortune to know Richard. He is very fine and uniquely talented.
nightcleaner
Dec28-04, 02:01 PM
Correct. It is difficult to find a topology course at many Universities, but Columbia University offers undergrad topology. My first topology professor at Columbia University was Michael Thaddeus. Amazing speaker!
Yes I am of good fortune to know Richard. He is very fine and uniquely talented.
Aw, shucks.
Anyway it was only one class, not the whole semester.
I am concerned that Marcus may be mad at us for hijacking his two worlds thread. So I am starting a new thread called hypervisions. Invite all to join in there. Sorry Marcus. hope you are not too angry at me. i am still trying to follow your posts, don't give up on me yet, ok?
nc
Aw, shucks.
Anyway it was only one class, not the whole semester.
Some people gain a whole degree... ner do anything with it.
And another can attend one class and produce amazing advances.
"A word to the wise is sufficient"
I am concerned that Marcus may be mad at us for hijacking his two worlds thread.
nc
Are you sure Marcus is not interested in helping you on this thread?
I never saw anyone so devoted to lending a hand and sharing expertise as Marcus has been to you.
Many of us have been watching the two of you and hoping for some wonderful sharing to reveal new things???
nightcleaner
Dec28-04, 11:40 PM
Thank you Shoshana, and I am grateful to you, and to Marcus, and to others, for helping my understanding.
nc
...
I am concerned that Marcus may be mad at us for hijacking his two worlds thread. So I am starting a new thread called hypervisions. Invite all to join in there. Sorry Marcus. hope you are not too angry at me. i am still trying to follow your posts, don't give up on me yet, ok?
...
hi Cleaner,
I didnt experience any vexation on that (or any other) account
rather the opposite (you and Sho and Kea socializing seems appropriate
in a thread partly devoted to combinatorics)
but I have been busy with another bit project away from PF
and havent felt an urge to join in discussion
I think starting "hypervisions" thread was a good idea----sharing ways of visualizing stuff in more than 3D----I hope you get some wider response.
BTW we might be in a lull until after the 1st of the year.
nightcleaner
Dec29-04, 01:44 PM
Thanks Marcus. Happy New Year. nc
this was a congenial thread (with selfAdj, Richard the nc, Kea, Shoshana, and others occasionally taking part) from around December 04.
I want to keep tabs on it, and perhaps add to it, because of the discussion of CDT (causal dynam. triang.) Monte Carlo moves
following this AJL paper
http://arxiv.org/hep-th/0105267
a new AJL paper just came out, a short one about the running of the spacetime dimension---a surprising result that they discovered in the course of simulating the evolution of many universes on their computer.
in the new AJL shortie they tell us to expect a long paper dated May 05 which will give details on their computer simulations, how they generate random spacetimes, random worlds.
the basic thing is to put a quarter of a million simplexes into the computer and have them be glued together randomly and then have the whole spacetime shebang be shuffled repeatedly by "Monte Carlo moves". there are some halfdozen or so of those simple moves which are adequate, if repeated enough, to explore all possible spacetime histories.
this new May 05 paper will be called "Reconstructing the Universe" and it will probably do a lot of the same things as http://arxiv.org/hep-th/0105267
except it will be more up-to-date. It will probably also show pictures of the various Monty moves and describe step by step how a simulation works.
An interesting historical note is that people (AJL and others) have been trying to make a simplicial model of the 4D universe, and do this kind of computer simulation, for something like 15 years without it working!
Jan Ambjorn has been especially stubborn and there is a long history of partial successes in 2D and 3D, and of repeated failures to get 4D.
then last year, in 2004, it must have felt, for Ambjorn, like someone finally stops beating against a wall and the wall vanishes.
They wrote a paper around April 2004 called "Emergence of a 4D world"
You can get all the links to earlier papers from the biblio in their current paper, should you wish. So I will just give the current one:
http://arxiv.org/hep-th/0505113
Getting back to a question asked in post #1 of this thread,
...I would like to know why the research output in DT is essentially flat. Given its apparent promise and the recent (2004) success, why arent more people getting into DT?
just to review. CDT seems to have the required low-energy limit and it has a hamiltonian that they construct explicitly in, for example, that 2001 paper I'm always citing.
and you can do calculations, which you cannot easily do in several other approaches to QG.
(calculations galore, done by the barbaric means of Monte Carlo, which I regard as having the proper empirical spirit.)
so it's got
1. classical limit
2. an interesting (Hartle Hawking wavefunction) semiclassical limit
3. dynamics
4. opportunities to calculate and do computer simulations
but look, while LQG is having 100 some papers a year and has a vigorous kind of quadratic-looking growth curve, the output in CDT is esssentially flat. This roughly tracks the output since 1998 when CDT was invented or began as a research line:
http://arxiv.org/find/grp_physics/1/OR+OR+abs:+AND+triangulations+AND+Lorentzian+dynam ical+abs:+AND+triangulations+AND+causal+dynamical+ ti:+AND+gravity+AND+Lorentzian+quantum/0/1/0/1998/0/1
http://arxiv.org/find/grp_physics/1/OR+OR+abs:+AND+triangulations+AND+Lorentzian+dynam ical+abs:+AND+triangulations+AND+causal+dynamical+ ti:+AND+gravity+AND+Lorentzian+quantum/0/1/0/1999/0/1
http://arxiv.org/find/grp_physics/1/OR+OR+abs:+AND+triangulations+AND+Lorentzian+dynam ical+abs:+AND+triangulations+AND+causal+dynamical+ ti:+AND+gravity+AND+Lorentzian+quantum/0/1/0/2000/0/1
http://arxiv.org/find/grp_physics/1/OR+OR+abs:+AND+triangulations+AND+Lorentzian+dynam ical+abs:+AND+triangulations+AND+causal+dynamical+ ti:+AND+gravity+AND+Lorentzian+quantum/0/1/0/2001/0/1
http://arxiv.org/find/grp_physics/1/OR+OR+abs:+AND+triangulations+AND+Lorentzian+dynam ical+abs:+AND+triangulations+AND+causal+dynamical+ ti:+AND+gravity+AND+Lorentzian+quantum/0/1/0/2002/0/1
http://arxiv.org/find/grp_physics/1/OR+OR+abs:+AND+triangulations+AND+Lorentzian+dynam ical+abs:+AND+triangulations+AND+causal+dynamical+ ti:+AND+gravity+AND+Lorentzian+quantum/0/1/0/2003/0/1
http://arxiv.org/find/grp_physics/1/OR+OR+abs:+AND+triangulations+AND+Lorentzian+dynam ical+abs:+AND+triangulations+AND+causal+dynamical+ ti:+AND+gravity+AND+Lorentzian+quantum/0/1/0/2004/0/1
Last 12 months (abbreviated LTM):
http://arxiv.org/find/grp_physics/1/OR+OR+abs:+AND+triangulations+AND+Lorentzian+dynam ical+abs:+AND+triangulations+AND+causal+dynamical+ ti:+AND+gravity+AND+Lorentzian+quantum/0/1/0/past/0/1
1998 3
1999 3
2000 5
2001 4
2002 6
2003 4
2004 4
LTM 4
selfAdjoint
May15-05, 01:13 PM
There may be a perception in the community that using fixed triangulations is "clunky" compared to networks and spin foams. And we must never discount the effect of John Baez's advocacy over the years!
But these AJL papers will be sure to change things. As one string physicst said back when the mirror dualities were discovered, maybe even some grad students will become interested!
... we must never discount the effect of John Baez's advocacy over the years!
...
actually it was John Baez's advocacy in FAVOR of CDT last year after the Marseille conference in May that galvanized me and may have made CDT a lot more visible to others as well. so that can work both ways
the analogy with a Feynman path integral is very strong
as if instead of one fixed jagged piecewise linear path that a particle might take, one imagines instead the continuum limit of a whole
"blur" or billions of possible zigzag paths
so likewise with CDT one may imagine a glittering blur of millions of triangulations
(not one fixed triangulation)
that averages out to a smooth spacetime shape
there is a common element in the last 3 CDT papers. since they carefully repeat it each time maybe we should listen extra carefully:
---from hep-th/0404---
Causal dynamical triangulations are a framework for defining quantum gravity nonperturbatively as the continuum limit of a well-defined regularized sum over geometries. Interestingly, and in complete agreement with current observational data is the fact that the physical cosmological constant \Lambda in dynamical triangulations is necessarily positive....
---from hep-th/0411---
Causal Dynamical Triangulations constitute a framework for defining quantum gravity nonperturbatively as the continuum limit of a well-defined regularized sum over geometries. We reported recently on the outcome of the first Monte Car..."
---from hep-th/0505---
"In the CDT approach, quantum gravity is defined as the continuum limit of a regularized version of the nonperturbative gravitational path integral . The set of spacetime geometries to be summed over is represented by a class of causal four-dimensional piecewise flat manifolds (“triangulations”). Every member T of the ensemble of simplicial spacetimes can be wick-rotated to a unique Euclidean piecewise flat geometry, whereupon the path integral assumes the form of a partition function ...
All geometries share a global, discrete version of proper time. In the continuum limit, the CDT time \tau becomes proportional to the cosmological proper time of a conventional minisuperspace model..."
... compared to networks and spin foams...
you prompted me to compare CDT with spin foams and it occurred to me that they look rather alike, there is a (superficial?) visual resemblance
but CDT is less dependent, I believe, on coordinates
I think of Georg Riemann around 1850 offering a way to do geometry without prior commitment to a particular metric---but only with some coordinate patches: a shapeless "manifold"---Riemann had to have coordinates.
And then around 1950 came Tulio Regge offering a way to do "general relativity without coordinates" (I think that was his paper's title or something like it).
so then you did not even need to make a prior commitment to coordinate patches (and the DIMENSION always sneaks in as the number of coordinate functions, the "n" in the local resemblance to Rn.)
So CDT as a Regge offspring makes less prior commitments. It doesnt have an essential need for coordinates because the "geometry is all in the gluing". the geometry is all in how the many identical cells are stuck together.
and although one chooses at the outset cells of a particular dimension (for instance simplices which are pieces of Minkowski space) when they are glued together it turns out that the dimension can be determined in some empirical ways (e.g. by running diffusion) and it does not need to come out exactly 4, or even to be an integer. it is, after all, not a DIFFERENTIABLE manifold that one makes by gluing the cells together. this seems very strange to me but it does look like one is going further in the direction of not having prior commitments to anything.
and yet they are able to define a transfer matrix and a hamiltonian and to calculate and to do computer simulations, which a lot of people working with stuff on differentiable manifolds do NOT seem able to do. so it is paradoxical----they throw out---they make less prior commitment to structure---and yet they seem able to calculate MORE rather than less----like they even get statistics about their spacetimes----having made so many of them that they must record statistics.
utrecht interests me, now 't Hooft and Robbert Dijkgraaf, and Renate Loll, and Jan Ambjorn are there and the last 3 of them are invited speakers at the October "Loops 05" conference. I am thinking that Utrecht work will be fairly noticeable at this year's conference
selfAdjoint
May16-05, 08:54 AM
It's an interesting and I believe deep toplogical question how much you actually predetermine the coordinate stuff with this "gluing"? There was always this uneasy feeling that AJL had presetermined pseudo-Riemannian with their "causality" - see also causal sets which seems to do that without the gluing.
Back in the grad school day, I was always impressed by the fact that the left side of the Riemann-Roch index theorem was pure toplogy and the right side was analytic. Of course since then we have the same thing much bigger with Atiyah-Singer. And look at the run BRST has had! Save your Jordan curves boys, topology will rise again!
It's an interesting and I believe deep toplogical question how much you actually predetermine the coordinate stuff with this "gluing"? There was always this uneasy feeling that AJL had presetermined pseudo-Riemannian with their "causality" - see also causal sets which seems to do that without the gluing.
Back in the grad school day, I was always impressed by the fact that the left side of the Riemann-Roch index theorem was pure toplogy and the right side was analytic. Of course since then we have the same thing much bigger with Atiyah-Singer. And look at the run BRST has had! Save your Jordan curves boys, topology will rise again!
I'll talk around the issues you raise here, which are interesting ones, though unable to address them head-on. Another way to think of the gluing of 4-simplices together is as carving up the S3 x R carcass.
AJL start by putting a topological S3 x R into the computer.
for technical reasons they specify that there are going to be, say, 80 time steps and some total number of simplexes, like 200,000.
this S3 x R gets repeatedly divided up into around 200,000 4simplexes, and they takes their statistical measures of what results.
and then they start again, using a different total number, like 300,000
their idea is that the CDT theory is the continuum limit
they say spacetime is not discrete (or that they have seen no indication of a fundamental discreteness, in that sense they are at odds with LQG which has seen a discrete spectrum of area and volume operators)
they say they have seen no indication of a minimum length
their spacetime, their universe, is a continuum limit of quantum geometries that are established in a way that does not use coordinates or metrics or connections, but only uses simplexes (instead of coordinates)
but they do not think the world is made of simplexes, that is just what they use to describe a sequence of finer and finer geometries.
now I have to think how to relate what I've just said to the issues you raised
I'll talk around the issues you raise here, which are interesting ones, though unable to address them head-on. Another way to think of the gluing of 4-simplices together is as carving up the S3 x R carcass.
AJL start by putting a topological S3 x R into the computer.
...
they put S3 x R into the computer and carve it up, in the case where they are using 4-simplexes as the basic relational cell (I think of the 4-simplex, which is a tiny piece of minkowski space, as an ATOM OF RELATIONSHIP, or as an atom of spacetime relation and causality)
in the cases where they have used 2-simplexes or 3-simplexes as the basic relational buildingblock, then they used a different topological carcass to carve up. they used S1 x R, or else S2 x R.
now the interesting thing, or one very interesting, is that if you dont use the right carving rules (dont make it causal or Lorentzian but only euclidean) then you can get any dimension. Like with S2 x R
dividing up into 3-simplexes it does not have to come out 3D. You can get a beast that is 2D or that is INFINITE dimensional. unbounded (infinite in the limit) anyway.
the infinite dimensionality comes from too much connectivity
like think of the earth surface with the N hemisphere carved into 360 skinny triangles each one with its vertex at the N pole. this is too connected because every spot is in the same triangle as the N pole. One has to have a carving rule that makes this kind of thing unlikely.
the point I am making is that DIMENSIONALITY MUST EMERGE and it is not predetermined by the fact that you started with a topological space that is
S3 x R
(because a mere topological space does not have preordained dimension, and when it gets some structure that permits dimension to be defined then that dimension does not have to be the same at every point)
and the dimensionality that emerges is also not predetermined by the fact that we happen to be using 4-simplexes to carve the S3 x R up in.
Because the topological space can be partitioned into 4-simplexes in such a way as to give very high dimension or very low dimension to various locations in it (the S3 x R carcass doesnt have any geometry until the 4-simplexes come and partition it)
I had better quote AJL about this, because it is a key point, and I will repost the basic CDT reading list in case anyone wants to refer to the original papers
This is from the introductory paragraph of "Emergence of a 4D World", reference 2 below:
"... a particular case of the more general truth, not always appreciated, that in any nonperturbative theory of quantum gravity dimension will become a dynamical quantity, along with other aspects of geometry. (By dimension we mean an effective dimension observed at macroscopic scales.)..."
Dimension is just another aspect of geometry. AJL are working with a topological continuum which has no prior geometry and they put the geometry on in a quantum way, and dimensionality emerges, and it is different at different scales, and down near planck scale it tends to be around D = 2, or even less. But happily enough at big scale, where we like to build our houses and go snow-boarding etc, it is right around D = 4.
Here is the short list of CDT references
1.
http://arxiv.org/hep-th/0105267
Dynamically Triangulating Lorentzian Quantum Gravity
J. Ambjorn (NBI, Copenhagen), J. Jurkiewicz (U. Krakow), R. Loll (AEI, Golm)
41 pages, 14 figures
Nucl.Phys. B610 (2001) 347-382
"Fruitful ideas on how to quantize gravity are few and far between. In this paper, we give a complete description of a recently introduced non-perturbative gravitational path integral whose continuum limit has already been investigated extensively in d less than 4, with promising results. It is based on a simplicial regularization of Lorentzian space-times and, most importantly, possesses a well-defined, non-perturbative Wick rotation. We present a detailed analysis of the geometric and mathematical properties of the discretized model in d=3,4..."
2.
http://arxiv.org/abs/hep-th/0404156
Emergence of a 4D World from Causal Quantum Gravity
J. Ambjorn (1 and 3), J. Jurkiewicz (2), R. Loll (3) ((1) Niels Bohr Institute, Copenhagen, (2) Jagellonian University, Krakow, (3) Spinoza Institute, Utrecht)
11 pages, 3 figures; final version to appear in Phys. Rev. Lett
Phys.Rev.Lett. 93 (2004) 131301
"Causal Dynamical Triangulations in four dimensions provide a background-independent definition of the sum over geometries in nonperturbative quantum gravity, with a positive cosmological constant. We present evidence that a macroscopic four-dimensional world emerges from this theory dynamically."
3.
http://arxiv.org/abs/hep-th/0411152
Semiclassical Universe from First Principles
J. Ambjorn, J. Jurkiewicz, R. Loll
15 pages, 4 figures
Phys.Lett. B607 (2005) 205-213
"Causal Dynamical Triangulations in four dimensions provide a background-independent definition of the sum over space-time geometries in nonperturbative quantum gravity. We show that the macroscopic four-dimensional world which emerges in the Euclidean sector of this theory is a bounce which satisfies a semiclassical equation. After integrating out all degrees of freedom except for a global scale factor, we obtain the ground state wave function of the universe as a function of this scale factor."
4.
http://arxiv.org/abs/hep-th/0505113
Spectral Dimension of the Universe
J. Ambjorn (NBI Copenhagen and U. Utrecht), J. Jurkiewicz (U. Krakow), R. Loll (U. Utrecht)
10 pages, 1 figure
SPIN-05/05, ITP-UU-05/07
"We measure the spectral dimension of universes emerging from nonperturbative quantum gravity, defined through state sums of causal triangulated geometries. While four-dimensional on large scales, the quantum universe appears two-dimensional at short distances. We conclude that quantum gravity may be "self-renormalizing" at the Planck scale, by virtue of a mechanism of dynamical dimensional reduction."
...This is from the introductory paragraph of "Emergence of a 4D World", reference 2 below:
"... a particular case of the more general truth, not always appreciated, that in any nonperturbative theory of quantum gravity dimension will become a dynamical quantity, along with other aspects of geometry. (By dimension we mean an effective dimension observed at macroscopic scales.)..."
Dimension is just another aspect of geometry. AJL are working with a topological continuum which has no prior geometry and they put the geometry on in a quantum way, and dimensionality emerges, and it is different at different scales, and down near planck scale it tends to be around D = 2, or even less. But happily enough at big scale, where we like to build our houses and go snow-boarding etc, it is right around D = 4.
Here is the short list of CDT references...
Now #5 on the short list has appeared
5.
http://arxiv.org/hep-th/0505154
Reconstructing the Universe
and we have another new title in the "in preparation" or "to appear" category, which is the first CDT paper AFAIK about BLACK HOLES.
This is to be a collaboration of Renate Loll with BIANCA DITTRICH.
The two of them co-authored a CDT paper back in 2002 but since then Dittrich has been doing stuff with Thiemann "Master Constraint" program and also with partial-observable, relational-time. In effect she has been helping Thiemann rake his chestnuts out of the fire.
Dittrich work on Master Constraint has been major. She is an interesting case because she is both active currently in LQG/LQC and also has a significant interest in CDT.
Also it seems like a high priority job to see how CDT deals with black holes
(we already know how CDT does the big bang non-singularity, or seem to know, all the AJL simulations portray that, but what AJL show is a vacuum state universe so far with no black holes)
the Dittrich-Loll paper to appear is called
Counting a Black Hole in Lorentzian Product Triangulations
It would not surprise me if Dittrich moved from AEI-Potsdam to Utrecht. The Utrecht group seems to me to be growing.
[EDIT in case anyone is curious the 2002 paper of Dittrich-Loll was
http://arxiv.org/abs/hep-th/0204210 ]
the basic incompatibility is that LQG is based on a smooth continuum and CDT on an unsmooth. CDT is the limit of piecewise flat manifolds, like a Feynman path is a superposition of piecewise linear paths and the result is not differentiable
neither CDT nor LQG is DISCRETE. but in LQG the underlying space is a differentiable manifold (smooth although without prior metric)
and in CDT the underlying space is not smoothly coordinatized, it is NOT a differentiable manifold. it is all jig-jaggy piecewise flat. and it does not have any coordinates.
in 1850 Riemann invented how to do a continuum without prior metric (but with smooth functions as coordinates, the point is the patches)
in 1950 Tulio Regge invented how to do a continuum and write Einstein Gen Rel on it WITHOUT COORDINATES, no differentiable functions.
So this is a major disconnect---it looks as if Renate Loll is A HUNDRED YEARS MORE ADVANCED, in the history of mathematics, than, say Lee Smolin and Carlo Rovelli. this could be serious.
But maybe it can be fixed. Maybe one can do LQG on a series of finer and finer piecewise flat manifolds. simplicial manifolds. whatever.
(here "flat" means 4D minkowski, the usual special relativity 4D flat)
the thing that Renate Loll can say (and the Utrecht gang have their knives out and mean business these days) is what I quoted a post back:
in any nonperturbative theory of quantum gravity dimension will become a dynamical quantity, along with other aspects of geometry.
Because LQG is built on a smooth manifold of some chosen dimension, with exactly 3 coordinate functions, or exactly 4, it is so-to-say stuck with that choice of dimension. But a piecewise flat manifold can be so wrinkled and kinky and crumpled and frazzled that it loses track of what its dimension is.
at least at small scale and maybe the real world is like that
this gives the Utrecht people a strong card to play
Now #5 on the short list has appeared
5.
http://arxiv.org/hep-th/0505154
Reconstructing the Universe
...
Here is the first paragraph of Reconstructing the Universe:
"Nonperturbative quantum gravity can be defined as the quest for uncovering the true dynamical degrees of freedom of spacetime geometry at the very shortest scales. Because of the enormous quantum fluctuations predicted by the uncertainty relations, geometry near the Planck scale will be extremely rugged and nonclassical. Although different approaches to quantizing gravity do not agree on the precise nature of these fundamental excitations, or on how they can be determined, most of the popular formulations agree that they are neither the smooth metrics gmu,nu (or equivalent classical field variables) of general relativity nor straightforward quantum analogues thereof. In such scenarios, one expects the metric to re-emerge as an appropriate description of spacetime geometry only at larger scales."
Renate Loll can write English better than most of us nativespeakers, or else it is an AJL team phenomenon. Anyway the style of the AJL writings is usually strong and clear. This helps (them at least).
Now that String seems less promising, seems off track in fact, the other approaches to Quantum Gravity which have so far been mostly at peace under the big Loop tent, are becoming more disposed to controversy among themselves.
Here right away at the beginning, the AJL writer is playing the strong card against LQG. The suggestion is that LQG cannot be right because it is too smooth at small scale. How can LQG be right? since it is built on a smooth differentiable manifold of fixed dimension, and uses "classical field variables" equivalent to the metric, namely the connections. Actually LQG has made a lot of progress and will have accomplished a lot even if it eventually turns out to have been a PILOT STUDY for something else. It has tackled the big bang and inflation and black holes and produced a lot of new ideas like Freidel/Starodubtsev and for instance lately Perez/Rovelli about the Immirzi parameter. Probably there is some stuff it DOES get right. Thiemann has started confirming details of Bojowald's LQC big bang using the full LQG theory etc. Plenty is happening in LQG. But the AJL writer hints how can LQG be fundamentally correct? since "because of the enormous quantum fluctuations predicted by the uncertainty relations, geometry near the Planck scale will be extremely rugged and nonclassical."
Well maybe that is not as aggressive as it sounded to me. Maybe everybody inside the big Loop tent feels that he or she is enough rugged and nonclassical. Maybe spin networks are that. So one can hear this opening paragraph different ways.
There may be some fire at the Potsdam conference in October.
I think String has dwindled so that there is less of a common rival and less of a collective spirit. So people and ideas can wrangle with each other more openly, which I guess is traditional in science so must be OK
selfAdjoint
May18-05, 03:50 PM
I'll talk around the issues you raise here, which are interesting ones, though unable to address them head-on. Another way to think of the gluing of 4-simplices together is as carving up the S3 x R carcass.
AJL start by putting a topological S3 x R into the computer.
for technical reasons they specify that there are going to be, say, 80 time steps and some total number of simplexes, like 200,000.
this S3 x R gets repeatedly divided up into around 200,000 4simplexes, and they takes their statistical measures of what results.
and then they start again, using a different total number, like 300,000
their idea is that the CDT theory is the continuum limit
they say spacetime is not discrete (or that they have seen no indication of a fundamental discreteness, in that sense they are at odds with LQG which has seen a discrete spectrum of area and volume operators)
they say they have seen no indication of a minimum length
their spacetime, their universe, is a continuum limit of quantum geometries that are established in a way that does not use coordinates or metrics or connections, but only uses simplexes (instead of coordinates)
but they do not think the world is made of simplexes, that is just what they use to describe a sequence of finer and finer geometries.
now I have to think how to relate what I've just said to the issues you raised
Two responses to this:
I. Maybe CDT will wind up doing with LQG what the lattice does with the standard model: achieve geniune if spotty non-perturbative results while the lion's share of the physics is being done n-loop perturbative with the continuum theory. (n being a small integer).
II. Topology has lots of more general things than polyhedra (which is what triangulated manifolds are). CW-complexes, ANRs, lots of things in graduated sequences of generality. And there are all those lovely categories.
Hi selfAdjoint,
before I forget I must say I am looking out for a paper by Bianca Dittrich and Renate Loll that applies CDT to Schwarzschild black holes.
It has not been posted yet. but curiously I just now encountered a citation to it in an article by Arundhati Dasgupta
http://arxiv.org/find/gr-qc/1/au:+Dasgupta_A/0/1/0/all/0/1
Right at the end of his recent article about black holes (mostly from LQG and related perspective), actually it is the very last sentence of his conclusions, he says:
" There is a discretisation of the Schwarzschild space-time using dynamical triangulation techniques in [20], it shall be interesting to obtain the entropy in that formalism. "
and his [20] is
"[20] B. Dittrich, R. Loll, Dynamical Triangulations of Black Hole Geometries, in preparation. "
However in the AJL article they give a different title, by coincidence it is also their reference [20]
"[20] B. Dittrich and R. Loll: Counting a black hole in Lorentzian product triangulations, preprint Utrecht, to appear. "
So far we know Bianca Dittrich mainly from her work with Thiemann, in particular on the Master Constraint programme.
Two responses to this:
I. Maybe CDT will wind up doing with LQG what the lattice does with the standard model: achieve geniune if spotty non-perturbative results while the lion's share of the physics is being done n-loop perturbative with the continuum theory. (n being a small integer).
II. Topology has lots of more general things than polyhedra (which is what triangulated manifolds are). CW-complexes, ANRs, lots of things in graduated sequences of generality. And there are all those lovely categories.
I am not so optimistic about LQG. I value its contribution enormously it has beat a track into the bush. unearthing interesting stuff like the Immirzi parameter and focusing attention on form theories of gravity as in the case of Freidel/Starodubtsev (who almost seem to have a background indep perturbative approach, resolving singularities like BH and BB, finding an automatic generic mechanism for inflation. many of its discoveries will probably persist and mutate in other contexts. BUT. my intuitive feeling is that CDT is now entering an exponential growth phase and remember
CDT HAS a hamiltonian, and looks like it HAS a reasonable chance of correct classical and semiclassical limiting behavior. and it also has the
DYNAMIC DIMENSION card.
and it looks more like a Feynman path integral to me. So I am apprehensive about the longrange prospects of Loop. I see that it can be a very valuable set of PILOT STUDIES, but I see a possibility that it could be eclipsed by CDT.
of course there is always the possibility of convergent evolution and the eventual proof of an equivalence theorem mapping one model onto the other.
(sorry, I have just been idly speculating, usually a waste of time----what we need to do, I suspect, is not speculate but quickly understand everything we can about CDT as it is right now)
I am not so optimistic about LQG. I value its contribution enormously it has beat a track into the bush. unearthing interesting stuff like the Immirzi parameter and focusing attention on form theories of gravity as in the case of Freidel/Starodubtsev (who almost seem to have a background indep perturbative approach, resolving singularities like BH and BB, finding an automatic generic mechanism for inflation. many of its discoveries will probably persist and mutate in other contexts. BUT. my intuitive feeling is that CDT is now entering an exponential growth phase and remember
CDT HAS a hamiltonian, and looks like it HAS a reasonable chance of correct classical and semiclassical limiting behavior. and it also has the
DYNAMIC DIMENSION card.
If all there is so far is space without particles, how do they avoid scale invariance since there seems to be nothing with respect to measure distance with? What is small distance compared to large distance in a universe without particles to measure with respect to?
selfAdjoint
May18-05, 08:44 PM
Here's a new paper http://www.arxiv.org/abs/hep-th/0505165
by Mohammed Ansari & Fotini Markopoulo:
We rewrite the 1+1 Causal Dynamical Triangulations model as a spin system and thus provide a new method of solution of the model
with lots of pictures.
Here's a new paper http://www.arxiv.org/abs/hep-th/0505165
by Mohammed Ansari & Fotini Markopoulo:
with lots of pictures.
selfAdjoint, thanks for catching this. I will add it to the surrogate sticky links thread. Good for Fotini for venturing into a new field herself and for getting her gradstudent Ansari into a promising research line at the ground floor
If you like pizza, look at footnote #3 on page 8.
I checked the list of grad students at Perimeter. It includes two that have recently co-authored CDT papers:
Tomasz Konopka
Mohammad Ansari
and one we know of from some posts here by John Baez and Kea, namely
Artem Starodubtsev
If I were a grad student, i would want to be in Utrecht (with Ambjorn and Loll), but if not there then maybe Perimeter would not be too bad because they seem to stay engaged. I wish some Utrecht code could be transplanted to a Canadian computer
nightcleaner
May19-05, 01:44 PM
Hi
This is great stuff. I wish I had more time here.
Marcus, do you think the mathematics of fractals as demonstrated by the Mandlebrot and Julia sets will apply to these CDT fractal dimensions? I am still looking for my books on fractals.
Thanks,
Richard
... do you think the mathematics of fractals as demonstrated by the Mandlebrot and Julia sets will apply to these CDT fractal dimensions? ...
I can only guess that yes, some of what has been learned about fractal sets WILL be applicable to spacetime at very small scales, but since i am not very knowledgeable about those things, I would not know what to expect.
It seems to me that some things about the familiar beautiful fractals would NOT apply. They are self-similar (proportions the same at all scales) and so their dimensionality is the same all the way down. But the CDT people seem to be saying that spacetime is almost just ordinary cliche 4D spacetime at large scale but gets more frizzy as you go down scale
so that at small sizes it gets quite cheesy and flakey.
So this is not self-similar or scale-invariant behavior at all.
It's early days for understanding these things (at least for me)
and also CDT could be wrong.
I like it that AJL have not been afraid to go ahead with something that is radically innovative and doesnt even have an underlying smooth continuum (such as strings live in, or such as the paraphernalia of LQG is built on)
instead of a smooth differentiable continuum, they have an extremely kinky continuum, without even coordinate patches. this is a moral satisfaction to me and resonates with my deep inner perversity, thus having a calming effect. it almost makes me happy. I hope you too
nightcleaner
May19-05, 02:22 PM
Yes, quite so.
I suspect the universal set is not discrete in itself, but any observer is a limited system and therefore the interaction of the observer with the universe is limited...hence the fact that, to any observer, the universe appears to have limits. These limits necessarily receed, as the observer develops.
R
Nightcleaner, I am coming to recognize the paper I call "Dynamically..."
as a tutorial.
(short for hep-th/0105267 "Dynamically Triangulating Lorentzian Quantum Gravity)
Lorentzian Q. G. is one of their old names for CDT. they finally settled on CDT permanently in 2004. But when you take that change in terminology into account, then the title tells you what it is. This is a HOW TO do it paper. they take you thru the 3D case partly because it is easier and once you have been thru the 3D case the 4D case feels better.
A tutorial type paper is one that it is worthwhile working thru at least part of it, equation by equation.
For someone who likes triangles and tetrahedra, you may at first be confounded by the fact that there are two kinds of triangles spacelike and timelike and they have different areas!
For example look on page 6, equation (4) near the bottom.
area of spacelike triangle = \frac {\sqrt{3}}{4}
I suspect it was Rafael Sorkin who made up the word "bones" for the D-2 simplexes. So if you are in 3D case, the "bones" are just line segments, the edges of the tets. the reason I think this is because he grew up in Chicago.
anyway a spacelike triangle is just an equilateral with sides equal ONE.
so naturally the area = \frac {\sqrt{3}}{4} we did this in middleschool or 9th grade
BUT THE SQUARE LENGTH OF A TIMELIKE EDGE IS MINUS ALPHA.
In CDT you allow the timelike length to be an imaginary number oi veh oi veh, and you give it some freedom so that its square length does not have to be exactly minus one, but can be - \alpha
So imagine a timelike triangle, in the sandwich, with its base in one spacelike layer. So its base has length one! OK but the other two sides are timelike. so the square of one of those sides is - \alpha .
It is an ISOSCELES triangle with the two equal sides IMAGINARY (whoops :yuck:) so what is the HEIGHT? well you just do pythagoras and square the hypoteneuse (minus alpha) and subtract 1/4 (the square of the side)
the square of the height = - \alpha - 1/4 = - \frac{4\alpha + 1}{4}.
the height = i \frac {\sqrt{4\alpha + 1}}{2} .
now the base = 1, remember it is spacelike and all in one layer,
so one half base times height = \frac {\sqrt{4\alpha + 1}}{4} .
where I dropped a factor of i because I thought you might not be watching.
and hey, an area or volume should be a real number.
what I was calling "height" in the previous post was just the height of a timelike triangle (timelike means it spans two layers, we worked with layers before, the CDT universe is foliated in spacelike layers)
but suppose now we have a TETRAHEDRON of the (3,1) sort, that has a spacelike base that it is sitting on. an equilateral triangle with side one.
And it has three timelike isosceles triangles as sides.
now what is the (timelike) height of that tetrahedron?
well you draw a picture and see that it is the vertical leg of a right triangle whose squarehypoteneuse is minus alpha and whose other square leg is 1/3
so (ignoring a factor of i) the height of the tetrahedron is
\frac {\sqrt{3\alpha + 1}}{\sqrt{3}} .
and the volume of a cone or pyramid is always 1/3 the base area times the height, and the base of this tet is just \frac {\sqrt{3}}{4}
cause its an equilateral triangle! so you multiply the base area times the height and the sqrt{3} cancels and you get
\frac {\sqrt{3\alpha + 1}}{4}
and one third of that (because it is a pyramid type thing) is
\frac {\sqrt{3\alpha + 1}}{12}
now let's see if that is what AJL say. YES!!!!! just look at their
equation (5)
so these are the stretches they have you do before anything aerobic happens. it is a tutorial.
selfAdjoint
May19-05, 07:55 PM
I suspect it was Rafael Sorkin who made up the word "bones" for the D-2 simplexes. So if you are in 3D case, the "bones" are just line segments, the edges of the tets. the reason I think this is because he grew up in Chicago.
Is there some sly angeleno dig at Chicago here? What is the connection of Chitown with 'bones"? And what about LA then? I saw Chinatown, and it was (loosely) based on real events. :biggrin:
americans like short visual-type words
(this is one reason much of our language is still strong even if the populace gets fat and fatuous)
even when they are doing misguided empty physics, american physicists will think up good words to call things
please remember I am not an angeleno, I do not do angeleno "digs"
I am proud that Rafael Sorkin grew up in Chicago.
Look at the list of organizers of the Loops 05 conference!
Almost no one there was born in the States!
Rafael Sorkin is one of the very few.
Theoretical physics in the US has lost ground and gotten off track probably. When we hear about really new developments they seem to be coming from Utrecht! It didnt used to be that way, IIRC.
I think "bones" is a good strong-visual monosyllable coinage, for something that needed a name. (the D-2 simplices are important in Regge calculus)
i dont know who named them bones. But I would bet an american
and Sorkin in 1975 wrote
"Time-evolution in Regge calculus" which I have not seen.
So I dont know who coined it, but he is as likely as anyone i can think of.
there is also an Englishwoman, ruth williams. she might have.
It was not a German, who would have said
\mathfrak {Knochen}
instead
Sometimes I think you are sensitive about the Midwest. We coasters like the Midwest. We listen to Garrison Kiellor Prairie Home Companion.
and sometimes i cant tell if you are joking, like when you say angeleno (which to N calif ears sounds derogatory)
selfAdjoint
May19-05, 10:36 PM
Marcus I think the "bones" came from the fact that what you have been calling a tet (for tetrad) is also known as a vierbein (= "four bones" or perhaps "quadruple bone"). In string theory where they do 1+1 GR on the string worldsheet, sometimes you see the term "zweibein" for the corresponding thing, and sometimes by a stated abus de langage they still say vierbein.
Marcus I think the "bones" came from the fact that what you have been calling a tet (for tetrad) is also known as a vierbein (= "four bones" or perhaps "quadruple bone"). In string theory where they do 1+1 GR on the string worldsheet, sometimes you see the term "zweibein" for the corresponding thing, and sometimes by a stated abus de langage they still say vierbein.
words are fascinating, arent they? and they are soaked in history which is fascinating too
In German Bein is "leg"
and a "Dreibein" is the same as a Tripod (word constructed the same way, as "three-leg" or "three-footed thing")
there is no connotation of "bone" as far as I know
and yet Bein does certainly sound like bone!
I suppose that Dreibein is an ancient German word, like the Greeks had Tripods for burning incense, and in Homer you have catalogs of X number of goblets and Y number of gold plates and Z number of Tripods, catalogs of things that one gave to the priest of apollo or in recompense to appease wrath etc.
I could be wrong. I will try to look it up
It is important to notice that in simplicial manifolds Regge calculus a "bone" has nothing obvious to do with a Bein or a Vierbein or Dreibein.
where the basic building block is a 4-simplex, the bones are the TRIANGLES that form the sides of the Tetrahedra that form the faces of the 4-simplex.
where the basic building block is a 3-simplex, or tetrahedron, then the bones are the EDGES of the triangles that form the sides of the 3-simplex.
The bones are, by definition, the (D-2)-simplexes.
By contrast, a "Dreibein" at some point in a manifold can be pictured as a little X,Y,Z reference frame made of 3 tangent vectors. You can stick out your thumb and 2 fingers to make XYZ axes and give the idea.
In a 4-manifold, a "Vierbein" at some point is the analogous thing made of 4 tangent vectors.
The reason that the concept bone is a natural is that in 3D Regge calculus, the edges play the same role and the triangles do in 4D.
IN 3D, To make the Regge version of the Einstein action one is measuring the DEFICIT ANGLE of the tetrahedra around some EDGE
IN 4D one is measuring the deficit angle of the 4-simplexes which are joined around some TRIANGLE.
THE BONE IS WHERE THE THINGS COME TOGETHER THAT YOU ADD UP THEIR DIHEDRAL ANGLES TO FIND OUT THE DEFICIT ANGLE
the reason you need the "bone" concept is so that you can speak in general about 3D and 4D and 5D gravity all at once, and so you can treat them as analogs of each other.
the deficit angle of the simplexes joined around a bone is, of course, how Regge discovered he could measure curvature and in this way, using simplexes, he could implement General Relativity.
it is only gradually dawning on me how beautiful this approach is
nightcleaner
May20-05, 02:32 PM
Hi Marcus
I am way behind on my reading and there are strawberries to put in and trees to plant, as well as dogs, rabbits, and a horse to care for. The good news is the sun has made an appearance and I have managed to get my laptop setup on my friend Peg's table, a rather crowded place, but with a nice view of the bird feeder, apple trees, sauna, and horse eating hay in the pasture. I have coffee, and a quiet idyllic setting for contemplation of the mysteries of alpha. Oh, lucky man.
I am reading the triangulation paper, Dynamically Triangulating Lorentzian Quantum Gravity, arXiv:hep-th/0105267v1, and I even think I can follow some of it. The idea that there are time-like and space-like edges, faces, volumes and so on, seems contrary to spacetime equivalence, but if it produces results I am willing to play along.
I am currently trying to understand Figure 1. Part (a) and part (b) seem to me to be identical tetrahedrons except for a small rotation. Part (a) is said to be a (3,1) tetrahedron, and part (b) is said to be a (2,2) tetrahedron. I would have thought (3,1) and (2,2) are (space,time) notations, but I can't seem to make sense of the figures from that perspective. Perhaps you or selfAdjoint or someone will help me find what it is I am missing here.
I have to use daylight to do chores, but hope to return to this tonight.
Thanks,
Richard
(later....)
Hi Marcus and all
I am working offline while the rice cooks.
Returning to the triangulation paper, I see that a point is defined as having volume one. This seems to me a generalization of the idea of volume, but the authors say that it is conventional and so I accept it, even though my sense of geometry tells me that a point has no volume at all. Then in eq. (3) Vol(space-like link)=1 and Vol(time-like link) = sqrt(alpha). Here a link is also given the attribute of volume. Further in equation (4) a volume of a space-like and a time-like triangle is defined. So the authors have generalized the use of the term volume to apply to point-like objects, line-like objects, and surface-like objects. I am not sure what the purpose of this generalization is. Then the authors go on to introduce the idea of space-like and time-like bones. Evidently they need a new term other than line, edge, and link to describe spacetime connectivity. The distinctions between these terms seem vague to me, but perhaps all will be made clear.
Richard
...
I am way behind on my reading and there are strawberries to put in and trees to plant, as well as dogs, rabbits, and a horse to care for. The good news is the sun has made an appearance and I have managed to get my laptop setup on my friend Peg's table, a rather crowded place, but with a nice view of the bird feeder, apple trees, sauna, and horse eating hay in the pasture. I have coffee, and a quiet idyllic setting for contemplation of the mysteries of alpha. Oh, lucky man.
...
The picture of happiness. the laptop and the coffee make it complete. I was curious about what kind of tree you'd be planting, whether fruit or a windbreak line of trees, or for stove-wood. People who plant trees think ahead. My wife and I planted a dozen redwoods in a creekbed near here that is owned by various bureaucratic agencies who seem to have forgotten it exists.
I am reading the triangulation paper, Dynamically Triangulating Lorentzian Quantum Gravity, arXiv:hep-th/0105267v1, and I even think I can follow some of it. The idea that there are time-like and space-like edges, faces, volumes and so on, seems contrary to spacetime equivalence, but if it produces results I am willing to play along.
I am currently trying to understand Figure 1. Part (a) and part (b) seem to me to be identical tetrahedrons except for a small rotation. Part (a) is said to be a (3,1) tetrahedron, and part (b) is said to be a (2,2) tetrahedron. I would have thought (3,1) and (2,2) are (space,time) notations, but I can't seem to make sense of the figures from that perspective. Perhaps you or selfAdjoint or someone will help me find what it is I am missing here.
Yes! I too am willing to play along. BTW that "Dynamically..." paper has the hard nuts and bolts. I have only calculated a very few of the volumes and checked a few of the sines and cosines. there is a limit to how much of that I want to do---the humble nittygrit as opposed to understanding the ideas. maybe it is a swinging pendulum: gnaw on the hard details a little while and then visualize and think about what it might mean.
the (3,1) and (2,2) refer to the two ways a tetrahedron can sit in the foliation. Foliation means "leaf-ing" or layering. their spacetime is made of layer upon layer of spacelike leaves or sheets connected by layers of tetrahedra.
there is also the (1,3) tetrahedron which is just the (3,1) upsidedown standing on its point with its base in the next layer up
figure 1 is talking about the D = 3 = 2+1 case. So the spatial sheets are 2D and they are divided up into actual triangles (this is the one case where a "triangulation" is actually what it sounds like)
time is just the discrete numbering of spacelike leaves---an integer layer-count
in between any two 2D spacelike leaves is filled in by a layer of tetrahedrons. The triangles that triangulate a layer of space are the BASES of (3,1) and (1,3) tetrahedrons. In that way the tetrahedrons actually CONNECT THE TWO TRIANGULATIONS of two adjacent spacelike sheets. A mosaic of tetrahedra in the 3D bulk sandwich, joins the mosaic of triangles covering each 2D leaf. I dont know whether to call these 2D things leaves (which "foliation" as in leafy foliage suggests) or sheets, so I may alternate.
The (3,1) and (2,2) notation refers to the number of vertices in each of two adjacent spatial sheets. A (2,2) tetrahedron has one top ridge (with two endpoints) and one bottom ridge (with two endpoints). With two spacelike sheets each covered by triangles, and a the tetrahedrons fitted in between to make a perfect connection between the two triangulations, then every side of every triangle must be the top ridge or bottom ridge of some (2,2) tetrahedron
Now you can erase the paper-thin spatial leaves, if you want, and just imagine a 3D spacetime that is PACKED SOLID WITH TETRAHEDRA, that is all the approximating 3D triangulation is. It is a 3D spacetime packed solid with tets but dont forget that the packing was arrived at in a special way that respects causality. If you like dectective work you could look at the tets and find the spacelike sheets and DEDUCE the plan of each one.
Each space-sheet consists of events which could have CAUSED events in the next layer up, and which could have BEEN CAUSED by events in the next layer down.
So this particular way of packing 3D spacetime solid with tets EMBODIES in it a primitive idea of causality. that things cause other things and not the other way around. that is the reason for the layers
nightcleaner
May21-05, 12:38 PM
Hi Marcus I'm online for an hour or so if you want to try real-time on this BB.
I'll work in edit and refresh the screen every few.
Causality seems to be an important item in this landscape of idylls. I read the image of foliation like a book. Write what you want on the pages. If you are clever, you can make a flip-vid (ok now I KNOW that one is a first use!).
Causality is placed in the idyll by two hands, one provides the sequence of the pages, the other provides something clever written upon them. Now we are considering fractal patterns in the idyll. A flip-book of fractals, chosen to foliate into a semblance of motion. The pages do not move and the patterns do not move, but the observer flips through the sequence......
Richard
Something to keep in mind. the CDT spacetime is not made of simplexes but is the CONTINUUM LIMIT of approximating mosaic spacetimes with smaller and smaller building blocks.
the quantum mechanics goes along with this process of finer and finer approximation. so at each stage in the process of going for the limit you have an ensemble of many many mosaic geometries
so there is not just one continuum which is the limit of one sequence of mosaics (mosaic = "piecewise flat", quite kinky manifold, packed solid with the appropriate dimension simplex)
there is a quantum jillion of continuums each being the limit of a quantum jillion of sequences of mosaics.
or there is a blur of spacetime continuums with a blur of different geometries and that blur is approximated finer and finer by a sequence of simplex manifold blurs
BUT DAMMIT THAT IS TOO INVOLVED TO SAY. So let us just focus on one of the approximating mosaics. Actually that is how they do it with their computer model. they generate a mosaic and study it and measure things, and then they randomly evolve it into another and study that, one at a time, and in this way they get statistics about the set of possible spacetime geometries. One at a time. One definite concrete thing at a time. Forget about the blur.
I am working offline while the rice cooks.
Returning to the triangulation paper, I see that a point is defined as having volume one. This seems to me a generalization of the idea of volume, but the authors say that it is conventional and so I accept it, even though my sense of geometry tells me that a point has no volume at all. Then in eq. (3) Vol(space-like link)=1 and Vol(time-like link) = sqrt(alpha). Here a link is also given the attribute of volume. Further in equation (4) a volume of a space-like and a time-like triangle is defined. So the authors have generalized the use of the term volume to apply to point-like objects, line-like objects, and surface-like objects. I am not sure what the purpose of this generalization is. Then the authors go on to introduce the idea of space-like and time-like bones. Evidently they need a new term other than line, edge, and link to describe spacetime connectivity. The distinctions between these terms seem vague to me, but perhaps all will be made clear.
...
yes, in mathtalk there is always this tension between wanting to use general terms so that you can JUST SAY IT ONCE and have that apply to all cases and all dimensions, and on the other hand wanting to use very concrete words that mean something to the reader.
So they have decided to use the word VOLUME to mean length of 1D things, and area of 2D things, and ordinary volume of 3D things, and hyperspace volume of 4D things.
the word BONE deserves a whole essay by itself. As you know the world evolved by itself without the help of any divine creator. However God did create mathematics (he just didnt bother to create the world, that happened spontaneously by some curious accident for which noone is responsible).
so by his infinte grace and glory and kindness, God arranged that in a 3D spacetime, packed solid with tetrahedra, the CURVATURE of the frigging thing can actually be measured by counting the tetrahedra around each LINE SEGEMENT. this sounds like a damnable lie but Regge discovered it, essentially. this is a CDT version of a truth Regge found in 1950.
And again by the infinite grace and mercy of the Creator (of mathematics) in a 4D spacetime the curvature of the frigging thing can be measured by counting the 4-simplexes around each TRIANGLE.
So to GENERALIZE the terminology, as mathematicians believe the Lord wishes them to do, so that they dont have to say edge in one case and triangle in another case, they invented the word BONE. a bone is the
D-2 thing.
so in the 3D case, the bone is the 1D thing
and in the 4D case, the bone is the 2D thing (the triangle)
I have a hard time imagining in 4D how a bunch of simplexes can surround a triangle.
but in 2D I can picture how 5 or 6 or 7 equilateral triangles can surround a POINT (which is the bone in 2D)
and obviously if the count is exactly 6 then the manifold is flat at that point, and if the count is 5 then the manifold has positive curvature at that point etc etc etc.
and in 3D I can picture how various numbers of tets can surround an edge (which is the 3D bone), and how that relates to curvature
but like I say it is hard for me to imagine in 4D how various numbers of 4simplexes can surround a triangle, which is the bone in 4D
my chorus is doing a concert in a few hours, so I must practice some
nightcleaner
May21-05, 01:12 PM
Hi
guest stopped in...will continue later, thanks. Great stuff....R
Something to keep in mind. the CDT spacetime is not made of simplexes but is the CONTINUUM LIMIT of approximating mosaic spacetimes with smaller and smaller building blocks.
the quantum mechanics goes along with this process of finer and finer approximation. so at each stage in the process of going for the limit you have an ensemble of many many mosaic geometries
so there is not just one continuum which is the limit of one sequence of mosaics (mosaic = "piecewise flat", quite kinky manifold, packed solid with the appropriate dimension simplex)
there is a quantum jillion of continuums each being the limit of a quantum jillion of sequences of mosaics.
or there is a blur of spacetime continuums with a blur of different geometries and that blur is approximated finer and finer by a sequence of simplex manifold blurs
BUT DAMMIT THAT IS TOO INVOLVED TO SAY. So let us just focus on one of the approximating mosaics. Actually that is how they do it with their computer model. they generate a mosaic and study it and measure things, and then they randomly evolve it into another and study that, one at a time, and in this way they get statistics about the set of possible spacetime geometries. One at a time. One definite concrete thing at a time. Forget about the blur.
This all sounds like a numerical method for the calculation of some calculus. Do they have a differential or integral equation for this process that they are doing with a numerical algorthm? Do they show that there is something pathalogical with the calculus to justify the numerical approach with computers? Thanks.
This all sounds like a numerical method for the calculation of some calculus. Do they have a differential or integral equation for this process that they are doing with a numerical algorthm? .
yes they have a rather nice set of equations, look at the article
http://arxiv.org/hep-th/0105267
equation (2) gives the action integral
the discrete version (following Regge) is (38) on page 13
from thence, in the next section, a transfer matrix and a hamiltonian
You may not realize this but the Einstein equation of classical gen rel is normally solved NUMERICALLY IN A COMPUTER these days because one cannot solve it analytically. that is how they do differential equations these days, for a lot of physical systems. It is not pathological, it is normal and customary AFAIK.
the Einstein eqns are only solvable analytically in a few highly simplified cases. so be glad they have a model that they CAN implement numerically---in the real world that is already something of a triumph
:smile:
Do they show that there is something pathalogical with the calculus to justify the numerical approach with computers? Thanks
as I say, they dont have to justify using numerical methods, and it is not pathological----its the customary thing to do if you are lucky
yes they have a rather nice set of equations, look at the article
http://arxiv.org/hep-th/0105267
equation (2) gives the action integral
the discrete version (following Regge) is (38) on page 13
from thence, in the next section, a transfer matrix and a hamiltonian
So from (2) it would seem that they are integrating over the various possible metrics on a given dimension. It would seem that the demension is given a-priori as 4D. I don't get, then, what this talk is about 2D at small scales.
Edit:
Just a moment, does all possible metrics include those that give distance only in 1, 2, and 3 dimensions. If so, is this the way to integrate over various dimensions as well?
as I say, they dont have to justify using numerical methods, and it is not pathological----its the customary thing to do if you are lucky
Isn't it more desirable to find an analytic expression? Or are they just taking it from experience that these path integrals are generally not analytic and require numerical methods to solve? And why Monte Carlo method. Is this to avoid even the possibility that the other methods of numerical integration can be pathological? Thanks.
...Isn't it more desirable to find an analytic expression? Or are they just taking it from experience that these path integrals are generally not analytic and require numerical methods to solve? And why Monte Carlo method. Is this to avoid even the possibility that the other methods of numerical integration can be pathological? Thanks.
It doesnt seem efficient for me to just be repeating what they say much more clearly and at greater length in their paper, mike. I hope you will read more in the paper.
There is also this recent paper hep-th/0505154 which IIRC has some discussion of what they are actually integrating over, and why numerical, and why Monte Carlo. They are the ones you should hear their reasons from rather than me trying to speak for them. thanks for having a look-see at the papers
cheers
Here is a relevant quote from right near the beginning of
http://arxiv.org/hep-th/0505154
the most recent CDT paper.
It addresses some of the issues raised in Mike's post, such as why numerical, why Monte Carlo. To save reader trouble i will copy this exerpt in, from bottom of page 2, the introduction section.
----quote from AJL----
In the method of Causal Dynamical Triangulations one tries to construct a theory of quantum gravity as a suitable continuum limit of a superposition of spacetime geometries [6, 7, 8]. In close analogy with Feynman’s famous path integral for the nonrelativistic particle, one works with an intermediate regularization in which the geometries are piecewise flat (footnote 2.) The primary object of interest in this approach is the propagator between two boundary configurations (in the form of an initial and final spatial geometry), which contains the complete dynamical information about the quantum theory.
Because of the calculational complexity of the full, nonperturbative sum over geometries (the “path integral”), an analytical evaluation is at this stage out of reach. Nevertheless, powerful computational tools, developed in Euclidean quantum gravity [9, 10, 11, 12, 13, 14, 15] and other theories of random geometry (see [16] for a review), can be brought to bear on the problem.
This paper describes in detail how Monte Carlo simulations have been used to extract information about the quantum theory, and in particular, the geometry of the quantum ground state (footnote 3) dynamically generated by superposing causal triangulations.
It follows the announcement of several key results in this approach to quantum gravity, first, a “quantum derivation” of the fact that spacetime is macroscopically four-dimensional [17], second, a demonstration that the large-scale dynamics of the spatial volume of the universe (the so-called “scale factor”) observed in causal dynamical triangulations can be described by an effective action closely related to standard quantum cosmology [18], and third, the discovery that in the limit of short distances, spacetime becomes effectively two-dimensional, indicating the presence of a dynamically generated ultraviolet cutoff [19]....
FOOTNOTES:
2. These are the analogues of the piecewise straight paths of Feynman’s approach. However, note that the geometric configurations of the quantum-gravitational path integral are not imbedded into a higher-dimensional space, and therefore their geometric properties such as piecewise flatness are intrinsic, unlike in the case of the particle paths.
3. Here and in the following, by “ground state” we will always mean the state selected by Monte Carlo simulations, performed under the constraint that the volume of spacetime is (approximately) kept fixed, a constraint we have to impose for simulation-technical reasons.
--------end quote from AJL----
I want to highlight something from the above quote:
by “ground state” we will always mean the state selected by Monte Carlo simulations
the ground state of the geometry of the universe (is not made of simplexes but is the limit of finer and finer approximations made of simplexes and) IS A WAVE FUNCTION OVER THE SPACE OF ALL GEOMETRIES that is kind of like a probability distribution covering a great bunch of possible geomtries
and we find out about the ground state wavefunction, find out things like what kind of geometries make up the bulk of it and their dimensionality etc, we STUDY the ground state by doing Monty simulations.
that's an interesting way of approaching it, I think. it's clear what they mean and operationally defined. If anyone is scandalized by this way of defining the quantum ground state, it would be lovely if they would tell us about it. contribute to the conversation etc.
Love this stuff. Renate Loll is keen as a knifeblade. they do not mess around. often seem to take original approaches.
Oh PLEASE CORRECT ME IF I AM WRONG. I think that before 1998 NO MATHEMATICIANS EVER STUDIED A PL MANIFOLD THAT WAS CAUSALLY LAYERED like this. The PL manifold, or simplicial manifold, is an important object that has been studied for many decades, I dont know how long but I encountered it already in grad school a long time ago. But what Ambjorn Loll invented to do was to MAKE THE SIMPLEXES OUT OF CHUNKS OF MINKOWSKI SPACE and to construct a NECCO WAFER foliation of spacelike sheets with a timelike FILLING in between. So you have a PL manifold which is 4D but it has 3D sheets of tetrahedrons.
and in between two 3D sheets of tets there is this yummy white frosting filling made of 4-simplexes which CONNECT the tets in one sheet with the tets in the next sheet up
and of course another layer of filling that connects the tets in that sheet with those in the one below it.
HISTORY: for a few years after 1998, Ambjorn and Loll tried calling that a "Lorentzian" triangulation. So it was going to be, like, a "Lorentizian" quantum gravity using Lorentzian PL manifolds. But the nomenclature didnt work out----the initials would have had to be LQG , for one thing, causing endless confusion with the other LQG.
So then in around 2003 or 2004 they started saying "causal" instead of "Lorentzian"
So here we have a geometric structure which AFAIK has not been studied in mathematics. A CAUSAL PL manifold.
TERMINOLOGY: The traditional "PL" means "piecewise linear" and it could be misleading. the thing is not made of LINES but rather building blox, but simplexes are in a very general abstract sense linear. So a simplicial manifold assembled out of simplex pieces has (for decades) been called "piecewise linear" or PL and the MAPPINGS BETWEEN such things are also piecewise linear (which is very important to how mathematicians think, they like to think about the mappings, or the morphisms of a category)
A NEW CATEGORY: we now have a new category with new mappings. the CAUSAL PL category. it will be studied. the Ambjorn Loll papers are the ground floor.
----quote from AJL----
In the method of Causal Dynamical Triangulations one tries to construct a theory of quantum gravity as a suitable continuum limit of a superposition of spacetime geometries [6, 7, 8]. In close analogy with Feynman’s famous path integral for the nonrelativistic particle, one works with an intermediate regularization in which the geometries are piecewise flat (footnote 2.) The primary object of interest in this approach is the propagator between two boundary configurations (in the form of an initial and final spatial geometry), which contains the complete dynamical information about the quantum theory.
Because of the calculational complexity of the full, nonperturbative sum over geometries (the “path integral”), an analytical evaluation is at this stage out of reach. Nevertheless, powerful computational tools, developed in Euclidean quantum gravity [9, 10, 11, 12, 13, 14, 15] and other theories of random geometry (see [16] for a review), can be brought to bear on the problem.
This paper describes in detail how Monte Carlo simulations have been used to extract information about the quantum theory, and in particular, the geometry of the quantum ground state (footnote 3) dynamically generated by superposing causal triangulations.
It follows the announcement of several key results in this approach to quantum gravity, first, a “quantum derivation” of the fact that spacetime is macroscopically four-dimensional [17], second, a demonstration that the large-scale dynamics of the spatial volume of the universe (the so-called “scale factor”) observed in causal dynamical triangulations can be described by an effective action closely related to standard quantum cosmology [18], and third, the discovery that in the limit of short distances, spacetime becomes effectively two-dimensional, indicating the presence of a dynamically generated ultraviolet cutoff [19]....
FOOTNOTES:
2. These are the analogues of the piecewise straight paths of Feynman’s approach. However, note that the geometric configurations of the quantum-gravitational path integral are not imbedded into a higher-dimensional space, and therefore their geometric properties such as piecewise flatness are intrinsic, unlike in the case of the particle paths.
3. Here and in the following, by “ground state” we will always mean the state selected by Monte Carlo simulations, performed under the constraint that the volume of spacetime is (approximately) kept fixed, a constraint we have to impose for simulation-technical reasons.
--------end quote from AJL----[/QUOTE]
OK so we have a new category where the objects are CPL manifolds and the morphisms are CPL mappings (causal piecewise linear)
there are only two basic papers so far
http://arxiv.org/hep-th/0105267
http://arxiv.org/hep-th/0505154
IMO it is a good time to get into the field.
AFAIK the CPL category has not been studied
and it will be studied.
(it is the basis of a new approach to quantum gravity which has really come alive in the past two years)
math grad students in Differential Geometry should know about CPL manifolds and consider proving some of the first easy theorems, picking the "low hanging fruit" is not a bad idea in math---where really new things are uncommon.
Or maybe i should not say "differential geometry" anymore, I should be saying "combinatorial geometry" or "simplicial geometry" I dont know---language fashions change---fields change terminology as they develop, sometimes.
a CPL mapping has to be a PL mapping (takes simplexes to simplexes, piecewise linear) and it has to respect the causal ordering as well.
I wonder if these things are going to be interesting. maybe and maybe not, can't tell much ahead of time. wouldnt have guessed the physics results of the past two years would be so exciting.
Matter fields have to be laid onto these CPL manifolds. I wonder how that will be done and what kind of new mathematical structure will appear when that is done?
these manifolds do NOT have coordinate patches----they are not locally diffeomorphic to Rn because they don't have a differentiable structure. you could put one on, maybe, but it wouldnt fit well, like a bad suit of clothes.
these CPL manifolds already have curvature. but the curvature is defined combinatorially by COUNTING the number of simplexes clustered around a "bone" (this is a very curious idea, the face of a face is a "bone" or "hinge")
I suspect Rafael Sorkin of promoting the word "bone" (or possibly even coining it). BONES ARE WHAT YOU ADD UP THE DIHEDRAL ANGLES AROUND so that you can know the deficit or surplus angle. I would have to go back to hard copy, a 1975 article by Sorkin in Physics Rev. D, to find out and that seems too much bother. His 1975 article is "Time-evolution in Regge Calculus". I suspect, but do not know, that the delightful word "bone" occurs in this article.
what one wants to be able to do in the CPL category is take LIMITS, going with finer and finer triangulations. I picture it as somewhat like taking "projective limits". the idea of a limit of finer and finer 4d causal triangulations would be defined. it would be some kind of manifold, maybe a new kind of manifold.
when you go to the limit, the bones disappear, but you still have the curvature. how?
I want to highlight something from the above quote:
by “ground state” we will always mean the state selected by Monte Carlo simulations
the ground state of the geometry of the universe (is not made of simplexes but is the limit of finer and finer approximations made of simplexes and) IS A WAVE FUNCTION OVER THE SPACE OF ALL GEOMETRIES that is kind of like a probability distribution covering a great bunch of possible geomtries
Do they have a "cononical" field equation like position and cononical momentum operators on the "wave function"?
It seems to me that they are supposing without explanation a Lorentzian metric. Can this metic form be derived? Is it necessary to do any calculations at all?
How does one pronounce "simplicial"?
Thanks.
How does one pronounce "simplicial"?
Hi Mike2
Pronounce "simplicial" with the stress on the second syllable "pli" with a short 'i' as in sim-PLi-shawl
Of course, I'm a minority accent English speaker!
Cheers
Kea
Hi Mike2
Pronounce "simplicial" with the stress on the second syllable "pli" with a short 'i' as in sim-PLi-shawl
Of course, I'm a minority accent English speaker!
Cheers
Kea
I expect we all would like your accent very much if we could hear it, let us adopt sim-PLi-shawl
as per Kea
(except that I am apt to say shul instead of the more elegant shawl)
all vowells have a tendency become the schwa
and be pronounced "uh"
Chronos
May22-05, 11:46 PM
If I may chime in a comment on statistics, the usual reason for using monte carlo methods is to give an unbiased representation of all possible [or at least reasonable] initial states of the system under study. The 'outlier' states are the ones you worry about - the ones where the model collapses and lead to unpredicatable outcomes. It is vitally important to find the boundary conditions, where the model works and where it does not. This is not necessarily a continuum, where the model always works when x>y, x<z. There may, instead, be discrete intervals where it does not work. You need to run the full range of values to detect this when you do not have a fully calculable analytical model. Interestingly enough, this kind of problem often arises in real world applications - like manufacturing - where you have complex interactions between multiple process variables.
If I may chime in a comment on statistics, the usual reason for using monte carlo methods is to give an unbiased representation of all possible [or at least reasonable] initial states...
Chronos, thanks for chiming in. I expect it has different associations. For some people, "Monte Carlo method" is a way of evaluating an integral over a multidimensional space, or more generally a way of evaluating the integral of some function which is defined over a very LARGE set, so that it would be expensive in computer time to do ordinary numerical integration.
what one does is to consider the integral as an average, or (in probabilistic terms) an EXPECTATION VALUE. And then one knows that one can estimate the expectation value empirically by sampling. So one picks some RANDOM points in the large set, and evaluates the function at each point in that random sample, and averages up the function values----and that "monte carlo sum" is a stab at the true value of the integral.
(I may be just repeating something you said already in different words. Cant be sure. but want to stress the application of M.C. to evaluating integrals over large sets where other methods inapplicable or too costly)
Naturally the more random points one can include in one's sample the better the value of the integral one is going to get.
Ambjorn et al (AJL) approach to quantum gravity is a PATH INTEGRAL approach, where the "path" is AN ENTIRE SPACETIME.
It is like a Feynman path integral except Feynman talks about the path of an actual particle as it goes from A to B, and AJL talk about the PATH THE UNIVERSE TAKES IN THE SPACE OF ALL GEOMETRIES AS IT GOES FROM BIGBANG TO BIGCRUNCH or from beginning to end whatever you want to call them. And for AJL a "path" is a possible spacetime or a possible evolution of the geometry. Well that is not such a big deal after all. It is just a Feynmanian path integral, but in some new territory.
And they want to study various properties like dimension. So they want to find expectation values, essentially, but the set of all paths is a BIG SET. So it is not practical to do the whole integral (over the range of all spacetimes, all evolutions from A to B or beginning to end). So what they are doing with their Monte Carlo is this:
they found a clever way to pick random spacetimes that are paths of geometry from beginning to end. So they pick many many, a large random sample, and they evaluate the function they want to study.
they evaluate the function they want to study for each of a random sample of spacetimes and they AVERAGE UP and that is using Monty method to evaluate the "path integral"
for now, the functions they are evaluating at sample points are very basic functions like "overall spacetime hausdorff dimension" or "spatial slice dimension" or "smallscale diffusion dimension, in the spacetime" , or in the spatial slice, or in a "thick slice". they have a lot of ways to measure the dimension and they are studying the general business of dimensionality.
but the functions could be more sophisticated like "number of black holes" or "density of dark energy" or "abundance of lithium" (maybe? I really cant guess, I only know that this is probably only the beginning)
With Monty path integral method it should be possible to evaluate many kinds of interesting functions (defined on the ensemble of spacetimes).
this is early days and they are studying dimensionality, but they can study a lot of other aspects of the world this way and i expect this to be done. they say they are now working on putting matter into the picture.
they are going to need more computer time.
the present spacetimes are ridiculously small (on order of a million blox) and shortlived.
Have you had a look at a computergenerated picture of a typical one of their spacetimes? if so, you know what i mean
Chronos, thanks for chiming in. I expect it has different associations. For some people, "Monte Carlo method" is a way of evaluating an integral over a multidimensional space, or more generally a way of evaluating the integral of some function which is defined over a very LARGE set, so that it would be expensive in computer time to do ordinary numerical integration.
I had to do a numerical calculation in a graduate course I took years ago to see the difference between the Monte Carlo method and some other traditional algorithms of numerical integration. What I learned was that most of the other numerical integration schemes rely on predictable algorithms that make some integrals impossible to evaluate. They blow up to infinity. Or there is always a great difference when you increase the resolution; they don't converge. It seems the algorithm used in traditional methods to divide the interval of integration into subdivisons itself actually contributes the the pathological nature of that numerical method. But the Monte Carlo method introduces a measure of randomness in the algorithm to help avoid any pathologies introduced by more predictable algorithms. Monte Carlo still equally divides the interval of integration, but picks at random where in each interval to evaluate the integrand.
I suspect that it is now common place to evaluate integrals in physics using Monte Carlo just to avoid even the possibility of other methods being pathological. Maybe someone else could confirm or deny this suspicion of mine.
I suspect that it is now common place to evaluate integrals in physics using Monte Carlo just to avoid even the possibility of other methods being pathological. Maybe someone else could confirm or deny this suspicion of mine.
Just occurred to me what Monty Carlo means:
it is Carlo Rovelli in his birthday suit.
I was just now replying to a post of selfAdjoint in the "1-to-10" thread and the thought occurred to that a lot of people may not have realized that getting a new model of the continuum may turn out to be THE FASTEST WAY TO A TOE.
Causal Dynamical Triangulations has a limited goal of merely arriving at a quantum model of spacetime that reproduces Gen Rel at large scale.
(but is based on quantum spacetime dynamics at microscopic scale)
Once people have a new continuum, and start working on it, and building theories of matter on that instead of Minkowski space, then it can be argued that the new features of the continuum are likely to inspire and enable new matter physics.
The CDT quantum continuum is still very new and the preliminary results on it are still coming in, so it involves guesswork to look ahead.
Suppose people start to reconstruct the Standard Model on the CDT spacetime foundations instead of on static flat txyz Minkowski, or a smooth-metric manifold.
CDT continuum has extremely non-classical geometry at small scale, but classical geometry at large scale. Or so appears (so early it is hard to be sure)
what new ideas about matter are going to be inspired on the way to putting matter fields into the "triangulations" picture, and what new mathematics enabled?
we should keep clearly in mind that the CDT is not a LATTICE approach where you take some tame classical geometry (mostly flat) and cover it with a grid.
Triangulations has the blocks assembled everywhichway, in arrangements you could not embed into a conventional flat txyz space, and results in extremely wild non-classical geometries.
so what happens when people start painting matter into the triangulations?
I suspect it would be foolish to try to put preconceptions on the outcome---they would just blind us.
And BTW let's keep in mind the general concept "nonperturbative quantum gravity" theory for something that gives you a new quantum picture of spacetime with Gen Rel in the large scale.
CDT is one possible "nonperturbative quantum gravity".
It happens to be one where they have reached the point of computer
simulations of the new model spacetime, and where they are getting interesting results.
But if there was a broad effort focussed on getting this kind of thing there could be other "nonperturbative quantum gravity" approaches also running computer models of spacetime and getting interesting results about dimensionality and bigbang cosmology and soforth.
spinfoams and loopcosmology does some of that, though CDT seems to have moved ahead of the pack, at least for now.
it just happens that this thread topic is CDT (small and fast-developing compared with LQG)
Looks like in this thread I never gave the abstract for the new CDT paper
"Reconstructing the Universe". I is good to examine the abstract of a Loll paper because she carefully articulates what she is doing and it can convey some perspective
http://arxiv.org/abs/hep-th/0505154
Reconstructing the Universe
J. Ambjorn (NBI Copenhagen and U. Utrecht), J. Jurkiewicz (U. Krakow), R. Loll (U. Utrecht)
52 pages, 20 postscript figures
"We provide detailed evidence for the claim that nonperturbative quantum gravity, defined through state sums of causal triangulated geometries, possesses a large-scale limit in which the dimension of spacetime is four and the dynamics of the volume of the universe behaves semiclassically. This is a first step in reconstructing the universe from a dynamical principle at the Planck scale, and at the same time provides a nontrivial consistency check of the method of causal dynamical triangulations..."
To me this suggests that NONPERTURBATIVE QUANTUM GRAVITY regardless whether you do it with triangles has these features.
There may be various ways to do this, and arrive at quantum models of the continuum. And they may CONVERGE on a picture with
1. 4D and Gen Rel in largescale limit
2. Semiclassical (HawkingHartle) wavefunction of the scalefactor or size of universe
3. some picture of quantum spacetime dynamics at very small scale, which cumulatively and collectively generates expected largescale behavior
The abstract does not start off by saying "CDT", it comes to that only later.
So it is putting forward a rather bold claim
It says the authors have evidence that IT DOESNT MATTER WHETHER YOU USE OUR EXACT METHOD OR NOT, we have found out something about spacetime
and if you do some OTHER method of nonpert. QG and get a quantum spacetime dynamics that reproduces Gen Rel at large scale then quite possibly you will get similar results because THAT IS HOW SPACETIME IS.
this is a bold claim and they dont present it as a certainty, but something that they offer "detailed evidence" for.
And in fact the paper in question is full of detailed evidence.
so it is not saying "our CDT is the unique only approach, you all have to change to our method", it is saying that however you do the approximation whether or not with triangles and pathintegral, or whatever, if you can open a quantum spacetime dynamics window on the small scale that reproduces Gen Rel spacetime at the large, then you will see similar things!
So please try a slew of other approaches! We will see you at the finish line.
It's confident, and seems at the same time to have a clear modest reasonableness.
Well, I started this CDT thread well before "Reconstructing" appeared, and it appeared right around post #67 of this thread. I quoted some from the first paragraph, in post #67. But I never quoted from the abstract yet in this thread, so it was high time to listen to it.
Let's remove references to any specfic method (in the above quote) and see what Loll's overall program might be:
http://arxiv.org/abs/hep-th/0505154
Reconstructing the Universe
"We provide detailed evidence for the claim that nonperturbative quantum gravity... possesses a large-scale limit in which the dimension of spacetime is four and the dynamics of the volume of the universe behaves semiclassically. This is a first step in reconstructing the universe from a dynamical principle at the Planck scale,..."
So the program can be called "quantum spacetime dynamics" in that you start with a dynamical principle at very small scale, and all of spacetime is supposed to GROW from the action of that principle at very small scale.
You dont make global assumptions----like it is a smooth manifold of some fixed chosen dimension with some number of coordinates-----you only specify how something works at the subsubatomic level. the whole shebang is supposed to HATCH from that seed.
and what hatches has to look and act right on the large scale
that is her program,
and IT SHOULD NOT DEPEND ON THE PARTICULAR METHOD
anything that deserves to be called "nonperturb. quantum gravity" should POSSESS A LARGESCALE LIMIT---should be able to put in place some microscopic dynamical principle and have a familiar 4D spacetime grow from it. It should be able to because they did this in one example of a "nonperturb. quantum gravity" and surely they do not have a patent on spacetime!
So now I think the bar has been raised. It should be possible to model the spacetime continuum with many different methods of nonpert. QG and get this kind of result. Because it is it---different ways of looking at it should converge.
This is perhaps rather radical to say and might be wrong, but it says the map of QG is now changed and the game is redefined with a new price of admission. The candidate methods can show who they are by reproducing 4D and the semiclassical cosmology result that was mentioned.
Or something like that. i am still not certain how exactly things have changed but i do believe we have a new game.
http://arxiv.org/abs/hep-th/0505154
Reconstructing the Universe
"We provide detailed evidence for the claim that nonperturbative quantum gravity... possesses a large-scale limit in which the dimension of spacetime is four and the dynamics of the volume of the universe behaves semiclassically. This is a first step in reconstructing the universe from a dynamical principle at the Planck scale,..."
So the program can be called "quantum spacetime dynamics" in that you start with a dynamical principle at very small scale, and all of spacetime is supposed to GROW from the action of that principle at very small scale.
...
just understanding the terms in which the major players see what they are doing can be a project in itself
what does the overall goal of nonperturbative quantum gravity mean to the people who are driving toward it along these various approaches like CDT?
Fotini Markopoulou gave a short definition in a recent paper she did with Mohammad Ansari
<<The failure of perturbative approaches to quantum gravity has motivated theorists to study non-perturbative quantization of gravity. These seek a consistent quantum dynamics on the set of all Lorentzian spacetime geometries. One such approach which has led to very interesting results is the causal dynamical triangulation (CDT) approach[1, 2]. In the interest of understanding why this approach leads to non-trivial results, in this paper we study...>>
this is from the introduction of
http://arxiv.org/hep-th/0505165
A statistical formalism of Causal Dynamical Triangulations
Mohammad H. Ansari, Fotini Markopoulou
20 pages, 19 pictures, 1 graph
Abstract:"We rewrite the 1+1 Causal Dynamical Triangulations model as a spin system and thus provide a new method of solution of the model."
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