New Bojowald (elements of LQC)

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In summary, the AJL authors found a lower asymptotic spacetime dimension that was not actually D = 2. It was D = 1.8. They also found that topologically the AJL universe is S3 x R. Lastly, they found that the use of noncommutative geometry suggests that the base of the fiber bundle B (space-time) is the spectrum of the total space.
  • #1
marcus
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http://arxiv.org/abs/gr-qc/0505057
Elements of Loop Quantum Cosmology
Martin Bojowald
30 pages, 4 figures, Chapter contributed to "100 Years of Relativity - Space-time Structure: Einstein and Beyond", Ed. A. Ashtekar (World Scientific)
Report-no: AEI-2005-025

"The expansion of our universe, when followed backward in time, implies that it emerged from a phase of huge density, the big bang. These stages are so extreme that classical general relativity combined with matter theories is not able to describe them properly, and one has to refer to quantum gravity. A complete quantization of gravity has not yet been developed, but there are many results about key properties to be expected. When applied to cosmology, a consistent picture of the early universe arises which is free of the classical pathologies and has implications for the generation of structure which are potentially observable in the near future."
 
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  • #2
new one from AJL (the dynamical triangulation folks)

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."

I :!) Renate Loll.
 
  • #3
A 2-D quantum universe on the Planck scale presumably includes time as one dimension since it is a reduction from 4-d at larger scales. Doesn't a universe made of one space dimension and one time dimension just describe a string?
 
  • #4
yanniru said:
A 2-D quantum universe on the Planck scale presumably includes time as one dimension since it is a reduction from 4-d at larger scales. Doesn't a universe made of one space dimension and one time dimension just describe a string?

you should read the basic CDT papers and probe the similarities and differences for yourself

ordinarily, I think, string theorists imagine a string existing IN some higher dimensional "target" space where it can move around [EDIT: yaniru, don't get confused when I say higher. I am not talking "extra" dimensions so much as that a 2D worldsheet is already in ordinary 4D spacetime, plus whatever "extra" you want to imagine. The 1D string is in surroundings of AT LEAST 3 ordinary dimensions. this is not the case with the AJL universe where diffusion processes indicate a dimension like 1.8 at very small scale. that universe is NOT inside some larger target space, so is not in that sense analogous to a string. Hope I'm making sense and that this helps clarify.]

and they do not imagine stuff going on inside a string (the string is the primitive object)

AJL do not have their universe IN some larger universe where it can move around or vibrate etc. so in that sense their universe is NOT "like" a string

In the present paper the lower asymptotic spacetime dimension that they found was not actually D = 2. It was D = 1.8.
With a string, the worldsheet dimension is AFAIK exactly D = 2
Also topologically the AJL universe is S3 x R

Also in principle stuff can move around inside the AJL universe (even when its hausdorff dimension is D = 1.8 or D = 2.1 or whatever, or even when its spectral dimension measured by diffusion processes, as in this latest paper).

which is also rather different from a string, which nothing moves around inside of.

But you really have to read the papers and decide for yourself how analogous or how like or unlike you think it is :smile:
 
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  • #5
marcus said:
ordinarily, I think, string theorists imagine a string existing IN some higher dimensional "target" space where it can move around

and they do not imagine stuff going on inside a string (the string is the primitive object)

AJL do not have their universe IN some larger universe where it can move around or vibrate etc. so in that sense their universe is NOT "like" a string
...
Also in principle stuff can move around inside the AJL universe (even when its hausdorff dimension is D = 1.8 or D = 2.1 or whatever, or even when its spectral dimension measured by diffusion processes, as in this latest paper).

In string theory, the higher dimensions are grafted to space-time in a fiber bundle. This is the essence of Kaluza-Klein theory. The usual construction involves a Calabi-Yau space grafted to every point of our space-time manifold. The string has degrees of freedom in these local Calabi-Yau spaces, interpreted as "internal degrees of freedom" of the string.

Now, of course, this isn't the whole story, as Calabi-Yau spaces are just intersections in projective space. So imagine a projective space grafted to each point of space time.

The use of noncommutative geometry suggests that the base of the fiber bundle B (space-time) is the spectrum of the total space E, and that Yang-Mills gauge symmetry arises from automorphisms of algebras generating the projective spaces.

That AJL are finding space to locally seem two-dimensional, does not mean it is in fact two-dimensional. For instance, there is a 16-dimensional projective space that can be represented as two-dimensional. The only way to find the true dimension is by examining the Minkowski representation of the structure algebra generating the projective space. Obviously, this structure algebra must be robust enough to contain automorphisms that include the symmetry of the standard model SU(3) x SU(2) x U(1). Sadly, a genuine 2-dimensional real space does not contain complex isometries, so can never contain any symmetries of the standard model.

Regards,

Mike
 
  • #6
kneemo said:
In string theory, the higher dimensions are grafted to space-time in a fiber bundle. This is the essence of Kaluza-Klein theory. The usual construction involves a Calabi-Yau space grafted to every point of our space-time manifold. The string has degrees of freedom in these local Calabi-Yau spaces, interpreted as "internal degrees of freedom" of the string.

Now, of course, this isn't the whole story, as Calabi-Yau spaces are just intersections in projective space. So imagine a projective space grafted to each point of space time.

The use of noncommutative geometry suggests that the base of the fiber bundle B (space-time) is the spectrum of the total space E, and that Yang-Mills gauge symmetry arises from automorphisms of algebras generating the projective spaces.

That AJL are finding space to locally seem two-dimensional, does not mean it is in fact two-dimensional. For instance, there is a 16-dimensional projective space that can be represented as two-dimensional. The only way to find the true dimension is by examining the Minkowski representation of the structure algebra generating the projective space. Obviously, this structure algebra must be robust enough to contain automorphisms that include the symmetry of the standard model SU(3) x SU(2) x U(1). Sadly, a genuine 2-dimensional real space does not contain complex isometries, so can never contain any symmetries of the standard model.

Regards,

Mike

AFAIK, the standard model is a renormalizable theory and therefore by definition not expected to be valid down to the Planck length. If spacetime "runs" to four dimensions well before the minimum length supported by the standard model, then the relevant algebra will be the one appropriate to that. No? Think some more about the idea of dynamic dimensionality and see if the math you cite isn't all hung up on fixed dimensionality.
 
  • #7
selfAdjoint said:
AFAIK, the standard model is a renormalizable theory and therefore by definition not expected to be valid down to the Planck length. If spacetime "runs" to four dimensions well before the minimum length supported by the standard model, then the relevant algebra will be the one appropriate to that. No? Think some more about the idea of dynamic dimensionality and see if the math you cite isn't all hung up on fixed dimensionality.

that is really interesting.
it helps me imagine how the space of CDT might support the standard model.

the dynamic dimension at very very small scale, where it gets D = 2 or even less, like D = 1.8, seems to be mostly relevant to how the coupling constant G behaves. It also helped to read a piece about this by John Baez in TWF #139 from back in 1999. I will get the link.
http://math.ucr.edu/home/baez/twf_ascii/week139

I discussed some parts of that TWF here, and quoted relevant passages
https://www.physicsforums.com/showthread.php?p=569075#post569075
 
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  • #8
This is a fantastic deviation from LqG, a dynamic dimensional model, that can
solve the infrared problem, I think that they are on the right track, I doubt if
this will be the final solution, but one can hope.
 
  • #9
marcus said:
the dynamic dimension at very very small scale, where it gets D = 2 or even less, like D = 1.8, seems to be mostly relevant to how the coupling constant G behaves.


According to my calculations, and I may be incorrect :biggrin: , there is a 1.9 relevence in the running coupling constant?, its definitely not 1.8

SA may remember from another forum the 'Tri-Coupled-Phase' of dimensional boundary 'entangled' state function?

A three-dimensional volume can interact with a two-dimensional volume, by introducing a dimensional 'impurity', because 2-D is TOO PERFECT!..so the best fit for any 2-D space interacting with a 3-D space..is actually 1.9..not 1.8. When compacting a 3-D volume down to a 2-D volume, there will be imposed upon the perfect 2-D volume, a default impurity, an 'information-loss' if you want.

Simplistically, a 2-D cannot be achieved, its a too-perfect solution!

I will leave for now the data I have to back this up, I am interested in how this recent kerfuffle develops, keep up the good work.

I believe this has some relevence:http://arxiv.org/abs/gr-qc?0409006
 
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  • #10
this thread began to be about Bojowald's LQC review paper but I got distracted when I saw this short AJL paper

marcus said:
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."
...

Now the big paper from AJL that we have been expecting has been posted.
Here is a thread about it, in case anyone wants to comment, share impressions, or has questions.

https://www.physicsforums.com/showthread.php?t=75993
 

Related to New Bojowald (elements of LQC)

1. What is LQC and how does it differ from traditional quantum mechanics?

LQC stands for Loop Quantum Cosmology and it is a theoretical framework that applies the principles of loop quantum gravity to study the behavior of the universe at the smallest scales. It differs from traditional quantum mechanics in that it takes into account the discrete nature of space and time, rather than treating them as continuous. This allows for a more complete understanding of the dynamics of the universe, particularly in the early stages of its evolution.

2. How does Bojowald's work contribute to the field of LQC?

Bojowald's work has greatly advanced the field of LQC by providing a deeper understanding of the quantum behavior of the universe. He has developed mathematical models that are consistent with the principles of LQC and has made significant contributions in areas such as the study of quantum black holes, the big bang singularity, and the effects of quantum fluctuations on the evolution of the universe.

3. What is the significance of the "new elements" in Bojowald's approach to LQC?

The "new elements" in Bojowald's approach to LQC refer to his incorporation of new mathematical tools and techniques that allow for a more precise and accurate description of the quantum behavior of the universe. These elements include new variables and representations that have helped to resolve some of the long-standing issues in LQC, such as the problem of time and the behavior of quantum states near the big bang singularity.

4. How does Bojowald's work address the problem of time in LQC?

One of the major challenges in LQC is the problem of time, or how to properly define and measure time in a quantum theory of the universe. Bojowald's work has made significant progress in this area by developing a "quantum relational time" that takes into account the interactions between different parts of the universe. This approach has helped to resolve some of the paradoxes and inconsistencies that arise when trying to apply traditional notions of time to a quantum cosmological model.

5. What are the potential implications of LQC and Bojowald's work for our understanding of the universe?

LQC and Bojowald's work have the potential to greatly enhance our understanding of the universe, particularly in its early stages. By incorporating quantum principles into the study of cosmology, we may be able to gain insights into the fundamental nature of space, time, and matter. This could also have implications for areas such as black hole physics, the origin of the universe, and the behavior of matter at extreme energy scales.

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