The Quantum Configuration Space of Loop Quantum Cosmology

In summary: Velhinho paper could provide valuable insights into the mathematical underpinnings of loop quantum cosmology and its relationship to loop quantum gravity. It explores the concept of the Bohr compactification of the line and its role as the quantum configuration space in LQC. It also delves into the polymer representation, drawing analogies with LQG and providing a simpler context for understanding its mathematical structure. Overall, it offers a valuable perspective for those interested in the formal and rigorous aspects of LQC.
  • #1
jal
549
0
Hi marcus!
It seems to me that there are many different approaches, from around the world, that are converging at an accelerated pace.
Maybe not? … It might just be my desire to understand minimum length.
http://arxiv.org/PS_cache/arxiv/pdf/0704/0704.2397v1.pdf [Broken]
The Quantum Configuration Space of Loop Quantum Cosmology
J. M. Velhinho
18 April 2007
abstract
The article gives an account of several aspects of the space known as the Bohr compactification of the line, featuring as the quantum configuration space in loop quantum cosmology, as well as of the corresponding configuration space realization of the so-called polymer representation. Analogies with loop quantum gravity are explored, providing an introduction to (part of) the mathematical structure of loop quantum gravity, in a technically simpler context.
I even looked up the following to try to get a better idea of what is happening.
http://www.iop.org/EJ/article/0264-9381/20/1/103/q301l3.html
Polymer and Fock representations for a scalar field
Abhay Ashtekar, Jerzy Lewandowski and Hanno Sahlmann
11 Dec 2002
“Our choices will ensure that the polymer scalar field can `live' on quantum geometry.”
I also skimmed the thread “Ashtekar's "Shadow states" paper”
-----------------

Well … it’s not called the QMLS.
Can someone draw a figure of what they think it looks like. A polymer representation on a line with triads in a 3d configuration, (a lattice). All that come to my mind is a funny looking torus which I labeled “void” with the triads making the dual tetra.
http://www.geocities.com/j_jall/3dspace.gif
jal
 
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  • #2
Jal, the Velhinho paper studies the fundamental definitions of LQC from a sophisticated mathematical perspective.
I think the paper could be valuable to a mathematician who wanted to understand the formal rigorous basis of LQC.

If one wants to read LQC papers intuitively and follow partly on the basis of physics-sense, then I don't think one needs the Velhinho paper. But if one specializes in high abstract math, I suspect it could be very useful.

For the first 3 or 4 years of LQC, from 2001-2004, nobody mentioned "Bohr compactification of the Reals". Bojowald was just proceeding and building it up based on his physical grasp and working by analogy with the full theory. He, and others working with him, got several interesting results (including removal of the BB singularity---quantum corrections making gravity repel at very high density---bounce---a natural inflation phase).

But in 2004 a paper appeared co-authored with Ashtekar and Lewandowski that brought in the "Bohr compactification"----a different topology on the Reals.

By an odd coincidence this was invented by Niels Bohr's brother Harald who was a mathematician. And it just happened to be what Bojowald Ashtekar Lewandowski needed to put LQC on a mathematically more precise and rigorous footing.

But it was a kind of abstract or "moral" accomplishment---it didnt seem to change any of the practical results. In other words, more interesting mathematically than physically.

Velhinho's paper is a careful and lengthy clarification of that whole business, pointing out mathematical ramifications.

He has been doing this kind of thing for 3 or 4 years now. He follows LQG and LQC and from time to time, as a highly trained abstract mathematician, he illuminates some aspect in depth to show what is going on from HIS perspective. I think it is a valuable contribution. If there were some subtle inconsistency, his spotlight might find it. Or he might help other mathematicians get into the field. But at this point I don't expect his papers to help me personally understand better.
But
 
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  • #3


Hello jal,
Thank you for your question and interest in understanding the Quantum Configuration Space of Loop Quantum Cosmology. This is a complex topic that requires a deep understanding of mathematical structures and concepts, so I will try my best to provide a simplified explanation.

Firstly, it is correct that there are many different approaches to understanding the universe, and they are all converging towards a common goal of understanding the fundamental nature of reality. Loop Quantum Cosmology (LQC) is one such approach, which combines ideas from quantum mechanics and general relativity to study the behavior of the universe at the very beginning, known as the Big Bang.

Now, let's break down the article you have referenced. The main focus is on the Quantum Configuration Space (QCS) of LQC. This is a mathematical space that represents the possible configurations of the universe at the quantum level. It is called the Bohr compactification of the line because it is a compact (finite) version of the Bohr model of the atom, where the electron orbits are quantized. In LQC, the universe is also quantized, and the QCS represents the possible states of the universe at a particular moment in time.

The article also mentions the polymer representation, which is a mathematical technique used to represent the quantum states of the universe. This representation is based on the idea that space and time are made up of discrete units, or "grains," much like a polymer chain is made up of individual units. This approach allows for a more detailed understanding of the quantum behavior of the universe at the Planck scale, where classical physics breaks down.

The analogy with loop quantum gravity is also mentioned, as LQC is a specific application of the broader theory of loop quantum gravity. This theory aims to reconcile quantum mechanics with general relativity, and it provides a framework for understanding the behavior of space and time at the quantum level.

Finally, the article mentions the "shadow states" paper, which introduces a new mathematical tool called the "shadow representation." This representation helps to better understand the quantum states of the universe and their evolution over time.

As for the figure you have requested, it is challenging to provide a visual representation of the Quantum Configuration Space without a deep understanding of the mathematical concepts involved. However, I suggest looking at the figure in the paper by Ashtekar, Lewandowski and Sahlmann (2002) that you have referenced. It shows a visualization of the polymer representation and how
 

1. What is the Quantum Configuration Space of Loop Quantum Cosmology?

The Quantum Configuration Space of Loop Quantum Cosmology is a mathematical representation of the quantum states of the universe. It describes the possible configurations of the universe at a given time, taking into account the fundamental principles of quantum mechanics.

2. How is the Quantum Configuration Space of Loop Quantum Cosmology different from classical cosmology?

Classical cosmology describes the universe using classical physics, which assumes that everything can be described in terms of continuous variables. In contrast, the Quantum Configuration Space of Loop Quantum Cosmology takes into account the discrete nature of quantum mechanics and describes the universe in terms of discrete variables.

3. What is the significance of the Quantum Configuration Space in understanding the early universe?

The Quantum Configuration Space plays a crucial role in understanding the early universe because it allows us to study the universe at a level where quantum effects are dominant. This is important because the early universe was in a highly quantum state, and understanding this state is essential for understanding the origins of the universe.

4. How does Loop Quantum Cosmology address the problem of the singularity in classical cosmology?

In classical cosmology, the Big Bang singularity is a point where the universe is thought to have begun. However, this singularity is problematic because it implies that the laws of physics break down at this point. Loop Quantum Cosmology addresses this issue by replacing the singularity with a "bounce," where the universe contracts and then expands again, avoiding the singularity altogether.

5. How does the Quantum Configuration Space of Loop Quantum Cosmology relate to the concept of a multiverse?

The Quantum Configuration Space of Loop Quantum Cosmology allows for the possibility of multiple universes or a multiverse. This is because it allows for different quantum states of the universe, each representing a different universe. However, the existence of a multiverse is still a topic of debate and is not a definitive aspect of Loop Quantum Cosmology.

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