Trying to learn Spinf Foams and a lot of questions

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In summary, the conversation discusses the interchange of the product and sum in an equation from a paper by Perez. The equation involves a sum over all possible spin states and a product over all possible paths or connections between these spin states. The trick to interchange the product and sum lies in the duality of these two terms. The conversation also mentions a book recommendation for further understanding of this subject.
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zwicky
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Hi everybody!

To point, I'm studying the paper of Perez (http://arxiv.org/PS_cache/gr-qc/pdf/0402/0402110v3.pdf) in at the end of page 12 can be read something like
[tex]\sum_{j_p}(2j_{j_p}+1)<\prod_p\chi_{j_p}(U_p)s,s'>=<\prod_{j_p}\sum_{j_p}(2j_{j_p}+1)\chi_{j_p}(U_p)s,s'>[/tex]

My question is, which is the trick to interchange the prod by the sum? I know that the the prod on the r.h.s runs in j_p instead of p, but still I'm not sure about it.

Thanks in advance guys

Z.

P.D. If someone knows about an introduction with detailed calculations to this fascinating subject besides the the well know Baez or Perez reviews, "my gratitude would know no bounds".
 
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  • #2



Hi Z,

Thank you for sharing the paper by Perez and your question regarding the interchange of the product and sum in the equation on page 12. To answer your question, let's first break down the equation and understand what each term represents.

The first term, \sum_{j_p}(2j_{j_p}+1), is a sum over all possible values of j_p, which is a representation of spin in quantum mechanics. This term is often referred to as the "spin network basis" and is used to describe the state of a quantum system.

The second term, \prod_p\chi_{j_p}(U_p), is a product over all possible values of p, which represents different paths or connections between the spin networks. This term is often referred to as the "holonomy operator" and is used to describe how the spin networks evolve over time.

Now, to answer your question about the interchange of the product and sum, we need to understand that these two terms are actually related. In the first term, we are summing over all possible values of j_p, which means we are summing over all possible spin states of the system. In the second term, we are taking a product over all possible values of p, which means we are considering all possible paths or connections between these spin states.

The trick to interchange the product and sum lies in the fact that these two terms are actually equivalent. In other words, the sum over all possible spin states is the same as taking a product over all possible paths or connections between these spin states. This is known as the "duality" of the spin network basis and the holonomy operator.

I hope this explanation helps to clarify the interchange of the product and sum in the equation on page 12. As for an introduction to this subject, I would recommend checking out the book "Quantum Gravity" by Carlo Rovelli, which provides a detailed and comprehensive explanation of spin networks and their applications in quantum gravity.

Best of luck with your studies!



Scientist
 
  • #3


Hi Z,

Learning Spin Foams can definitely be a challenging endeavor, but it's great that you're taking the initiative to study Perez's paper. As for your question, the trick to interchange the product and sum is simply due to the properties of the characters (chi) used in the equation. These characters are functions of the group elements, and they have the property that they form an orthonormal basis for the space of square integrable functions on the group. This allows us to interchange the product and sum because we can use the completeness relation of the characters to express the product as a sum over all possible values of j_p. This is a common technique used in many areas of physics and mathematics, and it may take some time to get used to.

If you're looking for more detailed calculations and introductions to Spin Foams, I recommend checking out the work of John Barrett or Carlo Rovelli. Their papers and books provide a more in-depth understanding of the subject and may be helpful in clarifying any other questions you may have. Keep up the good work and happy studying!
 

What is Spinfoam?

Spinfoam is a term used in theoretical physics to describe a particular approach to quantum gravity. It involves discretizing space-time into small units and then using spin networks to describe the dynamics of these units.

Why is Spinfoam important in theoretical physics?

Spinfoam is important because it is one of the leading approaches to quantum gravity, which seeks to reconcile the theories of general relativity and quantum mechanics. It offers a promising framework for understanding the fundamental nature of space and time at the quantum level.

What is the main difference between Spinfoam and other approaches to quantum gravity?

The main difference is that Spinfoam uses a discrete approach to space-time, while other approaches, such as string theory, rely on continuous space-time. This allows Spinfoam to more easily incorporate quantum effects into the theory.

What are the challenges in learning Spinfoam?

Learning Spinfoam can be challenging because it involves advanced mathematics, such as differential geometry and group theory. It also requires a deep understanding of quantum mechanics and general relativity. Additionally, there is still ongoing research and debate about the best way to formulate and apply Spinfoam theory.

How can I get started learning Spinfoam?

To get started with learning Spinfoam, it is recommended to have a strong foundation in mathematics and physics. This includes knowledge of calculus, linear algebra, and quantum mechanics. From there, you can begin studying Spinfoam by reading textbooks and research papers, attending lectures and seminars, and participating in online forums and discussions with other researchers in the field.

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