Spin networks with different intertwiners

In summary, the conversation discusses the definition of spin networks and how they are equipped with SU(2) representations and intertwiners. The speaker asks for an example of two spin networks with the same number of nodes, links, and coloring but different intertwiners. They also mention their interest in assigning complex valued functions to the networks and how the intertwiners contract with the links. A specific example is given of taking the tensor product of two half integer SU(2) representations, resulting in a 4-dimensional matrix with two intertwiners. The speaker then asks about the action of these intertwiners on the four-valent node and mentions an issue with a LaTeX formula.
  • #1
Heidi
411
40
Hi Pfs
Spin networks are defined by the way their links and their nodes are equipped with SU(2) representations and intertwiners.
Could you give an example of two different spin networks with the same number of nodes, links between them, the same coloring of the links (and their orientations) but have different intertwiners?
I would like to see how to assign squarable complex valued functions to them. how the intertwiners contract with the links.
thanks
 
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  • #2
if i take the tensor product of two half integer SU(2) representation on 2 dimensional Hilbert space, i get a 4 dimensional matrix with
##1/2 \otimes 1/2 = 0 + 1## and if i tensor it by it self i get the (0 + 1) + (1 + 0 + 1 + 2) representation with dimension 4 + 12 = 16. we see that 0 appears twice so there are two intertwiners. How do they act in the four valent node?
and what is the problem with the latex formula?
 

1. What is a spin network with different intertwiners?

A spin network with different intertwiners is a mathematical representation of quantum states in a quantum mechanical system. It is a graph-like structure that describes the relationships between different particles and their quantum properties, such as spin, in a given system.

2. How is a spin network with different intertwiners used in physics?

Spin networks with different intertwiners are used to study the behavior of quantum systems, particularly in quantum gravity and loop quantum gravity theories. They provide a way to mathematically describe the quantum properties of particles and their interactions in a given system.

3. What is the significance of different intertwiners in spin networks?

The different intertwiners in spin networks represent the different ways in which particles can interact and exchange quantum properties, such as spin. They play a crucial role in understanding the dynamics of quantum systems and can provide insight into the fundamental nature of reality.

4. How are spin networks with different intertwiners constructed?

Spin networks with different intertwiners are constructed using mathematical tools such as representation theory and group theory. These networks are built by connecting nodes, which represent particles, with edges that represent the interactions between them. The intertwiners are then used to label the edges and describe the quantum properties of the particles.

5. What are the potential applications of spin networks with different intertwiners?

Spin networks with different intertwiners have potential applications in various fields, including quantum computing, quantum information theory, and quantum gravity. They can also be used to study the behavior of complex systems, such as black holes, and may provide insights into the nature of space and time at a fundamental level.

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