Linear Sigma Model Invariance Under O(N)

In summary, Einj showed that the linear sigma model from page 349, Chapter 11 is invariant under rotations in the orthogonal group O(N).
  • #1
dm4b
363
4
In addition to my Faddeev-Popov Trick thread, I'm still tying up a few other loose ends before going into Part III of Peskin and Schroeder.

I was able to show that the other Lagrangians introduced thus far are indeed invariant under the transformations given. But, I am hung up on what I think should probably be the easiest - the linear sigma model from page 349, Chapter 11:

L[itex]_{LSM}[/itex] = (1/2) ( [itex]\partial_{\mu}[/itex] [itex]\phi^{i}[/itex] )^2 + (1/2)[itex]\mu[/itex]^2 ( [itex]\phi^{i}[/itex] )^2 - ([itex]\lambda/4![/itex]) ( [itex]\phi^{i}[/itex] )^4

which is invariant under

[itex]\phi^{i}[/itex] --> R[itex]^{ij}[/itex] [itex]\phi^{j}[/itex],

or, the Orthogonal Group O(N).

To show this, I've been using:

[itex]\phi^{j}[/itex] ^2 --> R[itex]^{ij}[/itex] R[itex]^{ik}[/itex] [itex]\phi^{j}[/itex] [itex]\phi^{k}[/itex]
= [itex]\delta^{j}_{k}[/itex] [itex]\phi^{j}[/itex] [itex]\phi^{k}[/itex]
= [itex]\phi^{j}[/itex] ^2

but, I guess I haven't convinced myself. Seems contrived (with the indices)

Any help/clarification would be greatly appreciated.
 
Last edited:
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  • #2
ψ'j2=(Rjkψk)(Rjlψl)=δklψkψljlψjψlj2
 
  • #3
andrien said:
ψ'j2=(Rjkψk)(Rjlψl)=δklψkψljlψjψlj2

Thanks andrien.

Looks like that's exactly what I have above in the OP, so I guess you're confirming that's correct.

Don't know why it still leaves me uneasy. I'll probably work out some explicit examples next, as that usually clears things up.
 
  • #4
You can see it in a "vectorial way". O(N) are rotations and you know that these kind of transformations leave the value of the square of the vector unchanged. That's the same thing.
 
  • #5
Einj said:
You can see it in a "vectorial way". O(N) are rotations and you know that these kind of transformations leave the value of the square of the vector unchanged. That's the same thing.

Thanks Einj. I totally get it in a conceptual way like that.

It was just the notation with the math. Wasn't quite sure I had it right!
 

Related to Linear Sigma Model Invariance Under O(N)

1. What is the Linear Sigma Model?

The Linear Sigma Model is a theoretical model used in particle physics to describe the interactions between particles. It is based on the concept of spontaneous symmetry breaking, where a symmetry of the system is broken at low energies. The model is often used to study the properties of quantum chromodynamics, the theory of the strong nuclear force.

2. What does "Invariance Under O(N)" mean in the Linear Sigma Model?

Invariance under O(N) refers to the symmetry of the model under rotations in N-dimensional space. This means that the equations and properties of the model do not change when the coordinates of the particles are rotated, making it easier to study and analyze the system.

3. How does the number of dimensions affect the Linear Sigma Model?

The number of dimensions, represented by the variable N, plays a crucial role in the Linear Sigma Model. In three dimensions, the model is known as the "O(3) model" and is used to study the properties of the pion, a subatomic particle. In four dimensions, the model is known as the "O(4) model" and is used to study the properties of the Higgs boson.

4. What is the significance of invariance under O(N) in the Linear Sigma Model?

The invariance under O(N) symmetry is important because it allows for the simplification of calculations and equations in the model. It also allows for a better understanding of the properties and behavior of particles at different energy levels.

5. How does the Linear Sigma Model contribute to our understanding of particle physics?

The Linear Sigma Model is a useful tool for studying the behavior and interactions of subatomic particles. It has been used to make predictions and calculations for experiments in particle accelerators, and has helped to further our understanding of the fundamental forces and particles in the universe.

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