Linear Sigma Model Invariance Under O(N)

by dm4b
Tags: invariance, linear, model, sigma
 P: 315 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$_{LSM}$ = (1/2) ( $\partial_{\mu}$ $\phi^{i}$ )^2 + (1/2)$\mu$^2 ( $\phi^{i}$ )^2 - ($\lambda/4!$) ( $\phi^{i}$ )^4 which is invariant under $\phi^{i}$ --> R$^{ij}$ $\phi^{j}$, or, the Orthogonal Group O(N). To show this, I've been using: $\phi^{j}$ ^2 --> R$^{ij}$ R$^{ik}$ $\phi^{j}$ $\phi^{k}$ = $\delta^{j}_{k}$ $\phi^{j}$ $\phi^{k}$ = $\phi^{j}$ ^2 but, I guess I haven't convinced myself. Seems contrived (with the indices) Any help/clarification would be greatly appreciated.
 P: 987 ψ'j2=(Rjkψk)(Rjlψl)=δklψkψl=δjlψjψl=ψj2
P: 315
 Quote by andrien ψ'j2=(Rjkψk)(Rjlψl)=δklψkψl=δjlψjψl=ψj2
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.

P: 235

Linear Sigma Model Invariance Under O(N)

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.
P: 315
 Quote by Einj 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!

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