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dm4b

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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.

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.

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