- #1
Dixanadu
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Homework Statement
Hey guys, so this is a follow up from my previous post.
So I have this Lagrangian for two coupled fields:
[itex] \begin{split}
\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi_{1})(\partial^{\mu}\phi_{1})+\frac{1}{2}(\partial_{\mu}\phi_{2})(\partial^{\mu}\phi_{2})-\frac{m_{1}^{2}}{2}(\phi_{1}\phi_{1}) -\frac{m_{2}^{2}}{2}(\phi_{2}\phi_{2})-g(\phi_{1}\phi_{2})^{2}
\end{split}[/itex]
I was given the transformation [itex]\phi_{1}\rightarrow\phi_{1}+\epsilon\phi_{2}, \phi_{2}\rightarrow\phi_{2}-\epsilon\phi_{1}[/itex], which (if I've done it right) yields the following:
[itex] \begin{split}
\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi_{1})(\partial^{\mu}\phi_{1})+\frac{1}{2}(\partial_{\mu}\phi_{2})(\partial^{\mu}\phi_{2})-\frac{m_{1}^{2}}{2}(\phi_{1}^{2}+2\epsilon\phi_{1}\phi_{2}) \\
-\frac{m_{2}^{2}}{2}(\phi_{2}^{2}-2\epsilon\phi_{1}\phi_{2})-g(\phi_{1}\phi_{2}-\epsilon\phi_{1}^{2}+\epsilon\phi_{2}^{2})^{2}
\end{split}[/itex]Now I have to "Find the most general values for m_1, m_2 and g such that this transformation is a symmetry".
Homework Equations
Apparently there is no coordinate transformation, so I've been told that I need to impose the condition
[itex]\mathcal{L}(\phi',\partial_{\mu}\phi')=\mathcal{L}(\phi,\partial_{\mu}\phi)[/itex]
The Attempt at a Solution
So err...am I just meant to equate both Lagrangians? If I do that I end up with the equations:
[itex]\frac{m_{1}^{2}}{2}(\phi_{1}^{2}+2\epsilon\phi_{1}\phi_{2})=\frac{m_{1}^{2}}{2}(\phi_{1}\phi_{1})[/itex]
[itex]\frac{m_{2}^{2}}{2}(\phi_{2}^{2}-2\epsilon\phi_{1}\phi_{2})=\frac{m_{2}^{2}}{2}(\phi_{2}\phi_{2})[/itex]
[itex]g(\phi_{1}\phi_{2})^{2}=g(\phi_{1}\phi_{2}-\epsilon\phi_{1}^{2}+\epsilon\phi_{2}^{2})^{2}[/itex]
And I'm stuck lol...how do I use these to find the most general values for m_1, m_2 and g?
Thank guys
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