- #1
Astrofiend
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Homework Statement
I want to show explicitly that the Lagrangian...
[tex]
L_\Phi = (D_\mu \Phi)^\dagger (D^\mu \Phi) - \frac{m^2}{2\phi_0 ^2} [\Phi^\dagger \Phi - \phi_0 ^2]^2
[/tex]
where [tex] \Phi [/tex] is a complex doublet of scalar fields, and
[tex]
D_\mu = (\partial_u + i \frac{g_1}{2} B_\mu)
[/tex]
is the covariant derivative, with [tex]B_\mu[/tex] the introduced gauge field
...is invariant under the U(1) gauge transformation
[tex]
\Phi' = e^{-i(g_1 /2)\chi\sigma^0} \Phi
[/tex]
[tex]
B'_\mu = B_\mu + \partial_\mu \Chi
[/tex]
where [tex]\chi = \chi (x)[/tex] - i.e an arbitrary function of spacetime, [tex] \sigma_0[/tex] is the 2x2 identity matrix.
The Attempt at a Solution
I've had a couple of flaccid cracks at it, but I'm afraid that I'm pretty lost with this one. The problem is, most textbooks or internet resources that I can find just basically state that the Lagrangian is gauge invariant under such U(1) transformations, and that this can be shown quite easily. Not exactly helpful for someone like me trying to work out how the maths works in the first place! Is anybody able to help out, or point me to any resources where this sort of gauge invariance is demonstrated explicitly, so I can get a feel for how the maths works? Help would be greatly appreciated!
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