- #1
latentcorpse
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Verify that the Lagrangian density
[itex]L= \frac{1}{2} \partial_\mu \phi_a \partial^\mu \phi_a - \frac{1}{2} m^2 \phi_a{}^2[/itex]
for a triplet of real fields [itex]\phi_a[/itex] ([itex]a=1,2,3[/itex]) is invariant under the infinitesimal SO(3) rotation by [itex]\theta[/itex]
[itex]\phi_a \rightarrow \phi_a + \theta \epsilon_{abc} n_b \phi_c[/itex]
plugging this in i get:
[itex]L= \frac{1}{2} \partial_\mu ( \phi_a + \theta \epsilon_{abc} n_b \phi_c) \partial^\mu ( \phi_a + \theta \epsilon_{abc} n_b \phi_c ) - \frac{1}{2} m^2 ( \phi_a{}^2 + 2 \phi_a \theta \epsilon_{abc} n_b \phi_c + \theta^2 \epsilon_{abc} \epsilon_{abc} n_b{}^2 \phi_c{}^2)[/itex]
but now i don't know how to get rid of anything. any ideas?
thanks.
[itex]L= \frac{1}{2} \partial_\mu \phi_a \partial^\mu \phi_a - \frac{1}{2} m^2 \phi_a{}^2[/itex]
for a triplet of real fields [itex]\phi_a[/itex] ([itex]a=1,2,3[/itex]) is invariant under the infinitesimal SO(3) rotation by [itex]\theta[/itex]
[itex]\phi_a \rightarrow \phi_a + \theta \epsilon_{abc} n_b \phi_c[/itex]
plugging this in i get:
[itex]L= \frac{1}{2} \partial_\mu ( \phi_a + \theta \epsilon_{abc} n_b \phi_c) \partial^\mu ( \phi_a + \theta \epsilon_{abc} n_b \phi_c ) - \frac{1}{2} m^2 ( \phi_a{}^2 + 2 \phi_a \theta \epsilon_{abc} n_b \phi_c + \theta^2 \epsilon_{abc} \epsilon_{abc} n_b{}^2 \phi_c{}^2)[/itex]
but now i don't know how to get rid of anything. any ideas?
thanks.