- #1
ClaraOxford
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This is a problem from my theoretical physics course. We were given a solution sheet, but it doesn't go into a lot of detail, so I was hoping for some clarification on how some of the answers are derived.
For the Lagrangian L=1/2(∂μ∅T∂μ∅-m2∅T∅) derive the Noether currents and charges.
jμa=∂L/(∂(∂μ∅a))*[itex]\Phi[/itex]aα - [itex]\Lambda[/itex][itex]\mu[/itex]α
Here, the lambda term is zero, because the Lagrangian is invariant under SO(3).
∅a → ∅a + [itex]\Phi[/itex]aαεα
We were first told to show that the above Lagrangian satisfies SO(3) symmetry (this was fine). The solution sheet then states that infintessimal transformations can be written as ∅a → ∅a-itc(Tc)ab∅b, where (Tc)ab=-iεcab
I could not work out how to derive this though.
Using the above info, I can see that [itex]\Phi[/itex]ac = -i(Tc)ab∅b, taking εα = tc
Then I just need to calculate ∂L/∂(∂μ∅a)
Is this just ∂μ∅a?? I'm not sure how to calculate this when there's 2 derivatives, one with a superscript and one with a subscript. And does the transpose affect things?
Homework Statement
For the Lagrangian L=1/2(∂μ∅T∂μ∅-m2∅T∅) derive the Noether currents and charges.
Homework Equations
jμa=∂L/(∂(∂μ∅a))*[itex]\Phi[/itex]aα - [itex]\Lambda[/itex][itex]\mu[/itex]α
Here, the lambda term is zero, because the Lagrangian is invariant under SO(3).
∅a → ∅a + [itex]\Phi[/itex]aαεα
The Attempt at a Solution
We were first told to show that the above Lagrangian satisfies SO(3) symmetry (this was fine). The solution sheet then states that infintessimal transformations can be written as ∅a → ∅a-itc(Tc)ab∅b, where (Tc)ab=-iεcab
I could not work out how to derive this though.
Using the above info, I can see that [itex]\Phi[/itex]ac = -i(Tc)ab∅b, taking εα = tc
Then I just need to calculate ∂L/∂(∂μ∅a)
Is this just ∂μ∅a?? I'm not sure how to calculate this when there's 2 derivatives, one with a superscript and one with a subscript. And does the transpose affect things?