8.324 Relativististic Field Theory II, Assignment 1 Solution
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چکیده
(a) Since the generators Ta are Hermitian, the transformation laws are δψ = iǫaTaψ (2) δψ̄ = −i ̄ Ta ψǫa (3) The Lagrangian is quite trivially seen to be invariant δL = −iδ ψ̄(∂/ −m)ψ − iψ̄(∂/ −m)δψ = −iψ̄(−iǫaTa)(∂/ −m)ψ − iψ̄(∂/ −m)(iǫaTa)ψ = 0 , (4) where we used the trivial relation [Ta, γ ] = 0. (b) We use the so called Noether method (described in e.g. Di Francesco et al.: CFT Section 2.4.2) to find the conserved current, i.e., we pretend that the transformation parameter ǫ is spacetime dependent. Then ̄ δL = (∂μǫa)ψγTaψ (5) which implies J ̄ = ψγTaψ (6) a − (Note, that Lie algebra indices are up and down without any regard for their placement). Using the EoMs the current can be shown to be conserved. (c) The conserved charges are thus
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