Problem 8. Show that a scalar field transforms under an infinitesimal Lorentz transforma- tion as (2) 0(2) - W","0.0, and that its derivative transforms as 20(2) 0,0(2) - w","0.00+ w 2,0. Show then that these changes induce a change in the Lagrangian density that can be written as (asume the Lagrangian density does not depend explicitly on 2) (7) 8C = -0,(Wx"L). Using Noether's theorem, deduce the existence of 6 conserved currents, which define the angular momentum tensor: (74) 08 = 24T4B 2014. (8) This tensor is anti-symmetric in a and B. Obtain the 6 independent constants of motion that these currents define, and interpret them in physical terms. Problem 8. Show that a scalar field transforms under an infinitesimal Lorentz transforma- tion as (2) 0(2) - W","0.0, and that its derivative transforms as 20(2) 0,0(2) - w","0.00+ w 2,0. Show then that these changes induce a change in the Lagrangian density that can be written as (asume the Lagrangian density does not depend explicitly on 2) (7) 8C = -0,(Wx"L). Using Noether's theorem, deduce the existence of 6 conserved currents, which define the angular momentum tensor: (74) 08 = 24T4B 2014. (8) This tensor is anti-symmetric in a and B. Obtain the 6 independent constants of motion that these currents define, and interpret them in physical terms