3.1 Lorentz group. Recall from Eq. (3.17) the Lorentz commutation relations, (a) Define the generators of rotations and boosts as L' = jk jik K₁ =Joi where i, j, k = 1,2,3. An infinitesimal Lorentz transformation can then be writ- ten (1-10-L-iẞ-K)0. Write the commutation relations of these vector operators explicitly. (For exam- ple, [L, Liek L.) Show that the combinations J+=(L+K) and J = (L-iK) commute with one another and separately satisfy the commutation relations of angular momentum. (b) The finite-dimensional representations of the rotation group correspond precisely to the allowed values for angular momentum: integers or half-integers. The result of part (a) implies that all finite-dimensional representations of the Lorentz group correspond to pairs of integers or half integers, (j+j-), corresponding to pairs of representations of the rotation group. Using the fact that J =σ/2 in the spin- 1/2 representation of angular momentum, write explicitly the transformation laws of the 2-component objects transforming according to the (4,0) and (0.) representations of the Lorentz group. Show that these correspond precisely to the transformations of L and R given in (3.37). (c) The identity = -2002 allows us to rewrite the L transformation in the unitarily equivalent form where *(1+10+B), ². Using this law, we can represent the object that transforms as (,) as a 2 x 2 matrix that has the R transformation law on the left and. simultaneously, the transposed & transformation on the right. Parametrize this matrix as Vo+V3 V1-iv2\ +iV2 Vo-v3 Show that the object V transforms as a 4-vector.