"Generalization to four-dimensional Lorentz transformations" in Peskin and Schroeder

Question

Regarding equation 3.16 on page 39 of Peskin and Schroeder, we are in the middle of talking about the $SU(2)$ group and its representations. It is said that we can write the generators of the algebra as an antisymmetric tensor: $$J^{ij}=-i(x^i\nabla^j-x^j\nabla^i),\qquad i,j=1,2,3,$$ and that "the generalisation to four-dimensional Lorentz transformations is now quite natural" : $$J^{\mu\nu}=i(x^\mu\partial^\nu-x^\nu\partial^\mu),\qquad \mu,\nu=0,1,2,3. \tag{3.16}$$ "We will soon see that these six operators generate the three boosts and three rotations of the Lorentz group". They then go on to consider a particular $4\times 4$ representation given by the matrices: $$(\mathcal J^{\mu\nu})_{\alpha\beta}=i(\delta^\mu_\alpha\delta^\nu_\beta-\delta^\mu_\beta\delta^\nu_\alpha). \tag{3.18}$$ These are the generators of the Lorentz group in the four-vector representation, but what are the generators in equation 3.16? Are they written in a particular representation? In what way do they "generalise" the the generators of $SU(2)$ written above them?

Answer 1

I suspect you are unfamiliar with the (mercifully) loose language of physicists. Strictly matrix realizations of a Lie algebra, and hence group, are termed Representations, like (3.18); but anything else, including (3.16) and its SU(2) antecedent, are just termed Realizations: versatile maps (linear, in this case) which satisfy the Lie Algebra (3.17).

In this case, you see that (3.16) acting on a 4-vector $x^\beta$ amounts to the action of the 4×4 matrix (3.18), i.e., the 4-D representation. But acting on more general homogeneous functions of the coordinates (tensors), it would produce other representations.

The two realizations, up to signs and flukey noncompaction metrics, should be related by inspection: they are rotations in 3-D and 4-D spaces, respectively. My gut says you'd enjoy Robert Gilmore's and Brian Wybourne's books, illustrating this language relentlessly.