# “Generalization to four-dimensional Lorentz transformations” in Peskin and Schroeder

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?

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.
• I think I understand the distinction now. When we look at "representations" of, say, $\mathfrak{su}(2)$ in which the elements of the algebra become differential operators acting on a space of functions, these are technically just "realisations" of the algebra, since they can't be written as matrices? I guess my follow up on that would be is this true for all infinite dimensional representations? Since they can't be written as matrices. – Charlie Oct 28 '20 at 13:52