Commutation of position four-vector with spacetime derivatives

I am trying to understand a simple demonstration in Ashok Das' Lectures in QFT. He does the following on p. 134

$$[P_\mu,M_{\nu\lambda}]=[\partial_\mu ,x_\nu\partial_\lambda-x_\lambda\partial_\nu]=\partial_\mu(x_\nu\partial_\lambda-x_\lambda\partial_\nu)-(x_\nu\partial_\lambda-x_\lambda\partial_\nu)\partial_\mu\tag{4.37}$$

So far everything is fine, just replaced the expressions for $$P_\mu$$ and $$M_{\nu\lambda}$$. However, the next step is where I have trouble. I calculate

$$=\eta_{\mu\nu}\partial_\lambda-\eta_{\mu\lambda}\partial_\nu-x_\nu\partial_\lambda\partial_\mu+x_\lambda\partial_\nu\partial_\mu.$$

I understand where the $$\eta$$ come from in the first few terms, but according to the explanation, the 3rd and four terms must vanish. However, for them to vanish, it would have to mean that $$x$$ and $$\partial$$ commute, and I am not sure why that would be the case. If they commute, wouldn't that change the definition of $$M_{\nu\lambda}$$? After all, it's terms would commute and maybe even cancel out! I know that there is something here that I am understanding wrong, but I'm not sure what it is.

• I'm afraid that of all the books that I have looked into, Das' was the only one that went into detail into the calculations, all others so far just give you the result. – Nick Heumann Sep 5 '19 at 23:15
• This calculation is just really sloppy, he’s misapplying the product rule. Just act with both sides on a test function and you’ll see that your two unwanted terms actually each appear twice, and cancel each other. – knzhou Sep 6 '19 at 4:42

After expanding out \begin{align} [P_a,M_{bc}]&=[\partial_a ,x_b\partial_c-x_c\partial_b]\\ &=\partial_a (x_b \partial_c-x_c\partial_b)-(x_b\partial_c-x_c\partial_b)\partial_a \end{align} it is not formally correct to apply that first $$\partial_a$$ to only the $$x_{b,c}$$ terms (which should generate those $$\eta_{a\{b,c\}}$$ terms); if there were a test function in the mix this would miss several terms that apply $$\partial_a$$ directly to the test function, due to the product rule. So assuming that the product rule still applies here you instead would get \begin{align} [P_a,M_{bc}]&= (\partial_a x_b) \partial_c- (\partial_a x_c)\partial_b + x_b \partial_a \partial_c -x_c \partial_a \partial_b - x_b\partial_c\partial_a+x_c\partial_b\partial_a\\ &=\eta_{ab}\partial_c - \eta_{ac}\partial_b - x_c [\partial_a, \partial_b] - x_b [\partial_c, \partial_a]\\ &=\eta_{ab}\partial_c - \eta_{ac}\partial_b \end{align} and those two terms vanish because we get to deal with only those nice functions for which partials commute.
2. Alternatively. one may use the operator rule $$[A,BC]~=~ [A,B]C+ B[A,C],$$ and the fundamental commutators $$[\partial_{\mu},x_{\nu}]~=~\eta_{\mu\nu}, \qquad [\partial_{\mu},\partial_{\nu}]~=~0,$$ to correctly reduce the left-hand side of eq. (4.37). It seems that that was what Das had in mind. (It should be emphasized that the presentation in Das' textbook is correct in its present form.)