Questions tagged [commutator]

A mathematical construct quantifying the difference in effect of applying two operators in two alternate successions. It is the defining product of a Lie algebra, the efficient underlying description of Lie groups, of use in several areas of physics, most notably quantum field theory.

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Why $\int dx \partial_\mu\neq \partial_\mu \int dx$ but $\int dp \partial_\mu=\partial_\mu\int dp$?

It's well known that $\int dx \partial_\mu\neq \partial_\mu \int dx$. But I have a hard time understanding $\int dp \partial_\mu=\partial_\mu\int dp$, because $[p,x]\neq0$ do not commute. However, ...
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Proving that $$[\phi(\vec{x}, 0), \phi(\vec{x}, t)] \sim e^{-i m t}-e^{+i m t}$$ in QFT

So far, I get the following (for the left term in the integral, $d$=3): \begin{equation} \begin{aligned} \Delta_{+}(x) &= \int \frac{\mathrm{d} \vec{p}^{d}}{(2 \pi)^{d} 2 e(\vec{p})} \exp (-i t e(...
Show that when angular momentum $L_x$ and $L_y$ commute with operator $G$, then $L_z$ also commutes with $G$
I want to prove that if Angular momentum $L_x$ and $L_y$ commute with an operator $G$, angular momentum $L_z$ also commutes with $G$. if $[L_x , G] = [L_y, G] = 0$ then $[L_z , G] = 0$ I know that \$...