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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(...
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Heisenberg Picture from $\left[f(A), B\right] = \left[A, B \right]\frac{\partial f}{\partial A}$

I have a question about the equation below: $$\left[f(A), B\right] = \left[A, B \right]\frac{\partial f}{\partial A}$$ Is this equation valid in the Schrödinger picture, Heisenberg picture, or in ...
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Commutation relations, complex scalar field

Why for the complex scalar field $$\hat\phi = \int \frac{d^3p}{(2\pi)^{3/2}(2E_{\vec{p}})^{1/2}}\left(\hat{a}_{\vec{p}}e^{-p \cdot x} + \hat{b}_{\vec{p}}^\dagger e^{p \cdot x}\right),$$ the ...