# 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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### Fock space with mixed anti-commutation/commutation relations?

Let's say we have two modes, with the following labeling of occupation number states: $\lvert \Psi \rangle = \begin{pmatrix} 0,0 \\ 0,1 \\ 1,0 \\ 1,1 \end{pmatrix}$ An example of (what I assume to ...
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### Quantum Mechanics and Schur's lemma

Today i was studying on a textbook and i crossed a paragraph that confused me a little. Suppose you have an algebra generated by $\hat{X}$ and $\hat{P}$ and a function $f(\hat{X},\hat{P})$ that ...
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### Retrieving the non-relativistic Hamiltonian from Relativistic QM

I'm trying to follow section 15.5 here, which derives the low-energy limit of the Dirac equation for an electron in a EM-field. After some manipulations (which I think I follow alright) the author ...
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### Commutation Relation of Angular Momentum and a Radial Function

How to show that $$\langle\vec{r}|[L_{j},f(\hat{r})]|\psi\rangle = 0~?$$ note that $\hat{r}$ is a operator and not a unit vector What I know so far is that the commutation relation of a normal ...
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### In the Dirac equation, if the $\alpha$ is the mean velocity, why does it commute with $x,y,z,t$ if the velocity is related to the momentum?

In the Wikipedia talk page for the Dirac equation I found the following passage: The Dirac equation can be proved with the help of the correspondence principle. The energy and momentum of a ...
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### commutation relations for many-body permanents

I'm interested in understanding the many-body generalization of the canonical commutation relations. I.e. commutators of the form $$[a^\dagger_I, a_J]$$ where $I,J$ are multi-indices with the ...
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### Commutativity vs Compatibility

As far as I know, two compatible observables have a complete set of common eigenvectors, and using this fact, one can prove that their corresponding operators are commutative. Well now is the converse ...
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### Renormalization and canonical commutation relations

My question is whether canonical commutation relations hold for renormalized quantum fields. Below I show reasoning which caused by doubts. Consider a relativistic scalar QFT. We have spectral ...