# 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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### Commutator of an operator with its Hermitian adjoint in linear quantum systems

I have a commutator of a single-mode photon field operator $\alpha$ with its Hermitian adjoint $\alpha^{\dagger}$. [$\alpha$, $\alpha^{\dagger}$] When this commutator gives a negative value, $\alpha$ ...
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### Representation of The Poincare Group

I am currently trying to understand the representations of the conformal group. I am following the script by J.D Qualls. At page 29, the author finds the effect of $L_{\mu\nu}$ by "studying the ...
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### The Cartan sub algebra and Killing form of the Poincaré algebra

Doing some studies on Group theory, I worked Frederic Schuller's lectures on youtube where he classifies all semisimple Lie algebras by the Dynkin's diagrams; I should say it was interesting. Trying ...
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### How to bound the expectation value of a commutator?

Are there formal ways to bound the following quantity: $$\langle[[{S_x},{H}],{H}]\rangle$$ The expectation value is taken on an eigenstate of $S_x$. $H$ is a dipolar Hamiltonian acting on $N$ ...
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### General commutator $[f(A),g(B)]$ of functions

Let us have two hermitian operators $A,B$ and their commutator $[A,B]:=AB−BA$, then let us have two functions $f,g$ and we want to to calculate $[f(A),g(B)]$ (everything is still hermitian). I have a ...
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### Lorentz boost transformations form a group?

In the QFT book of Ryder, he states that Lorentz boost transformations do NOT form a group. This is due to the boost generators $\textbf{K}$, i.e. they do not form a closed algebra under commutation. ...
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### How would an operator $A$ in the Schrodinger and interaction picture be related if the commutation relation $[A_s,H_0] = 0$ holds?

How would an operator A in the Schrodinger $A_s$ and interaction $A_I$ picture be related if the commutation relation $$[A_s,H_0] = 0$$ holds where $H_0$ is a solved hamiltoniain.
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### Superoperator Commutation Relations

Let's say we are given a generic Master equation: $$\dot{\rho} = -\frac{i}{\hbar}[H_0 + H_I, \rho] - \mathcal{L}\rho$$ where $H_0$ is unperturbed Hamiltonian and $H_I$ is the interaction ...
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### Spacetime translation in QFT

I have a question about the field under the spacetime translation. For example, in page 26 of Peskin's textbook, they give the translation properties of the field. So consider the space translation, ...
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