# 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 is commutation relations the first step in quantization?

Why is commutation relations the first step in quantization?
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### Holonomic basis

Is the following definition correct? Given a differentiable manifold $M$ and an ordered basis $\{e_j^m\}$ of the tangent space $T_m M$ with $m\in M$ (they are vectors and not vector fields). An ...
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### Covariant derivative with respect to commutator

I have some confusion with the notion of $\nabla_{[A, B]}\bf{v}$, that expression, with a commutator of vector fields as the subindex of the connection appears for instance in the definition of the ...
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### Virasoro algebra commutation (part 2)

This was a sub-question in my previous post that I ask separately now. In Introduction to Conformal Field Theory by Blumenhagen and Plauschinn (springer link) the Virasoro algebra is introduced the ...
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### The charge given by a commutator

I saw in the text that $[Q,X]=cX$ and says the operator $X$ has charge $c$ under the generator $Q$. I tried to understand why the coefficient $c$ means the charge. So I used this relation to get the ...
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### What is the QFT state with two distinguishable fermions present?

I want to describe a system with two non-interacting and definitely different fermions, say an electron neutrino, $\nu_e$, and an electron, $e^-$. The state describing a single electron is given ...
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### Commutator of Lie Group Generators

This is from Maggiore's "A Modern Introduction to Field Theory", Page 15. I have a Lie group with matrix generators $$T^{a}_{R}$$ Where $a$ takes values from 1 to the dimension of the Lie group. ...
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### Commutator $\vec{L}$ with $\vec{X}\cdot\vec{P}$

Let $\vec{X}=(X_1,X_2,X_3)^T$ and $\vec{P}=(P_1,P_2,P_3)^T$. Define $\vec{L}=\vec{X}\times\vec{P}$. Then, I can calculate $\vec{L}=(X_2P_3-X_3P_2,\,X_3P_2-X_2P_3,\,X_1P_2-P_1X_2)^t$. For all ...
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### Commutation of differential operators with boundary conditions

First post ever. Let's see how this goes... My question concerns the commutation of differential operators in the presence of boundary conditions. If it is of any help, this is relevant to me in the ...
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### Conjugate of total spin operator

I got a lattice, and the total spin operator for x and for y, for that lattice. I know that the x component conmutes with an operator called staggered spin operator in y. I also know that the ...
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### Definitions of operators and commutativity in quantum mechanics

If $[\hat A,\hat B] = 0$, where $\hat A$ and $\hat B$ are operators, then the operators commute. This also means that, when applied to a wavefunction, that one can measure observables $A$ and $B$ in ...
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### Dirac field local observables

This is actually a continuation of calculation I've been working on. It is well known that, in the case of Dirac fields $\psi(x)$, they satisfy anticommutatation relationships since they're fermionic ...
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### Commutation relation, field and conserved charges

One more question on basic commutation relation for fields. Let $\phi(x)$ be a scalar field and $$P^\alpha = \int d^3x T^{0\alpha},$$ where $T^{\alpha\beta}$ is the energy momentum tensor. Now, ...
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### In which cases is the Schrodinger representation more useful?

Dirac in his brilliant book derived quantum mechanics using non-commuting operators $\hat{q}$ and $\hat{p}$. He related these to the Schrodinger representation using a wavefunction $\psi(x)$ and the ...
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### Problem with Wick's theorem (normal ordering of a contraction)

Taking the example of two bosonic fields, Wick's theorem is $$T\{\phi(x_1)\phi^\dagger(x_2)\} = N\{\phi\phi^\dagger\} + N\{(\phi\phi^\dagger)_c\}$$ where the subscript $c$ ...
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### Ladder functions and operators - commutation and product

I have been studying An introduction to QFT by Peskin and Schroesder, in my free time, to learn about the fascinating subject of QFT. I have a couple of basic ...
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### QCD Conjugate Variable Determination

This is not homework question. I have not been in school for over 40 years. Are there conjugate variables in QCD? How are they determined? I can relate to Heisenberg uncertainty in QM as the ...
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### Physical significance of non-commutativity of ladder operators in Quantum Harmonic Oscillator

If we apply the raising (creation) operator to $Ψ_n(x)$ and the apply to it the lowering (annihilation) operator, we get $Ψ_n(x)$ times a constant. Does it physically say something? Can we get any ...
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### The relation between commutation and quanta

This question discusses discretization in some sense, and this question talks about how quantization and Hilbert Spaces are related (the answer seems to to be not at all), but what I'm curious about ...
I have to prove that $[A_j, H] = 0$, with; $$\vec{A} = \frac{1}{2Ze^{2}m}(\vec{L} \times \vec{P} - \vec{p} \times \vec{L}) + \frac{\vec{r}}{r}$$ $$H = \frac{p^2}{2m} - \frac{Ze^2}{r}$$ And, $Z, e, m$...
To prove that the conservation of linear momentum doesn't apply to electron in H-atom, is it sufficient to show that angular momentum operator ($\hat L$) and momentum operator ($\hat p$) do not ...