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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208 views

Why is commutation relations the first step in quantization?

Why is commutation relations the first step in quantization?
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296 views

Change of QM Momentum operator under coordinate transformation

Can any one please let me know what is the general procedure to construct the momentum operator under some coordinate transformation? For example, I understand that if $${\bf{r}}\rightarrow{\bf{r'}}=...
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commutators in an uncertainty relationship derived from a partition function?

The maximum information principle for the discrete case gives rise to a partition function (>>> see details here) $$Z(\lambda_1,\ldots, \lambda_m) = \sum_{i=1}^n \exp\left[\lambda_1 f_1(x_i) + \cdots ...
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QFT basics for Klein-Gordon fields

I am teaching myself QFT from Peskin for next years maths course and I have two questions: What is a c-number? Is it a complex number, and if so why does it mean, $[\hat{\phi}(x),\hat{\phi}(y)]~=~<...
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Question : Commutation relation for Dirac field

In Quantum Field Theory by Peskin and schroeder I couldn't understand the commutation relation calculation for Dirac field (pg. 53) : $\psi(x) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2E_p}}e^{...
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37 views

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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40 views

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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94 views

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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40 views

Commutation of alpha dirac matrix

I want to calculate the commutation of $[\hat{x},\vec{\alpha}\;\vec{p}]$. This boils down to $$[\hat{x},\vec{\alpha}\;\vec{p}] = i\hbar\hat{\alpha_x}+\left[\hat{x},\hat{\alpha_x}\right]\hat{p_x} +\...
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77 views

How to efficiently compute the commutator $[\hat{r},\nabla^2]$?

Given a system with Hamiltonian $ \hat{H} = \frac {\hat{p} ^2}{2m} + \hat{V}(r)$ in a certain state $|\psi \rangle$, I want to find if $\langle r \rangle$ varies with time. From $$ i \hbar\frac {d ...
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34 views

Canonical commutation relations for a real scalar field

I am taking my first course in QFT and have come across this problem From the canonical commutation relations for a real scalar field $\hat{\phi}$ show that $$[\partial_i \hat{\phi} , \hat{\phi}...
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37 views

Derivation for the commutation relation of $\alpha_m^\mu$ and $\alpha_n^\nu$

I have been trying to derive the commutation relation of $\alpha_m^\mu$ and $\alpha_n^\nu$ in a closed-string mode expansion, but I found an extra factor of $2$ that ruins things out: Given $\dot X = ...
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Uncertainty Principle and Commutators

In preparation for an exam I stumbled upon a quantum mechanics task I can´t really solve right know. So i hope someone can maybe give me a hint or two how to understand this. Here is the task: Let $...
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48 views

Commutation in coupled Harmonic Oscillators

Starting with a coupled Harmonic Oscillator problem $$ H = \frac{p_1^2 + p_2^2}{2m} + \frac{K}{2}\left[x_1^2 + x_2^2 + \left(x_1 - x_2\right)^2\right] = \left(\frac{p_1^2}{2m} + \frac{2K}{2}x_1^2\...
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58 views

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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67 views

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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37 views

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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90 views

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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Computing the commutator of the potential and angular momentum

Assume the potential $V$ is not just a function of position. I'm trying to compute $[V, L_i]$. This is what I have so far: $$ [V, L_i] = [V, \epsilon_{ijk}x_jp_k] = \epsilon_{ijk}(x_j[V,p_k]+[V,x_j]...
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Defining the propagator using the positive and negative frequency of the field

I am currently reading some notes on QFT and in the notes it defined that $$ \phi(x)=\phi^+(x)+\phi^-(x) $$ where $$ \phi^+(x)=\int\frac{d^3k}{(2\pi)^32E}a_ke^{-ikx} $$ and $$ \phi^-(x)=\int\frac{d^3k}...
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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 \begin{equation} T\{\phi(x_1)\phi^\dagger(x_2)\} = N\{\phi\phi^\dagger\} + N\{(\phi\phi^\dagger)_c\} \end{equation} 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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Specific Commutator interpretation: $[c_p^\dagger c_p, c_{k+q}^\dagger c_{k} ]$

I have the problem of computing/understanding a commutator. The operators I'm working with fullfill the standard bosonic commutation relation: $[c_q,c_k^\dagger]=\delta_{q,k} $ and $ [c_q,c_k]=[...
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Fields commutator, a concise review

I am entering in the quantum filed theory world, and I am consulting many notes, but something is a bit unclear about commutators. Hence I would like some expert of you tell me what exactly (if ...
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241 views

Commutator of canonical fields in Quantum Field Theory

Let $\phi(\vec{x},t)$ denote the canonical fields and $\pi(\vec{x},t)$ denote the canonical impulses where they're given by: \begin{equation} \phi(x)=\int\frac{d^3\vec{p}}{(2\pi)^3\sqrt{2\omega_{\...
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101 views

How to show that two operators in the forms of vectors commute?

This is an exercise from Peskin&Schroeder's book. The exercise requires to show that $\textbf{J+}$ and $\textbf{J-}$ commute with each other. What is the exact meaning of commutation between ...
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Heisenberg's original Quantum Condition

In Heisenberg's paper Quantum Theoretical Re-interpretation of Kinematic and Mechanical Relations where he first derives Quantum Mechanics, using his postulates of Quantum Mechanics and the Old ...
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Why if 2 operators commute they have a common set of eigenvectors and what's the relation to 2 fold degeneracy?

I have the following sentence in my lecture notes "Dirac hamiltonian and helicity have a common set of eigenvectors, this is also the reason for the two fold degeneracy found for every energy ...
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280 views

Kronecker delta commutation relations for QFT

Setup: In many textbook treatments of canonical quantization (e.g., Peskin and Schroeder), one imposes canonical (equal time) Dirac delta commutation relations on the conjugate field operators. e.g., ...
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Non-abelian gauge theories, which factors commute?

I am following Peskin and Schroeder (1995). It seems that we are magically supposed to understand which factors commute and which do not in the theory. These are the factors which appear in the ...
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If the mean of the commutator is zero, then what we can say about the commutator itself?

Suppose we have \begin{equation} \langle[H,N]\rangle=0 \tag{1} \end{equation} where both $H$ and $N$ are hermitian. Under which assumption I can claim that then $$[H,N]=0~?\tag{2}$$
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Plausibility of Heisenberg equation for the canonical momentum:

In this question, I want to to know wether my reasoning on the plausibility of the Heisenberg equation is flawed: Let's say I want to describe my system in the quantum-mechanics framework: ...
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commutator for KG field

I am reading Peskin and Schroeder's book, and am wondering about the commutator $$[\phi(\boldsymbol{x}),\pi(\boldsymbol{y})] = i\delta^{(3)}(\boldsymbol{x}-\boldsymbol{y}).$$ I do not see the usual ...
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Commutation of ladder operators in reciprocal space

For simplicity, let's assume a hamiltonian of the form \begin{equation} H=\sum_{\vec{r}}\sum_{\vec{\delta}}a^+_{\vec{r}}a^-_{\vec{r}+\vec{\delta}} \end{equation} where the $\vec{r}$ 's are the sites ...
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How to show that $\langle p|x \rangle = Ae^{-ipx} $ using the canonical commutator alone?

I am working through an exercise to show that $\langle p|x \rangle = Ae^{-ipx} $ using $[\hat{x},\hat{p}] = i $ alone. The first part of the exercise is to use the commutator and show that $$ e^{-ia\...
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Commutation of currents in QED

In an outline of a proof of the Ward identities in QED, the authors Green, Schwarz, and Witten in their book "Superstring theory", vol. I, Section 1.5.1, claim that in the QED the electromagnetic ...
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Prove conservation law in quantum mechanics

I major in Math, and I am studying Quantum Mechanics (QM). I see the conservation law in QM as a mathematical theorem. Please check if my understanding is right, and help me to prove the theorem? ...
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From gauge anomaly to chiral anomaly

Suppose the theory of chiral Weyl fermion (say, left) $\psi_{L}$, which interacts with abelian gauge field. This theory contains gauge anomaly, which I write in the form $$ \frac{dQ_{L}}{dt} = \text{A}...
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Anomalous commutators and gauge anomaly

Suppose we know, that the dynamics of theory with chiral fermions (say, left) and gauge field (for simplicity, abelian) leads us to presence of anomalous commutator of canonical momentum $\mathbf E(\...
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379 views

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 ...
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296 views

Quantum Hamiltonian commuting with the Pauli-Runge vector

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, $Z, e, m$...
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Proving that conservation of momentum doesn't apply to electron in H-atom

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 ...