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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Quantum field theory, interpretation of commutation relation

Let $\phi$ be the quantum field $$ \phi(x) = \int \frac{d^3\mathbf{p}}{(2\pi)^3} \frac{1}{\sqrt{2E_\mathbf{p}}} \Big[ b_\mathbf{p}e^{-ip\cdot x} + c_\mathbf{p}^\dagger e^{ip\cdot x} \Big] $$ with ...
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Why do the $L_z$ and $L^2$ operators share eigenfunctions, but the $L_x$ and $L_y$ operators don't?

In my lecture notes the following was written: I would understand in the case of an applied field if there was some symmetry breaking feature which would allow for a preferred axis or something which ...
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Commutation rules between itinerant and localized electron operators in s-f Model

The s-f Model is a model who could describe the $\textbf{magnetic 4 $\textit{f}$ systems}$, i.e systems where we could identify localized electrons in $4\,\textit{f}$ orbitals and conductions ...
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Could there be any situation where two noncommuting observables are simultaneously considered?

As physicist Robert Griffiths (one of the founders of the 'Consistent Histories' formalism) says: "Two physical variables whose operators do not commute correspond to incompatible sample spaces, and ...
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69 views

Generalizing the Stone-von-Neumann theorem

The Stone-von-Neumann theorem states (in rough terms) that a pair of operators $\left(\hat{Q}, \hat{P}\right)$ satisfying the exponentiated canonical commutation relation $e^{is\hat{Q}}e^{it\hat{P}} = ...
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What does it mean for commutator of position and momentum to be equal to the metric tensor?

I was reading: https://arxiv.org/abs/hep-th/9505152 And got confused by the first line stating $$ [P^{\mu}, X^{\nu}] = ig^{\mu\nu} \tag{1.1}.$$ I want to believe that this a generalization of the ...
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Proving the Equal-time commutation relations if Schrodinger Fields [closed]

The Schrodinger fields can be represented as How we can prove the the commutation relations between psi and psi *? Also how could we write the expression of hamiltonian H and the total number ...
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Proof for $p$ and $q$ eigenstates using only commutation relations [duplicate]

I should proof that $$p \langle q;t\lvert p;t \rangle = \frac{1}{i}\frac{\partial}{\partial q} \langle q;t\lvert p;t \rangle $$ using ONLY the commutation relations and the fact that $q$ and $p$ ...
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When are the eigenstates of two operators the same?

Suppose $\{\left|{u_{n}}\right>\}$ is the set of energy (Hamiltonian) eigenstats. Are they also eigenfunctions of any physically observable operator $\hat{A}$? Can we use the following eigenvalue ...
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Commutation relations spin operators

How can I prove $S_{i}^{+}S_{i}^{-}+S_{i}^{-}S_{i}^{+} =1$ ? Could it be because if I am using spin-1/2 particles the spin operators obey fermionic anti commutation relations?
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Commutation relation between charges in current algebra

Consider the vector current and axial vector current like $$j^{a\mu} = \bar{\psi}\gamma^{\mu}\frac{\tau^{a}}{2}\psi,$$ $$j^{a\mu}_5 = \bar{\psi}\gamma^{\mu}\gamma_5\frac{\tau^a}{2}\psi,$$ where $\tau^...
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63 views

Canonical Commutation Relation representation

Is it always possible that for two operators $\hat A$ and $\hat B$, which have a commutator of $[\hat A, \hat B] = i \hbar$, we can write the action of these two operators to a function $f(a)$, which ...
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117 views

Commutator of rotation matrices

How do you compute the commutator of rotation matrices in two different directions by different angles? Let $R_{x}(\alpha)$ be the rotation matrix about the $x$-axis and $R_{z}(\beta)$ be the ...
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62 views

Annihilation and Creation operator for bosons properties in second quantization

In second quantization the commutation relation of annihilation and creation operators of bosons is \begin{equation} [b,b^\dagger]=bb^\dagger-b^\dagger b=1 \end{equation} I am wondering what the ...
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Commutator of $L^{2}$ and $L_{z}$

I'm trying to work through a proof of why $[L^{2},L_{z}]=0$, and am getting lost on this step: We can use the commutation relation $[\hat{L}_{z},\hat{L}_{x}]=i\hbar\hat{L}_{y}$ to rewrite the term as:...
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Commutation relations following from quantization of a complex scalar field

As someone who has recently started doing QFT I have some (algebraic) confusion about the following derivation. Starting with the Lagrangian of a complex scalar field $$\mathcal{L} =\partial_\mu \psi^...
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Commutator under unitary transformation

How can I prove that the commutators are invariant under unitary transformations? I'm studying quantum mechanics, so (maybe) my professor is talking about the commutator of hermitian operators.
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Does the number operator $N$ of the Quantum Harmonic Oscillator commute with $x$? [closed]

Does the number operator $N$ of the Quantum Harmonic Oscillator commute with $x$?
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Commutator of Position Operator and Generator of Translations

Problem: Let $|\psi' \rangle = \hat T(\delta x) | \psi \rangle$ for infinitesimal $\delta x$, show that $\langle x \rangle'= \delta x +\langle x \rangle$ Attempt at Solution: Using $\hat T(\delta ...
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Proof of commutator in Tong's notes on QFT

I am following David Tong's notes on QFT: http://www.damtp.cam.ac.uk/user/tong/qft.html . In equation 2.21, he tries to prove $$[\phi(\vec{x}),\pi(\vec{y})] = i\delta^{(3)}(\vec{x}-\vec{y}).$$ Here, $\...
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Completeness relation, commuting operators

I have a question about some formulars our professor wrote on the black board. Let $\hat{Q}_{1},...,\hat{Q}_{N}$ be operators, which are a CSCO. We know now that there exists a set of eigenvectors $\{...
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Why is the field commutator $[\phi(\vec x, t), \phi(\vec y, t)]$ related to causality?

It's well known that $$ \langle 0| \phi(\vec x, t) \phi(\vec y, t) |0 \rangle \neq \delta(\vec x - \vec y) . $$ It is then regularly argued that this is not a big problem since the commutator $$ \...
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Derive $\langle p|\hat x=i\hbar {\partial_p}\langle p|$ from $[\hat x,F(\hat p)]=i\hbar \partial_{\hat p} F(\hat p) $

Suppose that $p=-i\hbar \frac{\partial }{\partial x}$ was known, i.e. $\langle x|p=-i\hbar \frac{\partial}{\partial x}\langle x|$. Suppose the only other known condition was $[x,F(p)]=i\hbar \...
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Evaluation of Commutation Relations in Ballentine's book

I'm reading chapter 3 of Leslie Ballentine's book Quantum Mechanics : A Modern Development but there are a few derivations I don't understand. Question 1 : In the middle of page 74, it says ...
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Relations between correlation function and commutators of two observables

In quantum theory (QFT or Quantum statistical mechanics), are correlation of two observable related to their commutator? Is there any explicit bound of the correlation by the norm of the commutator? ...
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Comutator of multiple creation and annihilation operators [duplicate]

Knowing that commutator of the creation and annihilation operators $a^\dagger$ and $a$, is $$ [a,a^\dagger]=\mathbb{I}~,$$ what is the commutator of $n$ copies of these operators, i.e., what does the ...
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Klein Gordon Hamiltonian commutator with annihilation (creation) operator

Probably I'm missing something trivial here. When calculating a commutator of Klein Gordon Hamiltonian with annihilation/creation operator it seems that the operators are inserted under the integral, ...
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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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Simple question on computing commutation relation

In bosonization, one faces with the following commutator: $$[\phi(x_1), \theta(x_2)]=\sum_{q\neq 0} \frac{\pi}{Lq} e^{iq(x_2-x_1)-\alpha |q|}\tag{1}$$ where $q$ is an non-zero integer multiple of $2\...
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Can you argue without explicitly calculate the eigenenergies that one Hamiltonian is gapped and another is not?

Consider a pair of one dimensional four band model $H_1$ and $H_2$, which read as: $$ H_1 = \begin{pmatrix}k\sigma_x-E_0&0\\0&k\sigma_x+E_0\end{pmatrix} + \alpha \begin{pmatrix}0&\sigma_x\...
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Does an Operator that neither commutes with $\hat{X}$ or $\hat{P}$, nor can be expressed as a “function” of $\hat{X}$ and $\hat{P}$ make sense?

When you come from classical hamiltonian mechanics (which is based on the phase space), observables are introduced as functions $f$ on the phase space $(q, p)$. There can't be a classical observable ...
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If $E$ and $P$ don't commute, why could we have an $E$-$K$ diagram?

If $E$ (energy) and $P$ (momentum) only commute in constant potential, how could we have an $E$-$K$ diagram in a solid material? $[E,p] \neq 0$. Then we cannot prepare electrons whose $E$ and $P$ are ...
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Do Equal Time Commutation / Anticommutation relations automatically also apply for spatial derivatives?

The question is basically in the title. My naive thought was that when a commutation relation holds for all field operators $\Psi(\vec{x})$ (by "all" I mean "at all positions $\vec{x}$") on a fixed ...
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Generalized commutator/anticommutator via phase factor

We know that the commutator between two operators $A$ and $B$ reads $[A,B]_{-}=AB - BA$, while the anticommutator reads $[A,B]_+=AB + BA$. I am wondering if someone has ever used a generalized ...
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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): $$ \begin{align} \psi(x) &= \int \frac{d^3p}{(2\pi)^3} ...
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What is the value of the commutator $[\vec{S}, H]$?

What is the commutation relation between $[S, H]$ where Hamiltonian $H= - \vec{S} \cdot \vec{B}$ , $\vec{S}$ is the spin and $\vec{B}$ is the magnetic field. I am getting $0$ but it seems wrong.
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Commutation of position four-vector with spacetime derivatives

I am trying to understand a simple demonstration in Ashok Das' Lectures in QFT. He does the following on p. 134 $$[P_\mu,M_{\nu\lambda}]=[\partial_\mu ,x_\nu\partial_\lambda-x_\lambda\partial_\nu]=\...
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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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How creation operator pops out while expanding field operator?

While doing QFT when we try to canonically quantize the Klein Gordon equation $\Box \phi =0$ we promote the $\phi $ to an operator field and impose the commutation rule $[\phi(x,t),\pi (y,t)]=i\hbar\...
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The conmutator of the square of Pauli-Lubanski vector and the generators of Poincare group

I'm working on trying to solve the following problem: Using the following expressión for the square of Pauli-Lubanski vector:$$W^2=-\frac{1}{2}M_{\mu\nu}M^{\mu\nu}P_{\alpha}P^{\alpha}+M^{\mu\nu}M_{\...
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One question about BRST symmetry in reading Srednicki’s book: Why should the BRST charge $Q_B$ be nilpotent?

In p.453, Srednicki claims that since the BRST transformation of a BRST transformation is zero, $Q_B$, the BRST charge, must be nilpotent: $$Q_{B}^{2}=0.\tag{74.32}$$ I don't know why.
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Commutation with unspecified potential function

Instead of a potential given like $V(r) = k r^2$ or $V(r) = y^2$ , if the potential is given like in the form a function but not clearly specified, can we tell that if that commutes with the ...
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Common observables and associated operators: operator momentum [duplicate]

Starting from my previous question Commutators in quantum mechanics and considering that the commutator $$\left[i\hbar\frac{\partial}{\partial x},x\right]=i\hbar, \tag{1}$$ the associated linear ...
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If Poisson Bracket of Momentum and Position is non-zero, why no Uncertainty Principle?

In Hamiltonian classical mechanics, we have that the Poisson bracket of position and momentum satisfies $$\{q_i, p_j\} = \delta_{ij}$$ But this implies that momentum and position 'generate' changes ...
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The use of the commutators in quantum mechanics: explanations [duplicate]

Considering that I've never studied quantum mechanics before I have need to understand the operator commutator. My start is: $[A,B]=AB-BA \tag{a}$ Now, why must be $$\left[\frac{\partial }{\partial ...
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114 views

Quantum Harmonic Oscillator Raising and Lowering operators

The commutator of the operators, $[a,a^\dagger] = 1$ is useful in rewriting the Hamiltonian in a neat way in terms of the creation and annihilation operators. So my question is, Is there a physical ...
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85 views

Number operator - annihilation operator commutation

Is there a rigorous way to prove that $$ (N+1)^{-1/2} a = a N^{-1/2} $$ where $a$ is a bosonic annihilation operator and $N=a^\dagger a$ is the corresponding number operator?
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69 views

What is the implication of overlap between eigenstates of two operators in Quantum Mechanics?

For instance, what does it mean that a certain position eigenstate is also an energy eigenstate? I understand that measurable (Observables) in Quantum mechanics are the operators. Their eigenvalues ...
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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 ...