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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How do Lieb-Robinson Bounds talk about locality without the position operator?

So we know when one goes from QM to QFT Lieb Robinson bounds become micro causality. But micro causality is a statement on the commutators assuming they are space-like, time-like or light-like. ...
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Asymptotic States, Propagator and Commutation Relations

Following Fradkin's discussion in the book QFT Integrated Approach, the commutation relation for asymptotic states satisfies $$\left\langle 0\left|\left[\phi(x), \phi\left(x^{\prime}\right)\right]\...
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Commutators between momentum and total angular momentum [closed]

Do $p^2$ and $r^2$ commute with $J^2$? Where $p$ is the momentum, $r$ is position coordinate and $J=L+S$ is the total angular momentum. I think the answer is yes due to the fact that the angular ...
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Separability of an Hamiltonian with spin

I'd like to know if this Hamiltonian $\hat{H}=\frac{p^2}{2m}+\frac{1}{2}m\omega^2r^2+\frac{A}{\hbar^2}(J^2-L^2-S^2)$ is separable into two parts $H_1=\frac{p^2}{2m}+\frac{1}{2}m\omega^2r^2$ and $H_2=\...
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Criterion for stationary density matrix

A density matrix $\rho$ is time independent iff it commutes with the Hamiltonian $H$. I am wondering if there is a criterion to test whether $[\rho, H] =0$ using some trace condition. Specifically, I ...
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$[(\hat{a}^{\dagger})^2, \hat{a}] = -2\hat{a}^{\dagger}$?

I'm confused by a line in the following wikipedia article on the squeeze operator in deriving the action of the squeeze operator on Heisenberg basis, the article seems to imply that $$[(\hat{a}^{\...
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Polar decomposition of a complex scalar field theory

In the text I am referring to, the field was substituted in terms of a number density and phase: $$\psi(x) = \sqrt(ρ(x))e^{iθ(x)}.$$ While quantizing the field, a commutation relation was imposed: $$[\...
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Uncertainty principle when the expectation value of commutator is zero

I'm reading section 4.3 of Introduction to quantum mechanics written by David Griffits. The book states that the product of the standard deviation of two components of angular momentum is greater than ...
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Resonance level model: Commutator

As a small part of an exercise on the resonant level model (all fermionic (field-)operators, $\Psi(\vec{x}) = \sum\limits_{\vec{k}}e^{i\vec{k}\vec{x}}c_{\vec{k}} $, $V$ is a constant, $d$ and $c$ ...
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What is the significance of commutator relationships in physics, e.g. $qp-pq = i \hbar$, $R(X, Y)Z = \nabla_X\nabla_Y Z - \nabla_Y\nabla_X Z$, etc?

Quantum mechanics has the commutator relationship: $$qp-pq = i \hbar$$ In relativity the Riemann tensor is a measure of how much covariant derivatives along a path commute. $$ R(X, Y)Z = \nabla_X\...
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Commutator of $[\hat{x}, \hat{k}]$

We define the operators $$\hat{x} |x\rangle = x |x\rangle\tag{1}$$ and $$\hat{k} |k\rangle = k |k\rangle\tag{2}$$ where $$\sqrt{2\pi} \Psi(k)=\int dxe^{-ikx}\Psi(x)\tag{3}$$ and $$\langle x|k\rangle = ...
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Commutator of $V(\hat{\vec{r}})$ and $\hat{L_z}$

Can someone explain to me why this is true? to me I see that $$x[\hat{p_y},V(r)]- y[\hat{p_x},V(r)]$$ $$=x(\hat{p_y}V(r)- V(r)\hat{p_y} )- y (\hat{p_x}V(r)+ \hat{p_x}V(r)).$$ The only way this can ...
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How to calculate commutator between fermionic terms and bosonic conjugate momenta?

Consider a system which has a three bosonic scalar fields and non-relativistic fermionic part. Let the field operator for the bosonic part be $\phi_k(x)$, $k=1,2,3$, with conjugate momentum $\pi_k(x)$....
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What does it mean for two variables to be canonically conjugate?

The word "canonical" has been used in many of my classes (canonical ensemble, canonical transformations, canonical conjugate variables) and I am not really sure what it means physically. ...
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What is the fundamental reason for the imaginary unit in Heisenberg's commutator relations?

The well known Heisenberg commutator relation $$[p,q]=\cfrac{\hbar}{i} \cdot \mathbb{I}$$ introduces the imaginary unit $i$ into quantum mechanics. I ask for the deeper reason: Why does the ...
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Effect of Momentum Operator in Position Eigenstate [duplicate]

In Lectures on Quantum Mechanics by Steven Weinberg, section 3.5, he asserts that we can infer $$P_{j} \Phi_{\mathbf{x}}=i \hbar \frac{\partial}{\partial x_{j}} \Phi_{\mathbf{x}}\tag{3.5.11}$$ from ...
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Common eigenstate of incompatible observables

In many resources I have seen that incompatible observables cannot have a common eigenbasis set, but may share one or few eigen states. I followed the thread Can incompatible observables share an ...
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How to calculate the OPE of the $X_L(z_1)X_L(z_2)$ in the free boson theory from the mode expansion?

From the polchinski page 238, given \begin{equation} [x_L,p_L] =[x_R,p_R]=i\tag{8.2.14} \end{equation} and the mode expansion $$\begin{equation} \begin{split} X_L(z) = x_L -i\frac{\alpha'}{2}p_L \ln z ...
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Path integral - operator formalism and continuum limit

Correlator of the position and momentum operators in quantum mechanics $$\langle x_f, t_f|[\hat{x}(t), \hat{p}(t)]| x_i, t_i \rangle = i\hbar \langle x_f, t_f| x_i, t_i \rangle$$ since $[\hat{x}(t), \...
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Derivation of the Hypersurface Deformation Algebra

Let $({M},{g})$ be a smooth $4d$ spacetime manifold with lorentzian metric $g$ and local coordinates $\xi^{\alpha}$ and let further $({N},{q})$ be a smooth $3d$ manifold with metric $q$ and local ...
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Dilation operator acting on $x$-dependent field

I've been studying conformal field theory (CFT) and got the following "apparent" inconsistency. Let's take dilation ($D$) and translation ($P_\mu$, 4-momentum) generators that according to ...
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Perturbed hydrogen atom transition probabilities

Given the perturbed hydrogen atom having Hamiltonian $$ \hat{H} = \frac{\vec{p}^2}{2m}-\frac{e^2}{|\vec{r}|}+\lambda \vec{S}\cdot \vec{r} $$ (1) say which operators commute with the Hamiltonian among $...
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How do you get the Jordan-Wigner commutation relations for spin-1/2 fermions?

I am currently trying to figure out how the answer is derived here. I understand that commuation is $[A,B] = AB - BA$. However I am confused how the $\exp(-i\varphi$) acts on the operators $c$ and $c^\...
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Baker-Campbell-Hausforff Formula: Is this an error in the book?

Possible Error: I am reading Quantum Theory of Finite Systems by JP Blaizot. In particular, chapter 4: Wick Theorems. I believe there is a typo in Equation (4.4), which in the textbook is $$ e^{A_n+...
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Baker-Campbell-Hausdorff for Many Operators

I am trying to show that $$e^{A_n}e^{A_{n-1}}...e^{A_2}e^{A_1}=e^{\sum_iA_i}e^{\frac{1}{2}\sum_{i>j}[A_i,A_j]} \tag{1}$$ is true for a set of operators $A_1,A_2,...,A_n$ such that $[[A_i,A_j],A_k]=...
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Exponential of an operator shifted by the derivative operator

Let $p(x)$ and $f(x)$ be sufficiently smooth functions and $D=\frac{d}{dx}$. It is easy to show that $$e^{p(x)D}f(x)=f(e^{p(x)D}x).\tag{1}$$ If $p(x)=a \in \mathbb{R}$ , we have the shift operator as $...
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Commutator of positive semidefinite Hamiltonians

I have the following questions about the commutator of positive semidefinite Hamiltonians. Under what condition, the commutator will be positive semidefinite? Under what condition, the commutator ...
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Would it be correct to interpret $\hat{A} V \neq \hat{A} \hat{V}$?

Define $V=V(\mathbf{r})$ as the potential function, $\hat{A}$ is some differential operator, then for an arbitrary function $\psi$ $$\hat{A} \hat{V} \psi \iff \partial (V\psi) $$ whereas $$\hat{A} V\...
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How to derive the commutation coefficient from coordinate basis (GR)?

Given two vectors $U$ and $V$ \begin{align} [U, V] &= [U^\mu e_{(\mu)}, V^\nu e_{(\nu)}] \tag{1} \label{eq1} \\ &= [U^\mu e_{(\mu)}(V^\nu) - V^\mu e_{(\mu)}(U^\nu)]e_{(\nu)} + U^...
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Heisenberg's picture on complex field operators

I've been reading David Tong's lecture notes on QFT, and specifically on Lecture 2, he writes (section 2.6, eq. 2.8.3) $$e^{i\hat{H}t}\,\hat{a}_{\vec{p}}\,e^{-i\hat{H}t}\,=\,e^{-iE_{\vec{p}}t}\,\hat{a}...
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Commutation relation confusion of ladder operators in Quantum Mechanics

Suppose that $X$ and $N$ are operators such that they follow the commutation relation $$[N,X]=cX$$ for some scalar c. In this Wikipedia article it is shown that if $|n \rangle$ is some eigenstate of ...
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Unit issues with commutator relations with two dependent variables

Suppose I have the following commutator relation for an operator $a[x,\omega]$ which depends on position $x$ and frequency $\omega$ $$ \left[a[x,\omega],a^{\dagger}[x^{\prime},\omega^{\prime}]\right]=\...
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How does non-commutativity of observables lead to quantum speedup in solving algorithms in quantum computing?

The question might be misleading, but I'd like to understand a thing. By reading this really interesting question, one realises that the relevant thing in quantum mechanics and not reproducible in ...
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3 votes
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Is MTW's covariant commutator $\left[\nabla_{a},\nabla_{b}\right]$ really the same thing as their vector field commutator $\left[a,b\right]$?

I now realize my question can be stated very concisely. In Chapter 11 of MTW, will the meaning be changed if in every instance we make the replacement $$\left[\mathbf{a},\mathbf{b}\right]\mapsto\left[\...
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2 votes
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Commutation relations in quantised Yang-Mills

Consider Yang-Mills theory with gauge group $G$. Let $\{T^a\}$ be a basis for the Lie algebra $\mathfrak{g}$, so that the connection coefficients can be written as $A_\mu = A_\mu^aT^a$. In the ...
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4 votes
4 answers
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Why does a symmetry operator commute with the Hamiltonian?

Suppose that a symmetry operator $O$ leaves the Hamiltonian $H$ unchanged. From books I know that there should be the relation $OH=HO$. But I don't understand why it is not that $H=HO$ since when the ...
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Derivatives of exponential operator

I'm reading the paper (eq.(14) and eq.(10)) and got curious how the paper uses this equation: $\frac{\partial}{\partial c}\exp(-i\Delta t (X+cY)) = \exp(-i\Delta t (X+cY))(-iY\Delta t + \frac{\Delta t^...
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How can I reconcile the apparent all-zeroes diagonal of commutators with the canonical commutation relation? [duplicate]

Out of interest I was trying to derive some properties of commutators in quantum mechanics. I found that, in my calculations, will have an all-zeroes main diagonal. I must be making a mistake, because ...
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Confusion about ladder operators

Let´s consider a system, that consists out of $N$ bosonic particles, that are not interacting with each other. The Hamiltonian of this system would be given as $$H = \sum_{i=1}^N \frac{\hbar^2}{2m}\...
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Does the interaction picture assume that the Hamiltonians commute?

Suppose the Hamiltonian is $H+H_1(t)$. In the Schrodinger picture, the evolution is: $$|\psi(t)\rangle=e^{i(Ht+\int_0^tH_1(t)dt)}|\psi(0)\rangle$$. The interaction picture introduces a change of basis ...
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How does the Fourier picture relate to non-commutativity?

A compelling video by 3Blue1Brown visualizes the uncertainty principle with Fourier transforms. The gap I'm trying to bridge is between this "Fourier picture" and the matrix-based statement $...
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Uncertainty principle manifesting in $j(j+1)$ vs $j^2$

To motivate my question, please consider a system with total angular momentum $j$. The fact that the largest eigenvalue of $J_z$ is $j$, while $J^2 = J_x^2 + J_y^2 + J_z^2$ has all eigenvalues equal ...
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Why do $\psi, \psi^*$ and the potential energy operator commute here?

This is the one-dimensional time-dependent Schrodinger Equation: $$i\hbar \frac{\partial \psi}{\partial t}= -\frac{\hbar^2}{2m}\frac{\partial^2 \psi }{\partial x^2} +\displaystyle {\hat {V}\psi}$$<...
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Commutation relations interacting fields

I am reading Schwartz's "Quantum field theory and the standard model". I have a question on how he derives the Feynman rules for an interacting scalar field from a Lagrangian formalism. In ...
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An identity for nested commutators

Let $A,B$ be Hermitian operators on an arbitrary Hilbert space. Define nested commutators of $B$ with respect to $A$ as $\text{Ad}_{A}(B) = \left[A,B\right]$, $\text{Ad}_{A}^{2}(B) = \left[A,\left[A,B\...
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Automated symbolic commutator tool

Is there a software tool (perhaps Mathematica extension?) designed to help with doing algebraic manipulations of non-commutative objects? Specifically, what I'm after is an interface along the lines ...
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Commutator with identity [closed]

For an operator $A$, is $[A,\textbf{1}] = \textbf{1}$ or $\textbf{0}$ where $\textbf{1}$ is the identity operator? I think it is $\textbf{0}$ but I want to confirm this.
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Anticommutation and Bogoliubove transformation

I am given the following transformation: \begin{equation} \begin{bmatrix} ...
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Quantum canonical transformation which exchanges $q$ and $p$

In classical mechanics, there is a canonical transformation which exchanges $q\to Q=p$ and $p\to P=-q$, generated by $F(q,Q)=qQ$. I'm curious if there is an analogous canonical transformation in the ...
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Commutators as contour integrals in 2D CFT, and classical limits

In a 2D CFT, the commutator of two operators $$A_i=\oint a_i(z)dz$$ can be given by $$[A_1,A_2]=\oint_0dw\oint_wdza_1(z)a_2(w)$$ where the $z$ integral is taken over a contour around $w$ and the $w$ ...
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