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.

Filter by
Sorted by
Tagged with
0
votes
1answer
27 views

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, $\alpha$ ...
2
votes
0answers
58 views

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 ...
4
votes
1answer
62 views

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 ...
0
votes
0answers
27 views

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 $N$ ...
2
votes
1answer
134 views

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 ...
3
votes
1answer
97 views

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. ...
0
votes
1answer
56 views

Commutation relation of the second quantisation of an operator

Let $h$ be a self-adjoint operator on a Hilbert space $\mathfrak{h}$ and $H_N:=\sum_{i=1}^N h_i$ defined on $H(\mathfrak{h})^{\otimes N}\subset \mathfrak{h}^{\otimes N}$. Let $\overline{\text{d}\Gamma(...
0
votes
1answer
76 views

The $3×3$ representation of weak $SU(2)$

I'm reading chapter 11.2 of the Cheng and Li textbook 'Gauge theory of elementary particle physics'. It says that $T_+$, $T_-$ and $Q$ do not form a closed algebra. In order to fix this problem the ...
0
votes
0answers
49 views

Commutator involving the exponential of the integral of an operator

I have quite a problem handling the following commutator involving the exponential of the integral of an operator $$\Bigg[\hat{A},\exp\!\Bigg(\int_0^td\tau\,\hat{B}(\tau)\Bigg)\Bigg]$$ especially as ...
1
vote
0answers
21 views

If two operators commute, then is it any basis of one of them also a basis of the other? [duplicate]

According to the compatibility theorem, there exists a common eigenbasis for the eigenfunctions of two operators $\hat A$ and $\hat B$ representing the observables $A$ and $B$ if and only if they ...
0
votes
1answer
53 views

Questions on field operator in QFT and interpretations

For a real scalar field, I have the following expression for the field operator in momemtum space. $$\tilde{\phi}(t,\vec{k})=\frac{1}{\sqrt{2\omega}}\left(a_{\vec{k}}e^{-i\omega t}+a^{\dagger}_{-\vec{...
0
votes
1answer
27 views

Biconditionality of the compatibility theorem for commuting operators

The compatibility theorem states that, if the operators $\hat A$ and $\hat B$ representing the observables $A$ and $B$ do commute, then there exists a common eigenbasis for their eigenstates, and ...
2
votes
1answer
143 views

How to compute the unitary transformation with the operator on exponential function?

Consider an unitary transformation \begin{equation} \hat{D}(f) = e^{-\frac{i}{2\hbar}f(t)\hat{q}^2} \end{equation} from the book I find that $\hat{D}\hat{p}\hat{D}^{\dagger} = \hat{p} + f(t)\hat{q}$, ...
1
vote
0answers
45 views

Question on discrete commutation relation in QFT

Given the commutation relation $$\left[\phi\left(t,\vec{x}\right),\pi\left(t,\vec{x}'\right)\right]=i\delta^{n-1}\left(\vec{x}-\vec{x}'\right)$$ and define the Fourier transform as $$\tilde{\phi}(t,\...
0
votes
1answer
42 views

On commutation relations for real scalar free fields in QFT

Suppose the creation and annihilation operators are as follow: $$ a_\mathbf{p} = \frac{1}{2} \Bigg(\sqrt{2\omega_\mathbf{p}} \tilde{\phi}(\mathbf{p}) + i\sqrt{\frac{2}{\omega_\mathbf{p}}} \tilde{\pi}(\...
2
votes
0answers
67 views

On commutation relations in QFT

I am self learning “Quantum fields in curved space” and I am stuck at trying to show the commutation relations for the free real scalar field. I know that the usual way to show is to invoked the ...
1
vote
1answer
50 views

What is my misunderstanding in Wick's theorem?

Trying to understand Wick's theorem, I took most of my knowledge from the corresponding Wikipedia article. The statement is that given the definition of normal ordering of operators $A,B,C,\ldots$ any ...
1
vote
0answers
34 views

Commutators of covariant derivative and Yang-Mills Field Strength in curved spacetime

I am stuck with YM Field Strength and commutator. For example, in flat spacetime we have the commutator $$F_{\mu \nu}=[\partial_{\mu}+A_{\mu},\partial_{\nu}+A_{\nu}] .$$ But what is the thing in ...
0
votes
0answers
30 views

Is it possible to write the Coulomb Hamiltonian in terms of creation and annihilation operators similar to harmonic oscillator? [duplicate]

Is it possible to write the Coulomb Hamiltonian in terms of creation and annihilation operators similar to the harmonic oscillator so that we can have the same commutations relations similar to that ...
0
votes
1answer
42 views

How to perform commutation of $[A+B, C+D]$? [closed]

I want to know a logical method of performing a commutation for [$a\hat{A} + b\hat{B}, c\hat{C} + d\hat{D}$] where $a$, $b$, $c$ and $d$ are just constants. I know the rules for [$\hat{A} + \hat{B}, \...
1
vote
0answers
19 views

Do partial derivation respect to velocity and total derivation respect to time commute? [duplicate]

Imagine we have a function of position $x^i$ and velocity $v^i$ $f(x,v)$. Position and velocity are both functions of time $t$. If the function doesn't depend explicitely on time, then we have the ...
1
vote
0answers
53 views

Uncertainty principles counterexample

Let $\hat A, \hat B$ be two operators on ${L}^2[-1,1]$ such that $$ \hat A\psi(x) = x\psi(x)$$ $$ \hat B\psi(x) = -i\hbar \frac{d\psi(x)}{dx}$$ It's easy to see that $\hat A$ is bounded and self-...
1
vote
1answer
21 views

Magnetic Moment MIT Bag Model

I'm reading the book "Advances in Nuclear Physics vol 13" by J. W. Negele and Erich Vogt In chapter 3, one wants to calculate the magnetic moment for a current loop. In page 29 how does one ...
6
votes
2answers
389 views

Canonical commutation relationship with general operators?

If someone hands me over the position operator $x$ and the momentum operator $p$ I know their commutator is given by $[x,p]=i$ (natural units). Now say somebody hands me either $x$ or $p$ but not both ...
0
votes
3answers
84 views

Translation on $x$-operator

I need to show that given $$[x, p_x]=i\hbar$$ the following is true: $$ e^{iap_x/\hbar}f(x)e^{-iap_x/\hbar}=f(x+a) $$ for a general function $f(x)$. I've tried using Taylor Series for both ...
2
votes
1answer
64 views

Proof of the set of compatible observable have a bound

Sakurai states that We can obviously generalize our considerations to a situation where there are several mutually compatible observables, namely, $$[A,B]=[B,C]=[A,C]=\cdots=0$$ Assume that we have ...
0
votes
0answers
78 views

Commuting but not anti-commuting operators

Two Hermitian operators $\hat{A}$ and $\hat{B}$ are such that they commute but don't anti-commute. In this case, even they commute their uncertainty product will not be zero, is it right?
2
votes
0answers
72 views

Why are commutators the first choice in describing observables that cannot be measured simultaneously?

In quantum mechanics, we convert Poisson brackets to commutators for the observables to account for the uncertainty principle. However, I do not understand why do we do this. What motivates us to ...
2
votes
1answer
96 views

Peskin-Schroeder time derivative of canonical momentum in Klein-Gordon theory

In section 2.4 of the book, it seems that the commutation relation $[\pi(x,t),\phi(x',t)\nabla^2\phi(x',t)] = 2[\pi(x,t),\phi(x',t)]\nabla^2\phi$ is used to verify that $i\frac{\partial}{\partial t}\...
1
vote
0answers
49 views

Super Virasoro Algerbra in string theory [closed]

I'd like to calculate the algebra ${G_r,G_s}$ for the Ramond sector, or Neuveu-Schwarz Sector for super Strings. The problem is that I don't know how to evaluate the anti-commutator relation between ...
2
votes
2answers
133 views

Commutation relations inconsistent with constraints

In section $9.5$ of Weinberg's Lectures on Quantum Mechanics, he uses an example to explain the clasification of constraints. The Lagrangian for a non-relativistic particle that is constrained to ...
0
votes
2answers
95 views

Eigenstates for $\vec{L}^2, L_z, L_x$ and $L_y$? [closed]

I am asked to find states $|j,m\rangle$ that are simultaneously eigenstates for $\vec{L}^2, L_z, L_x$ and $L_y$. I know that the $L_i$ operators do not commute and hence you cannot have a state $|\phi\...
0
votes
1answer
35 views

Simultaneous measurement of multiple observables

Commutation relations tell us which observables are compatible and which ones are not. How is that extended to more than two observables being measured at the same time (or successively)? If $\hat A$, ...
0
votes
0answers
15 views

Lie Superoperator Algebra Commutation relations

I am trying to derive commutation relation of Lindblad Superoperator relations. They are defined as: $$\cdot a^{\dagger}a=\mathcal{L}$$ $$a^{\dagger}a\cdot =\mathcal{R}$$ $$a\cdot a^{\dagger}=\mathcal{...
3
votes
6answers
263 views

Angular momentum commutation relations [duplicate]

The operator $L^2$ commutes with each of the operators $L_x$, $L_y$ and $L_z$, yet $L_x$, $L_y$ and $L_z$ do not commute with each other. From linear algebra, we know that if two hermitian operators ...
0
votes
1answer
33 views

What is the physical meaning of commuting in quantum mechanics? [duplicate]

In quantum mechanics, if two observables commute, then perfect knowledge can be gained about both observables simultaneously. But what does the commutator actually, physically represent? Like ...
0
votes
1answer
48 views

Conmutator of $[x, f(p)]$ on quantum mechanics [duplicate]

I'm struggling trying to prove $[\overrightarrow{x}, f(\overrightarrow{p})]= i\hbar\frac{\partial f(\overrightarrow{p})}{\partial \overrightarrow{p}}$. I already proved that $[x_i,p_i^n]=i\hbar \frac{\...
0
votes
1answer
21 views

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.
0
votes
1answer
32 views

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 ...
0
votes
1answer
59 views

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, ...
2
votes
0answers
47 views

Feynman Propagator

I have learned two ways of representing Feynman Propagator, which are as follows: The first method is to represent it in the form of commutation relations: $$[a^{-}(x), a^{+}(y)] = iG(x-y)$$ Where $a^{...
0
votes
1answer
73 views

Anti-commutation of operators proof

How does one prove the anti-commutation of the operators $$e^{\hat{y}} , \hat{P}_y^2 $$ where $\hat{y}$ and $\hat{P}_y$ are the standard position operator and translation generator operator in quantum ...
4
votes
3answers
165 views

Commutator and Factorization of the Eigenfunctions

I have stumbled upon the following statement: Consider an Hamiltonian $H$ that is function of a multitude of operators: $H(\hat{O_1},\hat{O}_2,...,\hat{O}_n)$. If we can show that $H$ commutes with ...
1
vote
1answer
99 views

Quantum Field Theory Commutation Relations

Let $ϕ(x,t)$ denote the canonical fields and $π(x,t)$ denote the canonical momenta where they're given by: \begin{equation} \phi(x)=\int\frac{d^3\vec{p}}{(2\pi)^3\sqrt{2\omega_{\vec{p}}}}(a_{\vec{p}}e^...
1
vote
1answer
64 views

AQFT: Microscopic causality (local commutativity) of the free Klein-Gordon field

Let us consider the positive and negative frequency part of the free Klein-Gordon field operator $$ \hat\phi(x) = \hat\phi^-(x) + \hat\phi^+(x) \tag{1} $$ where $$ \begin{aligned} \hat\phi^-(x) = \hat\...
0
votes
0answers
44 views

Is it in general? $[\Lambda,\Omega]=i\hbar\{\lambda,\omega\}_{x,p}$ [duplicate]

In R. Shankar's book, He has written $$[X_i,P_j]=i\hbar\{x_i,p_j\}=i\hbar$$ Is there any specific reason to use the Poisson bracket? Is there any general relation which looks like? $$[\Lambda,\Omega]=...
1
vote
0answers
83 views

Hamiltonian commutators and the time evolution operator

I am reading through a quantum optics book at the moment in which the authors state that given the interaction Hamiltonian: $$ V(t) = \int \vec{J}(\vec{r},t) \cdot \hat{A}(\vec{r},t) \hspace{1mm} d^3r ...
6
votes
2answers
273 views

Is there a way to prove that different angular momentum components anticommute without using a specific matrix representation?

I know spin-1/2 Pauli matrices satisfy the anticommutation relationship $\{\sigma_i, \sigma_j\}=2\delta_{ij} \mathbb{I}$. I wonder how this can be proved without writing down the matrix representation ...
0
votes
0answers
42 views

Why does the unitary evolution of an operator involve a commutator with the Hamiltonian?

The equation for the unitary evolution of an operator $\hat{A}$ is given (as far as I understand) as $$\frac{\partial\hat{A}}{\partial t}=\frac{i}{\hbar}[\hat{H},\hat{A}].\quad\mbox(1)$$ It seems that ...
0
votes
0answers
28 views

Cross product identity [duplicate]

In our lecture today we took this identity: $ \vec p \times e\vec A = -e \vec A \times \vec p +(\hbar/i)e (\nabla\times \vec A) $ which the professor, couldn't bother explaining. I understand the ...

1
2 3 4 5
19