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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Is symmetrization $xp-px$ required for commutation $[H,x]=0$?

Given a Quantum Hamiltonian $\hat{H}=ax^2+bp^2$ does not commute with either x or p. Suppose we have a Hamiltonian $H = k \hat{p}\hat{x}$ , why do we need it to be $H = k (\hat{p}\hat{x} - \hat{x}\hat{...
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What is $[\hat{x},\hat{p}^2]$? [closed]

The way I calculated it was as follows: $$[\hat{x},\hat{p}^2] = [\hat{x},\hat{p}]\hat{p} + \hat{p}[\hat{x},\hat{p}] = i\hbar \frac{\hbar}{i} \frac{d}{dx} + \frac{\hbar}{i} \frac{d}{dx} i\hbar = i\...
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What does the comma mean in this commutation rule between quantum operators?

The Theorem about quantum operators commutation relation says: Consider pairs $(U, V )$ of unitary representations on a Hilbert space $H$, satisfying the commutation rule: $$U(x) V(y)=\exp (i \...
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Is the commutator relation $[\hat{x}, \hat{p}_x]=i\hbar 1\!\!1$ an *assumption* in the quantum theory?

This question is somewhat related to (but not by any means the same as) the question I asked recently. In his Lectures on Quantum Theory, Isham essentially says (reference given below) that if an ...
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How do you know if a operator commutes with the hamiltonian?

In the question there is a central potential within a Hamiltonian, and I have to find the appropriate quantum numbers. They say that $j, m, s$, $\ell$ are the appropriate quantum numbers to describe ...
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An alternative proof of the generalized uncertainty principle? [closed]

I am looking through some notes for a quantum chemistry course and I encountered this question: $\Delta A\Delta B\geq\frac{1}{2}\left|\left<[\hat{A},\hat{B}]\right>\right|$ (for given state $\...
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Why does Peskin and Schroeder move normal ordering move outside a commutator?

The equation trying to prove that Wick's theorem by induction in P&S on page 90 implies that normal ordering can be moved outside a commutator (at least with a positive frequency field), which I ...
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Three operators commuting with each other

It is well known that if two operators commute, the it is possible to find common eigenfucntions for them. What if we have 3 operators that commute with each other? Will we find common eigenfunctions ...
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Is every pair of conjugate variables associated with a Fourier transform?

For example, in quantum mechanics, the commutator of the position and momentum is $$[\hat{P_i} ;\hat{Q_j} ] =i\hbar\delta_{ij}\neq 0, i\neq j$$ I know that the position space representation of the ...
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What can be said about the commutator of an operator with itself at different times?

In general for some smooth and bounded $\hat{V}$ $$ \left[\hat{V}(t_1), \hat{V}(t_2) \right] \neq 0 \text{ if } t_1 \neq t_2 $$ But what more can be said about commutators of this type? I am ...
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Can current and voltage be linked by an uncertainty relation when electrons tunnel through a barrier?

Quantum tunneling has been shown to be linked to uncertainty relations for some observables involved in the system. For instance, if we consider electrons tunneling through a potential barrier it can ...
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Degeneracy and Complete Sets of Commuting Observables

I want to understand how the degeneracy of an operator is related to the existence of a complete set of commuting operators that includes it. I know that if a set of operators commute, they possess a ...
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Does the canonical commutation relation give a unique solution for the momentum operator? [duplicate]

So lets say we are in a 1d system and in the position basis just for simplicity. The CCR is: $$ [x,p]=i $$ and the momentum operator is $-i\partial_x$. Is this solution unique or are there other ...
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The commutation relation between the square components of angular momentum

So my question is as follows. I was reading about Angular Momentum from Griffiths, Introduction to Quantum Mechanics and it is a well known fact that the components of angular momentum do not commute- ...
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Commutation relations/sturcture constants for Lorentz algebra

I am trying to compute the curvature for a gauge theory based on the pure (local) Lorentz group. The final hurdle is working with monstrous structure constants. My objective is to show that \begin{...
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Commutation relation for deviation of two hermitian operators

On page 35, right after equation 1.4.60, Sakurai says that the commutator $$[\bigtriangleup A, \bigtriangleup B] = [A,B]$$ where $\bigtriangleup A = A - \langle A \rangle$, and $A$ is a hermitian ...
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Eigenfunctions of compatible observables that are not shared

I'm using D.J. Griffiths's Introduction to Quantum Mechanics (3rd. ed), reading about the angular momentum operators $\mathbf L=(L_x,L_y,L_z)$ and $L^2$ in chapter 4. The author discusses ...
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Virasoro modes commutation calculation

Given the commutation relations: $$ [\alpha_m,\alpha_n]=m\delta_{m+n,0} $$ and $$ L_m=\frac{1}{2}\sum_\rho\alpha_{m+\rho}\alpha_{-\rho} $$ I am trying to calculate the commutator between $L_m$ and $...
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Are all operators really associative?

Operators are associative as seen here. But when we try to calculate $[\hat{x}, \hat{p}]$ for example, we use a test function and apply $\hat{p}$ to both $\hat{x}$ and the function, instead of ...
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Product of non-commuting operators

I want to expand the product: $$\left(\hat{A}_{1}+\hat{A}_{2}\right)\left(\hat{B}_{1}+\hat{B}_{2}\right)$$ $\hat{A}_1$ and $\hat{B}_1$ are operators both working on the same particle, and do not ...
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Commutator of net position and net momentum

It is well known that $$[\hat{x},\hat{p_x}] = [\hat{y}, \hat{p_y}] = [\hat{z}, \hat{p_z}] = i\hbar$$ But what if instead we wanted to know the commutator of the net displacement $\hat{r} = \sqrt{\...
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How does preservation of the Lorentz algebra demonstrate Lorentz invariance of a QFT?

In his book "Quantum Field Theory of Point Particles and Strings", Brian Hatfield makes the following claim (on p. 46) after canonically quantizing the free scalar field theory: We started with a ...
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Why is this sum the delta function?

I am reading the first chapter from Fetter and Walecka, which is on second quantization, and I have understood everything up to this section. It seems to me that these field operators essentially ...
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For $[A,B]=0$, if an eigenfunction of $A$ not an eigenfunction of $B$, does that imply degeneracy of one operator?

When two operators $A$ and $B$ commute, there can be functions which are eigenfunctions of $A$ but not that of $B$. For example, in case of the one-dimensional harmonic oscillator, any linear ...
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If two operators commute, does it mean that every eigenfuction of one is also an eigenfunction of the other?

I have trouble interpreting the result of a problem. If we have a function $$\psi ( \theta , \phi) = e^{-3i\phi}cos \theta $$ and two operators $$A=\frac{\partial}{\partial \phi} $$ $$B=\frac{\partial}...
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How come the commutator of $J^2$ (momentum) and $J^{\pm}$ (ladder operator) is zero while they don't have simultaneous eigenfunctions?

We know that $J^2$ has eigenfunctions which are the spherical harmonics, and these spherical harmonics definitely aren't eigenfunctions for $J^+$, because $J^+$ acting on one spherical harmonic ...
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SUSY algebra generators commutation relations

Given the form of the supersymmetric generators below: $$ P_\mu=i\frac{\partial}{\partial x^\mu} $$ $$ Q_\alpha=i\frac{\partial}{\partial\theta^\alpha}-\sigma^\mu_{\alpha\dot{\alpha}}\bar{\theta}^{...
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Operators “carrying” momentum and particle number

TL;DR: What does it mean for a operator in QFT to "carry momentum"? During a QFT lecture (discussing the real scalar field) my Prof. stated that the operator $$P^\mu := \int \frac{d^d\vec{p}}{(2\pi)^...
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Why is the degeneracy in the eigenvalue representation of eigenkets always lifted when using a maximal set of commuting observables?

I don't see how this implicit theorem Sakurai states in his book on QM on page 31 can be proven in general Assume that we have found a maximal set of commuting observables; that is, we cannot add ...
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How to analyse physics for following commutator relation?

Suppose I define following commutation relation $$[\hat{x}^\alpha,\hat{x}^\beta]=\frac{i}{l_p}\epsilon^{\alpha\beta\mu\nu}\hat{x}_\mu\hat{x}_\nu$$ where $l_p$ is Planck's length. What changes will ...
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How did Heisenberg come up with CCR?

Usually it is pointed out that the relation $[x,p]=i\hbar$ comes from the promoting the poisson bracket to commutator but as I know this process of quantization is called deformation quantization ...
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Commutator between Angular Momentum J and Cross Product

I'm trying to show that a cross product $C$ of two vectorial operators (let's say $A$ and $B$) it's a vector by it's own, which means, I want to show $$\left [J_i,C_j \right ] = i\hbar \epsilon_{...
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Geometrical description of canonical commutation relation [duplicate]

Does there exist any geometrical description of canonical commutation relation of quantum mechanics $$[\hat{x},\hat{p}]=i\hbar$$ maybe in phase space? What I meant by geometry is along the lines ...
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Geometric description of canonical commutation relation

Does there exist any geometrical description of canonical commutation relation (CCR) of quantum mechanics $$[\hat{x},\hat{p}] = i \hbar \, ,$$ e.g. in phase space? The commutator, along with the ...
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Annihilation operator in the Heisenberg picture

I study the monograph An Introduction to the Standard Model of Particle Physics written by W. N. Cottingham and D. A. Greenwood. I can't understand the equation about the annihilation operator in the ...
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What is my mistake with field commuators here?

Take a two scalar quantum fields and commute them: $$[\phi(x),\phi(y)]$$ This must be zero if $x$ and $y$ are space-like separated. Write $y=x+a$ then expand this in a Taylor series like so: $$[\...
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How to prove that different squeezed vacua are the ground states of inequivalent CCR representations?

one can find on wikipedia articles on squeeze operators and squeeze coherent states these squeezed coherent states depend on a squeezed parameter r. the usual coherent states have r = 0 i have to show ...
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How do you write commutators in polar coordinates?

In quantum field theory we have the commutators of fields must be zero outside the light cone. $[\phi(x),\phi(y)]=0$ if $|x-y|^2<0$ How can one write this in polar coordinates or a general ...
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Commutator of the Pauli-Lubanski vector operator and the generator of translations $P^\alpha$

I'm trying to obtain the commutation relation between the Pauli-Lubanski vector operator and the generators of the Lorentz Group: $$[W^\mu,P_\sigma]=[\frac{1}{2}\epsilon^{\mu\nu\lambda\rho} P_\nu M_\...
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How to calculate $\left[ \vec{L}^2, x_i \right]$

I've been asked to prove $\left[ \vec{L}^2, x_i \right] = -2i\hbar \varepsilon_{ijk}L_j x_k -2\hbar^2 x_i $ and I don't seem to get it correctly. I propose $\left[ \vec{L}^2, x_i \right] = \left[ L_l ...
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Radial Commutativity in AdS/CFT

I am reading Daniel Harlow's Tasi lecture notes on the emergence of bulk physics in AdS/CFT, and the question that I have is regarding the radial commutativity puzzle given in these notes. First, I ...
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Proof that the commutator of angular momentum and 4-momentum is 0

I have this commutator $[P^2,J_{\mu\nu}]$ that I'm supposed to prove is zero. If we expand it (given that $[P_{\alpha}, J_{\mu\nu}] = i(g_{\mu\alpha}P_{\nu} - g_{\nu\alpha}P_{\mu})$ and $[P_{\alpha}, ...
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Commutator of the Pauli-Lubanski with a vector $P^{\mu}$

The Pauli-Lubanski vector is defined as $W_{\mu} = -\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}J^{\nu\rho}P^{\sigma}$, where $\varepsilon_{\mu\nu\rho\sigma}$ is the 4-dimensional Levi-Civita symbol. The ...
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Representations of Conformal Group

I want to work out the Representations of the Conformal Group. I work with Francesco's Conformal Field Theory. He stats in equation 4.30 that $$e^{i x^\rho P_\rho}K_\mu e^{-i x^\rho P_\rho}= K_\mu +...
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Trick with “functional” derivative to evaluate commutators between diagonal hamiltonian and creation fermionic operator

I found a theorem that states that if $A$ and $B$ are 2 endomorphism that satisfies $[A,[A,B]]=[B,[A,B]]=0$ then $[A,F(B)]=[A,B]F'(B)=[A,B]\frac{\partial F(B)}{\partial B}$. Now i'm trying to apply ...
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Commutator of two “CFT charges”

In class it was shown that $$ i[Q_\epsilon,T^{\mu\nu}] = -(\epsilon\cdot\partial)T^{\mu\nu} - \partial_\rho (\epsilon^\mu T^{\rho\nu}) + \partial^\nu(\epsilon_\rho T^{\rho\mu}) $$ with $$ Q_\...
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Particle picture in position space in quantum field theory

When I operate $a^{\dagger}_k$ on vacuum, $|0\rangle$, I get a particle created in momentum space with a 4-momentum equal to $(\omega_k, \vec{k})$ where $\omega_k=\sqrt{m^2+\vec{k}^2}$ here I'm only ...
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Is there an identity for anti-commutator $\{ A B, \, C D \}$ in terms of commutators $[\, , \,]$ only?

I'm looking for an identity that could express the anti-commutator $$\tag{1} \{ A B , \, C D \} \equiv A B C D + C D A B $$ expressed as a combination of commutators only: $[A,\, C]$, $[A, \, D]$, etc....
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Anti-commutator for annihilation and creation operators: ordering of indices

I'm trying to prove that $\{\tilde a_i,\tilde a_j^{\dagger} \}=\delta_{ij}$, by defining $\tilde a_i=\sum_j \bar U_{ji}a_j$. U is an unitary matrix and $a_i$ refers to an element of the operator $a$. ...
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SUSY $\mathcal{N}=1$ algebra

Given the definitions $$ P_\mu= -i\partial_\mu $$ $$ Q_\alpha=-i(\partial_\alpha-(\sigma^\mu\bar{\theta})_\alpha\partial_\mu) $$ $$ \bar{Q_\dot{\alpha}}=+i(\bar{\partial}_\dot{\alpha}-({\theta}\sigma^...

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