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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1answer
362 views

Fock space with mixed anti-commutation/commutation relations?

Let's say we have two modes, with the following labeling of occupation number states: $ \lvert \Psi \rangle = \begin{pmatrix} 0,0 \\ 0,1 \\ 1,0 \\ 1,1 \end{pmatrix} $ An example of (what I assume to ...
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2answers
349 views

Does Schroedinger equation depend on the sign of Poisson bracket?

Let's consider Poisson bracket $$\left\{ A,B\right\} =\alpha_{p} \left( \frac{\partial A}{\partial p_{k}}\frac{\partial B}{\partial q^{k}}-\frac{% \partial A}{\partial q^{k}}\frac{\partial B}{\...
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2answers
190 views

Why does the Heisenberg uncertainty principle refer to momentum rather than velocity?

I've been looking at the Heisenberg uncertainty relations, and something that sticks out to me is the use of momentum rather than velocity. Shouldn't electrons have the same mass? And if they do, why ...
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Causal propagator relation from canonical field commutator

In Minkowski spacetime, the commutator of the Klein-Gordon field operator with itself at different spacetime points evaluates to the advanced minus retarded Green's function (i.e., the "causal ...
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3answers
222 views

Physical intuition behind why $\hat{J}^2$ and $\hat{J}_z$ commute

In Townsend's Quantum Mechanics textbook, he shows that $\hat{J}^2$, the squared magnitude of the angular momentum, and $\hat{J}_z$, the generator of rotations about the $z$-axis should commute. I ...
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1answer
348 views

Quantizing the Dirac field using commutation relations leads to an unbounded Hamiltonian?

If we were to quantize the Dirac field using commutation relations instead of anticommutation relations we would end up with the Hamiltonian $$ H = \int\frac{d^3p}{(2\pi)^3}E_p \sum_{s=1}^2 ...
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79 views

Commutators and the OPE

Imagine we have a QFT (not a CFT) with some operator $\phi(\vec{x},t)$. Assume that an OPE exists and compute $\langle \phi(\vec{x},t)\phi(\vec{y},t)\phi(\vec{y}+\vec\epsilon,t)\rangle$ in the $\vec{\...
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1answer
91 views

Show eigenvalue does not depend on magnetic quantum number $m$

We have a scalar operator $A$, being invariant under rotations which commutes with the angular momentum, i.e. $$[A,J_i]=0 \text{ where } i=x,y,z$$ $$[A,J^2]=0 $$ So eigenfunctions of $A$ can be ...
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1answer
84 views

Heisenberg Uncertainty relation

The derivation of the uncertainty principle says that for any 2 observables $A$ and $B $, we act $$ A+i\lambda B $$ on a normalized state and demand the norm of the new state be greater than or ...
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1answer
203 views

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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Regarding the commutator of ladder operator in QFT

I am trying to verify the computation of the commutator of the ladder operator for Klein-Gordon solutions, but it seems like I am unable to do it properly. Here is what I do: For, $$ \varphi(x^\mu)=\...
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1answer
28 views

Commutator of intrinsic derivatives in NP formalism when timelike and null congruence are both given

Suppose we have a congruence of affinely parametrized null geodesics (light rays), with tangent vector $\ell^a$, and a congruence of timelike curves (observers), with tangent vector $u^a$, such that ...
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Derivation of Schwingers action principle from Heisenberg Equation and CCR - Why does it work with Anticommuting variations?

In the Book "Quantum Field Theory I" by Manoukian, in section 4.3, from what I understood, he derived the quantum-action-principle of Schwinger only by using unitary time-evolution of the field ...
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321 views

Canonical commutation relations and a scalar interacting field

Let $\hat\phi(x)$ be a quantum scalar hermitian field and $\hat \pi(x)$ its conjugated field. IF the scalar field evolves according to a FREE theory it is possible to write down the following normal ...
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1answer
62 views

Problem calculating commutator with Casimir

I am trying to verify that the Casimir of the Lie group $SO(3)$ is actually $N^2=N_iN_i$, but I have problems, with indices surely, and I was wondering if someone could help me figuring out how to ...
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2answers
229 views

Commutator of $[\hat p, F(\hat x)]$ without using $\hat p=-i\hbar\frac\partial{\partial x}$?

I have been able to prove this relation by using a certain method, but it uses the fact that $$\hat p=-i\hbar\frac\partial{\partial x},\tag{1}$$ which is a relation I have avoided so far, so I wish to ...
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1answer
732 views

On the generators of the Lorentz group

We already know that the generators of Lorentz group are three boosts and three rotations. We have the relation $$ \left[ M_{\mu \nu},M_{\sigma \lambda} \right] = ig_{\mu \sigma} M_{\nu\lambda} - ...
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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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2answers
405 views

Euclidean QFT commutator vanishes for all spacetime separations?

In Minkowski spacetime, the commutator of the Klein-Gordon field operator with itself at different spacetime points evaluates to the advanced minus retarded Green's function of the classical theory, ...
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75 views

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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Do fields describing different particles always commute?

Is it true that field operators describing different particles (for example a scalar field operator $\phi (x) $ and a spinor field operator $\psi (x) $) always commute (i.e. $ [\phi (x), \psi (y) ]=0, ...
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267 views

Quantum Mechanics and Schur's lemma

Today i was studying on a textbook and i crossed a paragraph that confused me a little. Suppose you have an algebra generated by $\hat{X}$ and $\hat{P}$ and a function $f(\hat{X},\hat{P})$ that ...
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1answer
83 views

Retrieving the non-relativistic Hamiltonian from Relativistic QM

I'm trying to follow section 15.5 here, which derives the low-energy limit of the Dirac equation for an electron in a EM-field. After some manipulations (which I think I follow alright) the author ...
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398 views

Commutation Relation of Angular Momentum and a Radial Function

How to show that $$\langle\vec{r}|[L_{j},f(\hat{r})]|\psi\rangle = 0~?$$ note that $\hat{r}$ is a operator and not a unit vector What I know so far is that the commutation relation of a normal ...
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359 views

In the Dirac equation, if the $\alpha$ is the mean velocity, why does it commute with $x,y,z,t$ if the velocity is related to the momentum?

In the Wikipedia talk page for the Dirac equation I found the following passage: The Dirac equation can be proved with the help of the correspondence principle. The energy and momentum of a ...
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3answers
308 views

Finite lorentz transform for 4-vectors in terms of the generators

One or two sets of notes (one of them by Timo Weigand) on QFT that I have come across state explicitly that a finite lorentz transformation for 4-vectors can be written in terms of the generators $J^{\...
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353 views

Commutator relation $[p, px^n]$? [closed]

This seems like a really straightforward problem that nevertheless stumps me. We have $\frac{i}{\hbar}[p,px^n] = \frac{i}{\hbar}(p[p,x^n] + [p,p]x^n)$ Now I have previously found that $[p,x^n] = - i ...
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commutation relations for many-body permanents

I'm interested in understanding the many-body generalization of the canonical commutation relations. I.e. commutators of the form $$ [a^\dagger_I, a_J] $$ where $I,J$ are multi-indices with the ...
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2answers
433 views

Proof of a basic commutation relation (Heisenberg Matrix Mechanics)

See the following relation: $$i\hbar \frac{\partial \hat{F}}{\partial \hat {x}} = [\hat{F}, \hat{p}],$$ assuming that $\hat{x}$ and $ \hat{p}$ are the position and momentum matrices and $\hat{F}(\...
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1answer
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Commutator question

I'm watching this video going through a non-rigorous explanation of the Einstein field equations: https://www.youtube.com/watch?v=foRPKAKZWx8 At around 1:17:50 he shows an identity involving the ...
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1answer
60 views

What mathematicaly exactly is an ordering prescription?

This has allready been asked, but I still have some issues with it: It has been established in this question that the ordering prescription is not a function that maps operators to operators, but ...
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99 views

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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2answers
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Is there no coordinate conjugate to $L_x$?

The conjugate momentum corresponding to $\phi$ (azimuthal angle in sp. polar coordinate) is $L_z$ (sometimes written $L_\phi$) which is frequently used in quantum mechanics. Why is there no coordinate ...
3
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1answer
90 views

Schwartz QFT coupling to the photon

I am reading Schwartz's QFT textbook. In Eq. (10.104) he writes: $$ \left[i\partial_\mu-eA_\mu,i\partial_\nu-eA_\nu\right]~=~-e i[\partial_\mu A_{\nu}-\partial_\nu A_{\mu}]~=~ -e i F_{\mu \nu}. \tag{...
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1answer
230 views

Why does every operator that commutes with $\hat{H}$ have an inverse?

I was reading M.S. Dresselhaus, Applications of Group Theory to the Physics of Solids (PDF), and, in chapter 5, she defines the group of Schrodinger Equation as the group of all operators $\hat{P_R}$ ...
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Commutator relation of EM Field Covariant?

I read that for quantization of the EM-Field, you demand the canonical equal-time commutation relations: $$[A^\mu(\vec{x},t), \pi^\nu(\vec{y},t)] = i \hbar g^{\mu \nu} \delta^3(\vec{x} - \vec{y}). $$ ...
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1answer
265 views

How does one find the wavefunction of a particle in its rest frame?

In classical mechanics, the orbital angular momentum of a particle is defined as $\textbf{L}=\textbf{r}\times\textbf{p}$. This is zero in the rest frame of the particle where $\textbf{p}=0$. Quantum ...
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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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1answer
676 views

Commutation between energy and momentum

The composition property of Lorentz transformations forces the commutation relation: $$[P^{\mu},P^{\rho}]=0$$ where $P$ is the four-momenta. The above seems to imply that Energy and 3-momentum ...
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342 views

Uncertainty Principle: Commutators [duplicate]

How are commutators the mathematical basis for uncertainty principle? What makes one say that commutators imply uncertainty principle or vice-versa?
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What is the significance of taking the commutator?

I'm an undergrad studying QM, and sometimes we use the method of taking the commutator of two operators to make some statement about the group. I always get lost because I don't know WHY were taking ...
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1answer
198 views

What is the commutator of a Poisson bracket and the covariant derivative?

Consider a classical vector field $V^\mu$ on a curved background. We make a 3+1 split of coordinates into $t,x^i$, where $x^i$ are coordinates on spatial hypersurfaces $\Sigma$ and $t$ the parameter ...
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2answers
500 views

Commutativity vs Compatibility

As far as I know, two compatible observables have a complete set of common eigenvectors, and using this fact, one can prove that their corresponding operators are commutative. Well now is the converse ...
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319 views

Renormalization and canonical commutation relations

My question is whether canonical commutation relations hold for renormalized quantum fields. Below I show reasoning which caused by doubts. Consider a relativistic scalar QFT. We have spectral ...
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2answers
813 views

Derive canonical commutation relations from Schwingers principle

The book of Dyson "Quantum-Field-Theory" states in section 4.4 that one can derive canonical commutation relations from Schwingers quantum action principle. However, doesn't give a calculation for the ...
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2answers
506 views

Commutator of scalar field and its spatial derivative

Consider the usual commutation relations of two scalar fields $$\left[\phi_{m}\left(t,\boldsymbol{x}\right),\pi_{n}\left(t,\boldsymbol{y}\right)\right]=\boldsymbol{i}\delta_{mn}\delta\left(\...
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1answer
359 views

What does it mean to quantize something in physics?

I'm studying quantum mechanics without ever properly having studied analytical mechanics. Thus, I lack some basics to build upon, which is something I'm trying to fix. In particular, I realized the ...
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1answer
173 views

Origin of the canonical equal time commutation relations

I stumbled upon this QFT exercise: And I must say I'm somewhat baffled. I always thought that the commutation relations are something that is postulated in making the shift from classical physics ...
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3answers
922 views

Schrödinger equation for time dependent Hamiltonian and conjugation

The Schrödinger equation for the evolution operator reads: $$ \frac{\partial U}{\partial t} = -\frac{i}{\hbar}HU $$ where for a time dependent Hamiltonian which need not commute with itself at ...
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370 views

Expected value of commutator using path-integral

Consider a real scalar field theory in finite temeperature. According to the book by Kapusta and Gale, Finite-Temperature Field Theory, its retarded Green's function is given by $$iD^R(x,x') = Tr\{\...