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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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6 votes
1 answer
133 views

Obscure Calculations in Foldy-Wouthuysen Transformation (electron in EM field)

I'm studying the Foldy-Wouthuysen Transformation on Bjorken-Drell's book and I got stuck strying to replicate some calculations. First of all, introducing the transformation $\psi'=e^{iS}\psi$ we get ...
0 votes
1 answer
40 views

What does "not applying the CCR" mean exactly?

I've seen mentioned in a number of posts that some relations do or do not apply depending on whether one is "applying the CCR". For example, In Relationship between normal-ordered vacuum ...
2 votes
1 answer
258 views

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 ...
0 votes
0 answers
34 views

An interpretation for Propagator $D(x-y)$

When I learn QFT I always see that when we consider the causality problem in QFT, at first we may try to compute the propagator $D(x-y)$ for spacelike distance $(x-y)^2<0$, which is nonzero. An ...
0 votes
1 answer
74 views

Do gamma matrices commute with 4-vectors?

One of my exercises was to prove the identity $$\gamma^\mu\displaystyle{\not}a\gamma_\mu=-2\displaystyle{\not}a.$$ Which is trivial if $\gamma^\mu a_\nu=a_\nu \gamma^\mu$, as follows $$\gamma^\mu\...
0 votes
0 answers
61 views

Quickest way to calculate the commutator $[x^2, p^2]$ [closed]

Of course this is an elementary problem. Using the formula $[A, BC] = [A,B]C +B[A,C]$ and $[AB,C]=[A,C]B +A[B,C]$, with a few lines, we can get the final result $$2 i (xp+px). $$ However, is there any ...
5 votes
3 answers
3k views

What's the expectation value of the normally ordered commutator between annihilation and creation operator, $:[a,a^\dagger]:$?

According to the commutation relation of annihilation and creation operators, $$[a,a^{\dagger}]=1. \tag{1}$$ I would like to calculate the vacuum expectation value of the normal order of this ...
-1 votes
1 answer
42 views

Knowing all spin components at the same time [duplicate]

You can't know all spin components simultaneously due to the commutation relation (& Heisenberg's uncertainty principle): $[S_x, S_y] = i\hbar S_z$ But what if you know that $S_z=0$? Then that ...
1 vote
0 answers
31 views

Correlation functions of exponentials of fields

I've been trying to solve for scattering amplitudes for 4 graviton scattering in string theory. However, while going through Schwarz, Witten and Green book for string theory, I come across the ...
2 votes
1 answer
218 views

Commutator in the derivation of the Runge-Gross Theorem

In the derivation of the Runge-Gross theorem, a theorem that states a one-to-one correspondence between potential and density in an evolving system, a commutator appears that I seem to have trouble ...
3 votes
1 answer
76 views

Marginal of Exponential Operator in terms of $x$ and $p$

I am effectively looking to compute the marginal, $\langle x| \exp(\hat x\hat p ) | p \rangle$. Since $[\hat x, \hat p] = i$, it is not simply solved by trivially acting $\hat x$ on to the state on ...
0 votes
2 answers
384 views

How to lead Bessel $K$ function from eq. (4.12) in Srednicki?

In Srednicki, $$\begin{align*} \left[\varphi^{+}(x),\varphi^{-}(x^\prime)\right]_\mp=&\int\widetilde{\mathrm{d}k}\int\widetilde{\mathrm{d}k^\prime}\mathrm{e}^{\mathrm{i}(kx-k^\prime x^\prime)}\...
1 vote
0 answers
44 views

General bosonic commutator result

I'm considering these functions $f,g,h,j$, which all share the same form, of the boson creation and annihilation operators: $$ f(\hat{a}^{\dagger} ,\hat{a}) = \sum_{n,m} c_{n,m}(\hat{a}^{\dagger})^n (\...
0 votes
1 answer
257 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, ...
0 votes
0 answers
33 views

Proof that commuting Dirac fields violate causality

What is the proof that commuting Dirac fields violate causality? All sources I could find just quote this result, but I couldn't find an explicit derivation anywhere. In particular, the case I am ...
1 vote
1 answer
34 views

Does the anticommutator of two spinors affect the transpose of their product?

My lecture notes claim that for an anticommutation relation $$[ \psi_{\mu}(\bf{x},t),{\psi_{{\nu}}^{*}}(\bf{y},t)] = \delta_{\mu \nu} \delta^3(\bf{x}-\bf{y})$$ between two spinors, the transpose of ...
3 votes
1 answer
95 views

What do you get when you Taylor expand a Magnus expansion?

The Magnus expansion and Dyson series are very similar to each other, in that they both give a way to approximate a time-evolution operator as a series expansion $$U(t) = \mathcal{T}\left(\exp\left[-i\...
0 votes
1 answer
47 views

How does $[J^2, J_+] = 0$, but $ | l, m \rangle $ is not an eigenstate of $J_+$? [duplicate]

I am currently working on the angular momentum part (chapter 3) of Sakurai's QM 2nd. From eq. 3.159 and eq. 3.154, the book says $[J^2, J_+] = 0$ and $[J^2, J_z] = 0$, but I don't understand why $| l, ...
0 votes
2 answers
51 views

Constant of Motion in Quantum Mechanics for explicit time-dependent Operators

I was studying constants of motion in quantum mechanics, and at first, I don't understand the condition to be a constant of motion. Generally, the temporal variation of an operator $A$ is given by the ...
2 votes
1 answer
389 views

Lie algebra: Proof that the commutator of infinitesimal motions is an infinitesimal motion

I am following Classical and Quantum Mechanics via Lie Algebras by Neumaier and Westra. Setup I am stuck at part of Thm 2.3.1. Consider the matrix group $\mathbb{G}$. The set of $\mathbb{G}$-motions ...
14 votes
4 answers
519 views

Why does $\hbar$ appear twice in the axioms of QM?

Physical theories have dimensionful constants. Each constant can be found via measurement, by fitting some equation to data. Mathematically, you would expect each constant to be "defined" in this way ...
0 votes
0 answers
26 views

How can interacting field operators in $2D$ still satisfy the canonical commutation relation?

Free fields in any dimensions are well-known to be Gaussian, act on the Fock space and satisfy the canonical commutation relations. By definition, interacting field operators are NOT such cases, as ...
0 votes
1 answer
50 views

Does the Hamiltonian always commute with the Time Evolution Operator?

The time evolution operator $U(t, t_0)$ is given as the solution of the equation $$ i\hbar \dfrac{\text{d}}{\text{d}t} U(t, t_0) = HU(t, t_0)$$ whether or not the system is conservative. When the ...
-1 votes
2 answers
111 views

Is there a physical cause of uncertainty? [closed]

The uncertainty principle is confusing me. Considering this image from the article: Is the particle believed to be physically moving with similar capriciousness in real space; and if so, what ...
0 votes
0 answers
77 views

Quantizing the electric field without quantizing vector potential

I am trying to quantize the electromagnetic field, without using the vector potential. I start with a Fourier expansion: $$\begin{equation} \vec{E}(\vec{r},t) = \sum_{\epsilon} \vec{\epsilon} \int \...
4 votes
1 answer
78 views

Where does the "arbitrary constant" in the $L_{0}$ Virasoro operator come from?

In the 2007 "String Theory and M-Theory" textbook by Becker, Becker, Schwartz there is the following claim about the canonical first quantization of a bosonic string: the quantization of the ...
1 vote
2 answers
98 views

Quantum angular momentum of a particle in an homogeneous magnetic field

In non-relativistic quantum mechanics, the canonical momentum of a particle is defined as $$\tag{1} p_i = - i \hbar \: \partial_i. $$ When there's an external magnetic field (suppose for simplicity ...
4 votes
3 answers
6k views

Eigenstates of Ladder Operators

According to Griffiths' Introduction to Quantum Mechanics (page 147), if some function $f$ is an eigenfunction of $L^{2}$, then $L_{-}f$ is also an eigenfunction of $L^{2}$. Is $f$ also an ...
2 votes
0 answers
63 views

What does the Jacobi identity *mean* statistically?

Given that the commutator of a pair of operators shows up explicitly in the lower bound of the Robertson-Schrodinger inequality, I am wondering what, if any, statistical meaning/significance one can ...
0 votes
1 answer
61 views

Conmutators and Jacobi's Identity

I've come across an exercise asking me to calculate: $$[[A,B],[C,D]]$$ knowing $[A,C]=[B,D]=0$ and $[A,D]=[B,C]=1$ I've already solved it by "brute force", separating the commutator as ...
10 votes
2 answers
885 views

Why don't we use Hamilton-Jacobi method in QM?

In classical mechanics, we usually try to find a set of coordinates by Hamilton-Jacobi method to transform the Hamiltonian to zero such that the coordinates are conservations. However, we never try ...
0 votes
0 answers
24 views

Derivation of the quantization of the EM field in a dielectrics

I'm currently studying the quantization of the EM field in a dielectric medium and trying to understand the quantization scheme of Huttner and Barnett (1992, see Phys. Rev. A 46, 4306). The system ...
1 vote
0 answers
41 views

Conjugate momenta in Radial Quantization

When we radially quantize a conformal field theory, is there at least formally a notion of a conjugate momentum $\Pi$ to the primary fields $O$ which would satisfy an equal radius commutation relation ...
0 votes
2 answers
285 views

Solving Periodic Time Dependent Hamiltonians

For a general time dependent Hamiltonian, if the Hamiltonian at two different times $t_1,\,t_2$ satisfies $$\left[ \hat{H}(t_1),\hat{H}(t_2) \right]=0,$$ then the time evolution operator is $\hat{U}(t)...
0 votes
0 answers
28 views

How to prove that the spin operator commutes with the position operator? [duplicate]

In the lecture notes on Quantum Mechanics I'm reading, the author claims that the position operator $\hat{q}$, the square spin operator $\hat{s}^2$ and the spin operator component $\hat{s}_0$ (in a ...
0 votes
2 answers
436 views

Why does $[L_i,\textbf{L}^2]=0$ and $[L_i,H]=0$ imply that $[\textbf{L}^2,H]=0$?

For a particle in a central potential, the orbital angular momentum magnitude operator $\textbf{L}^2$ commutes with the Hamiltonian operator $H$, i.e. $$[\textbf{L}^2,H]=0.$$ I read that one way to ...
1 vote
0 answers
66 views

Canonical commutation relation on the spatial boundary of the hypersurface

Consider the equal time commutation relation of a field given on a $d$ dimensional spacelike hypersurface $\Sigma$ of a $d+1$ dimensional manifold given by $$[\Pi(t, x), \Phi(t, x')] = i\hbar\delta^{(...
0 votes
0 answers
63 views

Commutation behavior of spinors in Feynman diagrams

I am currently playing around with computing cross sections of several simple interactions in QED like Bhabha and Compton Scattering and I have stumbled upon a question which I havent yet managed to ...
0 votes
2 answers
1k views

Velocity operators in quantum mechanics

According to the Heisenberg equation of motion, the velocity operator is given by $$\hat{v}=\frac{d \hat{r}}{dt} = \frac{1}{i\hbar}[\hat{r},\hat{H}].$$ Question 1: How can I find the velocity operator ...
2 votes
1 answer
424 views

Primary fields in di Francesco's CFT

In the CFT book by Di Francesco et al. they use conventions such that part of the conformal algebra (see eq. 4.19) is $$ [D,P_\mu]=iP_\mu, \\ [D,K_\mu]=-iK_\mu, \tag{1} $$ where $P_\mu$, $D$ and $K_\...
2 votes
0 answers
35 views

Double Discontinuity In CFT

In the paper Analyticity in Spin in Conformal Theories Simon defines the double discontinuity as the commutator squared in (2.15): $$\text{dDisc}\mathcal{G}\left(\rho,\overline{\rho}\right)=\left\...
-1 votes
2 answers
156 views

The question about commutator $[\hat{x},\hat{p}]=i\hbar$ at $\hbar\rightarrow 0$ seemingly can't match with Poisson bracket $\{x,\,p\}=1$ [duplicate]

At the limit $\hbar\rightarrow 0$, all "quantum" should tend to "classical", but why is the quantum commutator $[\hat{x},\hat{p}]=i\hbar$ at $\hbar\rightarrow 0$ equal to $0$, but ...
4 votes
1 answer
543 views

Commutation relations in terms of eigenstates scalar product

This question has caught my attention because I was unaware of the fact that the position-momentum canonical commutation relations could be derived out of the only assumption for $\langle x | p\rangle$...
0 votes
0 answers
56 views

How to generalize the (anti)commutation for spacelike separation to more than $2$ field operators?

Let $\phi_1$ and $\phi_2$ be quantum field operators of certain spin on $\mathbb{R}^4$. Then, the principle of locality dictates that if $x$ and $y$ are space-like separated, we have \begin{equation} \...
2 votes
1 answer
57 views

How to show causality for a Klein-Gordon field in 1+1 dimensions using field commutators?

For a non-interacting massive scalar field $\phi$ in an $n+1$ dimensional minkowskian spacetime, the field commutator between two event points is $$ [\phi(x),\phi(y)] = \int \frac{\mathrm{d}^n p}{(...
0 votes
0 answers
34 views

Normalization of one particle state wave function in fock space - commutator

In deriving the 1/$\sqrt{N!}$ normalization factor the first step is looking at the one particle state (see image below). I am confused about how we got from the first line to the second? Maybe I am ...
0 votes
1 answer
204 views

How does Sakurai reduce a product to a commutator?

The following section is from Modern Quantum Mechanics by Sakurai; can any one help me finding out how this is done? In contrast, if we follow approach 2, we obtain \begin{align} \vert\alpha\...
0 votes
2 answers
251 views

Why does $e^{-H}\partial_j e^{H} = \partial_j + \partial_jH$?

I apologize if this is a dumb question but I have really thought about this a while and I can’t understand it. I have tried to prove this using the power series of the exponential function but I did ...
0 votes
1 answer
58 views

Commutator of raising operator in angular momentum with partial derivative wrt z

While fiddling around with certain commutation relations, i noticed the following relation while using spherical coordinates. What could this relation mean intuitively? Let me know if any information ...
0 votes
0 answers
38 views

Poisson Bracket and commutators in quantum mechancs [duplicate]

how did they reach the conclusion that quantization of the Poisson brackets $ (A,B) $ was equal to the commutator $ \frac{1}{i\hbar}[A,B] $ in quantum mechanics? so the quantum equations of motion ...

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