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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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45 views

Why $\int dx \partial_\mu\neq \partial_\mu \int dx$ but $\int dp \partial_\mu=\partial_\mu\int dp$?

It's well known that $\int dx \partial_\mu\neq \partial_\mu \int dx$. But I have a hard time understanding $\int dp \partial_\mu=\partial_\mu\int dp$, because $[p,x]\neq0$ do not commute. However, ...
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Proving that $$ [\phi(\vec{x}, 0), \phi(\vec{x}, t)] \sim e^{-i m t}-e^{+i m t} $$ in QFT

So far, I get the following (for the left term in the integral, $d$=3): \begin{equation} \begin{aligned} \Delta_{+}(x) &= \int \frac{\mathrm{d} \vec{p}^{d}}{(2 \pi)^{d} 2 e(\vec{p})} \exp (-i t e(...
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1answer
39 views

Simple question on computing commutation relation

In bosonization, one faces with the following commutator: $$[\phi(x_1), \theta(x_2)]=\sum_{q\neq 0} \frac{\pi}{Lq} e^{iq(x_2-x_1)-\alpha |q|}\tag{1}$$ where $q$ is an non-zero integer multiple of $2\...
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Can you argue without explicitly calculate the eigenenergies that one Hamiltonian is gapped and another is not?

Consider a pair of one dimensional four band model $H_1$ and $H_2$, which read as: $$ H_1 = \begin{pmatrix}k\sigma_x-E_0&0\\0&k\sigma_x+E_0\end{pmatrix} + \alpha \begin{pmatrix}0&\sigma_x\...
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5answers
249 views

Does an Operator that neither commutes with $\hat{X}$ or $\hat{P}$, nor can be expressed as a “function” of $\hat{X}$ and $\hat{P}$ make sense?

When you come from classical hamiltonian mechanics (which is based on the phase space), observables are introduced as functions $f$ on the phase space $(q, p)$. There can't be a classical observable ...
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2answers
141 views

If $E$ and $P$ don't commute, why could we have an $E$-$K$ diagram?

If $E$ (energy) and $P$ (momentum) only commute in constant potential, how could we have an $E$-$K$ diagram in a solid material? $[E,p] \neq 0$. Then we cannot prepare electrons whose $E$ and $P$ are ...
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1answer
28 views

Do Equal Time Commutation / Anticommutation relations automatically also apply for spatial derivatives?

The question is basically in the title. My naive thought was that when a commutation relation holds for all field operators $\Psi(\vec{x})$ (by "all" I mean "at all positions $\vec{x}$") on a fixed ...
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1answer
41 views

Generalized commutator/anticommutator via phase factor

We know that the commutator between two operators $A$ and $B$ reads $[A,B]_{-}=AB - BA$, while the anticommutator reads $[A,B]_+=AB + BA$. I am wondering if someone has ever used a generalized ...
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44 views

Commutators of dot and cross products

I apologize if this question is too basic, but I am wondering if identities for commutators such as $[AB,C]=A[B,C]+[A,C]B$ also hold for dot and cross products within the commutator (i.e., $[A\times ...
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1answer
88 views

Commutation relation for Dirac field

In "Quantum Field Theory" by Peskin and Schroeder, I couldn't understand the commutation relation calculation for Dirac field (pg. 53): $$ \begin{align} \psi(x) &= \int \frac{d^3p}{(2\pi)^3} ...
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1answer
74 views

What is the value of the commutator $[\vec{S}, H]$?

What is the commutation relation between $[S, H]$ where Hamiltonian $H= - \vec{S} \cdot \vec{B}$ , $\vec{S}$ is the spin and $\vec{B}$ is the magnetic field. I am getting $0$ but it seems wrong.
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2answers
64 views

Commutation of position four-vector with spacetime derivatives

I am trying to understand a simple demonstration in Ashok Das' Lectures in QFT. He does the following on p. 134 $$[P_\mu,M_{\nu\lambda}]=[\partial_\mu ,x_\nu\partial_\lambda-x_\lambda\partial_\nu]=\...
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42 views

Holonomic basis

Is the following definition correct? Given a differentiable manifold $M$ and an ordered basis $\{e_j^m\}$ of the tangent space $T_m M$ with $m\in M$ (they are vectors and not vector fields). An ...
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1answer
60 views

How creation operator pops out while expanding field operator?

While doing QFT when we try to canonically quantize the Klein Gordon equation $\Box \phi =0$ we promote the $\phi $ to an operator field and impose the commutation rule $[\phi(x,t),\pi (y,t)]=i\hbar\...
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1answer
45 views

The conmutator of the square of Pauli-Lubanski vector and the generators of Poincare group

I'm working on trying to solve the following problem: Using the following expressión for the square of Pauli-Lubanski vector:$$W^2=-\frac{1}{2}M_{\mu\nu}M^{\mu\nu}P_{\alpha}P^{\alpha}+M^{\mu\nu}M_{\...
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2answers
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One question about BRST symmetry in reading Srednicki’s book: Why should the BRST charge $Q_B$ be nilpotent?

In p.453, Srednicki claims that since the BRST transformation of a BRST transformation is zero, $Q_B$, the BRST charge, must be nilpotent: $$Q_{B}^{2}=0.\tag{74.32}$$ I don't know why.
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2answers
59 views

Commutation with unspecified potential function

Instead of a potential given like $V(r) = k r^2$ or $V(r) = y^2$ , if the potential is given like in the form a function but not clearly specified, can we tell that if that commutes with the ...
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1answer
96 views

Common observables and associated operators: operator momentum [duplicate]

Starting from my previous question Commutators in quantum mechanics and considering that the commutator $$\left[i\hbar\frac{\partial}{\partial x},x\right]=i\hbar, \tag{1}$$ the associated linear ...
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1answer
141 views

If Poisson Bracket of Momentum and Position is non-zero, why no Uncertainty Principle?

In Hamiltonian classical mechanics, we have that the Poisson bracket of position and momentum satisfies $$\{q_i, p_j\} = \delta_{ij}$$ But this implies that momentum and position 'generate' changes ...
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3answers
252 views

The use of the commutators in quantum mechanics: explanations [duplicate]

Considering that I've never studied quantum mechanics before I have need to understand the operator commutator. My start is: $[A,B]=AB-BA \tag{a}$ Now, why must be $$\left[\frac{\partial }{\partial ...
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2answers
102 views

Quantum Harmonic Oscillator Raising and Lowering operators

The commutator of the operators, $[a,a^\dagger] = 1$ is useful in rewriting the Hamiltonian in a neat way in terms of the creation and annihilation operators. So my question is, Is there a physical ...
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1answer
67 views

Number operator - annihilation operator commutation

Is there a rigorous way to prove that $$ (N+1)^{-1/2} a = a N^{-1/2} $$ where $a$ is a bosonic annihilation operator and $N=a^\dagger a$ is the corresponding number operator?
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1answer
58 views

What is the implication of overlap between eigenstates of two operators in Quantum Mechanics?

For instance, what does it mean that a certain position eigenstate is also an energy eigenstate? I understand that measurable (Observables) in Quantum mechanics are the operators. Their eigenvalues ...
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44 views

Covariant derivative with respect to commutator

I have some confusion with the notion of $\nabla_{[A, B]}\bf{v}$, that expression, with a commutator of vector fields as the subindex of the connection appears for instance in the definition of the ...
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1answer
119 views

Quantum mechanics, Fourier transformation

Why do we use $p=-i\hbar\frac{\partial}{\partial x}$ in quantum physics? (I know the reason for $i\hbar$, quantization). Is this right to say we can't measure velocity and position of electrons at the ...
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24 views

Explicit form of Klein factors in Giamarchi

In Giamarchi, Quantum Physics in One Dimension, Appendix B, I don't understand how he did his last step in equation B.8, as shown below. If anyone has gone over the derivation, I would really ...
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2answers
120 views

How does one obtain $\hbar$ as $\frac{h}{2\pi}$?

I'm reading Dirac's Principles of Quantum Mechanics. He defines $\hbar$ to be the real number satisfying the following relation $$ uv - vu = i\hbar[u,v]$$ where $u$ and $v$ are dynamical variables, ...
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1answer
62 views

Why does equal commutator relation imply equal operator?

In 1d bosonization, Giamarchi (Quantum Physics in One Dimension) Chap 2, shows that fermionic Hamiltonian $$H_f=\sum_k k(R_k^* R_k -L_k L_k)$$ is equal to the bosonic representation $$H_b = \sum_k |...
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1answer
41 views

Commutation relations in Gupta-Bleuler quantization

Quantization of the free electro-magnetic field has essential differences in comparison to quantization of say scalar or massive vector fields. In fact there are different approches to it. One of ...
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0answers
79 views

Why is the Schrödinger field an annihilation operator?

The relativistic scalar field operator is not a ladder operator. Its commutation relations are $$\begin{align} \left[\hat{\phi}\left(\vec{x}\right), \hat{\phi}\left(\vec{y}\right)\right] = \...
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1answer
79 views

Baker-Hausdorff for normal ordering exponential

Let $A=A^+ +A^-$ where $A^+,A^-$ denote the creation and annihilation portion of the field. Then in Eduardo Fradkin, Field Theories of Condensed Matter Physics, equation (5.284), it states that $$ :e^...
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1answer
83 views

Show that when angular momentum $L_x$ and $L_y$ commute with operator $G$, then $L_z$ also commutes with $G$

I want to prove that if Angular momentum $L_x$ and $L_y$ commute with an operator $G$, angular momentum $L_z$ also commutes with $G$. if $[L_x , G] = [L_y, G] = 0$ then $[L_z , G] = 0$ I know that $...
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0answers
27 views

Calculating the Commutation Relation of the Generators of $SO(n)$ [duplicate]

I'm working through problems in the book Einstein Gravity in a Nutshell by Zee, and I'm stuck on one of the harder problems. The problem is Calculate $[J_{(mn)}, J_{(pq)}]$. We are given that $[J_{(...
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1answer
59 views

Ehrenfest theorem and correlation among observables at the quantum scale

I am studying quantum mechanics and I encountered the famous Ehrenfest Theorem, which states that given an observable $A$, its expectation value time evolution is governed by $\partial_t\langle A\...
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2answers
56 views

How to differentiate exponentials of operators?

Suppose we have $$e^{At}e^{Bt}=F(t),$$ where $$A, B$$ - operators that do not commute. Now I need to take the derivative $$dF(t)/dt.$$ In which order do I write the operators? $$dF(t)/dt = Ae^{At}e^{...
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101 views

Virasoro algebra commutation (part 2)

This was a sub-question in my previous post that I ask separately now. In Introduction to Conformal Field Theory by Blumenhagen and Plauschinn (springer link) the Virasoro algebra is introduced the ...
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1answer
107 views

Virasoro Algebra commutation

In Introduction to Conformal Field Theory by Blumenhagen and Plauschinn (springer link) the Virasoro algebra is introduced the central extension of the Witt algebra. They give the central extension $$\...
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1answer
50 views

Uncertainity relation of Kinetic energy with position

In R. Shankar's Principle of quantum mechanics book in the problem Now $$\Delta T = \frac{-\hbar^2}{2m} \Delta( p^2)$$ And I don't arrive anywhere using this, but I also know that $\Delta A \...
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65 views

Commutation relation Lorentz Algebra

Related question, which I don't understand either. I think is easier to get the Lorentz group algebra as defined by Maggiore, $$ [J^{\mu\nu},J^{\rho\sigma}] = i(\eta^{\nu\rho}J^{\mu\sigma} - \eta^{\...
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2answers
44 views

Vanishing correlation function

Mirror Symmetry p. 206, Eq. 10.192. I have an operator $\mathcal{O}$ that commutes with my supercharge $\overline{Q}_+ $, $\left[\overline{Q}_+, \mathcal{O} \right]=0$. Why does the correlation ...
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1answer
44 views

Commutation of alpha dirac matrix

I want to calculate the commutation of $[\hat{x},\vec{\alpha}\;\vec{p}]$. This boils down to $$[\hat{x},\vec{\alpha}\;\vec{p}] = i\hbar\hat{\alpha_x}+\left[\hat{x},\hat{\alpha_x}\right]\hat{p_x} +\...
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1answer
67 views

Generators of conformal transformations change of basis

I recently started going through Introduction to Conformal Field Theory by Blumenhagen and Plauschinn ( springer link ). On page 11, they glue together the generators of conformal transformations as ...
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0answers
30 views

Normal ordering of operators: Commutator or no commutator [duplicate]

My exercise for a quantum optics course tells me to "find a normal ordering" for an Operator $\hat{O}(\hat{a},\hat{a}^\dagger)$, which is given as a (rather complicated) string of $\hat{a}$'s and $\...
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2answers
72 views

Heisenberg Picture from $\left[f(A), B\right] = \left[A, B \right]\frac{\partial f}{\partial A}$

I have a question about the equation below: $$\left[f(A), B\right] = \left[A, B \right]\frac{\partial f}{\partial A}$$ Is this equation valid in the Schrödinger picture, Heisenberg picture, or in ...
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1answer
103 views

Commutation relations, complex scalar field

Why for the complex scalar field $$ \hat\phi = \int \frac{d^3p}{(2\pi)^{3/2}(2E_{\vec{p}})^{1/2}}\left(\hat{a}_{\vec{p}}e^{-p \cdot x} + \hat{b}_{\vec{p}}^\dagger e^{p \cdot x}\right), $$ the ...
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1answer
56 views

Question about commutators acting on wavefunctions

Consider a commutator acting on a 1D wavefunction: $$[\frac{\hbar}{i} \frac{d}{dx},x]\psi(x)=(\frac{\hbar}{i} \frac{d}{dx}x-x\frac{\hbar}{i} \frac{d}{dx})\psi(x).$$ Now does this mean $\frac{\hbar}{...
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3answers
460 views

Why symmetry transformations have to commute with Hamiltonian?

Let us consider a unitary or antiunitary operator $\hat{U}$, that associates with each quantum state $| \psi \rangle$ another state $\hat{U} | \psi \rangle$. I have read that to $\hat{U}$ be a ...
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2answers
42 views

General commutation question

If I have three general observables, $\hat{C}$, $\hat{H}$, and $\hat{L}$, and the commutation relation between $\hat{C}$ and $\hat{H}$ is given by, $$ [\hat{C}, \hat{H}] = \hbar \hat{L} $$ At the ...
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1answer
70 views

QFT Complex scalar field and commutators

The conserved charge is $$Q=i\int\ d^3x(\phi\pi-\phi^\dagger\pi^\dagger)$$ Expressing this in terms of creation annihilation operators gives $$Q=i\int d^3 x \frac{d^3p d^3k}{(2\pi)^3(2\pi)^3}\frac{i\...
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1answer
84 views

How to efficiently compute the commutator $[\hat{r},\nabla^2]$?

Given a system with Hamiltonian $ \hat{H} = \frac {\hat{p} ^2}{2m} + \hat{V}(r)$ in a certain state $|\psi \rangle$, I want to find if $\langle r \rangle$ varies with time. From $$ i \hbar\frac {d ...