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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Commutator as a time-ordered product

I'm reading through Seiberg and Witten's paper "String Theory and Noncommutative Geometry," and one part in $\S$2.1 isn't quite clear to me. (Sorry, in advance, for the length.) My question is about ...
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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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Do commutation relations exist between superfields?

To quantize a theory, Klein gordon field for example, commutation relations are stablished. Or anticommuting ones in the fermionic case. If I have the Wess.Zumino model or the free model: $$S~=~\int\...
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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\{\...
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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 $\langle x | p\rangle$...
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550 views

The correspondence between Poisson bracket and Commutators in Quantum Mechanics

I don't understand canonical quantization. In passing from classical to quantum, one replaces the Poisson brackets with the commutators. I don't really understand this. How can we generally show that ...
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Quantizing first class constraints

Let $\gamma$ denote a first class constraint. Then if there exists a function on phase space $f(q,p)$ for which the Poisson bracket with the constraint does not vanish $\lbrace f, \gamma\rbrace \neq 0$...
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119 views

Dyson series for Hamiltonian with $c$-number commutator

I am trying to derive the evolution operator for a time dependent Hamiltonian which satisfies the commutator $$[H(t_1), H(t_2)]=I f(t_1,t_2)$$ Where $I$ is the identity operator, and $f(t_1,t_2)$ is ...
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362 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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63 views

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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Structure constants within a conformal field theory

I am working out some commutation relations within a CFT. We know that the modes of the currents within a CFT commute as $$ [J_m^a,J_n^b] = if_{abc}J_{m+n}^c + m\delta_{ab}\delta_{m,-n}, $$ Now, we ...
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Number operator and, annihilation and creation operators

While reading Ryder chapter on quantization of Klein-Gordon field I got stuck at the following: It can be shown that, $$[a(k),a^{\dagger}(k')]=(2\pi)^32\omega_k \delta^3(\mathbf{k}-\mathbf{k'})$$ and ...
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Second quantization field derivative commutator

For the symmetrized quantum field theory Lagrangian of the free dirac field $$\mathcal{L} = i[\overline{\psi}_a,({\partial_\mu}\gamma^\mu \psi)^a] -m[\overline{\psi}_a,\psi^a ]$$ the terms are ...
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Commutator of Angular Momentum and Position

I'm trying to show that $[L_i,x_k]=i\hbar \epsilon_{ikl}x_l$. I seem to be off by a sign. Here's what I did: $[L_i,x_k]=[\epsilon_{ikl}x_kp_l,x_k]=\epsilon_{ikl}(x_k[p_l,x_k]+[x_k,x_k]p_l) = \...
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142 views

Is the vanishing commutator of observables outside the light cone only a necessary or also a sufficient condition for causality?

The equal-time commutator of observables in QFT has to vanish outside the light cone in order to ensure causality. Mathematically spoken, $[ \bar{\psi}(x)\Gamma_1\psi(x),\bar{\psi}(y)\Gamma_2\psi(y)]|...
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Non-commutativity of operators, simultaneous eigenstates and degeneracies over finite dimensional Hilbert space

Suppose, I have two operators $\hat{A}$ and $\hat{B}$ defined over a finite dimensional Hilbert space $H$. Also assume that both the operators have their own eigenstates that span $H$(existence of ...
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How does linearity of a measurement imply that the commutator of all measured observables are $c$-numbers?

I really don't understand with the linearity conditions I have where this comes from.
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107 views

commutator to entropy in an uncertainty relationship?

Question: Does there exist a commutator to entropy in an uncertainty relationship? Similar Energy and time for instance.
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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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73 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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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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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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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\...
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The products of powers of Hermitian operators

Let's say I have two operators, $\hat{x}^k$ and $\hat{p}_x^l$, where $\hat{x}$ and $\hat{p}_x$ are the ordinary position and momentum operators. It seems fairly straight forward to show that $\hat{x}^...
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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 ...
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Geometrical way to view discretization of energy in quantum mechanics. How commutation relation implies discreteness?

The relation from which discreteness in eigenvalue of the energy of bound state arises is $[x, p]=i\hbar$ followed by the rule that wavefunction should be normalizable. So my question is there a ...
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70 views

Finding a closed formula using Baker-Hausdorff formula for a unitary transformation; An endless commutator

Consider the Baker-Hausdorff formula for two operators $a_1$ and $iHt$: $$e^{iHt}a_1 e^{-iHt} =a_1+[iHt,a_1]+\frac{1}{2!}[iHt,[iHt,a_1]]+\frac{1}{3!}[iHt,[iHt,[iHt,a_1]]]+....,$$ where $[A,B]=AB-BA$. ...
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What is radial ordering?

In my String theory lecture radial ordering was introduced and I don't understand what it is. My first guess was $$R(A(z)B(w)) = A(z)B(w)\Theta(|z|-|w|) + B(w)A(z)\Theta(|w|-|z|).$$ But then we have ...
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152 views

Commutation relation in quantized electromagnetic field theory

I have a question regarding a proposed problem (Problem 4.8) in Rodney Loudon's book "The Quantum Theory of Light". Let $U(t)$ be an operator defined by $$ U(t)=\exp\left\lbrace\frac{i}{\hbar}\int\...
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172 views

Propagator Causality with commutators all the way

We know that two fields commute - by locality and causality - iff there is spacelike separation $\left[\phi_l^k(x) , \phi_m^{k'}(y)\right] = 0$ for $(x-y)^2<0$ In the canonical quantization of ...
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Simmons-Duffin CFT Lecture Notes: Exercise 3.3

Here is the exercise 3.3 in the Simmons-Duffin CFT lecture notes: Show that in $d \ge 3$, $$[Q_\epsilon,T^{\mu\nu}] = \epsilon^\rho \partial_\rho T^{\mu\nu} + (\partial^\rho \epsilon_\rho)T^{\...
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Tensor operator of rank 2

Statement of the problem: Given that a second-rank tensor operator transform as $$T'_{jk} = R_{jm}R_{kn}T_{mn}$$ where R is the three-dimensional rotation matrix, I need to find the commutation ...
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Implication of commutation relation of total momentum with a field operator

If we have a free scalar field theory, and if I expand the field $\phi$ in terms of $a_p$ and $a_p^\dagger$, and by observing how $P^i$ acts on $a_p$ and $a_p^\dagger$, I can do a calculation to show ...
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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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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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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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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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Commutator implies division of phase space in cells of area $h$?

There is a detail from a very well-written answer here that interested me. Unfortunately Springer is charging 41 Euros to access the paper cited by the author. I wonder if someone could elaborate on ...
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Domain space of compatible and incompatible operators (observables)

Sakurai (Modern Quantum Mechanics, by J.J. Sakurai) states in the section on compatible operators: Let us first consider the case of compatible observables A and B. As usual, we assume that the ket ...
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Does E commute with B (in second quantization)?

In second quantization, there's a standard procedure where we first find solutions to Maxwell's Equations. After doing so we apply quantum mechanical properties to these solutions. So for some ...
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Commutator of Dilatation operator and normal ordered scalar fields

I try to show the following relation $$ [D,:\phi^n:]=-i\left(x^{\mu} \partial_{\mu} + \frac{n}{2}\right):\phi^n: $$ where $D$ is the Dilatation operator which is given by $$ D = -~i \sum_{l,m} \...
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Product of expectation values for multimode operators

If $A_{n}, B_{n'}, C^{\dagger}_{n''}, D^{\dagger}_{n'''}$ are multimode field operators that obey the bosonic commutation relations, under which circumstances the product of expectation values $\...
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Commutation Relation between Annihilation & Creation Operators and Ascending & Descending Operators

I am currently working on a QD-Cavity system. After the point Heisenberg Equation of motion is obtained from corresponding Hamiltonian of the system, in order to find the expression for bosonic ...
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Uncertainty Principle with the corresponding operators

Why does the corresponding operator do not commute if there is uncertainty related to two observables A and B that states $\Delta A\,\Delta B > 0 $ ?
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Commutation between angular momentum and Hamiltonian

Consider the following Hamiltonian of a 3-dimensional system: $$H=\frac{p^2}{2m}+V(r)$$ If the components of the angular momentum, $L_i$, commute with $H$, then: $$[H,L_i]=0$$ This condition can ...
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When trying to see what symmetries an operator generates, how do you “decide” what coordinate to apply it to?

Suppose I have $\hat{O}_{1}=-i\hbar\partial_{x}$ then \begin{eqnarray} e^{-i\gamma\hat{O}_{1}/\hbar}x\,e^{i\gamma\hat{O}_{1}/\hbar}=x+\gamma \end{eqnarray} and \begin{eqnarray} e^{-i\gamma\hat{O}_{...
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207 views

Why is it “disconcerting” if the components of an operator do not commute?

A symmetrized operator for the operators $\hat{H}$ and $\hat{N}$ is given by $$\hat{R}=\frac{1}{2\hat{H}}\hat{N}+\hat{N}\frac{1}{2\hat{H}}.$$ When $\hat{H}$ is the Hamiltonian and $\hat{N}$ is the ...
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652 views

Simultaneous eigenket

J. J. Sakurai states in his "Modern Quantum Mechanics", this fact as a theorem ($\pi$ is the parity operator): Suppose $$[H,\pi]=0$$ and $| n>$ is a nondegenerate eigenket of $H$ with ...
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Do the norms of the total and the orbital angular momentums commute? If yes, why is there a problem with 2p_{1/2}?

Question: For $\vec L$ the orbital angular momentum of an electron, $\bar S$ its spin, and $\vec J:=\vec L+\vec S$ the sum, do $\vec J^2$ and $\vec L^2$ commute? I assume it does: $[\vec J^2,\vec L^...
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Why is commutation relations the first step in quantization?

Why is commutation relations the first step in quantization?