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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Question about canonical quantization of the open string ghost system

In section 3.1.3 of Green, Schwarz and Witten book on superstrings, it is stated that the canonical anti commutation relations for the fermionic ghosts are $$ \{ b_{++}(\sigma, \tau), c^+(\sigma', \...
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Deriving the angular momentum operator using passive transformations [closed]

So I'm following the solutions here. For exercise 12.2.2, I'm not quite sure why the commutator relations aren't $$[X,L_z]=i\hbar\frac{\partial L_z}{\partial p_x}-L_zi\hbar\frac{\partial}{\partial p_x}...
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BCH formula on $e^{\alpha a^\dagger−\alpha^*a}$ [closed]

My actual question is to evolve $|0,α⟩$ using $a†b+b†a$.This is how I did my calculation: Evolved state is $e{^{−iHt}}|0,α⟩$. Then I used displacement operator and then inserted $UU^\dagger$ where U ...
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Invariance of commutator relations under change of basis

Consider the following: Let for operators $\hat A$ and $\hat B$ the following commutation relation holds: $$[\hat A,\hat B]=\hat C \tag{1}$$ and now we know that this relation holds, $$[\hat A',\hat B'...
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Can incompatible observables share an eigenvector?

I was recently introduced to the concept of compatible and incompatible observables and specifically the generalized uncertainty principle, which is written in my textbook as: $$ \sigma_A^2\sigma_B^2 \...
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Uniqueness of Fock space

Given a single-particle Hilbert space, it's not hard to construct a Fock space using tensor products and symmetrization/anti-symmetrization projection operators, from which we can define creation/...
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How to show that if two operators $A$ and $B$ commute, then simultaneous accurate measurement is possible?

I have proved that if two operators commute then their simultaneous accurate measurement is possible using the uncertainty equation but I am unable to do so without using it. I have tried and reached ...
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The Different Propagators for the KG Field

I am currently working through the first few chapters of Peskin and Schroeder and have arrived at the sections where the different propagators are discussed. In regular Quantum Mechanics, the ...
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Complete set of common eigenfunctions of two commuting operators (reverse theorem) [closed]

Problem 3.16 of introduction to quantum mechanics by David J. Griffiths states that \begin{equation} [\hat{P},\hat{Q}] \neq 0 \implies \hat{P} \text{ and } \hat{Q} \text{ do not have a complete set of ...
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Does the total orbital angular momentum $L^{2}$ commute with the atomic Hamiltonian?

I would like to know if the total orbital angular momentum $\mathbf{L}^{2}$ commutes with the Hamiltonian of a many-electron atom when we consider the interactions between the electrons. $$H = -\frac{...
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How should I imagine a spinor commutator, or consecutive occurences of $\bar{\Psi}$ and $\Psi$ in general?

I'm having a hard time making sense of an expression like $$\left[\Psi(x), \bar{\Psi}(y)\right].$$ Up until now I imagined a spinor operator to be something like a column vector of operators, ...
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Commutator relations for Propagators

From Shankar, $$[P,H]=0\rightarrow [P,U(t)]=0$$ where $P$ is the momentum operator, $H$ is the Hamiltonian, and $U(t)$ is the propagator to the Hamiltonian. My first question is why does this follow? ...
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Coefficient of an 1-form in position-representation of momentum operator where configuration space is NOT $\mathbb{R}^m$

I found this in the book Geometric Phase in Quantum Systems by A. Bohm et al. Where the position space representation of the momentum operator carries a (Where exactly my doubt is) coefficient of 1-...
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Looking for differential operators satisfying a specific commutation relation with the Laplace operator

Consider the Laplace operator on some manifold, $\Delta=(\det g)^{-1/2} \frac{\partial}{\partial x^j} \left( (\det g)^{1/2}g^{jk} \frac{\partial}{\partial x^k}\right) $. I am looking for differential ...
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Taking the trace of the square of a commutator involving Lie-algebra valued scalar fields

I was calculating the following: $$ \operatorname{Tr}[\phi,\psi]^2 $$ where $\phi$ and $\psi$ transform in the adjoint of $SU(N)$. So I expanded the above as $$ \operatorname{Tr}[T^a\phi^a,T^b\psi^b][...
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Commutator of Dirac field operators

If we let field operators $$\psi(x)=\int\frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_p}}e^{ip\cdot x}\sum_s(a^s_pu^s(p)+b^{s}_{-p}v^s(-p)).$$ Then the commutator of field operators will be $$[\psi,\psi^{\...
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Higher order expansion term of Baker–Campbell–Hausdorff (BCH) formula [migrated]

I want to calculate unto 20th order, the expansion of Baker–Campbell–Hausdorff formula. In Wikipedia up to 4th order expansion is available. Is it possible to get the expansion term up to 20th order ...
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Trace of commutators with flavor indices

I want to explicitly write out the Lagrangian term $$\operatorname{Tr}\bigg( \sum_{I\neq J}[\phi^I,\phi^J]^2\bigg) ,$$ where $I,J$ are flavor indices and $\phi$ is a scalar field. Why doesn't this ...
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Generalizing the Uncertainty Principle for acceleration, jerk, etc

The Uncertainty Principle states that the more you know about a particles position, the less you know about its velocity and vice-versa, and there is an equality $\sigma_x\sigma_p \geq \frac{h}{4\pi}$ ...
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Sundarshan's proof of the spin-statistics theorem

The proof starts with the $\rm SU(2)$ symmetric Lagrangian \begin{equation} \mathcal L=\sum_{r,s=1}^4\frac{1}{2}K_{rs}^0(\xi_r\dot\xi_s-\dot\xi_r\xi_s) \end{equation} where the fields $\xi$ either ...
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Creation/annihilation operators relation in equation 2.46 of Peskin and Schoroeder

We can get the following relations from the creation/annihilation operators: $$ H^n a_p = a_p (H - E_p)^n, $$ and $$ H^n a_p^{\dagger} = a_p^{\dagger} (H + E_p)^n. $$ How do we get that $$ e^{iHt} a_p ...
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Doubt about commutator of operators

I have a doubt regarding commutator algebra. I've seen this expression $$ [A,B^n] = nB^{n-1} [A,B]\tag{1}$$ and have used this often for position and momentum operators. However, I want to know when ...
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A Mathematical Formulae for Peskin and Schroeder's Exercise 2.2 (a)

I am self studying Quantum Field Theory and I am using the book An Introduction to Quantum Field Theory by Peskin and Schroeder. Currently I am working on problem 2.2 (a). In the textbook problem, the ...
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Issue finding the time dependence of the conjugate momentum of the Klein-Gordon field

I am following Peskin and Schroeder's book, trying to evaluate the following commutator: $$ i \partial_t \pi(x,t) = \left[\pi(x,t),\int \frac{d^3 x'}{2} (\pi^2(x',t)+(\nabla \phi(x',t))^2 + m^2 \phi^2 ...
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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)}\...
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Creation and annihilation operators in 2D Dilaton Gravity matter fields

In these notes by Strominger sec 3.6 we are given the creation and annihilation operators $$ a_w = -\frac{i}{2\pi}\int\frac{dz^-}{\sqrt{2w}}f(z^-)\overleftrightarrow{\partial}_-e^{iwz^-} $$ $$ {a_w}^\...
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Why convention for Heisenberg algebra?

In C. A. Keller, “Introduction to Vertex Operator Algebras,” 2017. the Heisenberg Lie algebra is presented as a a Lie algebra spanned by vectors $\alpha_n$, with $n\in\mathbb{Z}$ and $k$, where the ...
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(Anti-)commutation relations of fermions in GUT [duplicate]

Suppose we found a great unification group (like SU(5)) and wrote a Lagrangian with all the known particles/fields (quarks, electrons, neutrinos...) with such symmetry. How would commutation rules be ...
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Hadamard formula in quantum mechanics

When studying symmetries in quantum mechanics, one often has to calculate $UBU^\dagger$ where $B$ is a self-adjoint operator and $U$ is a unitary operator. More often than not $U$ has an exponential ...
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Commutator of an operator with its Hermitian adjoint in linear quantum systems

I have a commutator of a single-mode photon field operator $\alpha$ with its Hermitian adjoint $\alpha^{\dagger}$. [$\alpha$, $\alpha^{\dagger}$] When this commutator gives a negative value, $\alpha$ ...
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Representation of The Poincare Group

I am currently trying to understand the representations of the conformal group. I am following the script by J.D Qualls. At page 29, the author finds the effect of $L_{\mu\nu}$ by "studying the ...
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The Cartan sub algebra and Killing form of the Poincaré algebra

Doing some studies on Group theory, I worked Frederic Schuller's lectures on youtube where he classifies all semisimple Lie algebras by the Dynkin's diagrams; I should say it was interesting. Trying ...
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How to bound the expectation value of a commutator?

Are there formal ways to bound the following quantity: $$\langle[[{S_x},{H}],{H}]\rangle$$ The expectation value is taken on an eigenstate of $S_x$. $H$ is a dipolar Hamiltonian acting on $N$ ...
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General commutator $[f(A),g(B)]$ of functions

Let us have two hermitian operators $A,B$ and their commutator $[A,B]:=AB−BA$, then let us have two functions $f,g$ and we want to to calculate $[f(A),g(B)]$ (everything is still hermitian). I have a ...
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Lorentz boost transformations form a group?

In the QFT book of Ryder, he states that Lorentz boost transformations do NOT form a group. This is due to the boost generators $\textbf{K}$, i.e. they do not form a closed algebra under commutation. ...
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Commutation relation of the second quantisation of an operator

Let $h$ be a self-adjoint operator on a Hilbert space $\mathfrak{h}$ and $H_N:=\sum_{i=1}^N h_i$ defined on $H(\mathfrak{h})^{\otimes N}\subset \mathfrak{h}^{\otimes N}$. Let $\overline{\text{d}\Gamma(...
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The $3×3$ representation of weak $SU(2)$

I'm reading chapter 11.2 of the Cheng and Li textbook 'Gauge theory of elementary particle physics'. It says that $T_+$, $T_-$ and $Q$ do not form a closed algebra. In order to fix this problem the ...
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Commutator involving the exponential of the integral of an operator

I have quite a problem handling the following commutator involving the exponential of the integral of an operator $$\Bigg[\hat{A},\exp\!\Bigg(\int_0^td\tau\,\hat{B}(\tau)\Bigg)\Bigg]$$ especially as ...
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If two operators commute, then is it any basis of one of them also a basis of the other? [duplicate]

According to the compatibility theorem, there exists a common eigenbasis for the eigenfunctions of two operators $\hat A$ and $\hat B$ representing the observables $A$ and $B$ if and only if they ...
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Questions on field operator in QFT and interpretations

For a real scalar field, I have the following expression for the field operator in momemtum space. $$\tilde{\phi}(t,\vec{k})=\frac{1}{\sqrt{2\omega}}\left(a_{\vec{k}}e^{-i\omega t}+a^{\dagger}_{-\vec{...
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Biconditionality of the compatibility theorem for commuting operators

The compatibility theorem states that, if the operators $\hat A$ and $\hat B$ representing the observables $A$ and $B$ do commute, then there exists a common eigenbasis for their eigenstates, and ...
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How to compute the unitary transformation with the operator on exponential function?

Consider an unitary transformation \begin{equation} \hat{D}(f) = e^{-\frac{i}{2\hbar}f(t)\hat{q}^2} \end{equation} from the book I find that $\hat{D}\hat{p}\hat{D}^{\dagger} = \hat{p} + f(t)\hat{q}$, ...
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Question on discrete commutation relation in QFT

Given the commutation relation $$\left[\phi\left(t,\vec{x}\right),\pi\left(t,\vec{x}'\right)\right]=i\delta^{n-1}\left(\vec{x}-\vec{x}'\right)$$ and define the Fourier transform as $$\tilde{\phi}(t,\...
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On commutation relations for real scalar free fields in QFT

Suppose the creation and annihilation operators are as follow: $$ a_\mathbf{p} = \frac{1}{2} \Bigg(\sqrt{2\omega_\mathbf{p}} \tilde{\phi}(\mathbf{p}) + i\sqrt{\frac{2}{\omega_\mathbf{p}}} \tilde{\pi}(\...
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On commutation relations in QFT

I am self learning “Quantum fields in curved space” and I am stuck at trying to show the commutation relations for the free real scalar field. I know that the usual way to show is to invoked the ...
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What is my misunderstanding in Wick's theorem?

Trying to understand Wick's theorem, I took most of my knowledge from the corresponding Wikipedia article. The statement is that given the definition of normal ordering of operators $A,B,C,\ldots$ any ...
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Commutators of covariant derivative and Yang-Mills Field Strength in curved spacetime

I am stuck with YM Field Strength and commutator. For example, in flat spacetime we have the commutator $$F_{\mu \nu}=[\partial_{\mu}+A_{\mu},\partial_{\nu}+A_{\nu}] .$$ But what is the thing in ...
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Is it possible to write the Coulomb Hamiltonian in terms of creation and annihilation operators similar to harmonic oscillator? [duplicate]

Is it possible to write the Coulomb Hamiltonian in terms of creation and annihilation operators similar to the harmonic oscillator so that we can have the same commutations relations similar to that ...
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42 views

How to perform commutation of $[A+B, C+D]$? [closed]

I want to know a logical method of performing a commutation for [$a\hat{A} + b\hat{B}, c\hat{C} + d\hat{D}$] where $a$, $b$, $c$ and $d$ are just constants. I know the rules for [$\hat{A} + \hat{B}, \...
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Do partial derivation respect to velocity and total derivation respect to time commute? [duplicate]

Imagine we have a function of position $x^i$ and velocity $v^i$ $f(x,v)$. Position and velocity are both functions of time $t$. If the function doesn't depend explicitely on time, then we have the ...

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