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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Eigenstate of commutator of displacement and sqeezing operator [closed]

Is there a state such that, $$S(ξ)D(α)|Ψ⟩ = A|Ψ⟩$$ and simultaneously $$D(α)S(ξ)|Ψ⟩ = B|Ψ⟩$$ where, $D(\alpha) = e^{\alpha a^\dagger - \alpha^*a}$ and $S(\xi) = e^{\xi^*a^2 - \xi a\dagger^2}$
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Relation between Schrödinger's equation and commutator of position and momentum [closed]

Schrödinger's equation is $$\hat H\vert\psi\rangle=i\hbar\ \partial_t \vert\psi\rangle$$ I was trying to motivate this equation in a very hand-wavy way and tried to get to $$\hat H\vert\psi\rangle=K\...
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Commutator of displacement operator and squeezing operator [closed]

I would like to know how to solve the commutator of squeezing operator S(ξ) and displacement operator D(α). where, $D(\alpha) = e^{\alpha a^\dagger - \alpha^*a}$ and $S(\xi) = e^{\xi^*a^2 - \xi a\...
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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 ...
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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 ...
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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 \...
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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 ...
Daigaku no Baku's user avatar
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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^{(...
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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 ...
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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\...
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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 ...
a Fish in Dirac Sea's user avatar
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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 ...
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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} \...
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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}{(...
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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 ...
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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 ...
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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 ...
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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 ...
Jose Javier Garcia Morata's user avatar
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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 ...
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Lorentz invariance (LI) of time ordering operation

At Srednicki after eq. (4.10), we have a discussion about that the time ordering operation. Have to be frame inv. I.e it has to be LI. He wrote that for timelike separation we don't have to worry ...
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Why does we quantize fields $\phi(t,x)$ and not $\phi$?

In classical mechanics, the action of a theory is determined by its Lagrangian: $$S(q) := \int L(q(t),\dot{q}(t),t)dt $$ In the following, let us assume that $L$ does not depend explicitly on time. ...
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Quantization of an Interacting Field Theory

The procedure to quantize free field theories is imposing a commutation/anticommutation relation with the field and its conjugate momentum, as $$\mathcal L = i\bar\psi\gamma^\mu\partial_\mu\psi\...
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Questions about computing the commutator of the Lorentz generator

I am computing the commutator of the Lorentz generators, from the Eqn (3.16) to Eqn (3.17) in Peskin & Schroeder. $$ \begin{aligned} J^{\mu\nu} &= i(x^\mu \partial^\nu - x^\nu \partial^\mu ) &...
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How should I calculate the commutator between the Belinfante stress tensor and the field operator?

As known, there is an ambiguity on the definition of the stress tensor (or energy-momentum tensor). The canonical stress tensor, defined as the Noether's current corresponding to space-time ...
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$\hbar$ in spin and Schrodinger Equation

The reduced Planck constant $\hbar$ occurs in multiple places in physics. For example, the spin operator of a spin-$\frac12$ particle is given by $$\bf{J}=\frac{\hbar}{2}\bf{\sigma}$$ where $\bf{\...
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Why does $[Q,P]=i\hbar$ work for fermion? Shouldn't fermion satisfy anticommuting relation?

For hydrogen, we use $[Q,P]=i\hbar$ for electron, which is a fermion. Does it have a deeper reason such as that we're really considering the proton + electron system, which might be of bosonic nature?
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Commutator of conjugate momentum and field for complex field QFT

In Peskin & Schroeder's Introduction to QFT problem 2.2a), we are asked to find the equations of motion of the complex scalar field starting from the Lagrangian density. I want to show that: $$i\...
Nick Heumann's user avatar
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Does $\alpha$ and $p$ commute where $\alpha$ is a Dirac matrix and $p$ is a momentum operator?

Can I write $\langle\psi\vert\alpha.p\vert\psi\rangle$ as $\langle\psi\vert p.\alpha\vert\psi\rangle$ ?
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Proof of Spin commutation relation for Holstein-Primakoff-Transformation

I have run into an issue while trying to prove the Holstein-Primakoff commutations \begin{align*} [S^+_i,S^-_j]=2 \delta_{ij} S^z_i, \ [S^z_i, S^-_j]=-\delta_{ij} S^-_i \end{align*} where \begin{...
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Do functional derivatives commute in a specific example?

Edit: this question is related to other already asked questions, like Symmetry of second functional derivatives , but a clear and definitive answer has never been given, and I am here giving a ...
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Why is commutation bracket used instead of anti-commutation bracket on page 61 of Peskin QFT?

Peskin&Schroeder was performing a trick where they used $$J_za^{s\dagger}_0|0\rangle=[J_z,a^{s\dagger}_0]|0\rangle\tag{p.61}$$ and claimed that the only non-zero term in this commutator would be ...
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Action of Conjugate momentum $\hat{\pi}$ on $\hat{\phi}$ eigenstate [duplicate]

So I am trying to solve ex. 14.3 in Schwartz textbook "Quantum Field Theory and the Standard Model" and in the second requirement, he wanted me to show that the action of the conjugate ...
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Help with commutator algebra with fermionic operators

I am struggling to understand how the following is true for the fermionic creation/annihilation operators $a^\dagger, a$: $$[a^\dagger a, a]=-a$$ If someone could walk me through the math derivation ...
photonica's user avatar
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2 answers
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CFT Radial Quantization Raising and Lowering Operators Sign Question [closed]

Following Slava Rychkov (Page 41, Or here on arxiv page 39), I am trying to show that the momentum operator raises the scaling dimension. I've seen the other related questions on this matter by Y. ...
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Canonical transformations in Quantum Mechanics

Heisenberg's famous commutator of a pair of conjugated canonical variables is formulated for position and conjugated momentum $$[q,p] = i\hbar.$$ Intuitively I would guess that it would also work ...
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How does Dirac prove that the canonical coordinates $q_r$ form a complete commuting set of observables?

I'm reading "The principles of quantum mechanics" by Paul Dirac, and I've reached the point where he introduces the momentum operator. After showing that the $p_r$'s commute with each other ...
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If an operator $A$ commutes with the Hamiltonian $H$ do they have common eigenstates?

In this book Intermediate quantum mechanics by Hans A. Bethe and Roman Jackiw they write at page 103: In the case of spin, the Hamiltonian commutes with the spin operator of each electron $\...
amilton moreira's user avatar
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Exponential operator approximation: Suzuki-Trotter Expansion

One cannot solve the transition amplitude $\langle{x}\vert e^{-iHt}\vert{y}\rangle{}$ with $H=H_0+V$ by just applying the operators one after another on the bra/ket, because the free hamiltonian $H_0$ ...
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Are partition functions invariant under Bogoliubov transformations?

Consider a Hamiltonian $H(a_i, a^{\dagger}_i)$ as a function of some ladder operators $a_i, a^{\dagger}_i$. Now, consider a partition function $H(a'_i, a'^{\dagger}_i)$ where $a', a'^{\dagger}$ are ...
Dr. user44690's user avatar
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2 answers
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Uncertainty on the sum of two non-commuting operators

Suppose that I have an observable $$ \hat{E} = \sin(\alpha) \hat{Q} + \cos({\alpha}) \hat{P} $$ with $\hat{Q}, \hat{P}$ being non-commuting operators satisfying $$ [\hat{Q}, \hat{P}] = i \hbar $$ It ...
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From Klein-Gordon equation to Dirac equation: a wrong "derivation" [closed]

So let us start with the Klein-Gordon equation $$\tag{KG} (-p^\mu p_\mu + m^2)\phi = 0 $$ The idea is to "factorize" the operator $-p^\mu p_\mu + m^2$. \begin{equation}\tag{1} -p^\mu p_\mu + ...
ric.san's user avatar
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2 votes
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Non-vanishing amplitude outside light cone doesn't violate causality? [duplicate]

I am following Peskin & Schroeder's QFT book. And on equation 2.51, we get an expression for the free Klein-Gordon propagator for timelike intervals $x^0-y^0=t$, $x-y=0$: $$D(x-y) \sim e^{-imt}\...
Nick Heumann's user avatar
1 vote
1 answer
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Commutation relation between pairs of conjugated variables

Suppose I have four operators $\hat{q}_1$, $\hat{p}_1$, $\hat{q}_2$ and $\hat{p}_2$ such that $[\hat{q}_1, \hat{p}_1] = [\hat{q}_2, \hat{p}_2] = i$ $[\hat{q}_1, \hat{q}_2] = [\hat{p}_1, \hat{p}_2] = [...
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Commutation relation of position and Hamiltonian operator [duplicate]

In the book Quantum Mechanics volume 2 by Cohen-Tannoudji, in Electric dipole approximation, it was written that $\left[\boldsymbol{Z}, H_0\right]=i \hbar \frac{\partial H_0}{\partial P_{\boldsymbol{z}...
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