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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Derivation of the Hypersurface Deformation Algebra

Let $({M},{g})$ be a smooth $4d$ spacetime manifold with lorentzian metric $g$ and local coordinates $\xi^{\alpha}$ and let further $({N},{q})$ be a smooth $3d$ manifold with metric $q$ and local ...
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Causality, branch cut choice and analytic continuation of Euclidean 2-pt. correlator in 2D CFT

In 2D CFT, the Euclidean two point correlator of a primary operator $\mathcal{O}$ with conformal weights $h$, $\bar{h}$ is given by $$ \begin{align} \langle\mathcal{O}(z,\bar{z})\mathcal{O}(0,0)\...
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Geometric description of canonical commutation relation

Does there exist any geometrical description of canonical commutation relation (CCR) of quantum mechanics $$[\hat{x},\hat{p}] = i \hbar \, ,$$ e.g. in phase space? The commutator, along with the ...
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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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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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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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A question about commutation relation and functional derivatives

In wikipedia https://en.wikipedia.org/wiki/Canonical_commutation_relation. In quantum mechanics the Hamiltonian ${\hat {H}}$, (generalized) coordinate $ {\hat {Q}}$ and (generalized) momentum ${\hat {...
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Defining particles by their commutation/anti-commutation relations

In my studies of many-body physics, I have encountered three types of particles, which can be defined based on their commutation/anti-commutation relations. Fermions, defined by raising/lowering ...
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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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Generalized momentum in terms of wavefunction: Is it always $-i\hbar \partial/\partial q$?

I saw this kind of derivation several times in different notes/review/educational articles. (For example https://arxiv.org/abs/1904.06560 or http://wcchew.ece.illinois.edu/chew/course/QMALL20121005....
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Calculation of commutation relations in the SYK model

I'm reading this paper (https://arxiv.org/abs/1604.07818). And I'm having trouble showing an equality. We consider the following $SL(2,R)$ generators. \begin{align} D=-t\partial_t-\frac{1}{4},\ P=\...
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Representations of the Poincaré 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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Background on the Stone-von Neumann theorem

I'm actually a mathematician. I'm required to give a lecture on the Stone-von Neumann theorem. I already have all the mathematical details figured out, but I wish to make the lecture more interesting ...
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Raising and Lowering Operators of a Hamiltonian

Lets say that I have a Hermitian Hamiltonian $H$ with a non-Hermitian raising operator operator $A$ which satisfies \begin{equation} [H,A] = \Omega A, \quad \Omega \in \ \mathbb{R}_{>0}. \end{...
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Why is the anticommutator of the uncertainty principle omitted if it serves to increase the accuracy of our "knowledge" of a quantum state?

The generalized uncertainty principle can be derived and shown to be this which is fine and rigorous. $\langle ( \Delta A )^{2} \rangle \langle ( \Delta B )^{2} \rangle \geq \dfrac{1}{4} \vert \langle ...
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Topological Descent Equation

Assume that we have a cohomological field theory, with an odd symmetry generated by an odd operator $Q$ and an exact energy momentum tensor $T_{\mu\nu}=[Q,G_{\mu\nu}]$. Then by integrating over an ...
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What can be said about the commutator of an operator with itself at different times?

In general for some smooth and bounded $\hat{V}$ $$ \left[\hat{V}(t_1), \hat{V}(t_2) \right] \neq 0 \text{ if } t_1 \neq t_2 $$ But what more can be said about commutators of this type? I am ...
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Can current and voltage be linked by an uncertainty relation when electrons tunnel through a barrier?

Quantum tunneling has been shown to be linked to uncertainty relations for some observables involved in the system. For instance, if we consider electrons tunneling through a potential barrier it can ...
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Commutator of net position and net momentum

It is well known that $$[\hat{x},\hat{p_x}] = [\hat{y}, \hat{p_y}] = [\hat{z}, \hat{p_z}] = i\hbar$$ But what if instead we wanted to know the commutator of the net displacement $\hat{r} = \sqrt{\...
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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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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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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 of currents in QED

In an outline of a proof of the Ward identities in QED, the authors Green, Schwarz, and Witten in their book "Superstring theory", vol. I, Section 1.5.1, claim that in the QED the ...
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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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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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How does one arrive at the relation of commutator $\left[M^{-i}, M^{-j}\right]$ of Lorentz generators $M^i$ in terms of the string modes $\alpha_n^i$?

I am reading the book "String theory demystified" by David McMahon. On page 149, the author discusses the "critical dimension" for superstrings. the number of spacetime dimensions ...
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Path integral - operator formalism and continuum limit

Correlator of the position and momentum operators in quantum mechanics $$\langle x_f, t_f|[\hat{x}(t), \hat{p}(t)]| x_i, t_i \rangle = i\hbar \langle x_f, t_f| x_i, t_i \rangle$$ since $[\hat{x}(t), \...
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Commutation relations in quantised Yang-Mills

Consider Yang-Mills theory with gauge group $G$. Let $\{T^a\}$ be a basis for the Lie algebra $\mathfrak{g}$, so that the connection coefficients can be written as $A_\mu = A_\mu^aT^a$. In the ...
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Commutation relations (QFT curved spacetime Wald)

So I'm currently reading Wald's QFT on curved spacetime. In section 2.3, he constructs the quantum theory of $n$ decoupled harmonic oscillators by choosing $\mathcal{H}$ as the set of positive ...
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Commutator of scale transformation generator in conformal quantum mechanics

In the following notes on CFT by Joshua D. Qualls. We're introduced to conformal quantum mechanics with lagrangian:$$L=\frac{1}{2}\dot{Q}^2-\frac{g}{2Q^2}\tag{1.11}$$ It's action is invariant under $...
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Why are commutators the first choice in describing observables that cannot be measured simultaneously?

In quantum mechanics, we convert Poisson brackets to commutators for the observables to account for the uncertainty principle. However, I do not understand why do we do this. What motivates us to ...
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Feynman Propagator

I have learned two ways of representing Feynman Propagator, which are as follows: The first method is to represent it in the form of commutation relations: $$[a^{-}(x), a^{+}(y)] = iG(x-y)$$ Where $a^{...
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Does microcausality plus the time-slice property imply local primitive causality?

In quantum field theory, observables are associated with regions of spacetime. One of the basic principles of relativistic quantum field theory is microcausality, which says that observables ...
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I'm trying to prove that $ [\hat L^2, \hat V(\hat r ,\hat p )] = -2i\hbar (i\hbar \hat V - \hat V \times \hat L) $

First I prove that: $$[\hat L, \hat H] = 0$$ with $$\hat H = \frac{\hat{p}^2}{2m}+\hat V(\hat r)$$ Then, with $\hat V(\hat r,\hat p)$ as a vector: $$ [\hat L^2, \hat V(\hat r ,\hat p )] = -2i\hbar (i\...
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Use the commutation relation to show that the conjugate momentum acts on eigenstates of $\hat{\Phi}$ as $ - i \delta / \delta\phi_a(\mathbf{x})$

This is part (b) of Schwartz's Problem 14.3 in his Quantum Field Theory and the Standard Model textbook. Suppose that we have a real scalar field operator $\hat{\Phi}(x^0,\mathbf{x})$ with conjugate ...
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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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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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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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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^{\mu\nu} -...
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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 inverse d'Alembertian operator

Is there a commutation relation for the inverse d'Alembertian operator in general relativity? i.e. if we define $\Box = g^{\mu\nu}\nabla_\mu\nabla_\nu$ and $\Box \Box^{-1}X_{\alpha_1,\alpha_2...}=X_{\...
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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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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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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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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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