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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Trace of a commutator is zero - but what about the commutator of $x$ and $p$?

Operators can be cyclically interchanged inside a trace: $${\rm Tr} (AB)~=~{\rm Tr} (BA).$$ This means the trace of a commutator of any two operators is zero: $${\rm Tr} ([A,B])~=~0.$$ But what about ...
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What is the physical meaning of commutators in quantum mechanics?

This is a question I've been asked several times by students and I tend to have a hard time phrasing it in terms they can understand. This is a natural question to ask and it is not usually well ...
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Is there something behind non-commuting observables?

Consider a quantum system described by the Hilbert space $\mathcal{H}$ and consider $A,B\in \mathcal{L}(\mathcal{H},\mathcal{H})$ to be observables. If those observables do not commute there's no ...
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What is the connection between Poisson brackets and commutators?

The Poisson bracket is defined as: $$\{f,g\}_{PB} ~:=~ \sum_{i=1}^{N} \left[ \frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} - \frac{\partial f}{\partial p_{i}} \frac{\partial ...
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What is the Physical Meaning of Commutation of Two Operators?

I understand the mathematics of commutation relations and anti-commutation relations, but what does it physically mean for an observable (self-adjoint operator) to commute with another observable ...
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In QFT, why does a vanishing commutator ensure causality?

In relativistic quantum field theories (QFT), $$[\phi(x),\phi^\dagger(y)] = 0 \;\;\mathrm{if}\;\; (x-y)^2<0$$ On the other hand, even for space-like separation $$\phi(x)\phi^\dagger(y)\ne0.$$ ...
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Understanding Poisson brackets

In quantum mechanics, when two observables commute, it implies that the two can be measured simultaneously without perturbing each other's measurement results. Or in other words, the uncertainty in ...
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What does the Canonical Commutation Relation (CCR) tell me about the overlap between Position and Momentum bases?

I'm curious whether I can find the overlap $\langle q | p \rangle$ knowing only the following: $|q\rangle$ is an eigenvector of an operator $Q$ with eigenvalue $q$. $|p\rangle$ is an eigenvector of ...
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Why uncertainty principle is not like this?

In Griffiths' QM, he uses two inequalities (here numbered as $(1)$ and $(2)$) to prove the following general uncertainty principle: $$\sigma_A^2 \sigma_B^2\geq\left(\frac{1}{2i}\langle [\hat A ,\hat ...
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Eigenstate of position+momentum?

I'm studying Quantum Mechanics on my own, so I'm bound to have alot of wrong ideas - please be forgiving! Recently, I was thinking about the quantum mechanical assertion (postulate?) that states with ...
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Motivating Complexification of Lie Algebras?

What is the motivation for complexifying a Lie algebra? In quantum mechanical angular momentum the commutation relations $$[J_x,J_y]=iJ_z, \quad [J_y,J_z] = iJ_x,\quad [J_z,J_x] = iJ_y$$ become, on ...
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Momentum as Generator of Translations

I understand from some studies in mathematics, that the generator of translations is given by the operator $\frac{d}{dx}$. Similarly, I know from quantum mechanics that the momentum operator is ...
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Does the commutator of anything with itself not vanish?

In a quantum mechanics exam one question was to write the commutator of a couple of operators. Everybody got points taken away since they did not write $[Q_i, Q_i] = 0$ for all the operators $Q_i$ in ...
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A question about causality and Quantum Field Theory from improper Lorentz transformation

Related post Causality and Quantum Field Theory In Peskin and Schroeder's QFT p28, the authors tried to show causality is preserved in scalar field theory. Consider commutator $$ [ \phi(x), \phi(y) ...
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How to replace $T$-product with retarded commutator in LSZ formula?

I am reading Itzykson and Zuber's Quantum Field Theory book, and am unable to understand a step that is made on page 246: Here, they consider the elastic scattering of particle $A$ off particle $B$: ...
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Why Don't the Ladder Operators Commute?

I have two problems with ladder operators. The first is that I feel they should somehow result in measurable things. The asymmetry of applying the plus operator versus the minus operator is very ...
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Commutator of Lorentz boost generators : visual interpretation

I have always struggled to visualize the correctness of the commutation relation for the generators of the boost in the Lorentz group. We have $$[K_i,K_j] = i \epsilon_{ijk} L_k$$ I fail to picture ...
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Does uncertainty imply noncommutativity?

We already know that non-commutativity of observables leads to uncertainty in quantum mechanics cf. e.g. this and this Phys.SE post. What about the opposite: Does uncertainty imply noncommutativity? ...
7
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How does the proof of operator commutativity work with non-continuous operators?

In some books, a proof that if two self-adjoint operators $A$ and $B$ share a common eigenbasis $\{\phi_n\}$, then they commute is given as follows : For any $\phi_n$, $$AB\ \phi_n = a_n\ ...
7
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Dilation operator in CFT viewed as 'hamiltonian'?

From the commutation relations for the conformal Lie algebra, we may infer that the dilation operator plays the same role as the Hamiltonian in CFTs. The appropriate commutation relations are ...
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Canonical quantization in supersymmetric quantum mechanics

Suppose you have a theory of maps $\phi: {\cal T} \to M$ with $M$ some Riemannian manifold, Lagrangian $$L~=~ \frac12 g_{ij}\dot\phi^i\dot\phi^j + \frac{i}{2}g_{ij}(\overline{\psi}^i ...
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Find $\hat{x}$ operator given $\hat{p}$ operator

This is problem $1.2$ of Molecular Quantum Mechanics by Atkins, 4th edition. I'm given the momentum operator $$p=\sqrt{\frac{\hbar}{2m}}(A+B)$$ with $$[A,B]=1$$ and I need to find $x$ in this ...
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Theories with non-vanishing commutators outside the lightcone

I'm reading Weinberg's new book on Quantum Mechanics, and in Chapter 8.7 "Time-Dependent Perturbation Theory" he derives the usual Dyson series for the $S$ matrix when the interaction Hamiltonian ...
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Commutator with exponential $[A, \exp(B)]$

How can I tell if $A$ and $\exp(B)$ commute? For $[A, B]$ it's simply $AB-BA$ and for $[\exp(A), \exp(B)]$ I think it'd be $\exp(A)\exp(B) - \exp(B)\exp(A) = \exp(A+B) - \exp(B+A) = 0$. Update: it's ...
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Does it mean anything if the commutator of an operator with the Hamiltonian is equal to the Hamiltonian?

Question says it all, really. I have $[\hat{H},\hat{O}]=-2i\hbar\hat{H}$. Does this mean that the operator $\hat{O}$ (an observable) is special in some way?
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Does the canonical commutation relation fix the form of the momentum operator?

For one dimensional quantum mechanics $$[\hat{x},\hat{p}]=i\hbar $$ Does this fix univocally the form of the $\hat{p}$ operator? My bet is no because $\hat{p}$ actually depends if we are on ...
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Canonical Commutation Relations

Is it logically sound to accept the canonical commutation relation (CCR) $$[x,p]~=~i\hbar$$ as a postulate of quantum mechanics? Or is it more correct to derive it given some form for $p$ in the ...
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Commutator with a square root

How to find the commutator $[a, \sqrt{a^\dagger a}]$? Here $a$ is a usual bosonic annihilation operator, and $[a, a^\dagger] = 1$. The first thing I tried is $$ [x,A] = [x, \sqrt{A}]\sqrt{A} + ...
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Proof of Canonical Commutation Relation (CCR)

I am not sure how $QP-PQ =i\hbar$ where $P$ represent momentum and $Q$ represent position. $Q$ and $P$ are matrices. The question would be, how can $Q$ and $P$ be formulated as a matrix? Also, what is ...
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Is uncertainty principle a technical difficulty in measurement? [duplicate]

Is the uncertainty principle a technical difficulty in measurement or is it an intrinsic concept in quantum mechanics irrelevant of any measurement? Everyone knows the thought experiment of measuring ...
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What is the commutator of an operator and its derivative?

Is it possible to calculate in a general way the commutator of an operator $O$ which depends on some variable $x$ and the derivative of this $O$ with respect to $x$? $${O}={O}(x)\\ ...
6
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Causality and Quantum Field Theory

I have a problem with proof of causality in Peskin & Schroeder, An Introduction to QFT, page 28. To avoid confusion I use three vectors notation, rewriting the Eq. (2.53) for $y=0$ as follows: ...
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Commutator algebra in exponents

Considering $X$ and $Y$ such that $[X,Y]=\lambda$, which is complex, and $\mu$ is another complex number, prove: $$e^{\mu(X+Y)}=e^{\mu X} e^{\mu Y} e^{-\mu^2\lambda/2}$$ My attempt (so far) is: ...
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Proving that $i\hbar\frac{\partial}{\partial \mathbf{p}}$ is the operator of $\mathbf{x}$ in momentum space

How can I prove that $i\hbar\frac{\partial}{\partial \mathbf{p}}$ is the operator of $\mathbf{x}$ in momentum space?
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Fundamental Commutation Relations in Quantum Mechanics

I am trying to compile a list of fundamental commutation relations involving position, linear momentum, total angular momentum, orbital angular momentum, and spin angular momentum. Here is what I have ...
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Tricky operator identity: $[L^2,[L^2,\vec{r}]]=2 \hbar ^2 \{ L^2, \vec{r}\}$?

This operator identity showed up in a course I was taking, and it was given without proof. $$[L^2,[L^2,\vec{r}]]=2 \hbar ^2 \{ L^2, \vec{r}\}$$ The curly brackets denote the anticommutator, $AB+BA$. ...
5
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Evaluating commutator of $[\operatorname{sign}(X),\, \operatorname{sign}(P)]$

I wish to evaluate the following commutator: $[\operatorname{sign}(X),\, \operatorname{sign}(P)]$. Is there a general method for evaluating $[\operatorname{f}(X), \operatorname{f}(P)]$? I thought of a ...
5
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Why are anticommutators needed in quantization of Dirac fields?

Why is the anticommutator actually needed in the canonical quantization of free Dirac field?
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Heisenberg picture of QM as a result of Hamilton formalism

Consider the formula for the total time-derivative of a physical value in Poisson's formalism: $$\tag{1} \frac{dA}{dt} = -\{H, A\}_{P.B.} + \frac{\partial A}{\partial t}, $$ where $\{A, B\}_{P.B.}$ is ...
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Does This Really “Prove” Spin-statistics Theorem?

In quantization of scalar field theory we impose commutation relation between the field operators by hand and similarly we impose anti-commutation relation between Dirac field operators by hand. As a ...
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Do we need to expand the potential in a power series to show $[x, V(x)] = 0$?

Today in class (Intro to QM) we went over a couple of commutators. Among them was $[x, V]$, where $V=V(x)$ is a potential. What the teacher said to prove this is zero was: let's assume $V$ is analytic ...
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Mutual or same set of eigenfunctions if two operators commute

If two operators commute, do they have "a mutual set of eigenfunctions", or "the same set of eigenfunctions"? My quantum chemistry book uses these as if they are interchangeable, but they do not seem ...
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What physical significance has the Heisenberg Group?

I read that the canonical commutation relation between momentum and position can be seen as the Lie Algebra of the Heisenberg group. While I get why the commutation relations of momentum and momentum, ...
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Finding the creation/annihilation operators

Using Minkowski signature $(+,-,-,-)$, for the Lagrangian density $${\cal L}=\partial_{\mu}\phi\partial^{\mu}\phi^{\dagger}-m^2\phi \phi^{\dagger}$$ of the complex scalar field, we have the field ...
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Translator Operator

In Modern Quantum Mechanics by Sakurai, at page 46 while deriving commutator of translator operator with position operator, he uses $$\left| x+dx\right\rangle \simeq \left| x \right\rangle.$$ But for ...
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Commutation relations in QFT and the principle of locality

My question is, given two space-time points $x^{\mu}$ and $y^{\mu}$, if the events that occur at these points are simultaneous, i.e. $x^{0}=y^{0}$, are the two events necessarily space-like separated? ...
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Do mutual eigenkets imply commutation of two operators?

I have been working on this question. I have solved it, and I would like to check whether my line of reasoning is right or wrong Question: Prove that if there exists a mutual complete set of ...
5
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Imposing anti-commutation relations on fermionic quasi-particles

In many theories of CMT, we assume the nature of quasi-particles (without giving proper justifications). For example, we assume nature of quasi-particles to be fermionic in case of a interacting ...
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Compatible Observables

My QM book says that when two observables are compatible, then the order in which we carry out measurements is irrelevant. When you carry out a measurement corresponding to an operator $A$, the ...
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The implication of anti-commutation relations in quantum mechanics

All the textbooks I saw are very clear about the implications of commutating operators in quantum mechanics. However, much less is said about anti-commutation relations. Does it have a general ...