# 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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### Fields commutator, a concise review

I am entering in the quantum filed theory world, and I am consulting many notes, but something is a bit unclear about commutators. Hence I would like some expert of you tell me what exactly (if ...
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### Why do position operators in orthogonal directions commute?

In three dimensions, we have $\hat x$, $\hat y$, $\hat z$ as the position operators in the three orthogonal directions. If the components of angular momentum don't commute, why must these all commute? ...
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### Non-Fock representation of quantum field theory

I cannot find reference, but I read that in curved spacetime, there exists representation that is not Fock satisfying CCR and unitarily inequivalent to a Fock representation. In usual understanding ...
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### Composition of Lorentz transformations using generators and the Wigner rotation

I solved this problem by painful calculations of Lorentz matrices. However, I heard that there is a much easier solution using the generators of boosts and rotations and their commutation relations, ...
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### Why goes $i\rightarrow-i$ under $\mathcal{PT}$-transformation?

Question in the title. What I understand is that under $\mathcal{PT}$ reversal $\hat{p}\rightarrow\hat{p}$ and $\hat{x}\rightarrow-\hat{x}$ and then since the commutation $[\hat{x},\hat{p}]=i\hbar$ "...
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### A question about a commutation relation

Here $a$, $b$,$c$, $d$ $u$, $v$ range from 0 to 4 and the metric $g^{ab}=\text{diag}(-1,1,1,1)$. 16 $4 \times 4$ matrices $M^{ab}$ are defined as follows: \begin{equation*} (M^{ab})_{uv} = -i(\delta^...
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### Commutator of canonical fields in Quantum Field Theory

Let $\phi(\vec{x},t)$ denote the canonical fields and $\pi(\vec{x},t)$ denote the canonical impulses where they're given by: \begin{equation} \phi(x)=\int\frac{d^3\vec{p}}{(2\pi)^3\sqrt{2\omega_{\...
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### How to show that two operators in the forms of vectors commute?

This is an exercise from Peskin&Schroeder's book. The exercise requires to show that $\textbf{J+}$ and $\textbf{J-}$ commute with each other. What is the exact meaning of commutation between ...
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### Commutation relations of the generators of the Lorentz group

$$J^{\mu\nu} = i(x^\mu\partial^\nu-x^\nu\partial^\mu). \tag{3.16}$$ We will soon see that these six operators generate the three boosts and three rotations of the Lorentz group. To determine ...
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### Anticommutator expression

I need to show that this expression is contradictory. The is no more information is given for $\hat{b}$. $$\hat{b}^{\dagger}\hat{b}+\hat{b}\hat{b}^{\dagger}=-I$$
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### Commutator $[\hat{A},\exp(\hat{A})]= 0$

In Equation (4) of this Physics.SE post, Qmechanic wrote, $$\tag{4}\frac{d}{dt}e^{t\hat{A}}~=~\hat{A}e^{t\hat{A}}~=~e^{t\hat{A}}\hat{A}.$$ How does one get this equation?
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### Simultaneous measurement of two observables

In quantum physics the configuration of a particle is fully defined by it's wave function. When a measurement of a particular observable ( eg. position, angular momentum etc.) is made on the particle ,...
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### Energy differences and matrix mechanics

I was reading a wonderful explanation of matrix mechanics on mathpages1. There we see that $\hat{q}_{mn}=q_{mn}\exp(i(E_m-E_n)t/\hbar )$ and consequently using Hamilton's classical equations we arrive ...
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### Why must fermion fields anticommute and bosons commute?

Fermion fields must satisfy anticommutation relation. But why? I know that unless they anti-commute the Pauli exclusion principle cannot be satisfied. But is there some other deeper/fundamental ...
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### How to prove that given operators form an algebra? [closed]

I am new to Group theory and representations and I'm having trouble with this problem in an exercise: Given the two oscillator algebra $$[a, a^†] = 1$$ $$[b, b^†] = 1$$ $$[a, b] = 0$$ show that ...
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### Heisenberg's original Quantum Condition

In Heisenberg's paper Quantum Theoretical Re-interpretation of Kinematic and Mechanical Relations where he first derives Quantum Mechanics, using his postulates of Quantum Mechanics and the Old ...
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### Normal ordering for several operators

I understand that if we consider a normal ordering of two operators $\mathcal{N}(a_{p_1}a_{p_2}^{\dagger})$, it will be $a_{p_2}^{\dagger}a_{p_1}$, where $a_{p_1}$ is an annihilation operator of a ...
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### How did Heisenberg come up with the Canonical Commutation relation ($\hat X \hat P-\hat P\hat X=i\hbar$)?

All answers to questions like this dodge the question by saying it's a postulate of Matrix Mechanics, so let me rephrase it. Instead of how to derive the CCR, how does it follow from Heisenberg's ...
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### Two different definition of quantum Poisson bracket, which one to use?

In Dirac's 'Principles of Quantum Mechanics' the definition of quantum Poisson bracket is $uv-vu=i\hbar[u,v]$ with $[q_r,p_s]=\delta_{rs}$. But in lots of other books the definition is $uv-vu=[u,v]$ ...
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### How to know if a set of commuting observables is complete?

We define a complete set of commuting observables as a set of observables $\{A_1,\ldots, A_n\}$ such that: $\left[A_i, A_j\right]=0$, for every $1\leq i,~j \leq n$; If $a_1,\ldots, a_n$ are ...
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### Does $[P_j,B_k]=i(Mc^2)\delta_{jk}$ imply particle number conservation?

From reading Weinberg's Quantum Theory of Fields, Vol. 1, I learnt that for the Galilean group $[P_j,B_k]=i(Mc^2)\delta_{jk}$, and for the Poincare group $[P_j,B_k]=iH\delta_{jk}$ where $P_j$ and $B_k$...
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### Why if 2 operators commute they have a common set of eigenvectors and what's the relation to 2 fold degeneracy?

I have the following sentence in my lecture notes "Dirac hamiltonian and helicity have a common set of eigenvectors, this is also the reason for the two fold degeneracy found for every energy ...
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### Do half integer spin fields commute or anti-commute with spin integer fields?

What are the fundamental commutation/anti-commutation relations between half integer and integer spin fields? For instance, in QED do we have \begin{equation} [\psi(x),A^{\mu}(y)]=0 \end{equation} or \...
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### Non-abelian gauge theories, which factors commute?

I am following Peskin and Schroeder (1995). It seems that we are magically supposed to understand which factors commute and which do not in the theory. These are the factors which appear in the ...
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### Index notation with vector operators [closed]

I'm new to index notation. I'm trying to show that for to vector operators $\vec{A}$ and $\vec{B}$ the following relation holds: $\left[ \vec{L},\: \vec{A} \cdot \vec{B}\right] = 0$ As $A$ and $B$ ...
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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 ...
351 views

### Fock space with mixed anti-commutation/commutation relations?

Let's say we have two modes, with the following labeling of occupation number states: $\lvert \Psi \rangle = \begin{pmatrix} 0,0 \\ 0,1 \\ 1,0 \\ 1,1 \end{pmatrix}$ An example of (what I assume to ...
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### Does Schroedinger equation depend on the sign of Poisson bracket?

Let's consider Poisson bracket \left\{ A,B\right\} =\alpha_{p} \left( \frac{\partial A}{\partial p_{k}}\frac{\partial B}{\partial q^{k}}-\frac{% \partial A}{\partial q^{k}}\frac{\partial B}{\...
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### Why does the Heisenberg uncertainty principle refer to momentum rather than velocity?

I've been looking at the Heisenberg uncertainty relations, and something that sticks out to me is the use of momentum rather than velocity. Shouldn't electrons have the same mass? And if they do, why ...
### Physical intuition behind why $\hat{J}^2$ and $\hat{J}_z$ commute
In Townsend's Quantum Mechanics textbook, he shows that $\hat{J}^2$, the squared magnitude of the angular momentum, and $\hat{J}_z$, the generator of rotations about the $z$-axis should commute. I ...