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.

Filter by
Sorted by
Tagged with
0
votes
0answers
50 views

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 ...
2
votes
1answer
251 views

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? ...
2
votes
1answer
122 views

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 ...
4
votes
1answer
250 views

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, ...
1
vote
2answers
71 views

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 $[\hat{x},\hat{p}]=i\hbar$ "...
0
votes
1answer
40 views

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^...
0
votes
1answer
220 views

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_{\...
0
votes
1answer
94 views

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 ...
3
votes
2answers
1k views

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 ...
0
votes
1answer
60 views

Anticommutator expression

I need to show that this expression is contradictory. The is no more information is given for $\hat{b}$. $$\hat{b}^{\dagger}\hat{b}+\hat{b}\hat{b}^{\dagger}=-I$$
2
votes
2answers
184 views

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?
-1
votes
1answer
460 views

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 ,...
0
votes
1answer
66 views

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 ...
2
votes
1answer
325 views

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 ...
1
vote
0answers
44 views

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 ...
0
votes
0answers
69 views

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 ...
0
votes
1answer
73 views

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 ...
5
votes
1answer
351 views

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 ...
1
vote
2answers
130 views

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 $uv-vu=[u,v]$ ...
8
votes
2answers
666 views

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 ...
2
votes
1answer
187 views

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 $B_k$...
0
votes
0answers
149 views

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 ...
0
votes
1answer
66 views

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 \...
1
vote
1answer
312 views

How can the Canonical Commutation Relation be derived from Heisenberg's Equation of Motion?

Heisenberg's Equation of Motion states that a Matrix's time evolution is determined by the following: $$\frac{d\hat A}{dt}=\frac{1}{i\hbar}\left[\hat A,\mathcal{\hat H}\right]$$ Where the Matrix $\hat ...
-2
votes
1answer
111 views

Common basis of Hamiltonian and another operator [closed]

I have a Hamiltonian of $N\times N$ matrix, when I diagonalize it , I observe that the $N$ different eigenstates of the hamiltonian are also eigenstates of another operator $\hat O$ where the ...
2
votes
2answers
799 views

Commutator with exponential $[\exp(A),\exp(B)]$

$A,B$ are quantum mechanical operators. $[A,B]\neq 0$ that is given. $e^{A}=\sum_{n=1}^{\infty} \frac{A^n}{n!} $ Is the following correct? $$[e^{A},e^{B}]=e^{A}e^{B}-e^{B}e^{A}=e^{A+B}-e^{B+A}=0 $$...
0
votes
1answer
299 views

Commutator identities and Fourier transform

Is it possible to derive one side of the arrow below from the other by using only the Fourier transform and its reciprocal? $$[\hat{p},f(\hat{x})]=-i\hbar f'(\hat{x}) \leftrightarrow [\hat{x},f(\hat{...
0
votes
1answer
109 views

Commutators in second Quantization [duplicate]

How do you calculate the commutator $$[b_i^n, (b_j^\dagger)^m]$$ where $b_i$ are annihilators and $b_j^\dagger$ are creators in second quantization. All annihalators should be swapped to the right ...
5
votes
3answers
535 views

Stone-von Neumann theorem

According to Stone-von Neumann theorem, any two canonically conjugate self adjoint operators following the relation: $$[\hat{q},\hat{p}]=i\hbar$$ cannot be both bounded. I am confused about how we ...
2
votes
3answers
394 views

The commutation of partial derivatives in curved spacetime

While following a lecture series on General Relativity, an argument was presented that in order the spacetime to be flat, a vector parallel transported along two different paths should yield the same ...
0
votes
3answers
440 views

Planck's constant and the Uncertainty principle

Why should the uncertainty in measurement of two conjugate variables, say $p$ and $x$, be of the order of Planck's constant or higher and not lower? What is so sacrosanct about $h$? I always think $h$ ...
3
votes
1answer
307 views

Is there any plausible reason for the existence of conjugate variables in quantum mechanics?

(I updated the question, the new last passage is the important one) If I assume the position of a particle to be represented by an operator $\hat{x}$, and the time evolution to be carried out by the ...
-2
votes
1answer
75 views

Proving $[L_i,R_j] = i\hbar\epsilon_{ijk}R_{k}$

I need to prove $[L_i,R_j] = i\hbar\epsilon_{ijk}R_{k}$. I have that $L_i = \epsilon_{ijk}R_{j}P_{k}$ , so the commutator becomes: $$[\epsilon_{ijk}R_{j}P_{k},R_{j}] = \epsilon_{ijk}\left(R_j[P_k,...
0
votes
0answers
257 views

Kronecker delta commutation relations for QFT

Setup: In many textbook treatments of canonical quantization (e.g., Peskin and Schroeder), one imposes canonical (equal time) Dirac delta commutation relations on the conjugate field operators. e.g., ...
12
votes
2answers
288 views

Why does $\hbar$ appear twice in the axioms of QM?

Physical theories have dimensionful constants. Each constant can be found via measurement, by fitting some equation to data. Mathematically, you would expect each constant to be "defined" in this way ...
0
votes
2answers
834 views

Hermitian operator followed by another hermitian operator – is it also hermitian?

Consider the two hermitian operators $\hat A$ and $\hat H$: I can prove that the operator $[\hat A,\hat H]$ is non-hermitian as follows: $$\begin{align} \int\phi^*[\hat A,\hat H]\psi\,dx&=\int\...
-2
votes
1answer
69 views

Commutator calculation - tips? [closed]

I have to find out the following commutator $$ [a^\dagger a^\dagger a, a^\dagger a a] $$ and after expanding it with $[A,B]=AB-BA$ $$ [a^\dagger a^\dagger a, a^\dagger a a] = a^\dagger a^\dagger a a^...
2
votes
2answers
476 views

Commutation relation under time ordering

Consider a quantum system with the following Hamiltonian: $$H(t)=H_0+H_1(t),\tag{1}$$ where $H_0$ is a noninteracting Hamiltonian and $H_1(t)$ a time-dependent perturbation. To formulate the linear ...
3
votes
4answers
89 views

Quantum State Representation with Commuting Operators

Let $[A,B]=0$. Then, we can find a set of eigenvectors $\{|a_n,b_n\rangle\}$ common to both $A$ and $B$. According to this, and my own understanding, it makes sense to write an arbitrary quantum state ...
0
votes
1answer
196 views

Canonical commutation relation field theory

This is a question on computation. If we set the quantum field canonical commutation relations as $$ [\hat{\pi}(x),\hat{\phi}(x')] = -i\hbar \delta(x-x'), $$ the goal is to find the Fourier ...
4
votes
1answer
149 views

Operators - how to motivate they must be linear ? Is this comment a hint? [duplicate]

Is there a way to motivate, retrospectively, that observables must be representable by linear operators on a Hilbert space? Specifically, there seems to be a hint to something in the accepted ...
0
votes
1answer
93 views

Quantum Vector Operators. Showing $\textbf{r}(\textbf{L} \cdot\textbf{p})-\textbf{p}(\textbf{L} \cdot\textbf{r})=0$

I'm asked to prove this relation: $$\left [\textbf{A},\textbf{L} \cdot \textbf{B} \right]=-i \hbar \textbf{A} \times \textbf{B} + L_i \left [\textbf{A},B_i \right].$$ Where $\textbf{A}$ and $\textbf{...
0
votes
0answers
35 views

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 ...
1
vote
0answers
190 views

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 $B$ ...
1
vote
0answers
171 views

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 ...
7
votes
1answer
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 ...
2
votes
2answers
349 views

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}{\...
0
votes
2answers
186 views

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 ...
1
vote
0answers
173 views

Causal propagator relation from canonical field commutator

In Minkowski spacetime, the commutator of the Klein-Gordon field operator with itself at different spacetime points evaluates to the advanced minus retarded Green's function (i.e., the "causal ...
4
votes
3answers
222 views

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 ...