# 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.

620 questions
Filter by
Sorted by
Tagged with
62 views

### Finding a closed formula using Baker-Hausdorff formula for a unitary transformation; An endless commutator

Consider the Baker-Hausdorff formula for two operators $a_1$ and $iHt$: $$e^{iHt}a_1 e^{-iHt} =a_1+[iHt,a_1]+\frac{1}{2!}[iHt,[iHt,a_1]]+\frac{1}{3!}[iHt,[iHt,[iHt,a_1]]]+....,$$ where $[A,B]=AB-BA$. ...
46 views

### The charge given by a commutator

I saw in the text that $[Q,X]=cX$ and says the operator $X$ has charge $c$ under the generator $Q$. I tried to understand why the coefficient $c$ means the charge. So I used this relation to get the ...
24 views

### Symmetry in Fock-space 2-body interaction

The simplest two body interaction term for fermions is $$H = \sum_{ijkl} U_{ijkl} a_i^\dagger a_j^\dagger a_k a_l$$ and I'm trying to determine the symmetries on $U$. Unfortunately I keep getting ...
164 views

### Geometric interpretation of the second Bianchi identity?

Assuming a torsion free Christoffel symbol, the covariant derivative can be shown to satisfy the second (differential) Bianchi identity: [[\nabla_a,\nabla_b],\nabla_c]+[[\nabla_c,\...
53 views

51 views

### Does $x$-component $\hat{L}_x$ of angular momentum commute with $\hat{x}$? [closed]

Question from lecture notes: What of the following operators does not commute with $\hat x$? A. $\hat L_x$ B. $\hat L_y$ C. $\hat L_z$ D. None of the above. The answer ...
36 views

### Commutator $\vec{L}$ with $\vec{X}\cdot\vec{P}$

Let $\vec{X}=(X_1,X_2,X_3)^T$ and $\vec{P}=(P_1,P_2,P_3)^T$. Define $\vec{L}=\vec{X}\times\vec{P}$. Then, I can calculate $\vec{L}=(X_2P_3-X_3P_2,\,X_3P_2-X_2P_3,\,X_1P_2-P_1X_2)^t$. For all ...
163 views

### “Commuting observables share common eigenstates”

I am struggling to find a precise definition of this line from my quantum mechanics textbook: If $[A,B] = 0$, then the operators commute, and "commuting operators share common eigenstates". This ...
90 views

128 views

49 views

### A question about the commutator

I'm self-studying quantum mechanics and have a question regarding the commutator. Since the commutator of two operators is defined as $[A,B]$ = $AB$ - $BA$ Assuming that these operators do not ...
54 views

### Commutator of $\hat {L}_x$ and $\hat{V}(\hat{r})$ [duplicate]

Consider the angular momentum operator $\hat{L_x}=\hat y\hat{p}_z-\hat{z}\hat{p}_y$ and the potential operator $\hat{V}$ where the potential $\hat{V}=\hat{V}(\hat{r})$ is spherically symmetric. It ...
93 views

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 ...
44 views

### 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 $\lbrace f, \gamma\rbrace \neq 0$...
121 views

### Commutator of a quark current

In Quantum Chromodynamics, when we take the limit in which the u, d and s quarks have no mass, there exists a global symmetry $G \equiv SU(3)_L \otimes SU(3)_R$ in flavour space. The corresponding ...
46 views

147 views

I have a question regarding a proposed problem (Problem 4.8) in Rodney Loudon's book "The Quantum Theory of Light". Let $U(t)$ be an operator defined by $$U(t)=\exp\left\lbrace\frac{i}{\hbar}\int\... 1answer 72 views ### What is the state of particle at time t if at t=0 it is in an eigenstate of \hat{A}, and \hat{A} commutes with \hat{H}? EDIT: added (assuming \lambda to be non-degenerate). Based on the specifics of the question, we don't in fact know whether this is the case, so it may be that \left|\lambda\right> is not an ... 1answer 89 views ### Commutators in Gupta-Bleuler formalism for quantization of the electromagnetic field In the Gupta-Bleuler formalism we have for the canonical momenta$$\pi_\mu=F_{\mu0}-g_{\mu0}\partial_\alpha A^\alpha. $$Every resource I find online say that the equal time canonical commutation ... 1answer 137 views ### Are there any ways to exclude uncertainty in the values of any non-commuting operators? [closed] If two similar systems are created and In the first system the position is measured with accuracy and in the second one the momentum is measured with accuracy can this avoid the uncertainty in the ... 1answer 201 views ### Proving that an operator is hermitian [closed] Let A be an operator that is the product of two hermitian operators. Am I at liberty to say that if those two hermitian operators commute and their commutator is zero, then A is hermitian? 0answers 80 views ### Definitions of operators and commutativity in quantum mechanics If [\hat A,\hat B] = 0, where \hat A and \hat B are operators, then the operators commute. This also means that, when applied to a wavefunction, that one can measure observables A and B in ... 1answer 118 views ### Do all well-measured observables effectively commute? Do all well-measured observables effectively commute with each other? The rest of this long post clarifies what I mean by that simple-looking question. Consider quantum field theory in Minkowski ... 1answer 99 views ### Canonical commutation relation from Pauli-Jordan function In Steinmann's book, Perturbative Quantum Electrodynamics and Axiomatic Field Theory is stated that the commutator of two fields \varphi, which satisfy the Klein-Gordon equation$$ (\square-m^2)\...
This is actually a continuation of calculation I've been working on. It is well known that, in the case of Dirac fields $\psi(x)$, they satisfy anticommutatation relationships since they're fermionic ...