Commutator Definition and 274 Threads

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.

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  1. Filip Larsen

    I Solving matrix commutator equations?

    I have the matrix relationship $$C = A^{-1} B^{-1} A B$$ I want to solve for ##A##, where ##A, B, C## are 4x4 homogeneous matrices, e.g. for ##A## the structure is $$A = \begin{pmatrix} R_A & \delta_A \\ 0 & 1 \end{pmatrix}, A^{-1} =\begin{pmatrix} R_A^\intercal & -R_A^\intercal\delta_A \\ 0 & 1...
  2. han

    Is the Lorentz Boost Generator Commutator Zero?

    Using above formula, I could calculate the given commutator. $$ [\epsilon^{\mu\nu\rho\sigma} M_{\mu \nu}M_{\rho\sigma},M_{\alpha\beta}]=i\epsilon^{\mu\nu\rho\sigma}(M_{\mu \nu}[M_{\rho\sigma},M_{\alpha\beta}]+[M_{\rho\sigma},M_{\alpha\beta}]M_{\mu \nu}) $$ (because...
  3. G

    A Question about commutator involving fermions and Pauli matrices

    Suppose ##\lambda_A## and ##\bar{\lambda}_A## are fermions (A goes from 1 to N) and ##\{ \lambda_{A \alpha}, \bar{\lambda}_B^{\beta}\} = \delta_{AB}\delta_{\alpha}^{\beta}##. Let ##\sigma^i## denote the Pauli matrices. Does it follow that ##[\bar{\lambda}_A \sigma^i \lambda_A, \bar{\lambda_B}...
  4. P

    Schwartz's Quantum field theory (12.9)

    I am reading the Schwartz's quantum field theory, p.207 and stuck at some calculation. In the page, he states that for identical particles, $$ | \cdots s_1 \vec{p_1}n \cdots s_2 \vec{p_2} n \rangle = \alpha | \cdots s_2 \vec{p_2}n \cdots s_1...
  5. C

    Calculate the following commutator [[AB,iℏ], A]

    I've seen this question in a textbook Calculate the following commutator [[AB,iℏ], A] I'm not to sure how you go about it i know [A,B] = AB-BA
  6. guyvsdcsniper

    Proving commutator relation between H and raising operator

    I am going through my class notes and trying to prove the middle commutator relation, I am ending up with a negative sign in my work. It comes from [a†,a] being invoked during the commutation. I obviously need [a,a†] to appear instead. Why am I getting [a†,a] instead of [a,a†]?
  7. K

    Can a DC motor work without either a commutator or a controller?

    I am a mechanical engineer and my experience with electrical systems is almost nil. The concept of a simple DC motor explained here was quite interesting, especially the need of a commutator part: And then I checked this DIY simple DC motor here and was confused because there was no...
  8. Samama Fahim

    I Deriving the Commutator of Exchange Operator and Hamiltonian

    In the boxed equation, how would you get the right hand side from the left hand side? We know that ##H(1,2) = H(2,1)##, but we first have to apply ##H(1,2)## to ##\psi(1,2)##, and then we would apply ##\hat{P}_{12}##; the result would not be ##H(2,1) \psi(2,1)##. ##\hat{P}_{12}## is the exchange...
  9. Clifford Williams

    Help with this Commutator question please

    Hello, In QM class this morning my Prof claimed that the commutator [𝑥(𝜕/𝜕𝑦), 𝑦(𝜕/𝜕𝑥)] = 0. However, my classmate and I arrived at x(d/dx) - y(d/dy). Can someone explain how (or if) our professor is correct?
  10. U

    Questions on field operator in QFT and interpretations

    For a real scalar field, I have the following expression for the field operator in momentum space. $$\tilde{\phi}(t,\vec{k})=\frac{1}{\sqrt{2\omega}}\left(a_{\vec{k}}e^{-i\omega t}+a^{\dagger}_{-\vec{k}}e^{i\omega t}\right)$$ Why is it that I can discard the phase factors to produce the time...
  11. U

    Question on discrete commutation relation in QFT

    Given the commutation relation $$\left[\phi\left(t,\vec{x}\right),\pi\left(t,\vec{x}'\right)\right]=i\delta^{n-1}\left(\vec{x}-\vec{x}'\right)$$ and define the Fourier transform as...
  12. JD_PM

    Dirac-Hamiltonian, Angular Momentum commutator

    We want to show that ##[\hat{ \vec H}, \hat{ \vec L}_T]=0##. I made a guess: we know that ##[\hat{ \vec H}, \hat{ \vec L}_T]=[\hat{ \vec H}, \hat{ \vec L}] + \frac 1 2 [\hat{ \vec H}, \vec \sigma]=0## must hold. I have already shown that $$[\hat{ \vec H}, -i \vec r \times \vec \nabla]= -...
  13. JD_PM

    Working out ##\big[\varphi (x) , \varphi (0) \big]## commutator

    This exercise was proposed by samalkhaiat here (#9). I am going to work using natural units. OK I think I got it (studying pages 46 & 47 from Mandl & Shaw was really useful) . However I took the lengthy approach. If there is a quicker method please let me know :smile: We first Fourier-expand...
  14. A

    I Commutator of ##L^2## with ##L_x,L_y,L_z##...

    For a given state say ##{l,m_l}## where ##l## is the orbital angular momentum quantum no. and ##m_l## be it's ##z## component...a given state ##|l,m_l> ## is an eigenstate of ##L^2## but not an eigenstate of ##L_x##...therefore all eigenstates of ##L_x## are eigenstates of ##L^2## but the...
  15. A

    Universal motor with commutator, recovering the back EMF?

    So while thinking about motors this suddenly struck me, So as the universal series wound motor is spinning there is always some arcing going on around the place where the brushes contact the copper segments that slide past them, I assume this is at least partly because as each coil pair of the...
  16. RicardoMP

    Proof of the commutator ## [P^2,P_\mu]=0 ##

    I want to make certain that my proof is correct: Since ## P^2 = P_\nu P^\nu=P^\nu P_\nu ##, then ## [P^2,P_\mu]=[P^\nu P_\nu,P_\mu]=P^\nu[P_\nu,P_\mu]+[P^\nu,P_\mu]P_\nu=[P^\nu,P_\mu]P_\nu=g^{\nu\alpha}[P_\alpha,P_\mu]P_\nu=0 ##, since ## g^{\nu\alpha} ## is just a number, I can bring it...
  17. T

    A Exploring the Conditions for Evaluating Commutators with Fermionic Operators

    I found a theorem that states that if A and B are 2 endomorphism that satisfies $$[A,[A,B]]=[B,[A,B]]=0$$ then $$[A,F(B)]=[A,B]F'(B)=[A,B]\frac{\partial F(B)}{\partial B}$$. Now I'm trying to apply this result using the creation and annihilation fermionics operators $$B=C_k^+$$ and $$A=C_k$$...
  18. P

    I Proof of Commutator Operator Identity

    Hi All, I try to prove the following commutator operator Identity used in Harmonic Oscillator of Quantum Mechanics. In the process, I do not know how to proceed forward. I need help to complete my proof. Many Thanks.
  19. PeroK

    QFT Commutator for spacelike separation

    I got as far as: $$[\hat \phi(x), \hat \phi(y) ] = \int \frac{d^3p}{(2\pi)^{3}(2E_p)}(\exp(-ip.(x-y) - \exp(-ip.(y-x))$$ Then I simplified the problem by taking one of the four-vectors to be the origin: $$[\hat \phi(0), \hat \phi(y) ] = \int \frac{d^3p}{(2\pi)^{3}(2E_p)}(\exp(ip.y) -...
  20. Y

    Deriving commutator of operators in Lorentz algebra

    Li=1/2*∈ijkJjk, Ki=J0i,where J satisfy the Lorentz commutation relation. [Li,Lj]=i/4*∈iab∈jcd(gbcJad-gacJbd-gbdJac+gadJbc) How can I obtain [Li,Lj]=i∈ijkLk from it?
  21. B

    What is the value of the second term in the commutator for an N particle system?

    I have insertet the equations for H and P in the relation for the commutator which gives $$[H,P] = [\sum_{n=1}^N \frac{p_n^2}{2m_n} +\frac{1}{2}\sum_{n,n'}^N V(|x_n-x_n'|),\sum_{n=1}^N p_n] \\ = [\sum_{n=1}^N \frac{p_n^2}{2m_n},\sum_{n=1}^N p_n]+\frac{1}{2}[\sum_{n,n'}^N...
  22. Haynes Kwon

    I Compatible observables and commutator

    Is commutator being zero for two operators the same statement as the two observables are compatible?
  23. S

    Bra-ket of uncertainty commutator (Sakurai 1.18)

    It's easy to show that ##[\Delta A, \Delta B] = [A,B]##. I'm specifically having issues with evaluating the bra-ket on the RHS of the uncertainty relation: ##\langle \alpha |[A,B]|\alpha\rangle = \langle \alpha |\Delta A \Delta B - \Delta B \Delta A|\alpha\rangle## The answer is supposed to be...
  24. Quix270

    Are there any DC generators without brushes and a commutator?

    Are any generator that produce dc current without brushes or commutator?
  25. P

    I Quantum Computing - projection operators

    Assume ##P_1## and ##P_2## are two projection operators. I want to show that if their commutator ##[P_1,P_2]=0##, then their product ##P_1P_2## is also a projection operator. My first idea was: $$P_1=|u_1\rangle\langle u_1|, P_2=|u_2\rangle\langle u_2|$$ $$P_1P_2= |u_1\rangle\langle...
  26. JuanC97

    I Is my reasoning about commutators of vectors right?

    Hello guys, I have a question regarding commutators of vector fields and its pushforwards. Let me define a clockwise rotation in the plane \,\phi:\mathbb{R}^2\rightarrow\mathbb{R}^2 \,.\; [\,\partial_x\,,\,\partial_y\,]=0 \,, \;(\phi_{*}\partial_x) = \partial_r and \,(\phi_{*}\partial_y) =...
  27. W

    I Proving Commutator Identity for Baker-Campbell-Hausdorff Formula

    I'm having a little trouble proving the following identity that is used in the derivation of the Baker-Campbell-Hausdorff Formula: $$[e^{tT},S] = -t[S,T]e^{tT}$$ It is assumed that [S,T] commutes with S and T, these being linear operators. I tried opening both sides and comparing terms to no...
  28. Zhang Bei

    I The Commutator of Vector Fields: Explained & Examples

    Hi, I'm just starting to read Wald and I find the notion of the commutator hard to grasp. Is it a computation device or does it have an intuitive geometric meaning? Can anyone give me an example of two non-commutative vector fields? Thanks!
  29. A

    Commutator group in the center of a group

    Homework Statement [G,G] is the commutator group. Let ##H\triangleleft G## such that ##H\cap [G,G]## = {e}. Show that ##H \subseteq Z(G)##. Homework EquationsThe Attempt at a Solution In the previous problem I showed that ##G## is abelian iif ##[G,G] = {e}##. I also showed that...
  30. F

    A Invariance of Commutator Relations

    Does anybody know of examples, in which groups defined by ##[\varphi(X),\varphi(Y)]=[X,Y]## are investigated? The ##X,Y## are vectors of a Lie algebra, so imagine them to be differential operators, or vector fields, or as physicists tend to say: generators. The ##\varphi## are thus linear...
  31. W

    How to Prove the Commutator Relationship for Angular Momentum Operators?

    Homework Statement Show that ##[\hat{L} \cdot \vec{a}, \hat{L} \cdot \vec{b}] = i \hbar \hat{L} \cdot (\vec{a} \times \vec{b})## Homework Equations ##[\hat{L}_i, \hat{L}_j]= i \hbar \epsilon_{ijk} \hat{L}_k ## The Attempt at a Solution [/B] Maybe a naive attempt, but it has been a while. I...
  32. Rabindranath

    Angular momentum operator for 2-D harmonic oscillator

    1. The problem statement I want to write the angular momentum operator ##L## for a 2-dimensional harmonic oscillator, in terms of its ladder operators, ##a_x##, ##a_y##, ##a_x^\dagger## & ##a_y^\dagger##, and then prove that this commutes with its Hamiltonian. The Attempt at a Solution I get...
  33. C

    I Can [A,B^n] always equal 0 if [A,B] equals 0?

    This is not a homework problem. It was stated in a textbook as trivial but I cannot prove it myself in general. If [A,B]=0 then [A,B^n] = 0 where n is a positive integer. This seems rather intuitive and I can easily see it to be true when I plug in n=2, n=3, n=4, etc. However, I cannot prove it...
  34. Pencilvester

    I Commutator of two vector fields

    Hello PF, I was reading Carroll’s definition of the commutator of two vector fields in “Spacetime and Geometry”, and I’m having (I think) a simple case of notational confusion. He says for two vector fields, ##X## and ##Y##, their commutator can be defined by its action on a scalar function...
  35. binbagsss

    Delta/metric question (context commutator poincare transf.)

    Homework Statement Homework Equations [/B] I believe that ##\frac{\partial x^u}{\partial x^p} =\delta ^u_p ## (1) ##\implies ## (if ##\delta^a_b ## is a tensor, I'm not sure it is?) : ##\frac{\partial x_u}{\partial x^p} = g_{au} \delta ^a_p ## (2) The Attempt at a Solution [/B] sol...
  36. T

    Paschen back effect and commutator [J^2,Lz]

    Homework Statement I have been given a question on how the commutator relates to the paschen back effect the exact question is as follows Calculate the commutator ##[J^2,L_z]## where ##\vec{J}=\vec{L}+\vec{S}## and explain the relevance of this with respect to the paschen back effect Homework...
  37. P

    A Compute Commutator of Covariant Derivative & D/ds on Vector Fields

    Hi, let ##\gamma (\lambda, s)## be a family of geodesics, where ##s## is the parameter and ##\lambda## distinguishes between geodesics. Let furthermore ##Z^\nu = \partial_\lambda \gamma^\nu ## be a vector field and ##\nabla_\alpha Z^\mu := \partial_\alpha Z^\mu + \Gamma^\mu_{\:\: \nu \gamma}...
  38. L

    A Is [\vec{p}^2, \vec{p} \times \vec{L}] Equal to Zero?

    Is there some easy way to see that [\vec{p}^2, \vec{p} \times \vec{L}] is equal zero? I use component method and got that.
  39. SemM

    A The meaning of the commutator for two operators

    Hi, what is the true meaning and usefulness of the commutator in: \begin{equation} [T, T'] \ne 0 \end{equation} and how can it be used to solve a parent ODE? In a book on QM, the commutator of the two operators of the Schrödinger eqn, after factorization, is 1, and this commutation relation...
  40. A

    All possible inequivalent Lie algebras

    Homework Statement How can you find all inequivalent (non-isomorphic) 2D Lie algebras just by an analysis of the commutator? Homework Equations $$[X,Y] = \alpha X + \beta Y$$ The Attempt at a Solution I considered three cases: ##\alpha = \beta \neq 0, \alpha = 0## or ##\beta = 0, \alpha =...
  41. V

    I Commutator of p and x/r: Elegant Derivation in Position Basis

    This question came up in this thread: <https://www.physicsforums.com/threads/how-to-factorize-the-hydrogen-atom-hamiltonian.933842/#post-5898454> In the course of answering the OP's question, I came across the commutator $$ \left[ p_k, \frac{x_k}{r} \right] $$ where ##r = (x_1 + x_2 +...
  42. Milsomonk

    Commutator of the Dirac Hamiltonian and gamma 5

    Homework Statement Show that in the chiral (massless) limit, Gamma 5 commutes with the Dirac Hamiltonian in the presence of an electromagnetic field. Homework EquationsThe Attempt at a Solution My first question is whether my Dirac Hamiltonian looks correct, I constructed it by separating the...
  43. M

    A Commutation and Non-Linear Operators

    Suppose ##A## is a linear operator ##V\to V## and ##\mathbf{x} \in V##. We define a non-linear operator ##\langle A \rangle## as $$\langle A \rangle\mathbf{x} := <\mathbf{x}, A\mathbf{x}>\mathbf{x}$$ Can we say ## \langle A \rangle A = A\langle A \rangle ##? What about ## \langle A \rangle B =...
  44. tomdodd4598

    I Problem with Commutator of Gauge Covariant Derivatives?

    Hi there, I have just read that the gauge field term Fμν is proportional to the commutator of covariant derivatives [Dμ,Dν]. However, when I try to calculate this commatator, taking the symmetry group to be U(1), I get the following: \left[ { D }_{ \mu },{ D }_{ \nu } \right] =\left( {...
  45. binbagsss

    QFT Klein Gordon Theory, momentum commutator computation

    Homework Statement Homework EquationsThe Attempt at a Solution [/B] I think I understand part b) . The idea is to move the operator that annihilates to the RHS via the commutator relation. However I can't seem to get part a. I have: ## [ P^u, P^v]= \int \int \frac{1}{(2\pi)^6} d^3k d^3 k'...
  46. binbagsss

    Quantum Theory, propagator and causality, commutator

    Homework Statement Question: To find/ explain why there exists a continuous lorentz transformation that flips the sign for space-like separation but not time-like. Homework Equations Signature ## (-,+,+...) ## Definition of lorentz transformation: ##x^u=\lambda^u_v x^v ##...
  47. S

    How Does the Darwin Term Relate to Commutators in Quantum Mechanics?

    Homework Statement I am trying to fill in the steps between equations in the derivation of the coordinate representation of the Darwin term of the Dirac Hamiltonian in the Hydrogen Fine Structure section in Shankar's Principles of Quantum Mechanics. $$ H_D=\frac{1}{8 m^2...
  48. M

    Angular momentum commutation relations

    Homework Statement Show that ##|l, m\rangle## for ##l=1## vanishes for the commutator ##[l_i^2, l_j^2]##. Homework Equations ##L^2 = l_1^2 + l_2^2 + l_3^2## and ##[l_i^2,L^2]=0## The Attempt at a Solution I managed to so far prove that ##[l_1^2, l_2^2] = [l_2^2, l_3^2] = [l_3^2, l_1^2]##. I...
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