What is Commutation: Definition and 220 Discussions

In law, a commutation is the substitution of a lesser penalty for that given after a conviction for a crime. The penalty can be lessened in severity, in duration, or both. Unlike most pardons by government and overturning by the court (a full overturning is equal to an acquittal), a commutation does not affect the status of a defendant's underlying criminal conviction.
Although the concept of commutation may be used to broadly describe the substitution of a lesser criminal penalty for the original sentence, some jurisdictions have historically used the term only for the substitution of a sentence of a different character than was originally imposed by the court. For example, the substitution of a sentence of parole for the original sentence of incarceration. A jurisdiction that uses that definition of commutation would use another term, such as a remission, to describe a reduction of a penalty that does not change its character.A commutation does not reverse a conviction and the recipient of a commutation remains guilty in accordance with the original conviction. For example, someone convicted of capital murder may have their sentence of death commuted to life imprisonment, a lessening of the punishment that does not affect the underlying criminal conviction, as may occur on a discretionary basis or following upon a change in the law or judicial ruling that limits or eliminates the death penalty.In some jurisdictions a commutation of sentence may be conditional, meaning that the convicted person may be required to abide by specified conditions or may lose the benefit of the commutation. The conditions must be lawful and reasonable, and will typically expire when the convicted completes any remaining portion of his or her sentence. For example, the pardon may be conditioned upon the person's being a law-abiding citizen, such that if the beneficiary of the commutation commits a new crime before the condition expires the original sentence may be restored.

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  1. D

    A How Does Position Interact with Spin Angular Momentum in Quantum Mechanics?

    I know how position and momentum commute, but now I have the spin angular momentum operator involved as well as a dot product. Specifically, what would the commutation [x,S·p] be?
  2. T

    I Variant of Baker-Campbell-Hausdorff Formula

    Hi all, I was wondering if there was a clean/closed form version of the following expression: $$e^{X+Y}Ze^{-(X+Y)} - e^{Y}e^{X}Ze^{-X}e^{-Y}$$ where ##X,Y,Z## are matrices that don't commute with each other. I know of the BCH identity ##e^{X}Ye^{-X} = Y + [X,Y] + \frac{1}{2!}[X,[X,Y]] +...
  3. Hibarikyoya

    Prove the following commutation relations

    I tried in this way: $$[J^k, J^i] = \frac{1}{M^2}\ (L^{-1})^k_{\mu}\ (L^{-1})^i_{\nu}\ [W^{\mu}, W^{\nu}] $$ $$ = \frac{1}{M^2}\ (L^{-1})^k_{\mu}\ (L^{-1})^i_{\nu}\ (-i) \epsilon^{\mu \nu \rho \sigma} W_{\rho} P_{\sigma}.$$ At this point I had no idea how to going on with the calculation. Can...
  4. bella987

    Deriving the commutation relations of the Lie algebra of Lorentz group

    This is the defining generator of the Lorentz group which is then divided into subgroups for rotations and boosts And I then want to find the commutation relation [J_m, J_n] (and [J_m, K_n] ). I'm following this derivation, but am having a hard time to understand all the steps: especially...
  5. P

    Commutation relations between Ladder operators and Spherical Harmonics

    I've tried figuring out commutation relations between ##L_+## and various other operators and ##L^2## could've been A, but ##L_z, L^2## commute. Can someone help me out in figuring how to actually proceed from here?
  6. D

    A What happens when you commute Sx and Sz in spin operators?

    So we know [Sz, Sx] = ihbar Sy (S with hats on) so what happens if you get [Sx, Sz]? Is it the same result? Just trying to work out if I've gone wrong somewhere
  7. E

    I Commutation relations for an interacting scalar field

    Hi there, In his book "Quantum field theory and the standard model", Schwartz assumes that the canonical commutation relations for a free scalar field also apply to interacting fields (page 79, section 7.1). As a justification he states: I do not understand this explanation. Can you please...
  8. D

    I Commutation of Beam splitter operator with Displacement operator

    I have a beam splitter operator (a†)b +(b†)a. Does it commute with exp(αâ†-α*â). Here a and ↠are creation and lowering operator
  9. Ssnow

    Quantum References for generalized canonical commutation relations

    Hi to all, I ask if somebody of the Physics community know good references for article where the author works with generalized canonical commutation relations ( I mean that the author works with ##[x,p]=ic\hbar## with ##c## a real constant instead of ##[x,p]=i\hbar##). Thank you for the answers...
  10. 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...
  11. snypehype46

    Exercise involving Dirac fields and Fermionic commutation relations

    I'm trying to the following exercise: I've proven the first part and now I'm trying to do the same thing for fermions. The formulas for the mode expansions are: What I did was the following: $$\begin{align*} \sum_s \int d\tilde{q} \left(a_s(q) u(q,s) e^{-iq \cdot x}+ b_s^\dagger(q) v(q,s)...
  12. JD_PM

    Deriving Fourier Transform of Operators for Relativistic Quantum Field Theory

    I would say we first need to take the inverse Fourier transform of ##\chi## and associated quantities i.e. \begin{equation*} \chi_{\vec k} = \int d^3 \vec x \left( a_{\vec k} \chi e^{-i \vec k \cdot \vec x} + a^{\dagger}_{\vec k}\chi^* e^{i \vec k \cdot \vec x} \right) \tag{2}...
  13. J

    I Commutation between covariant derivative and metric

    First, we shall mention that it is known that the covariant derivative of the metric vanishes, i.e ##\nabla_i g_{mn} = 0##. Now I want tro prove the following: $$ \nabla_i A_k = g_{kn}\nabla_i A^n$$ The demonstration I encounter takes advantage of the Leibniz rule: $$ \nabla_i A_k = \nabla_i...
  14. Mr_Allod

    Commutation and Measurement of Observables

    Hello there, I am having trouble with part b. of this problem. I've solved part a. by calculating the commutator of the two observables and found it to be non-zero, which should mean that ##\hat B## and ##\hat C## do not have common eigenvectors. Although calculating the eigenvectors for each...
  15. abivz

    I Obtaining the Dirac function from field operator commutation

    Hi everyone, I'm new to PF and this is my second post, I'm taking a QFT course this semester and my teacher asked us to obtain: $$[\Phi(x,t), \dot{\Phi}(y,t) = iZ\delta^3(x-y)]$$ We're using the Otto Nachtman: Elementary Particle Physics but I've seen other books use this notation: $$[\Phi(x,t)...
  16. abivz

    I QFT - Field operator commutation

    Hi everyone, I'm taking a QFT course this semester and we're studying from the Otto Nachtman: Texts and Monographs in Physics textbook, today our teacher asked us to get to the equation: [Φ(x,t),∂/∂tΦ(y,t)]=iZ∂3(x-y) But I am unsure of how to get to this, does anyone have any advice or any...
  17. dykuma

    Hermitian Matrix and Commutation relations

    I think I roughly see what's happening here. > First, I will assume that AB - BA = C, without the complex number. >Matrix AB equals the transpose of BA. (AB = (BA)t) >Because AB = (BA)t, or because of the cyclic property of matrix multiplication, the diagonals of AB equals the diagonals of...
  18. R

    A Commutation between spin-operator and creation operator(QFT)

    Hi, so I'm currently reading the book "QFT for the gifted amateur", and doing the exercises. In exercise 14.2, which in itself is fine, the authors say that you can show using Noether's theorem that for a transverse polarized photon of momentum q, the z-component of the spin operator obeys the...
  19. JD_PM

    A Commutation relations between HO operators | QFT; free scalar field

    I am getting started in applying the quantization of the harmonic oscillator to the free scalar field. After studying section 2.2. of Tong Lecture notes (I attach the PDF, which comes from 2.Canonical quantization here https://www.damtp.cam.ac.uk/user/tong/qft.html), I went through my notes...
  20. Diracobama2181

    Commutation Problem: Eigenvectors, Basis & Hamiltonian

    a) This would be true whenever |a_n> is an eigenvector of B_i. b) If this holds true for each eigenvector, then B_i and B_j must share the same basis. Therefore, they must commute. Is this reasoning correct? C) Despite commuting with the hamiltonian. the energy states can be degenerate, which I...
  21. Diracobama2181

    I Hamiltonian Commutation Question

    Why is it the case that when some operators commute with the Hamiltonian (let's say A and ), it implies A and B commute, but even when each angular momentum component commutes with the Hamiltonian, it does not imply each the angular momentum components commute with each other?
  22. Diracobama2181

    Does $$S_1^x$$ commute with $$S^2$$?

    $$H$$ can be rewritten as $$H=\frac{1}{2}(S^2-S_{1}^2-S_{2}^2-S_{3}^2-S_{4}^2)$$. Let's focus on the x component, $$J^x=\sum_{i}S_i^x$$. Now $$S_1^x$$ commutes with $$S^2_1, S^2_2, S^2_3, S^2_4$$, but does it commute with $$S^2$$? If not, what is the exact relation between $$S^2$$ and $$S_1^x$$?
  23. binbagsss

    A Reading canonical commutation relations from the action (QHE)

    Hi I'm sure i understood this a week or so ago, and I've forgot the idea now. I'm just really confused, again, how you read the commutator relationships of from the action ? many thanks (source http://www.damtp.cam.ac.uk/user/tong/qhe/five.pdf)
  24. M

    How Do Lorentz Group Commutation Relations Apply to Spin Matrices?

    Homework Statement Prove that the sets ##(S_{\mu\nu})_L## and ##(S_{kl})_R##, where $$ \left( S _ { k \ell } \right) _ { L } = \frac { 1 } { 2 } \varepsilon _ { j k \ell } \sigma _ { j } = \left( S _ { k \ell } \right) _ { R } \quad\text{and}\quad \left( S _ { 0 k } \right) _ { L } = \frac {...
  25. Z

    What Are the Anti-Commutation Relations of SUSY Generators?

    I hope I put this in the correct section of this forum, I apologize if I didn't. Homework Statement :[/B] It is well known that the generators $$ Q_\alpha = \frac{\partial}{\partial \theta^\alpha} - i \sigma^\mu_{\alpha \dot \beta} \bar{\theta}^\dot{\beta} \partial_\mu $$ and $$...
  26. Another

    Question commutation in quantum mechanics

    Homework Statement Show that ##[L_{x}^2,L_{y}^2]=[L_{y}^2,L_{z}^2]=[L_{z}^2,L_{x}^2]## Homework Equations ##L^2 = L_{x}^2+L_{y}^2+L_{z}^2## ##L_x = yp_z-zp_y## ##L_y = zp_x-xp_z## ##L_z = xp_y-yp_x## ##[x_i,p_j]=iħδ_{ij}## ##[L_x,L_y]=iħL_z## ##[L_y,L_z]=iħL_x## ##[L_z,L_x]=iħL_y##...
  27. B

    A Covariance of equal time commutation relations

    I have a question regarding the covariance of the equal time commutation relations in relativistic quantum field theory. In the case of a scalar field one has that the commutator is (see Peskin, pag. 28 eq. (2.53) ) $ [\phi(0), \phi(y)] = D(-y) - D(y) $ is an invariant function, which is zero...
  28. H

    A Commutation relations for bosons and fermions

    For the free boson, the field operators satisfies the commutation relation, $${\varphi}_{x'}{\varphi}_{x} - {\varphi}_{x}{\varphi}_{x'} = 0$$ at equal times. While the fermions satisfies, $${\psi}_{x'}{\psi}_{x} + {\psi}_{x}{\psi}_{x'} = 0$$ at equal times. I interpret ##{\varphi}_{x}## and...
  29. A

    Quantum Field Theory, Momentum Space Commutation Relations

    Homework Statement Derive, using the canonical commutation relation of the position space representation of the fields φ(x) and π(y), the corresponding commutation relation in momentum space.Homework Equations [φ(x), π(y)] = iδ3(x-y) My Fourier transforms are defined by: $$ φ^*(\vec p)=\int...
  30. F

    Commutation Relation: Hi Parity Operator?

    hi, do the translation operator commute with parity operator?
  31. SemM

    A Does Commutativity Affect Linearity?

    Hi, I have in a previous thread discussed the case where: \begin{equation} TT' = T'T \end{equation} and someone, said that this was a case of non-linear operators. Evidently, they commute, so their commutator is zero and therefore they can be measured at the same time. What makes them however...
  32. 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...
  33. SemM

    I Solving the Schrödinger eqn. by commutation of operators

    Hi, I noticed that the raising and lowering operators:\begin{equation} A =\frac{1}{\sqrt{2}}\big(y+\frac{d}{dy}\big) \end{equation}\begin{equation} A^{\dagger}=\frac{1}{\sqrt{2}}\big(y-\frac{d}{dy}\big) \end{equation}can be used to solve the eqn HY = EY However I am curious about something...
  34. 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 =...
  35. C

    General commutation relations for quantum operators

    (This is not a homework problem). I'm an undergrad physics student taking my second course in quantum. My question is about operator methods. Most of the proofs for different commutation relations for qm operators involve referring to specific forms of the operators given some basis. For...
  36. TeethWhitener

    I Complex scalar field commutation relations

    I'm trying to derive the commutation relations of the raising and lowering operators for a complex scalar field and I had a question. Let's start with the commutation relations: $$[\varphi(\mathbf{x},t),\varphi(\mathbf{x}',t)]=0$$ $$[\Pi(\mathbf{x},t),\Pi(\mathbf{x}',t)]=0$$...
  37. F

    I Does a field operator always commute with itself?

    In quantum field theory (QFT), the requirement that physics is always causal is implemented by the microcausality condition on commutators of observables ##\mathcal{O}(x)## and ##\mathcal{O}'(y)##, $$\left[\mathcal{O}(x),\mathcal{O}'(y)\right]=0$$ for spacelike separations. Intuitively, I've...
  38. S

    I Understanding the Dirac Commutation Relations in QFT

    Hello! I am reading Peskin's book on QFT and at a point he wants to show that the Dirac field can't be quantified using this commutation relations: ##[\psi_a(x),\psi_b^\dagger(x)]=\delta^3(x-y)\delta_{ab}## (where ##\psi## is the solution to Dirac equation). I am not sure I understand the math...
  39. D

    I Kinetic and Potential energy operators commutation

    Hi All, Perhaps I am missing something. Schrodinger equation is HPsi=EPsi, where H is hamiltonian = sum of kinetic energy operator and potential energy operator. Kinetic energy operator does not commute with potential energy operator, then how come they share the same wave function Psi? The...
  40. 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...
  41. B

    Commutation Relations, 2D Harmonic Oscillator

    Homework Statement Consider a two-dimensional harmonic oscillator, described by the Hamiltonian ##\hat H_0 = \hbar \omega (\hat a_x \hat a_x ^{\dagger} + \hat a_y \hat a_y^{\dagger} + 1)## Calculate ##\hat H_0 \hat L | n_1, n_2 \rangle## and ##\hat L \hat H_0 |n_1, n_2 \rangle##. What does...
  42. B

    I Angular momentum operator commutation relation

    I am reading a proof of why \left[ \hat{L}_x, \hat{L}_y \right ] = i \hbar \hat{L}_z Given a wavefunction \psi, \hat{L}_x, \hat{L}_y \psi = \left( -i\hbar \right)^2 \left( y \frac{\partial}{\partial z} - z \frac {\partial}{\partial y} \right ) \left (z \frac{\partial \psi}{\partial x} -...
  43. B

    Expectation values and commutation relations

    Homework Statement I am trying to calculate the expectation value of ##\hat{P}^3## for the harmonic oscillator in energy eigenstate ##|n\rangle## Homework EquationsThe Attempt at a Solution [/B] ##\hat{P}^3 = (i \sqrt{\frac{\hbar \omega m}{2}} (\hat{a}^\dagger - \hat{a}))^3 = -i(\frac{\hbar...
  44. D

    Commutation relation using Levi-Civita symbol

    Homework Statement Hi,I have got a question as follow: Compute the commutation relations of the position operator R and the angular momentum L.Deduce the commutation relations of R^2 with the angular momentum L Homework EquationsThe Attempt at a Solution In fact I have got the solutions to...
  45. A

    Canonical commutation relation, from QM to QFT.

    Homework Statement This is a system of n coupled harmonic oscillators in 1 dimension. [/B] Since the distance between neighboring oscillators is ## \Delta x ## one can characterize the oscillators equally well by ## q(x,t) ## instead of ## q_j(t) ##. Then ## q_{j \pm 1} ## should be replaced...
  46. Safinaz

    I Commutation relation of hypercharge and SU(2) generators

    Hi all, I read in Cheng and Li's book "Gauge theory of elementary particle physics" Ch 11, specifically : Eq. (11.46) that the hypercharge commutes with the SU(2) generators, i.e., ##[Q-T_3,T_i]=0##, I'd like to understand what that mean and how this could be proved ?
  47. G

    I Common eigenfuctions for degeneracy

    Hello. I read the textbook and found that common eigenfunctions are even possible for degenerate eigenvalues. Let's say operators A and B commutes and eigenvalue a of operator A is N-fold degenerate, means that there are N linearly independent eigenfunctions having same eigenvalue a. These...
  48. ShadowMeson

    Declare a Commutation Relationship in FeynCalc

    Homework Statement I am a rookie to the QFT extension in Mathematica called FeynCalc, and tried to use that into solving some quiz. Soon I met a problem upon some condition presented in a problem which declares an relation of two same tensor with different indices results in some value when...
  49. Muthumanimaran

    Commutation of Lindblad Operators

    Homework Statement Let operator $$\mathcal{L}_{AD}(\rho)$$ and $$\mathcal{L}_{PD}(\rho)$$ is defined as $$\mathcal{L}_{AD}(\rho)=2a\rho{a}^{\dagger}-a^{\dagger}a\rho-\rho{a^{\dagger}}a$$ and $$\mathcal{L}_{PD}(\rho)=2a^{\dagger}a\rho{a^{\dagger}}a-(a^{\dagger}a)^2\rho-\rho(a^{\dagger}a)^2$$...