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  1. N

    Lie Bracket and Cross-Product

    OK. Thanks to you both. I think I understand it now.
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    Lie Bracket and Cross-Product

    Ok. Thanks for your response. The example I am using is from this video here starting at 12 mins and continuing here. Here he talks about tangents to the sphere with the Lie bracket being another tangent to the sphere which is at odds with the cross product which would produce a vector normal...
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    Lie Bracket and Cross-Product

    Prove that for a 2 sphere in R3 the Lie bracket is the same as the cross product using the vector: X = (y,-x,0); Y = (0,z-y) [X,Y] = JYX - JXY where the J's are the Jacobean matrices. I computed JYX - JXY to get (-z,0,x). I computed (y,-x,0) ^ (0,z,-y) and obtained (xy,y2,yz) = (z,0,x)...
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    Clebsch-Gordan Decomposition for 6 x 3

    Yes, I am familiar and recognize that Young tableaux is easier for practical purposes. However, this approach should give the same result, yes?
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    Clebsch-Gordan Decomposition for 6 x 3

    Thanks. Yes, I am familiar with using the ladder operators. I was more focused on the procedure outlined using the 3 ⊗ 8 by Georgi (LIe Algebras in Particle Physics page 143) and also here https://physics.stackexchange.com/questions/102554/tensor-decomposition-under-mathrmsu3 . I was trying...
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    Clebsch-Gordan Decomposition for 6 x 3

    Homework Statement [/B] I am trying to get the C-G Decomposition for 6 ⊗ 3. 2. Homework Equations Neglecting coefficients a tensor can be decomposed into a symmetric part and an antisymmetric part. For the 6 ⊗ 3 = (2,0) ⊗ (1,0) this is: Tij ⊗ Tk = Qijk = (Q{ij}k + Q{ji}k) + (Q[ij]k +...
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    Isospin Doublet Derivation Using Clebsch-Gordan Coefficients

    OK. Thanks. I think my problem was that I was trying to use the ladder operators to get those states. . Using the C-G coefficients from tables makes more sense.
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    Isospin Doublet Derivation Using Clebsch-Gordan Coefficients

    Homework Statement I am trying to improve my understanding of the Clebsch-Gordan coefficients. I am looking at page 5 of the following document https://courses.physics.illinois.edu/phys570/fa2013/chapter3.pdf Homework Equations I have derived the result for the I = 3/2 quadruplet but am...
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    Left invariant vector field under a gauge transformation

    Homework Statement For a left invariant vector field γ(t) = exp(tv). For a gauge transformation t -> t(xμ). Intuitively, what happens to the LIVF in the latter case? Is it just displaced to a different point in spacetime or something else? Homework Equations The Attempt at a Solution
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    Lie Bracket for Group Elements of SU(3)

    I think I may be confusing myself. I think what you are saying is that although the commutator of 2 vector fields results in a third vector field on the manifold, that field at a given point is, by definition, assigned to a tangent space (as are the original fields). In this sense trying to...
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    Lie Bracket for Group Elements of SU(3)

    Homework Statement Determine the Lie bracket for 2 elements of SU(3). Homework Equations [X,Y] = JXY - JYX where J are the Jacobean matrices The Attempt at a Solution I exponentiated λ1 and λ2 to get X and Y which are 3 x 3 matrices.. If the group elements are interpreted as vector...
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    SU(3) Cartan Generators in Adjoint Representation

    I am still struggling with interpreting the results I get. For ad( H1) and ad( H2) I get: Diagonalization using WolframAlpha gives: ad(H1): diag(-1,-1/2,-1/2,,0,0,1/2,1/2,1) ad(H2): diag(0,0,0,0,-√3/2,-√;-3/2,√;3/2,√;3/2) 1. All the weights are there but not in the correct order. 2...
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    SU(3) Cartan Generators in Adjoint Representation

    Yes, your point is well taken. I realize that this is a tedious approach and the eigenvalues can be found more easily using [Hi,Eα] = αiEα where Eα are the I, U and V spin operators. However, my approach should work correct? In the defining rep the 3 x 3 Cartan generators share the same...
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    SU(3) Cartan Generators in Adjoint Representation

    I am trying to work out the weights of the adjoint representation of SU(3) by calculating the 2 Cartan generators as follows: I obtain the structure constants from λa and λ8 using: [λa,λb] = ifabcλc I get: f312 = 1 f321 = -1 f345 = 1/2 f354 = -1/2 f367 = -1/2 f376 = 1/2 f845 = √3/2 f854 =...
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    Adjoint representation of SU(2)

    Thanks. I worked through your paper and understand how you got to part 2 (12) and (13). The part 1 am stuck on now is solving Ad(expX) = exp(ad(X)) for the matrices you have calculated. When I try to calculate the matrix exponential I get strange results. Any pointers you could give would be...
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    Adjoint representation of SU(2)

    Homework Statement [/B] I am looking at this document. http://www.math.columbia.edu/~woit/notes3.pdf Homework Equations [/B] ad(x)y = [x,y] Ad(X) = gXg-1 The Attempt at a Solution [/B] I understand how ad(S1) and X is found but I don't understand what g and g-1 to use to find Ad(X)...
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    Dirac notation for conjugacy class

    OK. Sorry about the formatting in the previous response. I think I get it now. The product of λ with a column vector gives another column vector. This column vector gets multiplied by the complex conjugate matrix which can be written as a column or row vector. Either way this operation...
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    Dirac notation for conjugacy class

    OK. I think I may be confusing things. \[ \left(\begin{pmatrix} r* & g* & b* \end{pmatrix}λi\begin{pmatrix} r \\ g \\ b \end{pmatrix}\right)\] appears to produce the QM superposition states for the gluons. if g has to be a 3 x 3 invertible matrix then this has to be the bra-ket...
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    Dirac notation for conjugacy class

    Sorry, x should be λ. Thus, Ad(g)λ = gλg-1. Ad is described here https://en.wikipedia.org/wiki/Adjoint_representation. Thanks.
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    Dirac notation for conjugacy class

    Is the RHS of the conjugate relationship Ad(g)x = gxg-1 from the Lie algebra equivalent to: <g|λ|g> In the Dirac notation of quantum mechanics? I am looking at this in the context of gluons where g is a 3 x 1 basis matrix consisting of components r,g,b, g-1 is a 1 x 3 matrix consisting of...
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