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...
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)...
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...
Homework Statement
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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 +...
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.
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...
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
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...
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...
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...
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...
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 =...
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...
Homework Statement
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I am looking at this document. http://www.math.columbia.edu/~woit/notes3.pdf
Homework Equations
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ad(x)y = [x,y]
Ad(X) = gXg-1
The Attempt at a Solution
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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)...
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...
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...
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...