I am looking at this document I do not understand how the author gets 5.12 and 5.13 on page 133. I think the matrix of partials should be the transpose of the one shown. Even so I still can't figure out how you get 5.13. Any help would be appreciated.
I am looking at the representation of D4 in ℝ4 consisting of the eight 4 x 4 matrices acting on the 4 vertices of the square a ≡ 1, b ≡ 2, c ≡ 3 and d ≡ 4.
I have proven that the 1-dimensional subspace of D4 in ℝ2 has no proper invariant subspaces and therefore is reducible. I did this in 2...
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)...
How does one go about finding a matrix, U, such that U-1D(g)U produces a block diagonal matrix for all g in G? For example, I am trying to figure out how the matrix (7) on page 4 of this document is obtained.
I am looking at page 2 of this document.https://ocw.mit.edu/courses/chemistry/5-04-principles-of-inorganic-chemistry-ii-fall-2008/lecture-notes/Lecture_3.pdf
How is the transformation matrix, ν, obtained? I am familiar with diagonalization of a matrix, M, where D = S-1MS and the columns of S...
Hello. I just came across this post. I understand how the 2nd term in (4) is to equivalent to the 1st term in the 1st equation i.e. uivjk - ujvik = εijlεlmnunvmk = (δimδjn - δjnδim)umvnk but I am having a mental block regarding how the first term in (4) is equivalent to the 2nd term in the...
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...
OK. So is it fair to generalize by saying that ene-1 = n applies to multiplicative groups where e = 1 and that e + n + (-e) = n applies for additive groups where e = 0? This makes intuitive sense but most of the literature I have read seems to focus on the multiplicative case.
Homework Statement
I am looking at the quotient group G = Z/3Z which is additive and abelian. The equivalence classes are:
[0] = {...,0,3,6,...}
[1] = {...,1,4,7,...}
[2] = {...,2,5,8,...}
I want to prove [0] is a normal subgroup, N, by showing gng-1 = n' ∈ N for g ∈ G and n ∈ N. Since G...
I interpret (gU)∩U = 0 to mean all non trivial elements of G move U outside of itself. Can I interpret this as being equivalent to the disjoint union of sets in the covering space?
I am confused about this statement.
OK. So covering maps are always quotient maps but quotient maps are not always covering maps. In the latter case I have read that q:X -> X/G is a covering map if and only if G acts properly discontinuously on X. Properly discontinuously means that for every x∈X there is a neighborhood U...
I am newbie to topology and trying to understand covering maps and quotient maps. At first sight it seems the two are closely related. For example SO(3) is double covered by SU(2) and is also the quotient SU(2)/ℤ2 so the 2 maps appear to be equivalent. Likewise, for ℝ and S1. However, I...
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
Thank you. I think this helps a lot. I realize now that modules and rings were somewhat of a red herring. However, I learned something that I didn't know before. Again, thanks for your patience and time.
Maybe abstract is the wrong term to use. The tangent space has the structure of the Lie algebra which seems more elaborate than just vector addition and scalar multiplication. That is the source of my confusion. I understand the relevance of the LIVF but I can't connect the dots between...
OK. I recognize the fact there is some 'backwards and forwards' between the 2 disciplines. I am a retired EE so am somewhat impartial. So the bottom line is that In the first case, forget about rings and modules since g, h ∈ G is not compatible with g ∈ G and h ∈ M. This makes perfect...
Thank you for your reply (and patience!).
My use of matrix ring may be incorrect. I wanted it to mean that the elements of the ring are matrices and these matrices act on vectors that are elements of the module. I don't know if that would effect your answer.
Generic means a space equipped...
OK. I forgot about the requirement for an Abelian group structure - a matrix ring operating on module elements that are vectors would be fine because the vectors form an Abelian group. I think I may also be confusing 'generic' vector spaces associated with abstract algebra with tangent...