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...
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...
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 +...
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
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 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
I have been reading about Rings and Modules. I am trying reconcile my understanding with Lie groups.
Let G be a Matrix Lie group. The group acts on itself by left multiplication, i.e,
Lgh = gh where g,h ∈ G
Which corresponds to a translation by g.
Is this an example of a module over a ring...
I am trying to improve my understanding of Lie groups and the operations of left multiplication and pushforward.
I have been looking at these notes:
https://math.stackexchange.com/questions/2527648/left-invariant-vector-fields-example...
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 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 =...
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)...
I am looking at the generators of the Lorentz group. The literature commonly refers to the generators as
Mij, Ji and Ki and defines:
Ji = (1/2)∈ijkMjk
I am confused about the factor of (1/2) in this equation as I thought that Mij is essentially the same as Ji
This also shows up in
Λ=...
I am trying to figure how one arrives at the following:
dxμ∂ν = ∂xμ/∂xν = δμν
Where,
dxμ is the gradient of the coordinate functions = basis of cotangent space
∂ν = basis of tangent space
I know that dual vectors 'eat' vectors to produce scalars. Is this demonstrated by absorbing d into ∂...
I am trying to figure how to get 1. from 2. and vice versa where the e's are bases for the vector space and θ's are bases for the dual vector space.
1. T = Tμνσρ(eμ ⊗ eν ⊗ θσ ⊗ θρ)
2. Tμνσρ = T(θμ,θν,eσ,eρ)
My attempt is as follows:
2. into 1. gives T = T(θμ,θν,eσ,eρ)(eμ ⊗ eν ⊗ θσ ⊗ θρ)...
I am trying to figure out the Feynman diagrams for the Hierarchy problem. Here's what I have.
(a) is the Higgs self interaction (φ4). It is quadratically divergent with additional logarithmic terms.
(b) is the 1 loop Yukawa correction to the Higgs where F might be the top quark and g is...
I am looking at Appendix A Equation 52 (Loop Integrals and Dimensional Regularization) in Peskin and Schroeder's book.
∫ddk/(2π)d1/(k2 - Δ)2 = Γ(2-d/2)/(4π)2(1/Δ)2-d/2 = (1/4π)2(2/ε - logΔ - γ + log4π)
Can somebody explain how this equation is derived? I would also like to know what the...
I am trying to understand the basics of Renormalization. I have read that β encodes the running coupling and can be expanded as a power series as:
β(g) = ∂g/(∂ln(μ)) = β0g3 + β1g5 + ...
However, I don't understand how this is derived.. I assume that the terms correspond to 1 loop, 2 loops...
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...
Not sure if this is the correct forum but here goes.
I am trying to prove [Ta,Tb] = ifabcTc
Where (Ta)bc = -ifabc and fabcare the structure constants for SU(3).
I picked f123 and generated the three 8 x 8 matrices .. T1, T2 and T3.
The matrices components are all 0 except for,
(T1)23 = -i...
I am confused about conformal transformations on Riemannian manifolds. Here's what I have so far.
1. Under a conformal transformation the metric changes by:
g' -> Ω2g
2. Under a Weyl transformation the metric changes by:
g' -> exp(-2f)g
3. Any 2D Riemann manifold is locally conformally...
I am trying to understand the derivation of the Dirac adjoint. I understand the derivation of the following identities involving Spinors, the Gamma matrices and Lorentz transformations:
(Sμν)† = γ0Sμνγ0
s[Λ] = exp(ΩμνSμν/2)
s[Λ]† = exp(Ωμν(Sμν/2)†)
The part I'm having trouble with is...
I am trying to my head around these two things in the context of string theory. The Polyakov action becomes simpler to solve in the conformal gauge which, as I understand it, makes the manifold locally Ricci flat in 2D. In Professor Susskind's lectures on String Theory he introduces the concept...
I have been watching Prof Susskind's lectures on String Theory. I am trying to understand a little more about D branes. I am trying to reproduce a basic quark-gluon interaction. I have the following:http://www.firstcoasttutors.com/images/pf.jpg [Broken]
I think this might describe the right...
How can I derive the running coupling for the weak interaction. I have found derivations for QED and QCD that involve the β function http://www.physics.umd.edu/courses/Phys741/xji/chapter1.pdf but I can't find anything specific for the WI. Thanks.