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    A Pullback of the metric from R3 to S2

    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.
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    I 2 and 3 dimensional invariant subspaces of R4

    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...
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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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    A Block Diagonalization - Representation Theory

    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.
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    I Block Diagonal Matrix and Similarity Transformation

    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...
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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

    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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    Proving Subgroup of Z/3Z

    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...
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    I Equivalence of Covering Maps and Quotient Maps

    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...
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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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    I Rings, Modules and the Lie Bracket

    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...
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    I Computation of the left invariant vector field for SO(3)

    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...
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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 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)

    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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    I Confused about Lorentz Generators

    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=...
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    A Tangent and Cotangent Bases

    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 ∂...
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    B Tensor Product, Basis Vectors and Tensor Components

    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)...
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    A Hierarchy problem and Feynmam diagrams

    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...
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    A Dimensional Regularization of Feynman Integrals

    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...
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    A Perturbative expansion of Beta function - Renormalization

    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...
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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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    I Bases for SU(3) Adjoint representation

    What are the bases for the adjoint representation for SU(3)?
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    I Adjoint representation of SU(3)

    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...
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    A Confusion regarding conformal transformations

    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...
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    A Spinor transformations

    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...
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    A Ricci Flow and Weyl Transformations

    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...
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    A D Branes and Gluon-Quark Interaction.

    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...
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    Canonical momentum for Dirac adjoint field

    I read that the canonical momentum for Dirac adjoint field is zero. Why is that?
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    Running Coupling for Weak Interaction

    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.
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