Read about group theory | 82 Discussions | Page 1

Summary:: Good introductory book about Group Theory? Hi, I am looking for a good introductory book about Group Theory for physicists.

The group ##\rm{O(3)}## is the group of orthogonal ##3 \times 3## matrices with nine elements and dimension three which is constrained by the condition, $$a_{ik}a_{kj} = \delta_{ij}$$ where ##a_{ik}## are elements of the matrix ##\rm{A} \in O(3)##. This condition gives six constraints (can be...

I am trying to get a foothold on QFT using several books (Lancaster & Blundell, Klauber, Schwichtenberg, Jeevanjee), but sometimes have trouble seeing the forest for all the trees. My problem concerns the equation of QED in the form $$ \mathcal{L}_{Dirac+Proca+int} = \bar{\Psi} ( i \gamma_{\mu}...

hi guys i saw this problem : if G is a group and a,b belongs to G and O(a) = e , b.a =a.b^2 then find O(b) , but i want to tackle this problem using Cayley diagrams , so my attempt is as following : $$ba =ab^{2}$$ then i might assume b as flipping , a as rotation : $$ fr = rf^{2}$$ then...

In Griffith's Introduction to Elementary Particles, he provides a very cursory introduction to group theory at the start of chapter four, which discusses symmetries. He introduces SO(n) as "the group of real, orthogonal, n x n matrices of determinant 1 is SO(n); SO(n) may be thought of as the...

I'm solving these problems concerning the SU(4) group and I've reached the point where I have determined the Cartan matrix of SU(4), its inverse and the weight schemes for (1 0 0) and (0 1 0) highest weight states. How do I decompose the (1 0 0) and (0 1 0) into irreps of SU(3) x U(1) using...

I was studying mathematical logic and came across this statement of group theory I'm having a hard time in understanding it. I have concluded that ##G## is any set but not an empty one, ##\circ## is a function having input as two variables (both variables are from set...

Consider the pseudoscalar and vector meson family, as well as the baryon J = 1/2 family and baryon J = 3/2 family. Within each multiplet, for each particle state write down its complete set of quantum numbers, its mass, and its quark state content. Furthermore, for each multiplet draw the (Y...

Summary: if we use up, down and staring quarks and their own antiparticle we can create the Eightfold way and understand mesons by the hyper charge and isospin projections. I don't understand how the conjugate representation of SU(3) allows us to create a vector space of dimension 3, while...

Hello, I am an undergraduate who has taken basic linear algebra and ODE. As for physics, I have taken an online edX quantum mechanics course. I am looking at studying some of the necessary math and physics needed for QFT and particle physics. It looks like I need tensors and group theory...

Hello, I am newish in group theory so sorry if anything in the following is not entirely correct. In general, one can anticipate if a matrix element <i|O|j> is zero or not by seeing if O|j> shares any irreducible representation with |i>. I know how to reduce to IRs the former product but I...

Looking into the infinitesimal view of rotations from Lie, I noticed that the vector cross product can be written in terms of the generators of the rotation group SO(3). For example: $$\vec{\mathbf{A}} \times \vec{\mathbf{B}} = (A^T \cdot J_x \cdot B) \>\> \hat{i} + (A^T \cdot J_y \cdot B)...

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 +...

Hi! I'm doing my master thesis in AdS/CFT and I've read several times that "Fields transforms in the adjoint representation" or "Fields transforms in the fundamental representation". I've had courses in Advanced mathematics (where I studied Group theory) and QFTs, but I don't understand (or...

Homework Statement Let ##G## be a group of order ##2p## with p a prime and odd number. a) We suppose ##G## as abelian. Show that ##G \simeq \mathbb{Z}/2p\mathbb{Z}## Homework Equations The Attempt at a Solution Intuitively I see why but I would like some suggestion of what trajectory I could...

Homework Statement [G,G] is the commutator group. Let ##H\triangleleft G## such that ##H\cap [G,G]## = {e}. Show that ##H \subseteq Z(G)##. Homework Equations The Attempt at a Solution In the previous problem I showed that ##G## is abelian iif ##[G,G] = {e}##. I also showed that...

Homework Statement Let ##G## be a group. Let ##H \triangleleft G## and ##K \leq G## such that ##H\subseteq K##. a) Show that ##K\triangleleft G## iff ##K/H \triangleleft G/H## b) Suppose that ##K/H \triangleleft G/H##. Show that ##(G/H)/(K/H) \simeq G/K## Homework Equations The three...

It goes without saying that theoretical physics has over the years become overrun with countless distinct - yet sometimes curiously very similar - theories, in some cases even dozens of directly competing theories. Within the foundations things can get far worse once we start to run into...

Homework Statement Let m ≥ 3. Show that $$D_m \cong \mathbb{Z}_m \rtimes_{\varphi} \mathbb{Z}_2 $$ where $$\varphi_{(1+2\mathbb{Z})}(1+m\mathbb{Z}) = (m-1+m\mathbb{Z})$$ Homework Equations I have seen most basic concepts of groups except group actions. Si ideally I should not use them for this...

I teach group theory for physicists, and I like to teach it following some papers. In general my students work with condensed matter, so I discuss group theory following these papers: [1] Group Theory and Normal Modes, American Journal of Physics 36, 529 (1968) [2] Nonsymmorphic Symmetries and...

Homework Statement I am translating so bear with me. We have two group homomorphisms: α : G → G' β : G' → G Let β(α(x)) = x ∀x ∈ G Show that 1)β is a surjection 2)α an injection 3) ker(β) = ker(α ο β) (Here ο is the composition of functions.) Homework Equations This is from a...

I do have a fair amount of visual/geometric understanding of groups, but when I start solving problems I always wind up relying on my algebraic intuition, i.e. experience with forms of symbolic expression that arise from theorems, definitions, and brute symbolic manipulation. I even came up with...

Why does the image of elements in a homomorphism depend on the image of 1? Why not the other generators?

I'm having a bit of an issue wrapping my head around the adjoint representation in group theory. I thought I understood the principle but I've got a practice problem which I can't even really begin to attempt. The question is this: My understanding of this question is that, given a...

Homework Statement Determine ##\phi(R_{180})##, if ##\phi:D_n\to D_n## is an automorphism where ##n## is even so let ##n=2k##. The solutions manual showed that since the center of ##D_n## is ##\{R_0, R_{180}\}## and ##R_{180}## is not the identity then it can only be that...

Homework Statement Decide all abelian groups of order 675. Find an element of order 45 in each one of the groups, if it exists. Homework Equations /propositions/definitions[/B] Fundamental Theorem of Finite Abelian Groups Lagrange's Theorem and its corollaries (not sure if helpful for this...

Homework Statement The SO(3) representation can be represented as ##3\times 3## matrices with the following form: $$J_1=\frac{1}{\sqrt{2}}\left(\matrix{0&1&0\\1&0&1\\ 0&1&0}\right) \ \ ; \ \ J_2=\frac{1}{\sqrt{2}}\left(\matrix{0&-i&0\\i&0&-i\\ 0&i&0}\right) \ \ ; \ \...

In Arnold's book, ordinary differential equations 3rd. WHY Arnold say Tg:M→M instead of Tg:G→S(M) for transformations Tfg=Tf Tg, Tg^-1=(Tg)^-1. Let M be a group and M a set. We say that an action of the group G on the set M is defined if to each element g of G there corresponds a...

I've been trying to understand representations of the Lorentz group. So as far as I understand, when an object is in an (m,n) representation, then it has two indices (let's say the object is ##\phi^{ij}##), where one index ##i## transforms as ##\exp(i(\theta_k-i\beta_k)A_k)## and the other index...

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