Representations Definition and 201 Threads

How powerful are continued fraction representations? From what I understand, they could be used to exactly represent some irrational numbers So, could they represent any root of an nth degree polynomial equation? Specially where n>4, since 5th degree roots are not guaranteed to have an...

I'm currently reading a book on relativistic field theory and I'm trying to understand spinors. After the author introduces the four parts of the Lorentz group he talks about spinors and group representations: "...With this concept we see that the 2x2 unimodular matrices A discussed in the...

Fredrik submitted a new PF Insights post Matrix Representations of Linear Transformations Continue reading the Original PF Insights Post.

My question concerns both quantum theory and relativity. But since I came up with this while studying QFT from Weinberg, I post my question in this sub-forum. As I gather, we first work out the representation of Poincare group (say ##\mathscr{P}##) in ##\mathbb{R}^4## by demanding the Minkowski...

When reading about GUTs you often come across the 'Standard Model decomposition' of the representations of a given gauge group. ie. you get the Standard Model gauge quantum numbers arranged between some brackets. For example, here are a few SM decompositions of the SU(5) representations...

I need to study in detail the rappresentations of the Poincare Group, i am interessed in the idea that particles can be wieved as irriducible representations of it. Do you have some references about it?

I'm reading a paper these days, How can I get 2.28? It seems for a D-dim SO(2) gauge field, we have spin2, spin1, as well as spin0 particles?

Hi their, It's a group theory question .. it's known that ## 10 \otimes 5^* = 45 \oplus 5, ## Make the direct product by components: ##[ (1,1)^{ab}_{1} \oplus (3,2)^{ib}_{1/6} \oplus (3^*,1)^{ij}_{-2/3} ] \otimes [ (1,2)_{ c~-1/2} \oplus (3^*,1)_{ k~1/3} ] = (1,2)^{ab}_{ c~1/2} \oplus...

Problem This is a conceptual problem from my self-study. I'm trying to learn the basics of group theory but this business of representations is a problem. I want to know how to interpret representations of a group in different dimensions. Relevant Example Take SO(3) for example; it's the...

Homework Statement (1+x2) y'' + 2xy' = 0 in powers of x Homework Equations y'' = \sum_{n=2}^{\infty} (n-1)na_nx^{n-2} y' = \sum_{n=1}^{\infty} na_nx^{n-1} The Attempt at a Solution (1+x2) y'' + 2xy' = (1+x^2) \sum_{n=2}^{\infty} (n-1)na_nx^{n-2} + 2x \sum_{n=1}^{\infty} na_nx^{n-1}...

Hello, I have some troubles understanding Hilbert representations for the standard free quantum particle On the one hand, we can represent Heisenberg algebra [Xi,Pj]= i delta ij on the space of square integrable functions on, say, R^3, with the X operator represented as multiplication and P...

Hello everybody, in Schwartz' QFT book it says (p. 483 - 484) In Problem 25.3 this is repeated asking the reader for a proof. I wonder though if this is really true. I know this can be proven for Lie algebras of compact Lie groups (or to be precise, every representation is equivalent to a...

I try to understand the statement "Every representation of SO(3) is also a representation of SU(2)". Does that mean that all the matrices of an integer-spin rep of SU(2) are identical to the matrices of the corresponding spin rep of SO(3)? Say, the j=1 rep of SU(2) has three 3x3 matrices, so...

Hi, I that <I|M|J>=M_{I}^{J} is just a way to define the elements of a matrix. But what is |I>M_{I}^{J}<J|=M ? I don't know how to calculate that because the normal multiplication for matrices don't seem to work. I'm reading a book where I think this is used to get a coordinate representation of...

The invariant of SL(2,C) is proven to be invariant under the action of the group by the following \epsilon'_{\alpha\beta} = N_{\alpha}^{\rho}N_{\beta}^{\sigma}\epsilon_{\rho\sigma}=\epsilon_{\alpha\beta}detN=\epsilon_{\alpha\beta} The existence of an invariant of this form (with two indices...

In the way of defining the adjoint representation, \mathrm{ad}_XY=[X,Y], where X,Y are elements of a Lie algebra, how to determine the components of its representation, which equals to the structure constant?

The group of SL(2,C) is sometimes defined (by physicists) as the group of 2 X 2 complex matrices of determinant = 1. But then we can talk about other representations of SL(2,C). So apparently the set of 2 X 2 complex matrices of determinant = 1 is but one representation of SL(2,C). If so...

I was rethinking about some things I learned but I came to things that seemed to be not firm enough in my mind. 1) When we want to find the unitary matrix that block-diagonalizes a certain matrix through a similarity transformation, we should find the eigenvectors of that matrix and stick them...

Can you give me an intuitive understanding of the following: "The spin states of massive and massless Majorana spinors transform in representations of SO(D-1) and SO(D-2), respectively". I see the similarity with vectors bosons, where massive vectors have d-1 degrees of freedom and massless...

The problem: Suppose G is Abelian with two representations as the internal direct product of subgroups: G=HxK1, G=HxK2. Assume K1 is a subset of K2 and show K1=K2. My attempted solution: I took the element (e_H, k_2), where e_H is the identity element of H and k_2 is an arbitrary element in K2...

Okay, so I am trying to understand on how to write Lagrangian in different representations. I know the formula of the SU(3) lagrangian in terms of the 3 and 3* rep. Now presume I have a model in the SU(3) 10 plet rep which includes exotic fermions not in the SM. How would I write out the...

I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series) In Chapter 2: Linear Algebras and Artinian Rings, Cohn introduces representations of k-algebras as follows: So, essentially Cohn considers a right multiplication: $$\rho_a \ : \ x \mapsto xa$$...

I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series) In Chapter 2: Linear Algebras and Artinian Rings we read the following on page 57: https://www.physicsforums.com/attachments/3149I am trying to gain an understanding of representations. I would...

Homework Statement There are two parts to this question... a)Which of the following are correct representations of The Momentum Principle? (assuming a small-enough Δt whenever it shows up) 1) \frac{Δ\vec{p}}{Δt} = \vec{F}|| + \vec{F}⊥ 2) For every action there is an equal and opposite...

Homework Statement Let V have dimension 3 and consider P_1(V ) = P(1,0,0) = span of {x,y,z}.Let I denote the subspace of all polynomials in P_1 of the form {rx+ry +rz|r any scalar}.Let W denote the subspace of all polynomials in P_1 of the form {rx+sy+tz|r+s+t = 0}. I and W are S_3 invariant...

Hi. I am currently studying about representations of Lie algebras. I have two questions: 1. As I understand, when we say a "representation" in the context of Lie algebras, we don't mean the matrices (with the appropriate Lie algebra) but rather the states on which they act. But then, the...

Hi there: I am reading a book (Atom-Photon interaction by Claude Cohen-Tannoudji, Page 448) and the following things gave a big headache. (1) Is there a density equation in Schrodinger Picture. because I encounter one, like: ##i \hbar \frac{d \sigma}{dt}=[\hat{H}, \sigma]## and...

The vectors \vec{\alpha}=\{\alpha_1,\ldots\alpha_m \} are defined by [H_i,E_\alpha]=\alpha_i E_\alpha they are also known to be the non-zero weights, called the roots, in the adjoint representation. My question is - is this connection (that the vectors \vec{\alpha} defined by the commutation...

To define spinors in QM, we consider the projective representations of SO(n) that lift to linear representations of the double cover Spin(n). Why don't we consider projective representations of Spin?

We have that A and B belong to different representations of the same Lie Group. The representations have the same dimension. X and Y are elements of the respective Lie algebra representations. A = e^{tX} B = e^{tY} We want to show, for a specific matrix M B^{-1} M B = AM Does it suffice to...

[FONT=tahoma]Let $\varrho :\mathbb{Z}\rightarrow GL_3(\mathbb{R})$ be the representation given by $\varrho (n)=A^n$ where A=$\begin{pmatrix} 2 & 5 & -1 \\ 2 & \frac{5}{2} & \frac{11}{2} \\ 6 & \frac{-2}{2} & \frac{3}{2} \\ \end{pmatrix}$ [FONT=tahoma]Does ρ have any 1-dimensional invariant...

Homework Statement Consider the following permutation representations of three elements in ##S_3##: $$\Gamma((1,2)) = \begin{pmatrix} 0&1&0\\1&0&0\\0&0&1 \end{pmatrix}\,\,\,\,;\Gamma((1,3)) = \begin{pmatrix} 0&0&1\\0&1&0\\1&0&0 \end{pmatrix}\,\,\,\,\,; \Gamma((1,3,2)) = \begin{pmatrix}...

I am reading a text about the splitting of the energy levels in crystals caused by the spin orbit interaction. In particular, the argument is treated from the point of view of the group theory. The text starts saying that a representation (TxD) for the double group can be obtained from the...

Hey guys. I'm trying to understand spinorial representations in relativistic QM/QFT. If I make any mistakes in my statements (drawing mostly from chapter 2 of Maggiore) please correct them. In QM things are simple enough. We have a different representation of ##SU(2)## for each fixed...

Correct me if I'm wrong, but exist 3 forms for represent periodic functions, by sin/cos, by exp and by abs/arg. I know that given an expression like a cos(θ) + b sin(θ), I can to corvert it in A cos(θ - φ) or A sin(θ + ψ) through of the formulas: A² = a² + b² tan(φ) = b/a sin(φ) = b/A...

Hello! I'm currently reading Peskin and Schroeder and am curious about a qoute on page 38, which concerns representations of the Lorentz group. ”It can be shown that the most general nonlinear transformation laws can be built from these linear transformations, so there is no advantage in...

The question is about the spinor representation decomposed under subgroups. It's a common technique in string theory when parts of dimensions are compactified and ignored, and we are only interested in the remaining sub-symmetry. I'm learning it from the appendix B in Polchinski's big book...

Say I have a 3x3 operator Q and I find its eigenvectors and eigenvalues. Now i know that those eigenvectors are the same as eigenfunctions so if i act on them with Q i will get the corresponding eigenvalue. What the question I am trying to solve asks is, Measure the quantity Q in state [b]...

First, greetings from newbie to "staff" Now, let's start: Since some days I'm struggling a little bit with this paper: http://jmp.aip.org/resource/1/jmapaq/v5/i9/p1204_s1?isAuthorized=no , especially with two questions: 1) On page 1205, II, A (right column): What does \tilde v B...

Homework Statement Write down the 3×3 matrices that represent the operators \hat{L}_x, \hat{L}_y, and \hat{L}_z of angular momentum for a value of \ell=1 in a basis which has \hat{L}_z diagonal. The Attempt at a Solution Okay, so my basis states \left\{\left|\ell,m\right\rangle\right\}...

This is discussed in Weinberg's Quantum Theory of Fields, in the chapter on Relativistic Quantum Mechanics. The point I am somewhat confused about occurs on page 63 - 64, if you have the book. He operates on a single particle state with the unitary homogeneous lorentz transformation...

Hello! Can someone explain to me, as clearly as possible, how one can find irreducible representations of Lie groups (and especially in the context of finding the spectrum of e.g. the bosonic string theory)? I am following BB&S and Polchinski but I cannot really understand how they...

Hi everyone, So I'm trying to basically generate a list of numbers between 20 and 20,000 (Hz) in log space that will give good resolution to parts of the audio spectrum that matter! After all that is the point of using log scale for frequency in the first place. The list generator I have...

Hi Everybody! I am working on QFT and learning representation theory from Coleman's lecture notes. Just the necessary stuff to go to the Dirac equation. To my question: From the generators of SO(3) I get through exponentiation an element of SO(3), this holds naturally for any Lie group...

Attached is a schematic of interest. Assume same voltage source and capacitance for each capacitor in each circuit. Correct me if I'm wrong, it appears all circuitry comprises two basic types of connection: series and parallel. I'm trying to figure out an approach to determining whether a...

Compute the coefficient of expansion α in terms of P and V... Homework Statement Compute the coefficient of expansion α in terms of P and V for an ideal Van der Waals gas Homework Equations (p+a/v^2)(v-b)=RT The Attempt at a Solution Is this as simple as solving for a? How...

Author: Brian Hall Title: Lie Groups, Lie Algebras, and Representations: An Elementary Introduction Amazon link https://www.amazon.com/dp/1441923136/?tag=pfamazon01-20 Level: Grad Table of Contents: General Theory Matrix Lie Groups Definition of a Matrix Lie Group Counterexamples...

I know how waves work, by the oscillation of particles along there mean position..fine I know how a simple oscillatory motion can be described by a sinusoidal wave , and what is its physical significance...okay but what I don't understand when waves (say a transverse) wave is represented by a...

I'm having some difficulty understanding the representation theory of the Lorentz group. While it's a fundamentally mathematical question, mathematicians and physicists use very different language for representation theory. I think a particle physicist will be more likely than a mathematician to...

Homework Statement Two equal masses (m) are constrained to move without friction, one on the positive x-axis and one on the positive y axis. They are attached to two identical springs (force constant k) whose other ends are attached to the origin. In addition, the two masses are connected to...