Representation Definition and 722 Threads

I have confusions about representation theory. In the following questions, I will try to express it as best as possible. For this thread say representation is given as ρ: L → GL(V) where L is the Lie group(or symmetry group for a physicist) GL(V) is the general linear...

When discussing how a rank two tensor transforms under SO(3), we say that the tensor can be decomposed into three irreducible parts, the anti-symmetric part, traceless-symmetric part, and a 1-dimensional trace part, which transforms as a scalar. How do we know that the symmetric and...

Being myself a chemist, rather than a physicist or mathematician (and after consulting numerous sources which appear to me to skip over the detail): 1) It’s not clear to me how one can go generally from a choice of basis vectors in real space to a representation matrix for a spatial symmetry...

I know that to go from a vector with coordinates relative to a basis ##\alpha## to a vector with coordinates relative to a basis ##\beta## we can use the matrix representation of the identity transformation: ##\Big( Id \Big)_{\alpha}^{\beta}##. This can be represented by a diagram: Thus note...

Hello so my doubt is when i are teaching about eletromagnétic filds the frist time a do a representatios is of a full field i Draw the two waves, ume perpendicular to another, and istill i know that's not the best representatios because i should have to rolate the wave while a draw. Later in...

In calculating the matrix elements for the raising operator L(+) with l = 1 and m = -1, 0, 1 each of my elements conforms to a diagonal shifted over one column with values [(2)^1/2]hbar on that diagonal, except for the element, L(+)|0,-1>, where I have a problem. This should be value...

Hello I hope you are all very well. I did a graph for a physics practical work and I would like to put the "error bar" (I don't know if it's called like this, I'm sorry). Something like this: I just don't know how long the margin should be. If someone could clarify to me the way to do the...

1) Likely an Einstein summation confusion. Consider Lorentz transformation's defined in the following matter: Please see image [2] below. I aim to consider the product L^0{}_0(\Lambda_1\Lambda_2). Consider the following notation L^\mu{}_\nu(\Lambda_i) = L_i{}^\mu{}_\nu. How then, does...

Homework Statement: Hi there, I'm currently taking an Optics course and the teacher is expecting us to have an understanding of the complex representation of waves. Although, hardly any of us have even heard of this yet. I've tried to google how to convert a cos(obj) and sin(obj) to an...

The Pauli matrices of the spin-1 representation are given by: ##T_{1}=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}##, ##T_{2}=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & -i & 0 \\ i & 0 & -i \\ 0 & i & 0 \end{pmatrix}## and ##T_{3}=\begin{pmatrix} 1 & 0 & 0 \\ 0...

Why representation of Lorentz group of shape (A,A) corespond to totally symmetric traceless tensor of rank 2A? For example (5,5)=9+7+5+3+1 (where + is dirrect sum), but 1+5+3+9+7<>(5,5) implies that (5,5) isn't symmetric ? See Weinberg QFT Book Vol.1 page 231.

Classical probability theory can be represented with measure spaces and functions over them. Quantum Probability is given as the theory of Hilbert spaces and operators over them. Both more abstractly are handled by the theory of C*-algebras and their duals. However I know of no structure for...

Hi, in the context of linear two-ports networks my (italian) textbook says that if its internal structure consists of passive one-port components with no independent current/voltage generators then the following (implicit) linear equations based representation always exist: ##\left[A\right] V...

Look, I am sorry for not being able to post any LaTeX. But I am stuck at a place where I feel I should not be stuck. I can not figure out how to correctly do this. I can't seem to recreate the Pauli matrices with that form using the 3 2-dimensional bases representing x, y, and z spin up/down...

Let (a, b, c) be some arbitrary positive integers such that: (q2^0 + q2^1+ . . . + q2^x), (q2^0 + q2^1+ . . . + q2^y), (q2^0 + q2^1 + . . . + q2^z), where: q = (1, 2), (x, y, z) = (1, 2, 3, . . ., n). In the case if and only if q = 2, we accept the following notation : [(q-1)2^0...

Hi, A function which could be represented using Fourier series should be periodic and bounded. I'd say that the function should also integrate to zero over its period ignoring the DC component. For many functions area from -π to 0 cancels out the area from 0 to π. For example, Fourier series...

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.

##nq = q(d_1 + d_2 q + d_3 q^2 + \dots + d_k q^{k-1})## ##= d_1 q + d_2 q^2 + d_3 q^3 + \dots + d_k q^k## ##= d_k d_{k-1} \dots d_1 0## Can someone please explain how to get from line two to line three. This is instructors solution and not sure I understand. Thanks!

I don't understand what the last paragraph of the attached page means. Why does the Kronecker delta being an invariant symbol mean that the product of a representation R and its complex conjugate representation has the singlet representation with all matrices being zero? Doesn't the number...

Homework Statement Given the expression s_{\pm}|s,m> = \hbar \sqrt{s(s+1)-m(m\pm 1)}|s,m \pm 1> obtain the matrix representations of s+/- for spin 1/2 in the usual basis of eigenstates of sz Homework Equations s_{\pm}|s,m> = \hbar \sqrt{s(s+1)-m(m\pm 1)}|s,m \pm 1> S_{+} = \hbar...

Bead sort is a not practical sorting algorithm that suppose to have a time complexity of ##O(\sqrt{n})##in the real world. I thought it would be great if a device that can take numbers from the computer,then turn it into analog stuff that the device can read,drop the beads,and return it...

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 n ≥ 2 Show that Dn = < a,b | a2, b2, (ab)n> Homework EquationsThe Attempt at a Solution I see that a and b are involutions and therefore are two different reflections of Dn. If we set set b = ar where r is a rotation of 2π/n And Dn = <a,r | a2, rn, (ab)2 > I am unsure...

90 years have gone by since P.A.M. Dirac published his equation in 1928. Some of its most basic consequences however are only discovered just now. (At least I have never encountered this before). We present the Covariant QED representation of the Electromagnetic field. 1 - Definition of the...

Hi, I am studying some material related to Grassmannians and in particular how to represent k-subspaces of ℝn as "points" in another space. I think understood the general idea behind the Plücker embedding, however, I recently came across another type of embedding (the "Projection embedding")...

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...

Hello! I read in many places the derivation of the representation for SU(2) using ladder operators and in all of the places they say that, due to the fact that we are looking for a finite dimensional representation, the ladder must end at a point, hence why we have an eigenvector of ##L_3##...

This one may seem a bit long but essentially the problem reduces to some matrix calculations. You may skip the background if you're familiar with Lorentz representations. 1. Homework Statement A Lorentz transformation can be represented by the matrix...

Is there a way to find the integer of a real number? Of course without using the [x] function. What I am looking for here is an algebraic formula.

Consider a three dimensional representation of ##U(1)\times SU(2)## with zero hypercharge ##Y=0##: $$ L= \begin{pmatrix} L^+ \\ L^0 \\ L^- \end{pmatrix} $$ Then the mass term is given by [1]: $$ \mathcal{L} \supset -\frac m 2 \left( 2 L^+ L^- +L^0 L^0 \right) $$ I am wondering where the...

When considering the position representation of a system with one degree of freedom endowed with canonical co-ordinates and momenta q and p, Dirac deduces that: which is the representation of d/dq in "matrix" form. But the derivative of a delta function is, I assume (from the definition of the...

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...

Hi, Is there a method to represent a known two-dimensional vector field w of two coordinates x and y with zero divergence and non-zero curl as $$ \vec{w}(x,y) = a \nabla b \, , \hspace{4mm} \nabla \cdot \vec{w} = 0 \, , \hspace{4mm} \nabla \times \vec{w} = f(x,y) \, ?$$ How would one proceed to...

I understand that because the rep. of quark is 3 and antiquark is ##\bar{3}## and we have ##3\otimes \bar{3} = 8\oplus 1##, that a gluon should be 8 or 1, but which one? and why? Thanks!

Ok, so my question is "Does an irreducible representation acting on operators imply that the states also transform in an irreducible representation?" and what I mean by that is the following. If I have an operator transforming in an irreducible transformation of some group, I get a corresponding...

Homework Statement Hi, guys. The question is: For a 3-state system, |0⟩, |1⟩ and |2⟩, write the matrix representation of the raising operators ## \hat A, \hat A^\dagger ##, ## \hat x ## and ##\hat p ##. Homework Equations I know how to use all the above operators projecting them on...

Suppose I have a hermitian ##N \times N## matrix ##M##. Let ##U \in SU(N)## be the matrix that diagonalizes ##M##: ##M = U\Lambda U^\dagger##, where ##\Lambda## is the matrix of eigenvalues of ##M##. This transformation can be considered as the adjoint action ##Ad## of ##SU(N)## over its...

I'm trying to understand diffeomorphisms, and I thought I basically understood them, but when I tried to work out a problem I created for myself, I realized I didn't know how to answer it. So let's consider a diffeomorphism generated by a vector field ##V##. If ##X## is a point on our manifold...

Question Form the canoncial partition using the following conditions: 2 N-particles long strands can join each other at the i-th particle to form a double helix chain. Otherwise, the i-th particle of each strand can also be left unattached, leaving the chain "open" An "open" link gives the...

Hi all I have a shallow understanding of group theory but until now it was sufficient. I'm trying to generalize a problem, it's a Lagrangian with SU(N) symmetry but I changed some basic quantity that makes calculations hard by using a general SU(N) representation basis. Hopefully the details of...

Homework Statement Please see attached file. I'm not quite sure if I'm on the right track here. I think the basis for F is throwing me off as well as T(f). Please advise. Thanks! Homework EquationsThe Attempt at a Solution

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 =...

Hello! I am reading some things about representation theory for SU(n) and I want to make sure I understand it properly. I will put an example here and explain what I understand out of it and I would really appreciate if someone can tell me if it is right or not. So for SU(2) we have ##2 \otimes...

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-1The 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). Also...

Hello Everyone.. Consider a small moving mass following a spiral path away from center and the body is rotating about its center as well. At some time the body impacts with a relatively bigger mass and transfers some of its energy to it. If we want to make an equivalent system with only linear...

I'm not at all involved in QG but from far away I noticed : Spin 2 representations are 5x5 matrices. But in gravity what mathematical objects are quantized ? If it's the metric then it's a 4x4 matrix so that cannot be that. Or : how does quantization reveal a 5x5 matrix ?

Arrays declared in java are stored in the memory in two ways: row-major-wise and column-major-wise. As per our teacher, the choice of the storage technique depends on how we enter the array elements. For example, for this code: int arr[][] = new int[10][10]; Scanner kb = new Scanner...

When we consider representation of the O(1,3) we divide the representation up into the connected part of SO(1,3) which we then extend by adding in the discrete transformations P and T. However, while (as far as I know) we always demand that the SO(1,3) and P be represented as linear operators...

Let U ∈ SU(N) and {ta} be the set of generators of su(N), a = 1, ..., N2 - 1. The action of the adjoint representation of U on some generator ta can be written as Ad(U)ta = Λ(U)abtb I want to characterize the matrix Λ(U), i. e., I want to see which of its elements are independent. It's known...