Representation theory Definition and 61 Threads

I'm trying to explicitly find a projective unitary scalar representation of the Galilean group. I'll denote a generic element of the group by ##(a, {\bf b},R, {\bf v})##, corresponding respectively to time translation, space translation, rotation and boosts. In a representation with central...

I would appreciate if someone could help me to understand what is happening in section 12.3 from the Howard George's book. First of all, the propose of the section is to show how $SU(3)$ decomposes into $SU(2) \times U(1)$. But i can't understand what is happening. First of all, i can't get the...

I have a couple of questions about classification of reductive groups over algebraically closed field (up to isomorphism) by so called root datum. In the linked discussion is continued that Obviously, a root datum ##(X^*, \Phi, X_*, \phi^{\vee})## contains full information ("building plan")...

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I am not quite sure of how this works, i.e. of what exactly I need to do with the hint. Any explanantion would be helpful!

I hope this is the right section as the question is about Lie groups and representations. First and foremost, in this post I'll be dealing with Dirac and Weyl spinor (not spinor fields) representations of the Lorentz algebra. Also, for simplicity, I'll use the chiral representation later on...

New to group theory. I have 3 questions: 1. Tensor decomposition into Tab = T[ab] +T(traceless){ab} + Tr(T{ab}) leads to invariant subspaces. Is this enough to imply these subreps are irreducible? 2. The Symn representations of a group are irreps. Why? 3. What is the connection between...

Hi Pfs, Please read this paper (equation 4): https://ncatlab.org/nla b/files/RedeiCCRRepUniqueness.pdf It is written: Surprise! P is a projector (has to be proved)... where can we read the proof?

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...

In Quantum Mechanics, by Wigner's theorem, a symmetry can be represented either by a unitary linear or antiunitary antilinear operator on the Hilbert space of states ##\cal H##. If ##G## is then a Lie group of symmetries, for each ##T\in G## we have some ##U(T)## acting on the Hilbert space and...

This is one problem from Robin Ticciati's Quantum Field Theory for Mathematicians essentially asking us to find Cartan subalgebras for the matrix algebras ##\mathfrak{u}(n), \mathfrak{su}(n),\mathfrak{so}(n)## and ##\mathfrak{so}(1,3)##. The only thing he gives is the definition of a Cartan...

I read this question https://physics.stackexchange.com/questions/95970/under-what-conditions-is-a-vector-spinor-gamma-trace-free . Also I read Sexl and Urbantke book about groups. But I don't understand why spinors is irreducible if these are gamma-tracelees. Also I read many papers about...

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.

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.

Consider two arbitrary scalar multiplets ##\Phi## and ##\Psi## invariant under ##SU(2)\times U(1)##. When writing the potential for this model, in addition to the usual terms like ##\Phi^\dagger \Phi + (\Phi^\dagger \Phi)^2##, I often see in the literature, less usual terms like: $$\Phi^\dagger...

Hey there, I've suddenly found myself trying to learn about the Lorentz group and its representations, or really the representations of its double-cover. I have now got to the stage where the 'complexified' Lie algebra is being explored, linear combinations of the generators of the rotations...

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...

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...

Dear All short explanation: I am trying to leverage my limited understanding of representation theory to explain (to myself) how many non-vanshing components of, for example, nonlinear optical susceptibility tensor ##\chi^{(2)}_{\alpha\beta\gamma}## can one have in a crystal with known point...

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...

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...

Hello guys, In 90% of the papers I've read about diferent ways to achieve generalizations of the Proca action I've found there's a common condition that has to be satisfied, i.e: The number of degrees of freedom allowed to be propagated by the theory has to be three at most (two if the fields...

Hi, this question is related to global and local SU(n) gauge theories. First of all, some notation: ##A## will be the gauge field of the theory (i.e: the 'vector potential' in the case of electromagnetic interactions) also known as 'connection form'. In components: ##A_\mu## can be expanded in...

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...

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...

In one General Relativity paper, the author states the following (we can assume tensor in question are tensors in a vector space ##V##, i.e., they are elements of some tensor power of ##V##) To discuss general properties of tensor symmetries, we shall use the representation theory of the...

Who really wrote the best introductory account of representation theory in QM that I've seen so far ? [Likely mis-attribution discussed here below; prefixed "Advanced" to reach lecturers who are more likely to know the answer to this question.] It's available via...

Hello! I am reading some representation theory (the book is Lie Algebra in Particle Physics, by Georgi, part 1.17) and the author solves a problem of 3 bodies connected by springs forming a triangle, aiming to find the normal modes. He builds a 6 dimensional vector formed of the 3 particles and...

Context The following is from the book "Ideas and methods in supersymmetry and supergravity" by I.L. Buchbinder and S.M Kuzenko, pg 56-60. It is about realizing the irreducible massive representations of the Poincare group as spin tensor fields which transform under certain representations of...

I had heard of adinkras but didn't realize that they were meant to play this role. Nor did I realize that the representation theory of supersymmetry is mathematically underdeveloped.

By space, I mean a vector space which could be a representation of a group or even have some expanded algebraic structure. So I am not sure if this question goes here or in the Algebra subforum. Consider the tensor square r\otimes r of an irreducible group representation r with itself, and...

Recently I read some comment on Sakurai's book (which I have not read) that the writer of said comment didn't understand part of the text until they understood irreducible representations. I do not know to what they were referring, but it piqued my interest in representation theory. My question...

Hello. If I represent a vector space using matrices, for example if a 3x1 vector, V, is represented by 3x3 matrix, A, and if this vector was the eigenvector of another matrix, M, with eigenvalue v, if I apply M to the matrix representation of this vector, does this holds: MA=vA? Also, if I...

Hello! I just started reading about SU(2) (the book is Lie Algebras in Particle Physics by Howard Georgi) and I am confused about something - I attached a screenshot of those parts. So, for what I understood by now, the SU(2) are 2x2 matrices whose generators are Pauli matrices and they act on a...

I recently got confused about Lie group products. Say, I have a group U(1)\times U(1)'. Is this group reducible into two U(1)'s, i.e. possible to resepent with a matrix \rho(U(1)\times U(1)')=\rho_{1}(U(1))\oplus\rho_{1}(U(1)')=e^{i\theta_{1}}\oplus e^{i\theta_{2}}=\begin{pmatrix}e^{i\theta_{1}}...

Hello everyone, I'm a undergraduate at UC Berkeley. I'm doing theoretical physics but technically I'm a math major. I really want to study quantum gravity in the future. Now I have a problem of choosing courses. For next semester, I have only one spot available for either representation theory...

Hi Everyone. There is an equation which I have known for a long time but quite never used really. Now I have doubts I really understand it. Consider the unitary operator implementing a Lorentz transformation. Many books show the following equation for vector fields: U(\Lambda)^{-1}A^\mu...

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...

Hi I am a physics graduate student. Recently I am learning representation theory of groups. I understand the basic concepts. But I need a good book with lots of examples in it and also exercise problems on representation theory so that I can brush up my knowledge.The text we follow is "Lie...

I am looking for a short pedagogical introduction to Young-tableaux and weight diagrams and the relationship between them, which contains many detailled and worked out examples of how these methods are applied in physics, such as in the context of the standard model and beyond for example. I am...

I want to learn how to write down a particle state in some inertial coordinate frame starting from the state ##| j m \rangle ##, in which the particle is in a rest frame. I know how to rotate this state in the rest frame, but how does one write down a Lorentz boost for it? Note that I am not...

On one hand, in reading Georgi's book in group theory, I comprehend the invariant tensor as a special "tensor", which is unchanged under the action of any generators. On the other hand, CG decomposition is to decompose the product of two irreps into different irreps. Now it is claimed that...

I'm self-studying representation theory for finite groups using "Group Theory and Quantum Mechanics" by Michael Tinkham. Most of it makes sense to me, but I'm having difficulty understanding what is meant by saying a function "belongs to a particular irreducible representation", or "has the...

It seems that there’s a loose way of discussing group representations that’s fine for the initiated, but confusing for the neophyte: I understand that the objects that represent group operations are usually (or always?) square matrices, which naturally can be described as “matrix...

I am reading Dummit and Foote on Representation Theory CH 18 I am struggling with the following text on page 843 - see attachment and need some help. The text I am referring to reads as follows - see attachment page 843 for details \phi ( g ) ( \alpha v + \beta w ) = g \cdot ( \alpha v +...

Hello My question is about the ground state of vibrations for a solid. I'm working with graphite and have found out that for k=0 (The Gamma symmetry point), the vibrational modes can be decomposed into irreducible represenations in the following way Vibration = 2 * E1u + 2 * E2g + 2 * A2u...

For some reason, diffeomorphism invariance seems to be treated like a second-class citizen in the land of symmetries. In nonrelativistic quantum mechanics, we consider Galilean invariance so important that we form our Hilbert space operators from irreducible representations of the Galilei...

I am familiar with the representation theory of finite groups and Lie groups/algebra from the mathematical perspective, and I am wondering how quantum mechanics/quantum field theory uses concepts from representation theory. I have seen the theory of angular momentum in quantum mechanics, and I...

Hi, I am currently studying Lie Algebra in Particle Physics' by Howard Georgi. I am finding the notes on Weights and Roots quite confusing. Can anyone suggest another book which explains this bit in a better fashion?

What's the use of it? Anyone show a simple but illustrative example of the usefulness of representation theory? I can see how faithful representations might be useful but not fully. What I can't imagine is how unfaithful representations can be of any use. Thanks