Representations Definition and 216 Threads

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 found J^2 and Jz, but I am not sure how to find Jx and Jy. I’m thinking maybe use J+-=Jx+-iJy ? But I get unclear results. Thanks!

How do various computer algebra systems (CAS) compare with respect to keeping multiple representations of the same mathematical object? For example, a polynomial could be represented by a list of coefficients, or a list of roots, or a list of factors with some factors non-linear. One design...

Hi Pfs Marcus wrote a huge bibliography during many years about LQG. but i do not see where to find an answer to a question. Penrose draw diagrams with edges labelled by numbers. What is the reason why later number were replaced by SU(2) representations?tÿ

___ Blaming people for "unhelpful answers" when you have an unclear question does not put the blame in the right place. Either your two 1's are distinct or they are not. If they are not distinct, you shouldn't have two of them. If they are, you shouldn't use the same symbol for them.

Every group needs to have that every element appear only once at each row and each column. But in the case of unfaithful representations of ##Z_2## sometimes we have ##D(e)=1##, ##D(g)=1##. When we write the Caley table we will have that one appears twice in both rows and in both columns. How is...

Suppose we have a group ##G = D_3## and its reducible representation ##T = E \oplus E## where ##E## is the two-dimensional irrep. In its own basis, this representation can be written like this (I provide the operator only for ##C_3^+## group element which is counter-clockwise rotation by...

We know from Lie representation theory that the Lie algebra is a vector space. Therefore a representation of the Lie group can be transformation of this vector space itself which we call the Adjoint Representation. An element of this vector space, is itself represented by a matrix. For example...

There are many representations of the delta function. Is there a place/reference that lists AND proves them? I am interested in proofs that would satisfy a physicist not a mathematician.

Can we consider the E and B fields as being irreducible representations under the rotations group SO(3) even though they are part of the same (0,2) tensor? Of course under boosts they transform into each other are not irreducible under this action. I would like to know if there is in some...

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Summary:: Looking for best literature or online courses on projective unitary representations of the Poincare Group. I'm watching an online course on relativistic QFT. I understand that because this theory deals with both QM and SR, there is a need to represent Lorentz transformations with...

I've been trying to continue my education by self-teaching during quarantine (since I can't really go to college right now) with the MIT Opencourseware courses. I landed on one section that's got me stuck for a while which is the second part of this problem (I managed to finish the first part...

Is is true that the dihedral group ##D_n## does not have an irreducible representation with a dimension higher than two?

I'm trying to 'see' what the generators of the Poincare Group are. From what I understand, it has 10 generators. 6 are the Lorentz generators for rotations/boosts, and 4 correspond to translations in ℝ1,3 since PoincareGroup = ℝ1,3 ⋊ SO(1,3). The 6 Lorentz generators are easy enough to find in...

How to prove that direct product of two rep of Lorentz group ##(m,n)⊗(a,b)=(m⊗a,n⊗b)## ? Let ##J\in {{J_1,J_2,J_3}}## Then we have : ##[(m,n)⊗(a,b)](J)=(m,n)(J)I_{(a,b)}+I_{(m,n)}⊗(a,b)(J)=## ##=I_m⊗J_n⊗I_a⊗I_b+J_m⊗I_n⊗I_a⊗I_b+I_m⊗I_n⊗J_a⊗I_b+I_m⊗I_n⊗I_a⊗J_b## and...

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Hi all I need to understand the following passage from Hall link page 78 : Some notation first: Basis for ##sl(2;C)##: ##H=\begin{pmatrix} 1&0\\0&−1\end{pmatrix} ;X=\begin{pmatrix} 0&1\\0&0\end{pmatrix} ;Y=\begin{pmatrix} 0&0\\1&0\end{pmatrix} ## which have the commutation relations...

(scroll to bottom for problem statement) Hello, I am wondering if someone could break down the problem statement in simpler terms (not so math-y). I am struggling with understanding what is being asked. I will try to break it down to the best of my ability Problem statement:Consider the inner...

Matrix representation of a finite group G is irreducible representation if \sum^n_{i=1}|\chi_i|^2=|G|. Representation is reducible if \sum^n_{i=1}|\chi_i|^2>|G|. What if \sum^n_{i=1}|\chi_i|^2<|G|. Are then multiplication of matrices form a group? If yes what we can say from...

I am confused. Look for instance cyclic ##C_2## group representation where D(e)= \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} and D(g)= \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix} and let's take invertible matrix A= \begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}. Then A^{-1}=...

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...

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...

Fock spaces use lists of integers (0 and 1 in the fermionic case) to describe set of particles. a list 0 1 1 0 1 0 0 0 ... for 1 fermion in the second third and fifth state may be associated to the real number 0.0110100000... so this set of list is not countable. a Fock space select a countable...

i am reading this paper. after equation 16 the author (Blasone) writes that In the thermodynamical limit this goes to zero, i.e. the Hilbert spaces con- structed over the respective vacuum states are orthogonal. From the second Schur’s lemma [33] it then follows that the two representations of...

Weinberg QFT book aside, what are good sources for the representation of the Poincare group used in physics?

Very cool paper just out relating fractal geometry to the representation of natural images in visual cortex. The main argument is that brain's representations are as high dimensional as they could possibly be without becoming fractal. https://www.nature.com/articles/s41586-019-1346-5 I'm...

One thing I was thinking about doing was to consider these representations as a basis for the gamma matrices vector space, then try to determine what the change of basis from one to the other would be. However I'm unsure if it's correct to treat the representations as a basis, or whether this is...

That's my attempting: first I've wrote ##e## in terms of the power series, but then I don't how to get further than this $$ \sum_{n=0}^\infty (-1)^n \frac {Â^n} {n!} \hat B \sum_{n=0}^\infty \frac {Â^n} {n!} = \sum_{n=0}^\infty (-1)^n \frac {Â^2n} {\left( n! \right) ^2} $$. I've alread tried to...

Let $\mu$ be a finite Borel measure on $S^1$. We have an action of $\mathbb Z$ on $L^2(S^1, \mu)$ defined by $n\cdot \varphi = e^{2\pi i n}\varphi$. The following is a standard theorem in functional analysis: Spectral Theorem. Let $\mathbb Z$ act unitarily on a Hilbert space $H$. Let $f$ be any...

Hello there, Given a Lie Algebra ##\mathfrak{g}##, its Cartan Matrix ##A## and a finite representation ##R##, is there a way of determining its highest weight ##\Lambda## in a simple way? In my course, we consider ##\mathfrak{g}=A_2= \mathfrak{L}_{\mathbb{C}}(SU(3))##. It is stated that the...

Greg Bernhardt submitted a new blog post Lie Algebras: A Walkthrough The Representations Continue reading the Original Blog Post.

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...

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...

Suppose I had some group G, and I could classify all of its irreducible K-representations for some K = R,C, or H. Given that information (how) can I classify its irreducible K-representations for all K. i.e. suppose I knew all the irreducible real representations of G, (how) could I then get...

Hello! I am reading some representation theory and I am a bit confused about some stuff. I read that SU(2) is the double covering of SO(3), so to each matrix in SO(3) corresponds one in SU(2). I am not sure I understand this. So if we have a 3D representation of SU(2), the 3D object it acts on...

Hello! Can someone recommend me some good reading about the Lorentz group and its representations? I want something to go pretty much in all the details (not necessary proofs for all the statements, but most of the properties of the group to be presented). Thank you!

I'm not asking about the math here, I'm interested in the wording physicists use in QP / QM / QFT. I'm frequently confused, when I'm reading threads here. They often start completely underdetermined and often also just wrong from a mathematical point of view, but seemingly, physicists know what...

Hello! I am reading some notes on Lorentz group and at a point it is said that the irreducible representations (IR) of the proper orthochronous Lorentz group are labeled by 2 numbers (as it has rank 2). They describe the 4-vector representation ##D^{(\frac{1}{2},\frac{1}{2})}## and initially I...

I'm reading "Division Algebras and Quantum Theory" by John Baez https://arxiv.org/abs/1101.5690 In the last paragraph of section 5 (Applications) he says the following "SU(2) is not the only compact Lie group with the property that all its irreducible continuous unitary representations on...

One day Steve (68 kg) rolls into class on a skateboard. When he rolls in on the skateboard, he and the skateboard move at 2 m/s toward the windows in the room. Steve then jumps off the skateboard and he ends up moving at 1.0 m/s toward the windows of room. How fast and in what direction is the 1...

Could anyone help with providing a simple example of a projective representation of a small finite group ( order of group not greater than six )? My understanding is that, if the group has N elements, then I should see N matrices in the projective representation. I would prefer the example to...

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...

I'm trying to get the hang of index gymnastics, but I think I'm confused about the relationship between rank-2 tensor components and their matrix representations. So in Hartle's book Gravity, there's Example 20.7 on p. 428. We're given the following metric: ##g_{AB} = \begin{bmatrix} F & 1 \\...

fresh_42 submitted a new PF Insights post How to Tell Operations, Operators, Functionals and Representations Apart Continue reading the Original PF Insights Post.

Hello! :smile: On page 51 where he want to invert $$\Lambda^{\mu}_{\nu} = \tfrac{1}{2} \text{tr}( \bar{\sigma}^{\mu}A \sigma_{\nu} A^{\dagger})$$ the person says we may use $$\sigma_{\nu} A^{\dagger} \bar{\sigma}^{\nu} = 2 \text{tr}(A^{\dagger})I.$$ to do that ... how do you prove this formula...

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...

Enjoy! http://www.math.columbia.edu/~woit/QM/qmbook.pdf

fresh_42 submitted a new PF Insights post Representations and Why Precision is Important Continue reading the Original PF Insights Post.