Questions tagged [representation-theory]

The systematic study of group representations, which describe abstract groups in terms of linear transformations of vector spaces, such that group elements or their generators are represented as matrices, reducing group-theoretic problems to linear-algebraic ones.

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48 views

Adjoint representation of a group vs of a Lie algebra

I just wanted some clarification on the adjoint representation. The definition of the adjoint representation of a Lie algebra seems absolute: taking the structure constants as generators. The ...
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Representations of the special conformal transformations

I am trying to follow the calculation of the special conformal transformations from these notes by Qualls on conformal field theory. In section 2.5 we use the following prescription to determine $K_\...
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GUT decomposition

How can we decompose a generic $SU(5)$ irreducible representation as a direct sum of irreducible $SU(3) \times SU(2) \times U(1)$ representations?
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Why Lorentz algebra is not represented by the basis of antisymmetric $4\times 4$ tensors? Confusion building Lorentz Lie algebra

I am very confused when building the Lie algebra of the Lorentz Group. In every books, they expand $\Lambda^{\mu}_{\nu} = \delta^{\mu}_{\nu} + \omega^{\mu}_{\nu}$ at the origin and you end up with the ...
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Wigner's classification

Could someone offer a very clear explanation of Wigner's classification of particles as infinite-dimensional unitary representations of the Poincare group?
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Anthony Zee's Proof of Schur's Lemma

In Anthony Zee's proof of Schur's lemma (in his book Group Theory in a Nutshell for Physicists, page 102), he used the following fact (summarized by myself) without proof: Proposition: Let $G$ be a ...
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A confusion about the notation in Ernst S. Aber Quantum Mechanics

I recently read the topic in chap 4 and chap 5 of Ernest S. Abers' book, Quantum Mechanics. In the section 4.2.5, he wrote: From Section 3.3.3 you know how the $D^{(j)}(J_i)$ acts on the $2j+1$ states ...
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Representations of $O(2)$ and related problems [migrated]

I'm currently studying Group Theory in order make a further application to physics and understand the math of some physical theories. I know that $SO(2)$ literally is a special case for $O(2)$ and ...
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How projective representations can lead to 't Hooft anomalies in quantum mechanics?

In Shao's talk https://youtu.be/2vTvHYYl1Qk?t=1554, he argues that in quantum mechanics "if a symmetry acts projectively on states, then we have a t' Hooft anomaly". But I'm having trouble ...
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Lorentz transformation of a (relativistic) wave and a field operator

My question follows chapter 10 of Tung's Group Theory book, in particular definitions 10.11 and 10.12. Let $\mathcal{R}[\Lambda]$ be an $n\times n$ matrix representation of $L_+^{\uparrow}$ and a wave-...
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Quasi-periodic potential and Bloch's theorem

Let's look at a physical system of a particle in a one dimensional periodic potential $V(x)$. When the potential satisfies the periodicity condition of the form $$ V(x + n b) = V(x),$$ this leads to ...
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Does the representation of the group also follow the properties of the group? [migrated]

Consider the finite group $\mathbb{Z}_2$ with two elements $\{e,g_1\}$ with $g_1^2=e$. We have been told in the class that there is a trivial representation of this group in which we have $$e\...
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Unitarity and boost

I wonder if anyone could shed some light on the representation theory of the Lorentz group. In particular, I would like to understand unitary and spinorial representations of boosts better. To my ...
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Possible charge for Abelian and non-Abelian theory

I am reading Tong's lecture note gauge theory. On page 6 in chapter 1 he writes Instead, the key distinction is the choice of Abelian gauge group. A $U(1)$ gauge group has only integer electric ...
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Selection rule on angular momentum quantum number for vector operators

According to the Wigner-Eckart theorem, the matrix element of a vector operator in the basis of eigenstates of the square of the magnitude of the angular momentum ($J^2$) and the z-component of the ...
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Relationship between multiplicity in the $k$-fold product of fundamentals and irrep dimension at large $N$

Equation 3.5 of this paper by Gross and Klebanov makes the following interesting claim. Take a group $U(N)$, with $N$ large, and consider the reducible representation $\mathcal{H}_{fund}^{\otimes k}$ ...
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Representation of Poincaré group and quantum field

How can we understand the quantum field $$\phi(x) = \int \frac{d^3p}{(2\pi)^3\sqrt{2E}} ( a_p e^{-ipx}+ a^\dagger_p e^{ipx}) ,$$ where $ a_p$ and $a^\dagger_p$ are creation annihilation operators, ...
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Where did the bases states go?

I wrote out the bases states for 4 1/2-spin particles in $|S,m\rangle$ representation. I know I should have a 16-dimensional Hilbert space, but I only have 9: $$|2,2\rangle,\dots,|2,-2\rangle; |1,1\...
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(Georgi chapter 14) Why does the $n=2$ states transform under the 5+1 of the angular momentum?

In Georgi's Lie algebra in particle physics, chapter 14, the 3D harmonic oscillator is studied. The systems exhibits $SU(3)$ symmetry in the energy levels, we can construct the 8 generators of the $SU(...
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Proving that the adjoint representation of a simple Lie Algebra satisfying $Tr(T_aT_b)=\lambda \delta_{ab}$ is irreducible

I am following the book "Lie Algebras in Particle Physics" by Howard Georgi and on page 51 he claims the statement above and goes on to prove it. I am new to this so my doubt might be ...
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1answer
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Commutator of scale transformation generator in conformal quantum mechanics

In the following notes on CFT by Joshua D. Qualls. We're introduced to conformal quantum mechanics with lagrangian:$$L=\frac{1}{2}\dot{Q}^2-\frac{g}{2Q^2}\tag{1.11}$$ It's action is invariant under $...
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Quantization of Helicity for massless particles [duplicate]

My understanding is that the quantization of Helicity for little group of massive particles comes from the fact that rotation in space leaves the 4-momentum $P^\mu=(m,0,0,0)$ invariant; we know that $...
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Symmetry Properties of Wigner's Matrices

I have an expression of the form $$S=\sum_{m,n=-j}^{j}(-1)^{m-n}D^{j}_{mn}(g)D^{j}_{mn}(g)$$ This is the end result of a long calculation, from which I am pretty confident that it is correct. For a ...
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Is there a non-trivial unitary representation of the general linear group?

Is there any interest in a quantum field theory using a unitary representation of the general linear group?
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Confusion about relationship between $\mathfrak{so}^+(1,3)$ and $\mathfrak{su}(2) \oplus \mathfrak{su}(2)$

Following from this question and the links within, I have a couple of questions about the use of $\mathfrak{su}(2) \oplus \mathfrak{su}(2)$ for the classification of finite real restricted Lorentz ...
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Final steps in obtaining the $(m,n)$-labelled finite-dimensional irreps of the restricted Lorentz group

I've been trying to understand how we get the $(m,n)$-labelled irreps of $SO^+(1,3)$ by reading posts such as this, this, this, this and links within, as well as the Wikipedia article on the matter, ...
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1answer
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Chiral spinor representations

I just want some suggestions because I cannot find anything relevant on the internet. I am working with $\mathfrak{so}(16)$ and $\mathfrak{so}(12)$. I know that their chiral spinor representations are ...
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Reasoning for projective representations of Lorentz group in the context of QFT

As far as I understand from posts such as this and this, when determining what is possible in a relativistic theory, Wigner's theorem tells us that we care about objects transforming under projective ...
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Is representation theory in QM a real physical thing or just a mathematical tool? [closed]

I was studying group theory and representation theory in Quantum Mechanics and I really don't understand yet if it is just a mathematical tool of seen the operators as a representation of a group ...
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Obtaining the unitary representation of the Lorentz group from infinitesimal transformations

My knowledge of Lie Groups and Lie Algebras is very limited, even more so when it comes to their representation theories. There is a difference that I can't quite understand, between the ...
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Orthogonality of basis functions of irreducible representations

Lets say I have two irreducible representations of a finite group $G$. Let the group have $l$ elements. The elements of the representation shall be linear operators $L$ that act on some function ...
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Is it correct to say that in QM Operators are a way of representing elements of a group acting on a state, as linear maps on a Hilbert space?

Sorry if my question is nonsense, but I was reading about representation theory in Quantum mechanics and I find it very interesting. As I understood, an example would be the charge operator, which is ...
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What are the $U(1)$ representations and their generators?

I previously asked a question about how it is possible for different fields in a gauge theory to have different $U(1)$ charges. I think the issue is that I do not actually know what the $U(1)$ irreps ...
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1answer
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Confusion about $U(1)$ representations and charge quantisation in the context of gauge theory

In a gauge theory, the fields transform under representations of the gauge group. When studying a special unitary group $SU(n)$, I've usually thought of the elements of a representation as being the ...
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How to determine the irreducible representations of a product of Clifford algebra generators

Suppose $\psi^a$, $\psi_{a}$ ($a=1,2,3,4$) are lowering and raising spinorial oscillators, respectively, and generate a Clifford algebra $\{\psi^a, \psi_{b} \} = \delta^a_b$, where an upper $a$ index ...
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Spin of Fundamental Particles

Is there any explanation/theorem which justifies that most fundamental particles have spin half or spin one? Apriori, studying representations of symmetry groups and their connection with spin of ...
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Equality of two formulas involving the Clebsch-Gordan coefficients

Consider the unique (up to unitary equivalence) unitary irreducible representation $(V_{j},D_{j})$ of $\mathrm{SU}(2)$ with dimension $2j+1$. Then, one usually defines the "Wigner D-matrices"...
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Confusion about tensor products of states in the fundamental representation of $\mathrm{SU}(2)$

This question is an extension of this question, I asked previously. Let us denote the unique irreducible unitary representations of $\mathrm{SU}(2)$ by $V_{j}$, where $\mathrm{dim}(V_{j})=2j+1$. It is ...
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Tensor product of representation of $\mathrm{SU}(2)$ Identity

I have troubles to understand an equation, which was stated in a lecture. Consider the spin-j representation $V_{j}$ of $\mathrm{SU}(2)$ with its standard basis $$\{\vert j,m\rangle\}_{-j\leq m\leq j}$...
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Does one have the freedom to impose a Euclidean metric in each finite dimensional representation of a finite dimensional semi-simple Lie Algebra?

This similar post led me to this question. In the adjoint representation, I can construct the killing form to act as a metric on the elements in the algebra. It is often convenient to choose a basis ...
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A strange requirement on projective representation

I am reading the QFT book by Weinberg. The first volume. On page 82, he discussed projective representation. $$ U(T_2) U(T_1) = \exp(i \phi(T_2, T_1)) U(T_2 T_1 ) .\tag{2.7.1} $$ Here $\phi(T_2, T_1 )$...
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Why does the adjoint action of $SU(2)$ on its own Lie algebra implement $SO(3)$?

This is a follow-up on this question. Is it obvious that the adjoint action, $Ad(g)$ of $SU(2)$ on its own Lie algebra implements $SO(3)$? I understand that the adjoint representation for a group is ...
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Rotation of plane wave spinor versus plane wave spinor with rotated momentum

Suppose I have a massless fermion with momentum $p^\mu = (E,\, E\cos\theta,\, 0,\, E\sin\theta)$. There are two ways to write the plane wave spinor $u(p)$ with respect to the spin $\xi$, and I seem to ...
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Normalizer of $SU(2)\times SU(2)$ in $SU(4)$

What is the normalizer of $SU(2)\times SU(2)$ in $SU(4)$ or how would I find it? Reason for the question: with 2 qubits, if I was interested in conjugation of 2-qubit gates with generic $SU(2)$ ...
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Commutation relation under an arbitrary Lie algebra representation

This is an exercise in Woit's book, B9, Problem 2: For the case of the Euclidean group $E(2)$, show that in any representation $\pi'$ of its Lie algebra, there is a Casimir operator $$ |\vec{P}|^2 = \...
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Invariant symbol, group representation

I have a question regarding the following passage from Srednicki's QFT book (p. 415) (https://web.physics.ucsb.edu/~mark/qft.html). Notations are $R$ = some representation of a lie group, $\bar{R}$ is ...
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Selection Rules criteria using Group Theory

When we are applying Group Theory to see whether the matrix element $\langle i|H'|f\rangle$ vanishes, we look at how the matrix element transforms. It can be shown that the matrix elements transforms ...
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Selection Rules using Group Theory

I was learning about the applications of Group Theory and one important application is looking at the selection rules in a weak EM field. We essentially want to see whether the matrix element $\langle ...
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Parametrizing a fermion bilinear for $N_f=2$

In https://arxiv.org/abs/hep-ph/9210253 the possible values for a fermion bilinear $ M=\bar{\psi_{L,i}} \psi_{R,j} $ with $i,j=1,2$ are parameterized in terms of pauli matrices as $M= \sigma \mathbb{I}...
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Infinite-dimensional representation of Lorentz algebra

In QFT, we need to use infinite-dimensional representations of the Lorentz algebra, because all the non-trivial finite-dimension al representations are not unitary, and we need unitary ...

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