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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Clebsch-Gordan Series and Rotation Matrices

I am referring to the first inequality in equation 3.390 on page 217 of Sakurai's "Modern Quatum Mechanics" textbook. The quantity $D^{(j)}(R)$ refers to a rotation operator in the ket space ...
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Apparent elimination of a 't Hooft anomaly in quantum spin system

The simplest system with a 't Hooft anomaly is the spin $\frac{1}{2}$ system with hamiltonian $\hat{H}=0$. The 't Hooft anomaly follows from the fact that such system has a trivial $SO(3)$ symmetry, ...
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From a quant-info perspective why are the reals indexing irreps of the Lorentz group less suspect than continuousness in space-time and general QM?

It is an opinion I occasionally hear, and perhaps hold myself, that the resolution to the 'infinities' that crop up in various bits of physics are artefacts of the approximation that space-time is ...
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A relation for adjoint representation of $U(N)$ acting on product of matrices and $SU(2)$ generators

Is the following relation true, and if so, what is the property that makes it so? \begin{align} \sum_{i=1}^3\mathrm{tr}\left([U^{-1}L_iU]\phi[U^{-1}L_iU]\phi\right) \stackrel{!}{=} \sum_{i=1}^3\...
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Basic Facts about Lie Algebras

I am reading P&S (Peskin's and Schroeder's book An Introduction to Quantum Field Theory) and in particular Chapter 15.4. At some point the authors say that any infinitesimal group element $g$ can ...
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Abstract definition of four-vector

It is a long time that I am looking for an abstract definition of four-vectors. This is the definition that I have reach to so far: A four-vector is an element of the representation space of the ...
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Why does general irreducible representation $(A, A)$ of a quantum field correspond to traceless symmetric tensor of rank $2A$? [duplicate]

I understand that under rotation, we will have components that transform like integer spin $(2A, 2A-1, ...... 0)$ from decomposition of $(A, A)$ representation. The scalar is the trace, therefore ...
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Invariants of inner product in pseudoreal representation of $SU(2)$

I am reading Peskin's and Schroeder (P&S), "An introduction to Quantum Field Theory", specifically the first paragraph on page 499 in section 15.4 "Basic Facts about Lie Algebras&...
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Importance of raising/lowering/ladder operators

While learning quantum mechanics I repetitively encountered the concepts of raising and lowering operators. Firstly in the harmonic oscillator $a_\pm \ $and now in angular momentum I am introduced to $...
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Confusion in group theory at the example of the Lie algebra $su(2)$

Please excuse me for this unstructured question: I have some problems with understanding group theoretical aspects relevant to physics and would like to make one of my confusions clear by discussing ...
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Constructing gauge invariants

Is there an efficient way for constructing gauge invariants given the number of operators one can use is fixed. For example, if I am given some boson in $\mathbf{3}$ of $SU(2)$, and I want to find ...
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Help needed in understanding $2\otimes 2=1\oplus 1\oplus 2$ of $SO(2)$

Vectors $\vec A=(A_1,A_2)$ and $\vec B=(B_1,B_2)$ are 2-dimensional representations of $SO(2)$. I want to understand the decomposition $$2\otimes 2=1\oplus 1\oplus 2.$$ I can easy identify that the ...
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Why does group representation theory look linear?

I'm reading first a few chapters of a physicist's group theory book and one naive question comes into my mind. I feel I probably missed something very basic and got bogged down in the details. My ...
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Non-Hermitian solution for $SU(2)$ Lie algebra?

In QFT textbooks, representations for Lorentz group is constructed from $A$-spin & $B$-spin discussion. The Lie albegra of Lorentz group is $[J_i, J_j] = i\epsilon_{ijk} J_k,~[J_i, K_j] = i\...
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Exponential of sum of spin operators

Let us consider a system of two spins, each with spin $S$. Let $\vec{S}_1$ and $\vec{S}_2$ be the two operators representing the $\mathfrak{su}(2)$ algebra. Consider now the exponential of the square ...
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Parameterization of $SU(4)$

I'm looking for a parameterization or general form for a $4\times 4$ special unitary matrix in closed form. I have looked at the answer to Good reference on the parametrization of $SU(3)$ and $SU(N)$ ...
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Given a representation of $su(3)$ labelled by $(p, q)$, is there a way to construct its state of greatest weight?

My current understanding of the representations of $\mathfrak{su}(3)$ is as follows: We can construct 3 $\mathfrak{su}(2)$ subalgebras with step operators $I_\pm, U_\pm, V_\pm$. These maybe be ...
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From the point of view of physics, why is it useful to know the irreps of rotation group?

In 3D, the rank two tensorial physical quantities, for example, the electric susceptibility, the conductivity, the stress tensor etc, are in general, not irreducible representations i.e. neither ...
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Is the concept of bicolored gluons mathematically precise/meaningful? Please explain

Each flavour of quark carries a colour quantum number: red, green or blue. I know what it means mathematically. But elementary textbooks (e.g, particle physics by Griffiths) also say that gluons are ...
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What does it mean for the $\textbf{B}$-field (Hypercharge) to be in the 0 representation within the SM?

I was reading through the wikipedia page for the mathematical formulation of the standard model and I noticed that it listed the representations of the vector bosons under the SM gauge groups as being ...
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Is every unitary representation of the Poincare group a direct sum of Wigner's irreducible representations?

Is Wigner's classification of unitary irreducible representations of the Poincare group [1] sufficient for constructing all unitary representations of the Poincare group by taking direct sums? The ...
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Quantum chromodynamics - why are there no $rrb$ or $ggr$ terms?

$$\Psi_{colour}^{qqq} = \frac{1}{\sqrt{6}}(rgb + gbr + brg -grb - rbg - bgr)$$ My lecturer stated that there cannot be any $rrb$ or $ggr$ terms in the expression above. I would like to understand what ...
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What does it mean by spin 1/2 or spin 2 field?

I see in common discussions people simply use the terminology spin 1/2 field or spin 2 fields as if it is some common term like hamiltonian. How to think about these fields and understand what it ...
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Extracting $\mathbf b$ from $M = a I + \mathbf b \cdot \mathbf S$, when $S_i$ are higher spin matrices?

This is a cross-post of a question that I posted on the Math SE, that did not get any answers there. It is fundamentally a mathematics question, but it pertains to spin matrices, which many Physics SE ...
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Orthogonality relations for matrix elements of irreducible representations

I am reading Howard Georgi's "Lie Algebras in Particle Physics" and have a question concerning the presented orthogonality relations for matrix elements of irreducible representations. To ...
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How does SUSY differ from just having standard model (SM) particles in new Lorentz Group representations?

Under the standard model (SM), it is the irreducible representations of the Lorentz group which allows us to classify particles according to their spin. We can then further label particles using the ...
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Fermionic oscillator and reducible representation

Consider the fermionic oscillators $\{a, a^\dagger\} = 1$, $\{a, a\} = \{a^\dagger, a^\dagger\} = 0$. The commonly used irreducible representation is given by $|0\rangle$, $a^\dagger |0\rangle$ where $...
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Tensor products in Howard Georgi's "Lie Algebras in Particle Physics"

My question is regarding eq.(3.39) in the second edition of Georgi's book (for those who have the book:)). The section deals with tensor product states where the states comprising the product ...
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The irreducible representation of rank $n$ spinor in 3D

I was reading Ref. 1, where it is asserted that the irreducible resolution of a rank $n$ spinor can be written as totally symmetric spinors. $$\Psi^{n}=\Psi^{\{n\}}+\zeta \Psi^{\{n-2\}}+\zeta \zeta \...
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What does this curly bracket notation mean?

I came across this $$\langle H_A^{i\{1,\,k\}}\rangle=f^{i\{1,\,k\}},\,\langle H_{A\{1,\,k\}i}\rangle=f_{A\{1,\,k\}i},\,i,\,k=2,\,3$$ curly bracket notation to denote VEVs (vacuum expectation values) ...
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Irreducible representations and Hilbert spaces

I am reading Howard Georgi's book "Lie Algebras in Particle Physics" where he writes the following (chapter 1.14:eigenstates): "... if some irreducible representation appears only once ...
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What does the Verma module in the reducible Virasoro algebra represent?

In the conformal field theory book by Francesco, Mathieu, Senechal, the Verma module is built from a primary field $|\phi\rangle$, and if one of the descendants is a singular vector $|\chi\rangle$, ...
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Does the location of the Hilbert space of momentum eigenstates in QFT change under time translations and boosts?

I have two questions concerning Wigner's transformation law for irreps of the Poincare group: \begin{equation} U[\Lambda,\vec{a}]\vert p,\sigma\rangle=e^{ip\cdot a}D_{\sigma'\sigma}[\Lambda;p]\vert \...
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What elements of my group am I missing and which group is it? [closed]

I am working on an exercise which is asking to find the elements of the symmetry group of the following figure given below: Note that the rectangular sides of the box all have the exact same pattern ...
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Spin as Poincaré invariant label

I was thinking about how we construct unitary representations for the Poincaré group in the case of massive particles. We move to a frame where the particle is at rest, and here the little group that ...
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On the physics of representations of the Lorentz group over real vs. complex vector spaces

After reviewing this question as well as this one, I am left with some confusion, mainly about the nature of complex and real representations of the Lorentz group and how we do physics. I understand ...
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How to make notation like $Y_{l m_{l}}(\theta, \phi)\chi_{m_s}$ more rigorous as a tensor product?

Sometimes in quantum mechanics we come across notation like $Y_{l m_{l}}(\theta, \phi)\chi_{sm_s}$ where $Y_{lm_l}$ is a spherical harmonic representing the spatial part of some particle wavefunction ...
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Why is the spin part of the wavefunction contained in a Hilbert space of dimension $2s+1$ for a particle with spin $s$?

The statement in the title of the question is in my lecture notes. I don't understand the reason for this. Furthermore, I'm having trouble understanding what spin is in the first place. I know that ...
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Does the Dirac Spinor live in the complexification of the Lorentz group?

In this question I learned that when working with quaternionic representations of (the double cover of) our relevant orthogonal group, we cannot avoid working in complex vector spaces. However it is ...
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Constructing irreducible Lie algebra representations

In physics, a (irreducible) representation $\rho:\mathfrak{g}\rightarrow\mathrm{End}(V)$ of a Lie algebra $\mathfrak{g}$ is often "constructed" by finding the weights of the representation, ...
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Dimension of spin zero Hilbert subspace

Consider a system of 4 different spin-5/2 particles, each transforming in the dimension-6 irreducible representation of $SU(2)$. The dimension of the total Hilbert space is $6^4$. Could anyone show ...
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Non-linear group representations

Different representations of the Lorentz group in infinite-dimensional vector spaces are related to the types of particles (classification by spin and mass). These representations are linear. Are non-...
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Computing $\left\langle\chi^{V}, \chi^{V}\right\rangle$ in $D_3$

I am having trouble computing $\left\langle\chi^{V}, \chi^{V}\right\rangle$ in $D_3$. The character table is: $$ \begin{array}{c|ccc} & (E) & (R) & (S) \\ \hline D^{V} & 3 & 0 &...
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$\mathcal{su}(3)$ irreps in the fermionic oscillator

Consider the fermionic oscillator, for $d=3$ degrees of freedom, with the Hamiltonian $$H=\frac{1}{2}\sum_{j=1}^3(a_{F_j}^{\dagger} a_{F_j}-a_{F_j} a_{F_j}^{\dagger})$$ Use fermionic annihilation and ...
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Hamiltonian spectrum with degeneracies equal to powers of 2?

Does there exist a (finite or continuous) group such that every irreducible representation has dimension equal to a power of 2? The context is that I have a quantum many-body Hamiltonian on a 1D spin ...
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2 votes
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$W_3$ algebra conformal block at level 1

I'm in the process of learning CFT theory, and I'm trying to reproduce the known result of the calculation of the $W_3$ algebra conformal block at level 1 given in Appendix A.2 of the paper https://...
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2 votes
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Finding the Hermitian generator of a representation of a Symplectic transformation

Consider a set of $n$ position operators and $n$ momentum operator such that $$\left[q_{i},p_{j}\right]=i\delta_{ij}.$$ Lets now perform a linear symplectic transformation $$q'_{i} =A_{ij}q_{j}+B_{...
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Sum of squared Wigner $3j$ symbols

Since Wigner $3j$-symbols for $m_1 = m_2 = m_3= 0 $ have the property $$ \sum_{l_3 = |l_1 -l_2|}^{l_1 + l_2}(2 l_3 + 1)\begin{pmatrix} l_1 & l_2 & l_3 \\ 0 & 0 & 0 \\ \end{pmatrix}^2 =...
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3 votes
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Operators in the addition of $N$ spin 1/2 particles

Set-up: We are given $N$ spin $\frac{1}{2}$ particles, with associated Pauli operators $\{X_i, Y_i, Z_i\}_{i=1}^N$. We can quickly verify that the operators $\bar{X}=X_1+X_2+\ldots X_n$, $\bar{Y}=Y_1+...
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Clebsch-Gordan Matrix Generalization

I am trying to obtain the Clebsch-Gordan matrix that changes from the coupled angular momentum basis to the decoupled basis when coupling several $\frac{1}{2}$ spins. So far, I have obtained the ...
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