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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Understanding a rule related to Young tableau

This is my first time learning Young tableau, and I am struggling with a specific rule. I am reading it from this lecture note. The rule I am confused about is the very last point at the end of page 2....
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What's the transform of the state when using Fourier transformation of operators to diagonalize a Hamiltonian?

The off-diagonal term of the 1-D Bose-Hubbard model is : $K=\sum_l a_l^\dagger a_{l+1}+a_{l+1}^\dagger a_l$. We can diagonalize it by introducing the Fourier transformation:$a_l=\frac {1}{\sqrt{N}}\...
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Identical Particles in Quantum Field Theory

In Quantum Field theory by M. Schwarz, the author in the introduction of chapter 12 on Spin Statistics theorem says, while describing identical Particles: Let $$|s_1p_1n_1,...,s_3p_3n_3\rangle \tag{1}$...
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Meaning of Representation

I am reading Schwartz's book "Quantum Field Theory and The Standard Model", in chapter 8, the author says "A set of objects $\psi$ that mix under a transformation group is called a ...
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Derivation of Raising operator of $\rm SU(2)$

I'm reading a paper called: "A Simple Introduction to Particle Physics Part I - Foundations and the Standard Model" and i have some questions regarding the derivation of the raising and the ...
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Relations between the spin of representations of Lorentz group and Poincare group

It is known that Finite dimensional irreducible representations of Lorentz group can be indexed by two half integers $(s_1,s_2)$ and the sum $s_1+s_2$ is called the spin. Infinite dimensional unitary ...
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What does it mean for particles to “be” the irreducible unitary representations of the Poincare group?

I am studying QFT. My question is as the title says. I have read Weinberg and Schwartz about this topic and I am still confused. I do understand the meanings of the words "Poincaré group", &...
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Help evaluating a trace identity of $SU(2)$ generators

In solving a problem I came across the need to evaluate the following: lets say that $J_\pm$, $J_z$ for an $N$ dimensional representation of $SU(2)$ and choose a $j<N$. In other words, the set of $...
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40 views

How do you observe “silent” quantum vibrations?

In the theory of quantum vibrations (aka phonons) it is useful to divide up the vibrational normal modes of a crystal based on their representation within the symmetry group of the crystal. The ...
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What is the cause of discrete spectra in quantum mechanics? Both compact Lie groups and bounded Hamiltonians?

What is the mathematical cause of the "quantum" in quantum mechanics? What causes some observables to take on discrete values? There seem to be two different causes, compactness of symmetry ...
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State orthogonal to a symmetric state must be antisymmetric

I am working out the exchange symmetry of the eigenstates of the total angular momentum operator of a system of two spin-1 bosons. I know that there must be a quintet, triplet, and a singlet state. ...
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Why does $A_\mu$ undergo an adjoint representation matrix transformation?

This question pertains to the following passage from Weinberg's second volume on QFT. It appears on page 4, section 15.1. To make the Lagrangian invariant, we need a field $A^\alpha_\mu$, whose ...
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Book on Representations and Group Theory in Particle Physics for Mathematicians

Is there a book for someone who already knows some group theory and theory of group representations on the mathematical side, and just wants something which explains the applications in particle ...
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When a Hilbert space's state vector becomes a spinor?

Given a Hilbert space $\mathcal{H}$, we pick up some state vector $| \psi\rangle$ which lives in the Hilbert space $\mathcal{H}$. The $| \psi\rangle$ is a vector of the Hilbert space satisfies the ...
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Proof of Rivlin–Ericksen representation theorem relies on arbitrary tensors

I am having philosophical difficulties with the use of arbitrary orthogonal tensors in the proof of the Rivlin–Ericksen representation theorem on page 6 of the these lecture notes (author unknown; ...
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How can we go from a 4-dimensional representation of $SO(4)$ to the 3-dimensional one of its proper subgroup $SO(3)$?

In Walter Greiner's book, "Relativistic Quantum Mechanics", when discussing infinitesimal tranformations: $$x^{\prime\nu}=a^{\nu}{}_{\mu}x^{\mu},$$ where the $a^{\nu}{}_{\mu}$ is are ...
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Three blocks and the representation of $S_3$

I've been studying chapter 1 of the famous group "Lie Algebras in Particles Physics" by Georgi. I am rather confused by section 1.16. The claim is the following. Consider a system of three ...
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Why are Casimir operators required to be Hermitian?

What is the physical significance of requiring Casimir operators (of e.g. Poincare group or the conformal group) to be Hermitian? What breaks down if we do not impose this condition? EDIT: To be ...
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Branching rules for coset construction: $\widehat{su}(4)_2 \rightarrow \frac{\widehat{su}(4)_2}{\widehat{su}(2)_4} \oplus \widehat{su}(2)_4$

I am trying to find the branching rules for the coset construction: $\widehat{su}(4)_2 \rightarrow \frac{\widehat{su}(4)_2}{\widehat{su}(2)_4} \oplus \widehat{su}(2)_4$, where the subscript indicates ...
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Matrix representation of the $\mathfrak{su}(1,1)$ $K$ operators

I am trying to find the matrix representation of the $\mathfrak{su}(1,1)$ $K_{-}$, $K_{+}$ and $K_0$ matrices commonly used in quantum optics defined as $$K_{-}=\frac{1}{2}\hat{a}\hat{a},\quad K_{+}=\...
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Infinite dimensional representation of generators of the Lorentz group

I was reading Schwichtenberg's "Physics from Symmetry 2nd Edition". In Section 3.7.11, there is the discussion on the infinite dimensional representations. If we consider a transformation of ...
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Why a group that describe rotations always have $su(2)$ Lie algebra?

I'm reading the book Physics from Symmetry by Jakob Schwichtenberg. In part II the author explain the Lie group theory and in particular he treat the $SU(2)$ group. At a certain point the author tells ...
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Physical significance of the reality of an ${\bf N}$ representation: how the nature of interactions is affected?

Background The fundamental representation of ${\rm SU(N)}$ is denoted by ${\bf N}$ and the conjugate of the fundamental is denoted by ${\bar{\bf N}}$. If the representation ${\bf N}$ is related to ${\...
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4 spin-1/2 particles representation [duplicate]

A follow-up question of $\frac{1}{2}\otimes\frac{1}{2}=0⊕1 $ : If I have 4 spin-1/2 particles in my system, how can I use a series of direct sums to represent $\frac{1}{2}\otimes\frac{1}{2}\otimes\...
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Clebsch-Gordan coefficients, spin networks and intertwiners

After spending some time with LQG books and articles i have still some problems regarding concepts of this theory. Spin network is built from lines labeled by spin label $j$ and since angular momentum ...
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A question about the equation $\frac{1}{2}\otimes\frac{1}{2}=1\oplus0$

I have a question about this equation: $$\frac{1}{2}\otimes\frac{1}{2}=1\oplus0.$$ I'm a bit confused by the right-hand side. Should '1' and '0' be interpreted as the total spin? If so, if there're ...
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Angular momentum singlet states for arbitrary representations

Suppose we have two-particle states $$ |0\rangle^{(j)} = \frac{1}{\sqrt{2}} \big(|+m,-m\rangle - |-m,+m\rangle \big) $$ based on one-particle states in the SU(2) $j$-representation, where I use the ...
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Do the spinor transformation matrices form a matrix representation of the corresponding Lorentz group?

Suppose $\Psi$ is a Dirac spinor, then let the transformation matrix $S$ be defined as usual: $\Psi'=S(\Lambda)\Psi$, where $\Lambda$ is the Lorentz transformation matrix. Then the questions is: for ...
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Why the little group of a massless spin-1 particle is ISO(2) rather than SO(2)?

As the title suggests, after going through a lot of Wikipedia, and references of books, I have learned that: The little group of a massive spin-1 particle is SO(3), while the little group of a ...
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82 views

What is the overlap between different spin component eigenstates?

I am trying to find an expression for the overlap between the eigenstates of different spin component operators in a spin-S system. Say I have operators $\hat{S}_i,~i=x,y,z$ with eigenvalue equations $...
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Effect of spectral flow on representations of a CFT

In this paper titled Spectral flow and conformal blocks in $AdS_3$ the authors states (See paragraph 2 on page 2), For WZNW models based on compact Lie groups, the spectral flow maps primary states ...
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What extra observable(s) are needed to label the basis states of a representation of the rotation group $N$-dimensional space?

In $3$-dimensional space, any given irreducible representation of the rotation group has a basis whose states are uniquely labeled by the eigenvalues $m$ of a single observable $J_z$, which is one of ...
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Triplet states, Dicke states and symmetric spin-1 states

When adding two spin-1/2 particles, it is well known that the basis of the composite system can be written in terms of the spin singlet and triplet states. These sates have a well defined symmetry ...
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Distinction between dual space inner product and inner product against which a representation is unitary

Every vector space $|\vec{v}\rangle$ over the field $\mathbb{R}$ or $\mathbb{C}$ contains a dual space, and so if we make an identification between elements in the dual space and the original vector ...
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What kind of matrix transformations are allowed in general relativity?

In special relativity, one can transform a 4-vector as follows: $$ x'=\Lambda x $$ Of course in this case, $\Lambda$ cannot be an arbitrary $4\times 4$ matrix of $\mathbb{M}(4,\mathbb{C})$. For ...
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Hadrons as tensors of flavor symmetry even though flavor symmetry is broken?

I will briefly summarize what I know and then ask my questions. If you spot mistakes in my summary, please tell me. The idea of flavor symmetry is that massless QCD is invariant under SU(6) ...
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Discreteness of the general angular momentum in quantum mechanics [duplicate]

When studying the general angular momentum $\textbf{J}$, which is defined as a vector operator with its components being Hermitian operators satisfying the commutation relations \begin{align*} \textbf{...
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What is the application of dimension $6$ representation of $SU(3)$ in particle physics?

As we know, the $uds$ transforms in fundamental representations of $SU(3)$. It has the antifundamental partner. According to representation theory, $$ \mathbf{3} \otimes \mathbf{\bar{3}}= \mathbf{8} \...
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Which properties of Pauli matrices can be derived and cannot be derived from their algebra (without explicit representation)? [closed]

Usually, Pauli matrices $\{ \sigma^i \} ~(i = 1, 2, 3)$ are defined as \begin{align} \sigma^1= \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix},~ \sigma^2= \begin{pmatrix} 0 & -i \\ i & 0 ...
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What's the relation between irreducible representation and $d$-vector?

In many papers, especially these related to the triplet superconductor, the superconductor order parameter, the d-vector, always appears together with the irreducible representation of the space group....
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What state has angular momentum zero among the tensor product $V_j \otimes V_j$? [closed]

While studying the representation theory, I came up with the following example, but it seems hard for me to solve. For an integer or half-integer $j$, let $V_j$ be a $(2j+1)$-dimensional complex ...
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93 views

Evaluating a matrix element of a $3\times 3$ Hamiltonian in terms of Gell-Mann matrices

A generic $3\times 3$ Hamiltonian can be expressed in terms of eight Gell-Mann matrices ($\lambda$) as \begin{align} {\cal H} &= h_{0} I + H= h_{0} I + \sum_{\alpha=1}^{8} h_{\alpha} \lambda_{\...
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In what way observables are a representation of the symmetry group?

I was studying a course about Lie groups, Lie algebras and their representations (and classifications) when I encountered this statement : When a physical system admits symmetry, the observable form ...
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Why use two spacetime indices to label Lorentz generators?

I've seen (e.g. in Srednicki) the following notation for the connection between a Lorentz transformation $\Lambda$ and the Lorentz generators $M^{\mu\nu}$: \begin{equation} {\Lambda^\mu}_\nu = {\left(...
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49 views

How to see if certain representation of $SU(2)$ is symmetric/antisymmetric/mixed?

I am looking at the tensor product of $n$ spin halfs (fundamental of $SU(2)$): \begin{equation} \left(\frac{1}{2}\right)^n = \frac{1}{2} \times \frac{1}{2} \times ... \times \frac{1}{2} = \frac{n}{2} +...
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1answer
61 views

Action of a massive spin-2 field

I'm reading about gravitational waves at the moment (mainly using Maggiore's textbook). In it he gives the Pauli-Fierz action for a massive spin-2 field and the action contains the trace of the field. ...
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Lorentz spinor in Lorentz $\rm Spin(3,1)$ signature and the real structure?

In this paper: J. Wang, X. Wen and E. Witten, "A new ${\rm SU}(2)$ anomaly", J. Math. Phys. 60 (2019) 052301, arXiv:1810.00844, it says the following in p.2, It says for $3+1$ dimensional ...
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62 views

Existences of Majorana spinors in $\rm Spin(4)$ and $\rm Spin(1,3)$

$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SU{SU}$ We know that $$ \Spin(1,3)=\SL(2,\mathbb C) $$ and $$ \Spin(4)=\SU(2) \times \SU(2). $$ It is also said $\Spin(1,...
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$ (5^* \times 5^*)_{asym}={10}$ in A. Zee's book p.409 versus PDG Sec.114

What is the mathematical or physical way to understand why the 4th and 5th components in the Georgi Galshow SU(5) model has the SU(2) doublet $(1,2,-1/2)$: $$ \begin{pmatrix} \nu\\e \end{pmatrix} $$ ...
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Quantum group symmetry in QFT

In Wikipedia article Coleman–Mandula theorem there's statement: Quantum group symmetry, present in some two-dimensional integrable quantum field theories like the sine-Gordon model, exploits a ...

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