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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Generators for the complexification of $\mathfrak{so}(1, 3)$

As far as I know $\mathfrak{so}(1,3) \cong \mathfrak{sl}(2,\Bbb C)$ and $\mathfrak{so}(1,3)_\Bbb C \cong \mathfrak{sl}(2,\Bbb C)_\Bbb C \cong \mathfrak{sl}(2,\Bbb C) \oplus \mathfrak{sl}(2,\Bbb C)$ I'...
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What do these energy level diagrams and notation mean in this Buckminsterfullerene dipole transitions question?

I have some questions about the authors' solution to the following homework style question: $C_{60}$ fullerene molecule question" /> I have added the number of electrons per level into the image, the ...
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Why being a non-compact group implies no finite unitary representation? Is this a mathematical theorem? [duplicate]

It is said that relativistic quantum mechanics is hard because unlike the $SO(3)$ group which is compact, the Poincare' group ($\mathscr{P4}$) is a non-compact group and from here it do imply that ...
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On finite-dimensional unitary representations of non-compact Lie groups

In this thread Lorentz transformations for spinors, V. Moretti made a claim as follows: "it is possible to prove that no non-trivial finite-dimensional unitary representation exists for a non-...
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How are $SL(2, \mathbb{C})$ and $SL(2, \mathbb{C}) \times SL(2, \mathbb{C})$ related to the Lorentz Group?

I know from Weinberg and Schwartz's book on Quantum Theory of Fields that $SL(2, \mathbb{C})$ double-cover $SO^{+}(1,3)$. However, moving to the Lie algebra, based on the following wiki: https://en....
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Massless representation of $so(3,2)$

I am studying the article by Flato and Fronsdahl (http://dx.doi.org/10.1007/BF00400170). And they say that in the massless case, if the spin $s$ = 0, then the representation $D(1.0)$ (formula (3)) is ...
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What are singlet representations for the electroweak gauge group $SU(2)\times U(1)$?

This question comes from Srednicki's textbook Quantum Field Theory. On page 532, the left-handed Weyl fields $\ell$ (a single lepton family, electron and its neutrino) and $\overline{e}$ are in the ...
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How to find explicit (finite-dimensional) representation of a group?

Given the field in a finite-dimensional vector space $V$, how do I explicitly find the generators of the group in the representation $R:G\rightarrow GL(V)$? I know the theory behind groups' ...
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Projective Representations of $SO(2)$?

For $SO(3)$ and $SO^+(3,1)$, their projective representations take one of two possible forms \begin{equation} D(\Lambda_1)D(\Lambda_1) = \pm D(\Lambda_1\Lambda_2) \end{equation} As far as I know, only ...
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The Vector representation of the Poincare group

The Poincare group is the semi-direct product of the translation group and the Lorentz group. Its generators are $P^{\mu}$ and $M^{\mu\nu}$, and a representation of these generators is important in ...
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Classifications of quasiparticles

Different particles can be represented as different irreducible representations of Poincare group. Can we classify quasiparticles using irreducible representations of some group? Also, quasiparticles ...
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How to decompose the representation of $\rm SU(5)$?

This question comes from Srednicki's textbook "Quantum Field Theory". On pages 514-515, it states: Under the unbroken $\rm SU(3)\times SU(2) \times U(1)$ subgroup, the $5$ representation of ...
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Tensor product of fundamental representation of $SU(5)$

In Srednicki's QFT book (see pg. 605) he writes that the fundamental representation of $SU(5)$ may be written as (97.2), $$ 5\to \left(3,1,-\frac{1}{3}\right)\otimes \left(1,2, +\frac{1}{2}\right).\...
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Thinking about spin triplet and singlet states in QFT

In the case of quantum mechanics, we can think of $SU(2)$'s 2-dimensional representation, which describes spin-1/2 space. This allows us to understand the spin state of a pair of spin-1/2 particles by ...
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Representations of the Poincaré group

I am currently trying to understand the representations of the conformal group. I am following the script by J. D. Qualls. At page 29, the author finds the effect of $L_{\mu\nu}$ by "studying the ...
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Does spin 3/2 imply 2/3 full rotations? [duplicate]

In this Wikipedia page it writes: ... a spin of 1/2 means that the particle must be rotated by two full turns (through 720°) before it has the same configuration as when it started. The animation in ...
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Spin-$n$ particle comes back to itself after $360/n$ degree rotation

On the Wikipedia page for spin, a claim is made that a spin-$n$ particle comes back to itself after a $360/n$ degree rotation. I quote: A spin-zero particle can only have a single quantum state, even ...
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Physical Significance of Irreducible representation of 3D rotation group

I read about the irreducible representation of groups in the book 'Lie Algebra in Particle Physics'. Since then, I have been wondering if there is a special physical meaning hidden in irreducible ...
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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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Lorentz transformation: unitary $U(\Lambda)$ on the state of the Hilbert space, but the boost $\Lambda_{1/2}$ is not unitary on $\psi$

In p.59 of Peskin and Schroeder QFT book, he mentioned that the operator $U(\Lambda)$ that implements the Lorentz transformations on the state of the Hilbert space is unitary, even though the boost $\...
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Irreducible Representations: Howard Georgi's book

I am currently studying the book Lie Algebras in Particle Physics by Howard Georgi (2nd edition) and I couldn't understand this part: A representation is completely reducible if it is equivalent to a ...
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Discrete series representation of $SO(2,1)$ [closed]

Since $so(2,1)$ is a noncompact Lie algebra, its unitary representations are all infinite-dimensional. Now, I have seen that the unitary representations of $so(2,1)$ can be divided into three ...
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Are there finite dimensional representations of the Poincaré algebra with non-nilpotent momentum generators?

The matrix representations of the Poincaré algebra that I am familiar with, have a nilpotent set of momentum generators: $p^\mu p^\nu = 0$. I am wondering whether it is possible to have finite ...
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Difference between $\phi(x)\to\phi'(x)=\phi(\Lambda^{-1}x)$ and $\phi(x)\to\phi'(x) =e^{-\frac{i}{2}\omega_{\mu\nu}\mathcal{L}^{\mu\nu}}\phi(x)$

A set of objects $\phi^\alpha$, with $\alpha=1,2,...n$, transforms as a representation $D(\Lambda)$ of dimension $n$ of the Lorentz group if, under a Lorentz transformation: $$\phi^\alpha(x)\to\phi^{\...
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Representation of $\mathrm{SO(3)}$ acting on 2-index tensors Zee's Group theory book

In the chapter IV.1 "Reducible or Irreducible?" of Zee's Group Theory book (p. 188-), the author breaks a 2nd rank tensor $T^{ij}$ into invariant subspaces with respect to the action of $\...
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Difference between Cartesian product $\times$ and tensor product $\otimes$ on groups

After a comment of John Baez to a question I asked on MathOverflow, I would like to ask what the difference between, for example, $SU(3)\times SU(2) \times U(1) $ and $SU(3) \otimes SU(2) \otimes U(1)$...
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How to prove Weyl spinors transform as a representation of Lorentz group?

In my QFT lecture notes, it is written that the Lorentz group elements can be written as \begin{equation*} \Lambda = e^{i\vec{\theta}\cdot\vec{J} + i\vec{\eta}\cdot\vec{K}} \end{equation*} where $\Big\...
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Symmetry action on $k$-space creation operator

Assuming I am working in a periodic system and performing a plane wave expansion, I generally have creation/annihilation operators given by $c_{\mathbf{k},\mathbf{Q}}$. The action of a symmetry $g$ is ...
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What does $\Lambda^{-1}_{\frac{1}{2}}\gamma^\mu\Lambda_{\frac{1}{2}}=\Lambda^\mu_{\phantom{\mu}\nu}\gamma^\nu$ mean?

\begin{equation} \Lambda^{-1}_{\frac{1}{2}}\gamma^\mu\Lambda_{\frac{1}{2}}=\Lambda^\mu_{\phantom{\mu}\nu}\gamma^\nu \end{equation} In P&S, p. 42: Equation (3.29) says that the $\gamma$ ...
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What does "operators on a Hilbert space form an algebra" mean?

I was reading some group theory notes and I am familiar with the concept of a Lie algebra, but I cannot imagine what the following formulation means: What is more, not only states, but also the ...
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Mixed symmetry of rank $3$ tensor

I have rank 3 tensor $T_{ijk}$ with following properties: $T_{ijk}=T_{jik}$ $T_{ijk}=-T_{kji}$ Is it true that there is the only one tensor of rank 3 with those properties and it is $T_{ijk}=0$. I'm ...
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How does an external electric field induce a symmetry lowering perturbation to the $D_{3h}$ point group?

I have the following homework style question which I will typeset 'word for word'. I am having trouble understanding the authors' solution to part $\mathrm{ii}.$ of the following questions: An ...
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Representation theory of $SU(2)$: Left-Right symmetric models

My question concerns the following excerpt of the article A Fermionic bi-Doublet Effective Field Theory for Dark Matter: I do not understand how the action of $SU_L(2) \times SU_R(2)$ on $Mat^{2 \...
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Group Theory Book

I am looking for a problem-and-solutions book that deals with group theory topics that are important in physics. Some topics I am looking for are as follows. (1) Group Representations (2) ...
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Gauge field and Lie group

I'm studying $SU(N)$ gauge theory, but I'm confused. Here(Gauge fields -- why are they traceless hermitian?), the reason why a gauge field is in the Lie algebra of a gauge group $G$ is that we have ...
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Physically, what is a pseudoreal representation?

There are three kinds of representations: real, complex, and pseudoreal. A complex representation is not equivalent to its conjugate, and a real one is, which is pretty straightforward. A pseudoreal ...
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Parameters in Lie group theory

This is from P.Woit's "Quantum Theory, Groups and Representations", p. 151. If the Hilbert space is the space of complex-valued square-integrable functions on the circle, we want to find the ...
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Translating an operator (generator of a symmetry) acting on a field

The representation of Poincare symmetry on fields at the origin, $\Phi(0)$, induces a representation of Poincare symmetry on a field at any point $\Phi(x)$. For Lorentz transformations, we define a ...
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$\rm SU(2)$ transformation of spinors

In the book QFT by Ryder on the topic $\rm SU(2)$ and the rotation group, it is stated that, The group $\rm SU(2)$ consists of $2\times2$ unitary matrices with unit determinant, $$UU^\dagger = 1, \...
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What happens when you replace an identity matrix with a matrix full of ones?

In physics, we often use resolutions of identity $$\sum_n |n\rangle\langle n|=\mathbb{I}$$ to simplify expressions. Sometimes, the "full matrix" (for lack of a better term) $$\sum_{m,n}|m\...
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Matrix elements from commutation relations

Suppose we are given commutators of the spin operators: $[S_X, S_Y], [S_Y, S_Z]$ and $[S_Z, S_X]$. Then can we completely determine the matrix representation of the operators? Can we do it in any ...
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Composition of angular momentum (quantum): how do we know that finding common eigenspace of $J^2$ and $J_z$ is enough for degeneracy?

I have some basic question on composition of angular momentum (actually spin in my case), I forgot some basis. The fundamental commutation relations between $J_x,J_y,J_z$ (the three components of the ...
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Rotations of spherical harmonics and Wigner $D$-matrices

I seem to be having trouble understanding how Wigner D-matrices rotate spherical harmonics. I asked this question on the Maths Stack Exchange but decided to cast my net a bit wider and ask the ...
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On mathematical representation of spin singlet

Is there a general way to express the singlet state? For instance, is this form $$ \frac{1}{\sqrt{2}}\left(\vert \uparrow \rangle \vert \downarrow \rangle - \vert \downarrow \rangle \vert \uparrow \...
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Nilpotency of conformal transformation matrices

I have recently found the statement that the generators of (infinitesimal) conformal transformations over the algebra of fields at the origin must be nilpotent. This claim is said to follow from the ...
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Gauge Field Transformation Properties

I'm a bit confused about the gauge transformation properties of non-abelian gauge fields, and I just wanted some clarification. I keep seeing the statement that "gauge fields transform in the adjoint ...
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How do we understand the ${\bf 3}$ of $Q_L({\bf 3}, {\bf 2})_{1/3}$?

A left-handed quark doublet of the Standard Model is specified as $Q_L({\bf 3}, {\bf 2})_{1/3}=(u,d)^T$. I have a problem understanding this quark doublet as a triplet of ${\rm SU}(3)$. Any help? I ...
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Why should the infinite-dimensional representation of Poincare group induced by the unitary representation of little group be unitary?

In Weinberg's Quantum Field Theory (Vol. I, pages 64-67) it is stated that a unitary representation of little group induces a unitary representation of the Poincare group. But I don't understand how ...
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How groups act on fields in QFT?

I read a lot a posts on how to verify what are the symmetries of a given Lagrangian but I really can't find what I need and can't even get it by myself, this because I don't actually understand how ...
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Connection between particle physics and weight diagrams

I have a hard time combining two topics that are often discussed in physics in a coherent way. In a lot of Introduction to particle physics-classes one will hear about "multiplets", which ...

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