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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0answers
44 views

Relation of Wigner $d$-matrix $d^l_{m',m} = d^l_{-m,-m'}$

How do you derive the symmetry relation of the Wigner $d$-matrix, i.e., $$ d^l_{m',m} = d^l_{-m,-m'} $$ I know how Wikipedia proves this using the fact that $(Y_l^m)^* = (-1)^m Y_l^{-m}$ (basically ...
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1answer
126 views

Angular momentum and rotation group representations

In Sakurai's book it's written that the operator $D_{m',m}^{(j)}=\left\langle{j,m'}\Big|\exp{\frac{-i \mathbf{ J\cdot \hat{n} } \phi}{\hbar}}\Big|{j,m}\right\rangle$ is the "$2j+1$-dimensional ...
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5answers
2k views

Spin operators in QM

In a text (Introduction to Quantum Mechanics by Griffiths) I am using it states without motivation that spin angular momentum has the same commutations relations as orbital angular momentum (these ...
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1answer
109 views

Arriving at the $\big(\pi_\ell,P_\ell(\mathbb{C}^2)\big)$ representation of $\mathfrak{su}(2)$

I think I'm really close, but confused on applying the multivariable chain rule and untangling the result. The $(\Pi_\ell,P_\ell(\mathbb{C}^2))$ representation of $SU(2)$ induced from the fundamental ...
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1answer
389 views

Symmetry of Clebsch-Gordan coefficients

The symmetry of clebsch-gordan coefficients $\left< j_1j_2;m_1m_2 \middle| j_1j_2;JM \right>$ under exchange of $j_1,m_1$ and $j_2,m_2$ is \begin{equation} \left< j_1j_2;m_1m_2 \middle| ...
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2answers
62 views

What is the vector representation of $\mathrm{SO}(6)$?

What is meant by the vector representation of $\mathrm{SO}(6)$? I have only encountered the term vector representation in the context of the Lorentz group $\mathrm{SO}(1,3)$, where it refers to the $...
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0answers
54 views

Representing a four-vector by a rank 2 tensor

When representing a 3-vector $(x,y,z)$ as a skew-symmetric matrix like this: $X=\begin{pmatrix} 0 & -z & y\\ z & 0 & -x\\ -y & x & 0 \end{pmatrix}$ $X$ transforms a a rank 2 ...
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1answer
72 views

Is there an easy way to compute $\exp(-i\pi J_2) |jm\rangle = (-1)^{j-m} |j,-m\rangle$?

Is there an algebraic way to compute $\exp(-i\pi J_2) |jm\rangle = (-1)^{j-m} |j,-m\rangle$. I know this is basically the Wigner $d$-matrix (which I can just look up), but how is it derived in this ...
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33 views

Rotation generators calculation $(J^{\pm})^2=s_{\pm}(s_{\pm}+1)$ [duplicate]

The generators of the Lorentz group $M_{\mu\nu}=-M_{\nu\mu}$ can be split into $$ M_{ij}=\epsilon_{ijk}J_k $$ and $$ M_{0i}=K_i $$ rotations and boosts respectively. These obey the commutation ...
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47 views

Explicit form of symmetry operators including spin

Symmetry operators like $\hat{1}$: unity operator $\hat{R}_{x,\pi}$: rotation around $\vec{e}_{x}$ with angle $\pi$ $\hat{M}_x$: mirror at a plane with normal in e.g. x direction $\hat{I}$: inversion ...
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2answers
43 views

Understanding the inverse in the definition $(\tilde{\Pi}_gf)(v)\equiv f(\Pi^{-1}_gv)$

I'm trying to understand the representation $\tilde{\Pi}$ induced from the fundamental representation $\Pi$, defined as $(\tilde{\Pi}_gf)(v)\equiv f(\Pi^{-1}_gv)$ for $g\in G,\hspace{1mm}f\in\mathcal{...
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1answer
161 views

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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1answer
108 views

$SO(3)$ 5×5 irrep: what does it look like explicitly? (Zee IV.1)

BACKGROUND. So my understanding is this: the $3\times3$ rep of $SO(3)$ are matrices $R$ that rotate a vector $x$, for example: $$ x \rightarrow Rx = \left(\begin{matrix} 1&0&0 \\ 0&C&...
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1answer
29 views

Definition of reducible representation

A reducible representation of a group $g \rightarrow D(g)$ is one which leaves a subspace $U$ invariant, i.e. $D(g)|u\rangle \in U, \space \forall |u\rangle \in U$.A completely reducible ...
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4answers
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$\mathrm{SU(3)}$ decomposition of $\mathbf{3} \otimes \mathbf{\bar{3}} = \mathbf{8} \oplus \mathbf{1}$?

I have a question about the tensor decomposition of $\mathrm{SU(3)}$. According to Georgi (page 142 and 143), a tensor $T^i{}_j$ decomposes as: \begin{equation} \mathbf{3} \otimes \mathbf{\bar{3}} = \...
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1answer
41 views

Representations of $SO(4) \times SO(2)$

In a paper I'm reading the representations of $\mathrm{SO(4)} \times \mathrm{SO(2)}$ play an important role. Specifically, the $(\mathbf{4},\mathbf{1})\oplus(\mathbf{1},\mathbf{2})$ is considered. I ...
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1answer
60 views

What are the irreducible representations (Clebsch-Gordan decomposition) of $\mathbf{10}\otimes \mathbf{3}$ in $SU(3)$?

Since a rank-3 tensor has 10 components and a rank-1 tensor has 3 components in $SU(3)$, I know that we are searching for the different irreducible representations of the tensor $v_{ijk}w_{l}$. The ...
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4answers
609 views

Position representation of spin states and spin operators

How can we represent a spin states $ \lvert S_x:+\rangle, \lvert S_y:+\rangle,\lvert S_z:+\rangle ,\lvert S_x:-\rangle, \lvert S_y:-\rangle $ and $\lvert S_z:-\rangle$ in position representation ...
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21answers
32k views

Comprehensive book on group theory for physicists?

I am looking for a good source on group theory aimed at physicists. I'd prefer one with a good general introduction to group theory, not just focusing on Lie groups or crystal groups but one that ...
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2answers
900 views

Spinors in 2+1 dimensions

I am trying to understand representations of the Poincare/Lorentz group, and in particular spinors, in 2+1 dimensions. I know some of the math, but I'm not sure about the physical interpretation of it ...
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1answer
92 views

Spinor Understanding: QFT vs pure Representation Theory

I have some questions on terminology used in QM & QFT and (pure mathematical) representation theory treating the concept of "spinor". Let us focus on Dirac spinor as described in https://en....
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1answer
63 views

What kind space does spinor lives in?

I'm trying to read some differential geometry these days and I just encountered orientable manifold. Quote: "If $M$ is nonorientable, $M$ has a two-sheeted orientable covering manifold $\tilde{M}$. ...
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0answers
59 views

Particles in curved space-time and group representation

It is well-known, that particles in Minkowski space can be constructed as unitary projective representations of the Poincaré group, i.e. isometry group of Minkowski space $M_d= \frac{Poincare_d}{SO(1,...
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1answer
55 views

Left and right Weyl representations are inequivalent representations

We introduce the two-component spinors in the following representations: $$ \psi_\alpha \rightarrow\psi'_\alpha=\mathcal{M}_\alpha^\beta\psi_\beta$$ $$ \bar\psi_\dot{\alpha}\rightarrow\bar\psi_\dot{\...
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3answers
212 views

Operator-valued vectors and representation theory

Let $G$ be a Lie group and $\pi : G\to GL(V)$ a finite-dimensional representation of $G$ in the vector space $V$. For every $g\in G$ we have a linear transformation $\pi(g) : V\to V$. Being linear, if ...
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1answer
70 views

Representing $su(2)$ Lie algebra on a torus

I've recently taken up the study of QFT (as a post retirement hobby), based on texts by David Tong and Anthony Zee. My question is based on the Lie Algebra of the $SU(2)$ group, and how this may ...
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1answer
94 views

Lie algebra/group/basis of the four gamma matrices along with the identity?

Do the four gamma matrices along with the identity element constitute a lie algebra? With real coefficients we have $$ \mathbf{v}_{\mathbb{R}}=aI+t\gamma_0+x\gamma_1+y\gamma_2+z\gamma_3 \tag{real ...
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1answer
233 views

How are unitary representations different from other representations?

I understand that unitary representations arise naturally in quantum mechanics when groups act on the Hilbert space in a way that preserves probability. I don't understand what details make unitary ...
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0answers
40 views

How do the Weyl spinors differ from dotted and undotted spinors? [duplicate]

As asked in the title, how do the Weyl spinors $(\frac{1}{2},0)$ and $(0,\frac{1}{2})$ differ from dotted and undotted spinors?
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0answers
37 views

Possible to have Dirac AND Majorana mass?

Supposing you have a lagrangian consisting of $(1/2,0)\oplus (0,1/2)$ representation. Writing in terms of Weyl fermions, the following terms are possible: $$-\frac{m_1}{2} (\psi_R^T \epsilon \psi_R -...
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1answer
61 views

Behavior of derivative on Lorentz transformations of spinors

I'm currently working through Supersymmetry Demystified by Patrick Labelle and one passage in particular confuses me. Specifically, if $\eta$ and $\chi$ are right and left Weyl spinors respectively, ...
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2answers
260 views

Are all representations of a finite group unitary?

I am reading Zee's Group theory in a nutshell for physicists and came across the following theorem (Page 96): Unitary representations The all-important unitarity theorem states that finite ...
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0answers
54 views

How do I show that the tensor product of $\mathbf{3} \otimes \mathbf{\bar{3}}=\mathbf{1} \oplus \mathbf{8}$? [duplicate]

It's often stated that the tensor product of the representations of $SU(3)$ satisfies $\mathbf{3} \otimes \mathbf{\bar{3}}=\mathbf{1} \oplus \mathbf{8}$, and that this implies that if flavour $SU(3)$ ...
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3answers
78 views

How am I to interpret $\text{Tr}(\text{ad}_X\text{ad}_Y)$?

I'm trying to show that the $(2,0)$ Killing tensor is invariant under the $\text{Ad}$ homomorphism: $K(\text{Ad}_A(X),\text{Ad}_A(Y))=K(X,Y),$ with $X,Y\in \mathfrak{g},\hspace{1mm}A\in G,$ and $K(X,Y)...
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2answers
487 views

Definition of a spinor and applications to GR

I understand the construction of the Clifford algebra $C(r,s)$ and in turn the corresponding $Pin$ and $Spin$ groups. I would like first to clarify that $Spin(r,s)^e$ is the universal covering group (...
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0answers
22 views

Notation of basis functions for irreducible representations

In character tables for symmetry groups, there are typically basis functions for each irreducible representation given. There are basis functions given like $xy$, $S_x$ or $R$. Could someone explain ...
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2answers
41 views

What is the Eigenvalue of $T^2$ ($SU(3)$ Casimir)?

For example, in $SU(2)$, $\hat{S}^2|s,m_s>=\bar{h}^2 s(s+1)|s,m_s>$. What about in $SU(3)$, $\hat{T}^2|T,m_3,m_8>=?|T,m_3,m_8>$ where $\hat{T}^2=\sum_i^8 T_iT_i $, $T_i = \frac{\lambda_i}...
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3answers
758 views

Representation under which Pauli matrices transform

In Peskin and Schroeder's Quantum Field Theory, there is an identity of Pauli matrices which is connected to the Fierz identity, (equation 3.77) $$(\sigma^{\mu})_{\alpha\beta}(\sigma_\mu)_{\gamma\...
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1answer
895 views

How do simple two-component Fierz identities follow from a property of the Pauli matrices?

On page 51 Peskin and Schroeder are beginning to derive basic Fierz interchange relations using two-component right-handed spinors. They start by stating the trivial (but tedious) Pauli sigma identity ...
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1answer
79 views

How does the adjoint of $SO(10)$ branch under $SU(5)$

We can split up $SU(5)$ into real and imaginary parts as $U=U_R+iU_I$ and in doing so embed this in $SO(10)$ as $\begin{pmatrix} U_R & -U_I \\ U_I &U_R\end{pmatrix}$. Hence we know that $SU(5)...
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0answers
44 views

Decompose $SU(4)$ into $SU(3) \times U(1)$

I'm solving these problems concerning the $SU(4)$ group and I've reached the point where I have determined the Cartan matrix of $SU(4)$, its inverse and the weight schemes for $(1 0 0)$ and $(0 1 0)$ ...
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0answers
22 views

Group theory and representation theory reference [duplicate]

Could anyone suggest some reference(s) on group theory and representation theory geared to physicist? The reference should be rigorous and not for a novice (but not for an expert either) it should ...
2
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1answer
76 views

Georgi - decomposition of representations into subgroups

I have long been unable to follow section 12.3 of Georgi - Lie algebras in particle physics. This section deals with how irreps of $SU(3)$ decompose as irreps of subgroups $H \subset SU(3)$ and is ...
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1answer
153 views

What is the weight system for these ${\rm SU}(5)$ representations?

I need to work out the weight systems for the fundamental representation $\mathbf{5}$ and the conjugate representation $\overline{\mathbf{5}}$. I'm not clear what this means. The $\mathbf{5}$ ...
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0answers
60 views

Surjective homomorphism between ${\rm SL}(2,\mathbb{C})$ and the restricted Lorentz group ${\rm L}_0$

I am reading "Group theory and physics" by Sternberg. Ch. 1.2 deals with homomorphism between ${\rm SL}(2,\mathbb{C})$ and the Lorentz group ${\rm L}$, respectively ${\rm L}_0$, the restricted Lorentz ...
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2answers
168 views

Computing the spin degrees of freedom for a massless particle in $D$ dimensions

According to the paper A Lagrangian formulation of the classical and quantum dynamics of spinning particles, a relativistic spinless particle in $D$ spacetime dimensions can be described by the ...

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