Questions tagged [group-theory]

Group theory is a branch of abstract algebra. A group is a set of objects, together with a binary operation, that satisfies four axioms. The set must be closed under the operation and contain an identity object. Every object in the set must have an inverse, and the operation must be associative. Groups are used in physics to describe symmetry operations of physical systems.

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45 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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221 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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51 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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1answer
39 views

Different definitions of commutator in operator theory/quantum mechanics vs. in group theory

In group theory, the commutator of two elements $g$ and $h$ in a group is defined as $$[g,h]=ghg^{-1}h^{-1}$$ However, in quantum mechanics, we always see commutator relation between two operators $A$...
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3answers
74 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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18 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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39 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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1answer
526 views

Why do Lorentz boosts not form a group?

An excerpt from my lecture notes on relativity (translated from Dutch): "Special (special in the notes indicates that the determinant of the representation matrix equals +1) Lorentz transformations ...
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2answers
159 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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62 views

If gravity is a gauge theory, what is the Lie group? [duplicate]

Here I asked a question. In one curious comment, I see a statement that gravity is a gauge theory. However, my definition (based on what I read till date) of a gauge theory is a field theory which is ...
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1answer
87 views

What are the $A_{\mu}{}^a$ fields in Yang-Mills theory?

At some point of the demonstration of Yang Mills theory we assume an ansatz that $A_{\mu}=t^a A_{\mu}{}^a$ where $a=1, \ldots,n^2-1$ and the $t^a$ are the generators of the $SU(n)$ symmetry in order ...
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1answer
72 views

Peskin Schroeder Higgs mechanism for an $SU(3)$ gauge theory with a scalar field $\varphi$ in the adjoint representation

In Peskin Schroeder pag.696 a Higgs mechanism for an $SU(3)$ gauge theory with a scalar field $\phi$ in the adjoint representation is presented. The covariant derivative of $\phi$: $$ D_{\mu}\phi_{a} =...
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43 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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1answer
42 views

Simultaneous diagonalization of Cartan generators of $SO(6)$

This question is naive but for some reason I'm not getting the expected result. The generators of $SO(6)$ can be written in this way: $$(J_{ab})_{cd}=i(\delta_{ac}\delta_{bd}-\delta_{ad}\delta_{bc}),...
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21 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 ...
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1answer
109 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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1answer
30 views

Equation for the multiplicity of a set of planes

The multiplicity (m) of lattice planes counts the number of planes related to (hkl) by symmetry. For example, the multiplicity of the {100} planes would be 6 because the following planes are all ...
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1answer
58 views

Is every unitary operator induced by a Hamiltonian?

Diving deeper into the mathematical inner workings of quantum mechanics: The set of unitary operators on the Hilbert space $\mathcal{H}$ forms a group. While for finite-dimensional Hilbert spaces, ...
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56 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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1answer
71 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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37 views

Fidelity of unitary operators: is $||U-\tilde U|| < \delta$ a *necessary* and *sufficient* condition?

There's a notion of fidelity of quantum states. However, is there a standard notion of the fidelity of unitary operators? Say, I wish to approximate a unitary operator $U$ acting on $n$ qubits with a ...
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33 views

Group action on $2$x$2$?

Group action of group $C_{3v}$ is defined for all $2$x$2$ matrices over field of complex numbers, for all $g$ from $C_{3v}$ $$D(g)A=E(g)AE(g^{-1})$$ where $E$ is $2D$ representation of $C_{3v}$. How ...
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1answer
88 views

What is the Lie group of gravity?

If the lie group of the three gauge forces are $SU(3)×SU(2)×U (1)$, then what is the symmetry group of gravity? $SL(2,C)$? Just a newbie in Lie groups.
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1answer
52 views

$\mathfrak{so}(2n+1)$ Cartan subalgebra

For the Lie algebra $\mathfrak{so}(n)$, $n^2$ $n \times n$ real and antisymmetric matrices can be introduced as $$(M_{pq})_{jk} = \delta_{pj}\delta_{qk}-\delta_{pk}\delta_{qj}, \qquad j,k=1, ..., n $$...
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34 views

How to find standard basis? [migrated]

How to find standard (symmetry adaptated) basis in representation of group $C_{3v}$ in representation $$D(C_{3v})=A_2\otimes E\otimes E=$$ $$ =A_2\otimes (A_1\oplus A_2 \oplus E)=$$ $$=A_2\oplus ...
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2answers
79 views

How to formulate with a tensor multiplication method $2\otimes 2\otimes 2$ in $SU(2)$ group?

In $SU(2)$ we can write $2\otimes 2=3\oplus 1$ or \begin{equation} q_iq^j=\left(q_iq^j-\frac{1}{2}\delta^i_jq^kq_k\right)+\frac{1}{2}\delta^i_jq^kq_k, \end{equation} where $q_i$ is a $SU(2)$ doublet, ...
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65 views

$SU(2)$ vs $SO(3)$ in Quantum Mechancs

When we're talking about spatial rotations is quantum mechanics, why do we need to resort to $SU(2)$? Why isn't $SO(3)$ enough? I've read that $SO(3)$ isn't simply connected, and I've read about ...
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2answers
83 views

Why are covering groups more 'fundamental'?

So I understand that the Lie algebra of $SO(1,3)$ is isomorphic to the Lie algebra of $SU(2)\oplus SU(2)$, and the Lie algebra of $SO(3)$ is isomorphic to one copy of $SU(2)$ (at the group level we ...
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1answer
49 views

Lorentz Invariance of Weyl Lagrangian

I have been reading 'Quantum Field Theory and the Standard Model' by Schwartz and have gotten stuck on a line of reasoning in Section 10.2.2. I understand that we can construct a (right-handed) four-...
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1answer
78 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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1answer
70 views

Origin of antisymmetric $\ell=2$ irrep in direct product of two symmetric second-rank tensors

In the excerpt below from Chapter 18 Section 6 of the textbook Group Theory -- Application to the Physics of Condensed Matter by Dresselhaus, Dresselhaus, and Jorio, the irreducible representations of ...
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48 views

Clebsch-Gordan coefficients with fields in tensor representation of $SU(N)$ and relation to Young diagram

I have a question on how to write down the basis of the irreducible representations in terms of the direct product of the fields of irreducible representations that expressed with the indices of the ...
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28 views

Definitions for transformations of fields

I'm reading Kaku's QFT book and got stuck at the concept of transformations of fields in chapter 2. I don't understand the following definitions for transformations of a scalar field under SO(2) group ...
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2answers
66 views

Branching of $SU(3)$ under $D_8$

The question is to work out the branching of $SU(3)$ representations of $\mathbf{3}, \mathbf{\overline{3}}, \mathbf{8}$ under the dihedral group $D_8 = \langle r,s \mid r^4 = s^2 = e, rs=sr^{-1} \...
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2answers
363 views

Reconstructing unitary representation of Lie group from its generators

This question is about the motivation for Weinberg's approach in "The Quantum Theory of Fields" to obtain unitary representations of Lie groups out of its generators. One is dealing with a Lie group $...
6
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1answer
85 views

Application of a non-matrix Lie group?

I'm studying Lie theory from Brian C. Hall's "Lie Groups, Lie Algebras, and Representations," in which he focuses on matrix Lie groups (defined as sets of matrices) rather than general Lie groups (...
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1answer
53 views

Weinberg's “Derivation” of Lie algebra commutation relations

I have a question regarding the evolution of Lie algebra conditions in Weinberg's The Quantum Theory of Fields vol. 1: Foundations, chapter 2. I will reproduce the text here and state my two ...
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3answers
50 views

Townsend's Infinitesimal Operators

I'm currently reading John Townsend's Modern Approach to Quantum Mechanics and the infinitesimal operators have me a bit puzzled: $$\hat R(d\phi \boldsymbol{k})=1-{i \over \hbar}\hat J_zd\phi$$ $$\...
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21 views

S-matrix elements for Nucleon-Pion decay

I want to compute nucleon-pion decay rates. I am a bit confused how I can compute the S-matrix. Let's say we have a Nucleon Pion scattering and I want to compute their corresponding S matrix: \begin{...
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1answer
42 views

$SU(3)$ Clebsch-Gordan Coefficient

I have a problem computing the ratio $$\frac{P(\pi^0 P\rightarrow\Delta^+)}{P(K^- P\rightarrow\Sigma^{*0})}.$$ The problem demands reducing the $S$-matrix first but I really don't see how to get this ...
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1answer
91 views

Adjoint representation of $SU(2)$

I'm trying to understand how the $SU(2)$ representations work. We know that the fundamental representation of $SU(2)$ is $\frac{1}{2} \sigma^{\alpha}$ where $\sigma^{\alpha}$ the Pauli matrices. These ...
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1answer
39 views

Scaling transformations, definitions and all that's not mentioned

If we transform the massless scalar field Lagrangian $$\mathcal{L}=\frac{1}{2}(\partial_\mu\varphi)^2-\frac{\alpha}{4!}\varphi^4$$ with the simultaneous transformations $$x\mapsto x^\prime= \lambda x,\...
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2answers
62 views

Why in an irreducible unitary representation of the Poincare group all momenta are on the same mass shell?

This is a question about the approach of Weinberg in "The Quantum Theory of Fields" to the irreducible unitary representations of the Poincare group in Chapter 2. Let $U(\Lambda,a)$ be such a ...
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71 views

Dynamics of a linear chain of harmonic oscillators

Let's consider a linear chain of particles with harmonic nearest neighbor interaction: Assuming all particles have the same mass, Equations of motion are (with periodic boundary conditions): $$m\ddot{...
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32 views

Tensor Method $SU(N)$

I'm working out the $SU(N)$ tensor method and reading Cheng-Li page 102, 103 (Sec. 4.3). I'm following the definition (4.94) which are $\psi^i=\psi_i^*$, $U_i^{.j}=U_{ij}$ and $U^i_{.j}=U_{ij}^*$ ...
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49 views

Groups with cardinality larger than the Reals in physics

In what physical theories are sets with cardinality larger than $\aleph_1$ used? There are plenty of examples of finite, countable, and uncountable vector spaces in physics, but do physicists ever ...
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53 views

Selfstudy Conformal Field Theory and Lie algebra from scratch roadmap [closed]

A couple of months ago, I stumbled upon Conformal Quantum Mechanics (CQM) which was really interesting yet confusing. I didn't have the prerequisites so I couldn't follow the paper. I want to first ...
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2answers
465 views

What exactly does the belt/plate trick demonstrate?

I am reasonably familiar with the math behind spinors, the fact that $SU(2)$ is the universal (double) cover of $SO(3)$, etc. I've often seen the "belt trick" and the "plate trick" used to motivate ...
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2answers
107 views

Generator of a rotation matrix

$$T(\phi)= \begin{bmatrix} \cos(\theta) &\sin(\theta) & 0 \\ \sin(\theta)\cos(\phi) & -\cos(\theta)\cos(\phi) & \sin(\phi)\\ \sin(\...
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1answer
28 views

Construct color octet

I'm reading about color octet, considering $r= \begin{bmatrix}1 \\ 0 \\ 0 \end{bmatrix}$ $b= \begin{bmatrix}0 \\ 1 \\ 0 \end{bmatrix}$ $g= \begin{bmatrix}0 \\ 0 \\ 1 \end{bmatrix}$ and $\lambda_{i}$ ...