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. ...

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254 views

Is the fundamental representation of $SU(3)$ irreducible?

I want to check if the fundamental representation of $SU(3)$ is irreducible. The algebra is $$\mathbb{su}(3) = \{ m \in Mat(3,\mathbb{C} )\ |\ m = -m^+,\ Tr[m] = 0 \}$$ and I've found the generators. ...
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2answers
110 views

How would I relate $\Lambda=e^{-i\omega_{\mu\nu}J^{\mu\nu}/2}$ to the Lorentz boost matrix?

$\omega_{\mu\nu}$ contains infinitesimal parameters and $J^{\mu\nu}$ contains generators of boost and rotation. Any 4-vector transforms as $p^\mu=\Lambda^\mu_\nu p^\nu$. Starting from given ...
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0answers
82 views

Relation between representations/classifications

Generally a quantum system can be characterized in the following way: its states form a representation space for every symmetry group of that system. The representation has to be unitary (or anti-...
8
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2answers
271 views

Is a spinor in some sense connected to space?

Spinors transform under the representation of $SL(2,\mathbb{C})$ which is the double cover of the Lorentz group $SO(1,3)$ - or in the non-relativistic case under $SU(2)$, the double cover of $SO(3)$. ...
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0answers
75 views

What is the difference between the groups $PSU(N)$ and $SU(N)$? [closed]

What is the difference between the groups $PSU(N)$ and $SU(N)$? For example how is $PSU(2,2|4)$ different than $SU(2,2|4)$?
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1answer
109 views

If isospin is conserved under strong interactions why it is represented by SU(2)?

As far as I know from my readings SU(2) is a representation group of isospin symmetry which shows deep symmetry of the strong force which conserves flavor. Isospin symmetry is broken under weak ...
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1answer
72 views

Why is the Poincaré group non-abelian?

Based on what I've learned, I gather the Poincaré group is the group of isometries of Minkowski spacetime and it is a non-abelian Lie group. Why is it non-abelian? Or perhaps rather, does the fact ...
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1answer
74 views

Why use class multiplication to describe topological entangling and merging?

I'm studying some references about topological defects in ordered media like Soft matter physics: An introduction by Kleman and the Review modern physics paper The topological theory of defects in ...
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3answers
908 views

Tensor product of two different Pauli matrices $\sigma_2\otimes\eta_1 $

I'm solving problem 3.D in H. Georgi Lie Algebra etc for fun where one is to compute the matrix elements of the direct product $\sigma_2\otimes\eta_1$ where $[\sigma_2]_{ij}\text{ and }[\eta_1]_{xy}$ ...
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3answers
178 views

$SO(3)$, $SU(2)$ and symmetries in quantum mechanics [duplicate]

A rotation in the vector space $\mathbb{R}^3$ is represented by the known 3x3-matrices. But at this point I'm really confused how to get from there to Quantum Mechanics. The group of $\mathrm{SO}(3)$...
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0answers
40 views

Simplification of matrix-element given the Wigner-Eckardt theorem and Clebsch-Gordon coefficients of a 1,1/2 system

How can I simplify the following matrix-elements $$\left\langle 1,1/2;m_1,m_2\left| S \right| 1,1/2;m_1^{'},m_2^{'} \right\rangle$$ given the Wigner-Eckard theorem $$\left\langle j,m|S|j^{'},m^{'} \...
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1answer
99 views

Does $GL(N,\mathbb{R})$ own spinor representation? Which group is its covering group? (Kaku's QFT textbook)

In Kaku's QFT textbook page 54, there is a saying: $GL(N)$ does not have any finite-dimensional spinorial representation. This implicates that $GL(N)$ owns infinite-dimensional spinorial ...
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0answers
168 views

Group theory and quantum optics

This is a question about application of group theory to physics. The starting point is the group $SU(n)$. I have a representation $R$ of $SU(n)$ that takes values on the unitary group on an infinite ...
2
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2answers
138 views

What's the degree of freedom of this kind of matrix?

We first have a unitary matrix $$\{a_{ij}\}\quad(n\times n)$$ I know how to calculate its degree of freedom, which is $n^2$ if we consider a real variable as one degree of freedom. Now we have a ...
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2answers
189 views

Subgroup of Lorentz Group Generated by Boosts

It is common knowledge that a composition of boosts is not a boost, but involves a rotation. Further, in discussions of Thomas precession, it is often stated that boosting in $x$, then $y$, then back ...
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1answer
270 views

How can we measure chirality in experiments?

Chirality is a concept quite different from helicity. These two concepts only happen to have the same numerical value for massless particles. I understand that we can measure helicity, but how can we ...
3
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1answer
103 views

Is there a general theorem stating why the restricted Lorentz group's exponential map is surjective?

The exponential map for the restricted Lorentz group is surjective. An outline of why is shown on the wiki page Representation Theory of the Lorentz Group. Is there a more general theorem that states ...
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2answers
262 views

Why is the $(\frac{1}{2},\frac{1}{2})$ representation of the Lorentz group realized as the vector space of complex $2\times 2$ matrices?

Why can we write an arbitrary object $v_{a \dot{b} }$ our transformations in this basis act on as $$ v_{a \dot{b} } = v_{\nu} \sigma^{ \nu}_{a \dot{b} } = v^0 \begin{pmatrix} 1&0 \\ 0&1 \...
3
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2answers
139 views

Identifying irreps of $SU(2)$

How does one verify that, the representations of $SU(2)$ corresponding to $j=1/2$ or $j=1$ is irreducible? I think showing the irreducibility (taking the representative matrices into a block-diagonal ...
11
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1answer
312 views

What exactly do we mean by symmetry in physics?

I'm referring here to invariance of the Lagrangian under Lorentz transformations. There are two possibilities: Physics does not depend on the way we describe it (passive symmetry). We can choose ...
9
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2answers
235 views

If $v_{a \dot{b}}$ transforms like a four-vector, what does $v_{a}^{\dot{b}}$ describe?

The $( \frac{1}{2}, 0)$ representation of the Lorentz group acts on left-chiral spinors $\chi_a$, the $( 0,\frac{1}{2} )$ representation on right-chiral spinors $\chi^{\dot a}$. The $( \frac{1}{2}, \...
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1answer
143 views

How to interpret spin observables constructed by non-standard phase choices?

If we try to find matrix elements of ladder operators ( $J_{\pm}$) for spin when they act on eigenstates of $J^2$ and $J_z$ ( $\newcommand{ket}[1]{\left|#1\right\rangle} \newcommand{avg}[1]{\left\...
9
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1answer
198 views

Group notation $\otimes$ and $\oplus$ used for representations of quarks and mesons

I've been trying to figure out this statement from the PDG quark model summary (PDF). Following $\mathrm{SU}(3)$, the nine possible $q\bar{q}′$ combinations containing the light $u$, $d$, and $s$ ...
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1answer
90 views

Physical interpretation of diffeomorphism from $SO(3)$ to $\mathbb R \mathbb P^3$

I am not good at picturing either $SO(3)$ or $\mathbb R \mathbb P^3$, the latter denoting the real projective space. Can someone give me a rough physical understanding of the geometry and implication ...
3
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0answers
131 views

Decomposing a representation under a subgroup [closed]

I am trying to understand what is the method for decomposing representations of a group under one of its subgroups. I already had a look in Slansky, but I could not extract a concrete set of rules/...
5
votes
1answer
311 views

Branching rules for $SU(3)$

How does one compute the branching rules for $SU(3)\to SU(2)\times U(1)$.? In particular, I do not know how to put the abelian charges. Take for example the adjoint $\mathbf{8}$ of $SU(3)$. I can ...
2
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1answer
162 views

Geometry, Group Theory, and Statistical Mechanics

During the course of my first statistical mechanics course we generally concerned ourselves with a bulk amount of our system and considered it in terms of a set of lattice sites that had a state. How ...
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0answers
34 views

Reference request: Relation between $Sp(N)$, $Spin(N) $, $SU(N)$ groups and physics [duplicate]

I want to understand the relationship of the so common $SU(N)$ and $SO(N)$ groups in physics with the symplectic group which I think is the double cover of the first and the Spin groups $Spin(N)$. Is ...
2
votes
1answer
113 views

Representations of subalgebra in the super virasoro algebra

In the Virasoro algebra, which is generated by $L_n$, one has the obvious subalgebra spanned by $L_{-1}$ ,$L_{1}$ and $L_{0}$ which is isomorphic to the Lie algebra $\mathfrak{sl}(2,\mathbb{R})$. The ...
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1answer
56 views

What conserved quantities does a one-dimensional non-symmetric lattice have?

When I asked what leads to degeneracy of eigenstates of free particle, the answer was parity. But it appears that even if we consider a lattice with non-symmetric cell, so the potential looks as shown ...
2
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1answer
113 views

How to diagonalise the Lagrangian mass term with SU(4) symmetry and self-dual tensors

I should write the mass term of the Lagrangian with global SO(4) symmetry in tensor representation with anti-symmetric tensors and then diagonalise this term with defining a new set of tensors (self-...
5
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0answers
346 views

How to calculate $3\otimes 3$ and $3\otimes 3\otimes 3$ in $SU(3)$? [closed]

EDIT: I have boiled my question down to How many independent components does a rank three totally symmetric tensor have in $n$ dimensions? A derivation would be nice too. OP: I know that I can ...
4
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2answers
275 views

SU(3) antiquark triplet transformation

I'm reading a rather elementary particle physics text, Modern Particle Physics by Thomson. He is staying away from the heavy group theoretic stuff. He derives the transformation law for an SU(2) ...
2
votes
1answer
196 views

Is there a connection between Lie Groups and observable quantities in physics?

Good evening everybody. I have some questions about the relation between Lie groups and observables in physics. Indeed, taking the example of spin formalism of Quantum mechanics I know that Pauli's ...
2
votes
1answer
111 views

Unitary representations of $SO(3)$ and $so(3)$

According to my skript: Quantum mechanic states $ψ ∈ \mathcal H$ changes under a rotation $R ∈ \text{SO(3)}, \vec{x} \rightarrow R\vec{x}$ according to $ψ \rightarrow U(R)ψ$, whereas $U(R)$ is a ...
3
votes
3answers
431 views

In physics, what is the importance of distinguishing between a matrix and a group? [closed]

On the topic of Pauli matrices, I have noticed that some authors tend to use the term matrix and group interchangeably. I am asking because I do not see see any profound difference referring to the ...
4
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0answers
71 views

Why the bosonic part of the superconformal group $SU(2,2|1)$ is $SO(4,1) \times U(1)_R$?

Why in $d=4$ $\mathcal{N}=1$ SCFT the bosonic part of the superconformal group $SU(2,2|1)$ is $SO(4,1) \times U(1)_R$? More generally how can I determine the such a thing in other theories? Is there ...
2
votes
2answers
257 views

Why is space isotropic in the vector particle's decay?

I come across one proof the Landau-Yang Theorem, which states that a $J^P=1^+$ particle cannot decay into two photons, in this paper (page 4). The basic idea is, the photon's wavefunction should be ...
1
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1answer
91 views

Question about global internal $SO(n)$ symmetry

I have the following Lagrangian (density) for bosons $$L = \partial_{\mu} \phi^i \partial^{\mu}\phi^i+ m^2\phi^i \phi^i$$ and I am trying to understand why this Lagrangian is invariant under ...
2
votes
2answers
260 views

Quadratic Casimir operator of higher dimensional $\mathfrak{su}(3)$ representations

In higher dimensional representations of $\mathfrak{su(3)}$, what will be the quadratic Casimir operator? Is it same as in lower dimensions or different?
3
votes
1answer
331 views

How can the Gallilean transformations form a group?

In class my professor said the Galilean transformations form a group of order 10. $$ x'=x-vt\\ y'=y\\ z'=z\\ t'=t\\ $$ But how do these form a group? I don't see 10 things to interpret as elements. I ...
3
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0answers
111 views

Different ways of derivation of Gell-Mann-Okubo mass formula

Recently my teacher have told me that there are many ways of derivation of Gell-Mann-Okubo mass formula by using group representation theory (by using dynamical group etc). Where can I read about ...
10
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2answers
471 views

Vector spaces for the irreducible representations of the Lorentz Group

EDIT: The vector space for the $(\frac{1}{2},0)$ Representation is $\mathbb{C}^2$ as mentioned by Qmechanic in the comments to his answer below! The vector spaces for the other representations remain ...
2
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3answers
97 views

What is “a vector of $SO(n)$”?

I'm watching (or trying to watch) this lecture from NPTEL on classical field theory. I've understood everything in the series up till this point, including the first half of the lecture on elementary ...
3
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0answers
138 views

Why is only the third component of weak isospin used as a conserved quantity?

Using Noether's theorem \begin{equation} \partial_0 \int d^3x \left(\frac{\partial L}{\partial(\partial_0\Psi)} \delta \Psi \right) = 0 \end{equation} we get three conserved quantites $Q_i$ from ...
5
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1answer
133 views

Is the Standard Model an invariant subgroup of $SU(5)$?

It is well known that the Standard Model (SM) gauge group is a subgroup of $SU(5)$: \begin{equation} SU(3) \times SU(2)\times U(1) ~\subset~SU(5) \end{equation} This can be easily checked using the ...
2
votes
0answers
82 views

Difference between the 1/2 representation of $SU(2)$ and the (1/2,1) representation of $SU(2)\times SU(2)$? [closed]

What's the difference between the $j = 1/2$ representation of $SU(2)$ and the $(j,j') = ( 1/2 , 1 )$ representation of $SU(2)\times SU(2)$?
3
votes
2answers
392 views

How to understand non-associative composition of velocities in STR?

In STR the composition of non parallel movements is in general non-associative. The formula is $\displaystyle\bar{u}\oplus\bar{v}= \frac{\bar{u}+\bar{v}_{\|}+\bar{v}_\bot/\gamma}{1+\bar u\cdot\bar v/...
11
votes
2answers
741 views

Lie algebra in simple terms [closed]

My question is regarding a vector space and Lie algebra. Why is it that whenever I read advanced physics texts I always hear about Lie algebra? What does it mean to "endow a vector space with a lie ...
0
votes
1answer
163 views

Coordinate system for crystallographic groups

In the International Tables for Crystallography for each crystallographic group an asymmetric unit is supplied (mathematicians call this a fundamental domain of the group). This region is a bounded ...