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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3
votes
2answers
138 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
votes
1answer
309 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
votes
2answers
234 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}, ...
4
votes
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} ...
9
votes
1answer
192 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 $s$ ...
1
vote
1answer
88 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
votes
0answers
125 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 ...
4
votes
1answer
286 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
votes
1answer
156 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 ...
1
vote
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 ...
1
vote
1answer
109 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 ...
1
vote
1answer
55 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
votes
1answer
112 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 ...
5
votes
0answers
344 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
votes
2answers
265 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
187 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
106 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
430 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
votes
0answers
70 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
250 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
vote
1answer
90 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
251 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
316 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 ...
2
votes
0answers
102 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
votes
2answers
465 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
votes
3answers
94 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
votes
0answers
136 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
votes
1answer
130 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
81 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
378 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 ...
10
votes
2answers
704 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
158 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 ...
6
votes
2answers
195 views

Does the $\bf{1+3}$ representation of $SU(2)$ also represent $SU(2)\times SU(2)$?

I'm a bit confused about this following issue concerning representations of $SU(2)$. Denote by 1 the 1-dimensional representation of the group $SU(2)$ (=the spin 0). Similarly, denote by 2 and 3 the ...
5
votes
1answer
681 views

How to get result $3 \otimes 3 = 6 \oplus \bar{3}$ for $SU(3)$ irreducible representations?

Let's have $SU(3)$ irreducible representations $3, \bar{3}$. How to get result that $$ 3\otimes 3 =6 \oplus \bar{3}~? $$ I'm interested in $\bar{3}$ part. It's clear that for $3 \otimes 3$ we can use ...
9
votes
2answers
967 views

Dirac spinors under Parity transformation or what do the Weyl spinors in a Dirac spinor really stand for?

My problem is understanding the transformation behaviour of a Dirac spinor (in the Weyl basis) under parity transformations. The standard textbook answer is $$\Psi^P = \gamma_0 \Psi = ...
2
votes
1answer
100 views

Solution space of a differential equation with 3D rotational symmetry

We know that the space of solutions will be invariant under 3D rotations, but why can we say that the space of solutions will constitute a representation of the rotation group $SO(3)$? We know that a ...
2
votes
1answer
100 views

How to break a irreducible representation into its subgroups

In Grand Unified Theories (though I'm sure this a general group theory result) people write the irreducible representations of a group (i.e., the gauge bosons) using a sum of irreducible ...
5
votes
1answer
262 views

Is time reversal operator not a representation of Lorentz group?

I'm puzzled why every book says that time reversal operator is a representation of full Lorentz group. Because of physical consideration, time reversal is an antilinear operator. While the definition ...
4
votes
1answer
321 views

Is General Relativity based on a Symmetry?

In short: Is there any kind of symmetry one can start with to derive general relativity (GR)? Longer: Einstein had the opinion that GR was the generalisation of special relativity, because instead of ...
2
votes
0answers
35 views

Measure of interaction of two quarks and Casimir operators [closed]

Let's have two quarks, which refers to representations of $r_{1}$ and $r_{2}$ of color symmetry group. They create bounded state which refers to the representation $r$. There is a statement that ...
2
votes
1answer
270 views

What is the four-dimensional representation of the $SU(2)$ generators?

Recently, I have been learning about non-Abelian gauge field theory by myself. Thanks @ACuriousMind very much, as with his help, I have made some progress. I am trying to extend the Dirac field ...
3
votes
2answers
105 views

Unification of the electroweak theory

Can the electroweak theory be described by the spontaneous symmetry breaking of $SU(3)$ to $SU(2)\times U(1)$?
6
votes
1answer
227 views

Symmetries of AdS$_3$, $SO(2,2)$ and $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$

Basically, I want to know how one can see the $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ symmetry of AdS$_3$ explicitly. AdS$_3$ can be defined as hyperboloid in $\mathbb{R}^{2,2}$ as $$ ...
1
vote
0answers
53 views

Uncharged Gluon Representation

I'm currently learning for an exam and in an old exam the question was posed: If Gluons were uncharged, under which representation would they transform? The ...
3
votes
1answer
86 views

Two different transformation laws for Quantum Fields

I found a nice answer to a problem that was bothering me for quite a while in a lecture script (unfortunately in german). The first step of the answer, is what remains unclear to me. The script states ...
11
votes
3answers
282 views

Group representations as vectors and isomorphism between weights and matrix generators

This might be something basic, but it is unclear to me. So I am used to work with representations of groups as matrices. These matrices represent the structure of the Lie algebra by satisfying the ...
1
vote
0answers
45 views

What is the difference between the compact U(1) group and non-compact U(1) group? [duplicate]

Is compact $U(1)$ and non-compact $U(1)$ just two different representations for the same group or the same gauge theory? If not, what the difference of them? Or are there any properties that are ...
1
vote
1answer
62 views

Correspondence between one-parameter subgroups of $G$ and $T_eG$

I am reading the proof of this theorem from Andreas Arvanitoyeorgos and I cannot get some points in it, highlighted below. Theorem. The map $\phi \to d\phi_0(1)$ defines a one-to-one correspondence ...
3
votes
1answer
116 views

Permutations of two identical particles in two dimensions

In three spatial dimensions there are only two possible statistics: Bose-Einstein and Fermi-Dirac. This is the fact related with the statement that first homotopic group of 3-dimensional configuration ...
2
votes
1answer
205 views

Group theoretical reason that Gluons carry color-charge and anti-colorcharge

I was wondering how it is possible to see from the $SU(3)$ Gauge Theory alone that Gluons carry two charges colors: $g\overline{b}$ etc. Some background: The W-Bosons (pre-symmetry breaking) ...