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. ...

learn more… | top users | synonyms

3
votes
0answers
71 views

What is physically irreducible representation?

When I use bilbao crystallographic server recently, I noticed a notation called physically irreducible representation. Paper says it is a direct sum of two complex conjugate representations (if $\...
1
vote
0answers
24 views

Water dipole by symmetry argument [closed]

I'm a mathematician and I'm studying Group and Representation theory and I came across with an interesting exercise involving physics, although I don't know physics, since I'm a mathematician, I found ...
4
votes
0answers
88 views

Completely positive maps and symmetric states

Let $\mathcal{N}$ be a completetely positive trace preserving map (aka a quantum channel) acting on a finite dimensional system $\mathrm{A}$, and let $\pi$ denote the maximally mixed state on $\mathrm{...
1
vote
2answers
146 views

Symmetry and degeneracy in quantum mechanics

If an operator commutes with the Hamiltonian of a problem, does it always must admit degeneracy? For example, parity operator commutes with the Hamiltonian in case of a free particle and we have two ...
0
votes
0answers
25 views

$R$-Symmetry Group

On p238/239 of the Freedman and van Proeyen book on Supergravity, they show how the $R$-symmetry group must be $U(\mathcal{N})$ for $\mathcal{N}$-extended supersymmetry in $d=4$. At the bottom of ...
1
vote
0answers
23 views

Help to verify (numerically) invariant Haar measure on unitary group

Sorry if this question is not appropriate for the forum. From the paper http://gemma.ujf.cas.cz/~brauner/files/Haar_measure.pdf I am interested to understand and verify equation (3). Can anyone please ...
1
vote
1answer
76 views

Why do people care about Mathieu groups and related things? (Something about monstrous moonshine)

Before I begin, let me say I don't know anything about what I am asking. This morning for somewhat random reasons I decided to google moonshine and related things. As it were I discovered my ignorance ...
0
votes
1answer
51 views

Coleman Mandula theorem and translations

I don't know what Coleman Mandula theorem is, however if I were forced to say something about it, I will say it is a statement that suggests that internal and spatial symmetries have no unique ...
1
vote
1answer
36 views

Structure constant of the commutators of generators in broken symmetry

When I read a paper related to spontaneously global symmetry breaking, I cannot understand a statement: If we use the notation $T^i$ for the unbroken group generators in $H$ and $X^a$ the broken ...
4
votes
1answer
122 views

$SO(4,2)$ symmetry of the hydrogen atom

The hydrogen atom with Hamiltonian obviously has $SO(3)$ symmetry since it just depends on the radius. $$ H = \frac{\mathbf{p}^2}{2m} - \frac{k}{r}$$ This is generated by angular momentum $\mathbf{L}...
0
votes
1answer
48 views

Why are half integer and full integer spin properties of elementary particles, not of all points in space?

Tensors and spinors arise mathematically from the representation of the rotation group $SO(3)$ as a ball in 4D with all antipodal points on the surface identified. In this picture it is shown that ...
0
votes
1answer
55 views

Which groups can be lattice gauge groups?

Let me state first off that in this question I am most interested in lattice gauge theories, and not necessarily with Fermion couplings. But if Fermions and continuum gauge theories can also be ...
1
vote
1answer
75 views

Why do we say that gluons carry color charge?

We know that gluons are Lie algebra $su(3)$-valued one-form fiels $A_{\mu}$. And because of $[A_\mu,A_\nu]$ does not vanish generally for the non-Abelian case, gluons have self-interactions. Now how ...
0
votes
2answers
64 views

Good reference on the parametrization of $SU(3)$ and $SU(N)$

For the 2-dimensional $SU(2)$ matrices, there is a fairly general parametrization formulation: $s_2=\begin{bmatrix} e^{i\alpha}\cos(\theta) & -e^{-i\beta}\sin(\theta) \\ e^{i\beta}\sin(\...
2
votes
0answers
68 views

Use Cartan subalgebra in spinor representation to find weights of vector representation

For $SO(2n)$ we can construct the lie algebra elements by using antisymmetric combinations of $\gamma_\mu$ which obey the Clifford algebra. Up to some prefactor the elements $ S_{\mu \nu} = \alpha [\...
0
votes
1answer
33 views

Implication of rotational symmetry on scattering matrix/ scattering cross-section [closed]

How does the rotational invariance helps simplifying Non-relativistic quantum scattering problems? Is there any any additional information that can be extracted about the scattering amplitude? It ...
0
votes
1answer
91 views

Susy transformation for gauge multiplet

How can the supersymmetrie transformation $\delta A_\mu = \frac{1}{2} \overline{\epsilon}\gamma_\mu \psi $ be derived from the susy algebra ( or group ). Where $ (A_\mu , \psi)$ are in a gauge ...
1
vote
1answer
60 views

Simple concept question about the dimensionality of a representation in point group

Concept question about the dimensionality of a representation in group theory here: Look at 3.1(c) of problem set, from group theory application to the physics of condensed matter of M.S.Dresselhaus: ...
2
votes
1answer
77 views

Symmetry and Group theory book

I would like to start learning about symmetries in physics and how they affect physical quantities. As far as I know, the mathematical language that describes symmetries is the Group Theory. So, I ...
0
votes
1answer
61 views

What is the difference between the $Spin(3,1)$ group and the $SO(3,1)$ group?

What is the difference between the $Spin(3,1)$ group and the $SO(3,1)$ group?
1
vote
1answer
94 views

Mesons and Young Tableaux

I need some help conecting Young Tableaux with actual particles. I think I have some feel for using Young Tableaux for instance: a baryon in SU(3) where the states are u,d,s can be represented by $\...
0
votes
0answers
27 views

Do tensor product tables for irreducible representations apply for non-symmorphic space groups?

I'm reading Dresselhaus's book on group theory for solid-state physics, but I'm having trouble understanding how to get irreducible representations for phonons away from $\mathbf{k} = \mathbf{0}$ for ...
3
votes
1answer
72 views

How to build tetra-quark mesons symmetry group?

I was thinking about the issue while reviewing my group theory notes. One can construct mesons with a nonet as an octet and a singlet, $SU(3)\otimes SU(3) = 8\oplus \bar{1}$. In a same way but for $qq$...
1
vote
0answers
45 views

Is it proper definition of the free motion? The orbit of free motion is a free group [closed]

That's what I wrote in my notes but I don't understand this definition, I've studied group theory but free groups were not included. Can someone explain this definition, please?
3
votes
1answer
167 views

Why is there no 1/3 spin? [duplicate]

Why do no particles have a 1/3 spin? Why are all particles' spin either a half-integer or integer? How would a particle with such a spin behave, as a fermion, boson, or neither?
3
votes
0answers
44 views

What is $\mathrm{U(1)}$ vector and axial?

In hadron physics we talked about $\mathrm{U(1)_V}$ (vector) and $\mathrm{U(1)_A}$ (axial) as well as $\mathrm{SU(3)_L}$ (left) and $\mathrm{SU(3)_R}$ (right). There are certain relations between them ...
1
vote
0answers
82 views

Mathematical definition of reversible processes

If I label an initial thermodynamic state as $\psi$ and the final thermodynamic state as $\xi$ then can I say that under a reversible process the two states are related to each other by a continuous ...
1
vote
1answer
52 views

Lorentz group in SUSY

Why do we carry Lorentz group to be included also in supersymmetry? That is after we extend our symmetry to supersymmetry, we carry with us the Lorentz group. Why not other group instead?
1
vote
0answers
40 views

Is it possible to define a symmetry group for the Einstein metric?

I was just wondering if there exists a group of transformations that act on the metric such that the EFE are invariant. At first I thought it would be the group of 2nd roots of unity. That is, the set ...
1
vote
1answer
64 views

Is spin angular momentum conserved?

According to the Noether theorem, we only have the conserved quantity $$J+S,$$ where $J$ is the orbital angular momentum and $S$ is the spin angular momentum. But I am always impressed that the spin ...
3
votes
1answer
182 views

Invariant tensors in a general representation and their physical meaning

I'm trying to use tensor methods to find invariant elements of representations. Specifically I'm looking at representations of $SU(5)$. I can show that the invariant element in $5\otimes\bar{5}$ (or ...
1
vote
0answers
20 views

How to introduce symmetry of particles to a layperson? [closed]

I want to introduce the concept of special unitary symmetry and how it is important in particle physics to a layperson. Without being technical, is there a way to explain the fundamental concept?
3
votes
2answers
101 views

AdS/CFT Group Theory

I have a two part question about AdS/CFT: Is the only necessary ingredient that the isometry group of AdS matches the conformal group in one dimension less or are there other prerequisites to build ...
1
vote
1answer
67 views

How to normalize matrix representations properly?

In the convention, where the Dynkin index $Tr(T_a T_b)$ of the lowest-dimensional representation is $\frac{1}{2} \delta_{ab}$, how can I normalize a given set of matrices properly? For example, given ...
1
vote
1answer
75 views

Poincare group representation and complete set

In Weinberg's book of Qft, chapter 2 of volume 1, he uses the eigenstates of the four-momentum to construct the unitary irreducible representations of the Poincare group. My question is, since $P^\mu,...
0
votes
1answer
125 views

Enhancing the QED $U(1)$ gauge symmetry

QED is a gauge theory based on $U(1)$ gauge symmetry, which gives rise to photon as the gauge boson mediating the interaction. Mathematically, I think it is perfectly allowed to implement a $U(1)\...
1
vote
1answer
63 views

Correct Yukawa Term with a SU(2) Higgs Triplet?

Given $SU(2)$ doublet fermions $\Psi^1$ and $\Psi^2$ and a $SU(2)$ triplet Higgs $H$, how does the correct Yukawa term look like in tensor notation? Schematically, we have $$ 2 \otimes 2 \otimes 3 \...
3
votes
1answer
70 views

Charge Conjugation for $SU(N)$?

For $SU(2)$ the charge conjugation operator $C$ reads explicitly $$ C \Psi = i \sigma_2 \Psi^\star ,$$ where $\sigma_2$ is a Pauli matrix. What is the generalized charge conjugation for $SU(N)$?
1
vote
1answer
48 views

determining electrostatic field using only symmetries

As an exercise, I'm trying to (rigorously) determine as much as possible about the electrostatic field due to a infinite line of charge (along the z-axis) without using Maxwell's equations or any of ...
1
vote
0answers
45 views

Spacetime as a coset of a symmetry group

In the introduction to his nice PNAS paper on symmetry, David Gross said Einstein’s great advance in 1905 was to put symmetry first, to regard the symmetry principle as the primary feature of ...
3
votes
1answer
132 views

Diffeomorphism group vs. $GL(4,\mathbb{R})$ in General Relativity

I am quite confused with the groups Diff$(M)$ and $GL(4,\mathbb{R})$ in the context of general relativity. I understand that the symmetries of GR are the transformations that leave the equations ...
4
votes
1answer
137 views

Why are one-particle states called representations of Poincaré group?

The one-particle states in the Hilbert space of a quantized relativistic field theory are said to form representations of the Poincaré group. Why is that? I mean, popular texts in QFT do not ...
4
votes
3answers
79 views

In most physical cases, the elements of a group can be represented by unitary matrices. Why no time-reversal?

In Dresselhaus's group theory page 19, a theorem writes: Every representation (of a Hamitonian's group) with matrices having non-vanishing determinants can be brought into unitary form by an ...
1
vote
1answer
106 views

Direct Sum representation of multiple particles in Quantum Mechanics

Suppose that I have three non-interacting spin-1/2 particles such that I can represent the combined system in a basis of \begin{align} D^{(1/2)}_1 \otimes D^{(1/2)}_2 \otimes D^{(1/2)}_3 & =(D^{(...
0
votes
1answer
54 views

spin representations and polynomials

I'm reading Group Theory and General Relativity by Moshe Carmeli and his discussion of spin representations of SU(2) and the isomorphism to the space of homogenous polynomials is confusing me. I'll ...
1
vote
1answer
49 views

Matrices belonging to orthochronous Lorentz group

My Professor says that all members of the orthochronous Lorentz group may be written as $e^\Gamma$, where $$ \Gamma^{\mu}_{\nu}=\Lambda^{\mu \rho} \eta_{\rho \nu}$$ Here $\Lambda$ is an antisymmetric ...
5
votes
0answers
63 views

$(\frac{1}{2},\frac{1}{2})$ and $(\frac{1}{2},0)\bigoplus (0,\frac{1}{2})$ [duplicate]

I am confused about the notation. What's the differences between $(\frac{1}{2},\frac{1}{2})$ and $(\frac{1}{2},0)\bigoplus (0,\frac{1}{2})$, or maybe $(\frac{1}{2},0)\bigoplus (\frac{1}{2},0)$ ? (...
1
vote
2answers
151 views

The anticommutator of $SU(N)$ generators

For the Hermitian and traceless generators $T^A$ of the fundamental representation of the $SU(N)$ algebra the anticommutator can be written as $$ \{T^A,T^{B}\} = \frac{2N}{d}\delta^{AB}\cdot1_{d} + d_{...
2
votes
1answer
91 views

Space group setting of a crystal structure

I was reading a PDF of a crystal phase in order to draw its structure, when I noticed that it was, apparently, ambiguously described. The PDF lists two descriptions of the monoclinic structure: 1) ...
5
votes
1answer
219 views

Is a $SU(2)$ supergauge theory really a $SU(2)$ gauge theory?

Consider $SU(2)$ supergauge theory with $A$, a doublet of two chiral superfields in the fundamental representation. $$A= \begin{pmatrix} \Phi_1\\ \Phi_2 \end{pmatrix}$$ where $\Phi_1$ and $\Phi_2$ ...