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

0
votes
0answers
33 views

Pauli matrices and $SU(4)$ [on hold]

together with the 2-dim identity matrix, the pauli matrices build a basis of the vector space of the 2x2 hermitian matrices. The set of i*pauli matrices build a basis for the SU(2). Assuming, we ...
5
votes
1answer
71 views

Where does the matching condition for $U(1)$ subgroups come from in unified models?

The matching conditions for a breaking $G \rightarrow \prod_i G_i$ are $$\omega_G-\frac{C_2(G)(\mu)}{12 \pi}=\omega_{G_i}-\frac{C_2(G_i)(\mu)}{12 \pi} ,$$ where $C_2(g)$ denotes the quadratic ...
4
votes
1answer
99 views

Integrating elements of a Lie group with respect to parameters of the corresponding Lie algebra

I am working with an operator $\textbf{M}$ that is represented by the Lie group SO(1,3), thus it can be written as, $$ \textbf{M} = \exp{\textbf{L}} $$ where, $$ \textbf{L} = \begin{bmatrix} ...
2
votes
0answers
66 views

What is the meaning of SU(2) triplet scalar field? [closed]

The following is an about a Left-Right Symmetric model. $SU(2)\otimes SU(2)$ $(2\otimes 2=3\oplus 1)$ will generate a triplet, which in Left-Right Symmetric model is ...
4
votes
1answer
51 views

Non coherence of Fermions and Bosons through $U(1)$

I "know" the textbook answer why we cannot write, $$ |\psi\rangle = a|j=\tfrac{1}{2}\rangle + b|j=1\rangle $$ as "each term in the quantum superposition transforms differently under $U(1)$", $$ ...
3
votes
1answer
82 views

How to write the Clebsch-Gordan decomposition in tensor notation

Let be $G$ a Lie Group and $\textbf{N}$ its complex representation. It is known that any state $|\ ab\ \rangle\in \textbf{N}\otimes\textbf{N} = \oplus_I\textbf{r}_I$ may be decomposed through the ...
7
votes
2answers
108 views

Lie groups with same algebra

I had a problem when considering symmetry breaking in an SO(4) gauge theory: $\mathcal{L} = \left| D_\mu\phi \right|^2$ where $D_\mu$ is the SO(4) covariant derivative. Then assuming there is some ...
3
votes
2answers
73 views

Bilinears in adjoint representation

Below are two statements from my notes and I am trying to verify them explicitly. In both cases the fields are assumed to transform under the fundamental representation of $O(N)$ - --'The kinetic ...
0
votes
0answers
26 views

Construct any Hamiltonian that is the linear combination of existing constructable Hamiltonians

In the paper Quantum Computation over Continuous Variables, it states that since $$e^{iAt}e^{iBt}e^{-iAt}e^{-iBt} = e^{-[A,B] t^2} + O(t^3)$$ when $t\rightarrow 0$, if one can apply a set of ...
1
vote
1answer
20 views

Bifundamental representations [closed]

Can someone give me explicit examples (in matrix form) of bifundamental representations? Illustrative would be for instance: a) SU(3) x SU(2) b) SO(4) x U(1) c) E6 x U(1) but other you may have ready ...
3
votes
1answer
73 views

Does there exist finite dimensional irreducible rep. of Poincare group where translations act nontrivially?

I read several textbooks of QFT and find that there are two ways to classify the particles or fields. The first one is to study the irreducible representation of Lorentz group (or exactly the ...
1
vote
0answers
30 views

Ordering of basis elements of a Lie-group representations tensor product [migrated]

Let's consider a Lie Group $G$ and its complex representation $\textbf{N}$. Let's consider the decomposition $$ \textbf{N}\otimes\bar{\textbf{N}} = \oplus_{J}\textbf{r}_J $$ where $\textbf{r}_J$ are ...
1
vote
0answers
42 views

Example of a symmetry and the group with which it is modelled? [duplicate]

Could you please provide a specific example of a symmetry and the group with which it is modelled? I am beginner to study symmetry in physics, please answer with just an example. This question is ...
0
votes
0answers
35 views

Calculating Clebsch-Gordan coefficients through Racah's formula

So Clebsch-Gordan coefficients are found in tables, but I need to calculate them using Racah's formula, which reads as following: $c_+ (J,M) f_{m_1}^{M+1}=c_+(j_1,m_1-1) f_{m_1-1}^M + c_+(j_2,M-m_1) ...
2
votes
2answers
100 views

Why representations instead of just groups?

This question is essentially asking for a clarification on what has already been said in this one. What I don't understand is why it is the representations that are important in Quantum Field Theory ...
2
votes
0answers
42 views

Non-physical representations of double group

In group theory, to account for electron spin, double group is introduced. The key difference between an ordinary point group and a double group is an extra element $\bar{E}$ with the meaning of a ...
0
votes
1answer
22 views

Decomposition of the adjoint representation of a spontaneously broken compact group

Let be $G$ a compact group, symmetry of the theory I am working with. $G$ is broken into one subgroup $H$. I define the generators of G as $T_A = \{T_a,T_\hat{a}\}$, where the first are the unbroken ...
2
votes
0answers
41 views

Lorentz invariance of the Heaviside function [duplicate]

Consider the Heaviside function $\Theta(k^{0})$. This function is Lorentz invariant if $\text{sign}\ (k^{0})$ is invariant under a Lorentz transformation. I have been told that only orthochronous ...
0
votes
1answer
47 views

Homework-lile questions about Poincare transformation [closed]

Here is a page from a paper which I am currently reading. This page mainly talk about Poincare symmetry. Now I can not understand how is Eq.(3.32) is derived. Also Eq.(3.28) looks odd to me. Why ...
3
votes
0answers
62 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
23 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
80 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 ...
1
vote
2answers
90 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
21 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
19 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
69 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
46 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
32 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
99 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 ...
0
votes
1answer
42 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
44 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
68 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
59 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) \\ ...
2
votes
0answers
64 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
30 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
87 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
59 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
62 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
58 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
67 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
25 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 ...
2
votes
0answers
31 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 ...
1
vote
0answers
43 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
123 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
40 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
72 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
48 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
38 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
58 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 ...
2
votes
1answer
160 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 ...