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

2
votes
1answer
161 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
19 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
91 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 ...
2
votes
1answer
58 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
65 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 ...
0
votes
1answer
115 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 ...
1
vote
1answer
52 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 ...
1
vote
0answers
39 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
47 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
44 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
117 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
131 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
78 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
83 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 & ...
0
votes
1answer
52 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
48 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
60 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
118 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} + ...
2
votes
1answer
75 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
214 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 $\Phi_2$ ...
0
votes
0answers
15 views

Can i extend the non-linear realization of the chiral group to $U$ with complex pions?

In chiral perturbation theory we build a Lagrangian invariant under $SU(2)_L\times{}SU(2)_R$ which acts on the matrix $U$ that accommodates the pion degrees of freedom in the following way ...
0
votes
0answers
70 views

Irreducible representations of $SU(2)$, Tensor-operators under rotations

First of all: this is my first question so please give feedback to the way I'm formulating the question! The question is about an exercise I have to solve, but I simply get nowhere. It is given the ...
1
vote
0answers
48 views

The universal covering group of a symmetry group [duplicate]

In Weinberg QFT Vol.1, it says one can enlarge the symmetry group $H$ to the universal covering group $C$ such that one obtains a trivial cocycle or $C$ is simply connected whereas $H$ is not. I get ...
2
votes
1answer
142 views

What does Addition of Angular Momenta tell us about Group Theory?

I've come across this a lot, but I've never understood it. I do know basic Group Theory including Lie Groups. In Introduction to Quantum Mechanics, Griffiths ends the chapter on spin with the remark ...
0
votes
0answers
26 views

What is therelation between nonlinear sigma model, complex projective group?

The O(N) nonlinear sigma model has topological solitons only when N=3 in the planar geometry. There exists a generalization of the O(3) sigma model so that the new model possess topological solitons ...
1
vote
2answers
337 views

Operator vs. Matrix in quantum formalism

We use in Dirac formalism of QM the tool of operators and kets in spatial and spin spaces to obtain eigenvalues and eigenkets. But the operation here is simply that of a matrix multiplication. Now ...
0
votes
0answers
33 views

Textbooks in group theory for budding theoretical particle physicists [duplicate]

I am looking for textbooks in group theory. I have just finished my undergraduate studies in Physics and am looking to specialise in theoretical high-energy physics. Therefore, textbooks in group ...
1
vote
1answer
104 views

Matrix represenation of total angular momentum operator

I see that for total ket in QM of hydrogen atom we define a tensor product of kets of spatial and spin spaces, upon which spatial and spin operators, operate respectively. For the total angular ...
1
vote
0answers
37 views

Does Bose-Statistics mean that we are only allowed to write symmetric tensor products of Higgs fields in the Lagrangian?

Given Higgs fields, transforming as the representation $R$ of a group $G$. Does the physical fact that scalars=boson commute mean that we are only allowed to write symmetric tensor products of the ...
2
votes
3answers
98 views

Showing a mapping between $SU(2)$ and $SO(3)$

I know this has been done on this site in a different manner but I'm wondering if it's possible to show the 2:1 Lie group homomorphism between $SU(2)$ and $SO(3)$ using exponentials of the generators ...
0
votes
1answer
59 views

OPE of parity even theories in CFT.

If I consider an OPE of some operators, which belong to a theory where parity is not violated, will I have a constraint on the kind of operators appearing in the right hand side ? For example, I ...
2
votes
1answer
85 views

Why are there only two linearly independet quartic Higgs terms for the adjoint $24$ in $SU(5)$ GUTs?

I've read the statement in countless papers, for example, here Eq. 4.2 or here Eq. 2.1 without any further explanation or reference, that the "most general renormalizable Higgs potential" for an ...
3
votes
1answer
165 views

Relationship of symplectic group (Hamiltonian structure) to unitary group in quantum mechanics?

Wikipedia claims here that the 2 out of 3 property is the following relationship between unitary, orthogonal, symplectic, and general linear complex groups: $U(n)=Sp(2n,R)∩O(2n)∩GL(n,C)$ Intuitively ...
2
votes
1answer
30 views

If Hamiltonian is invariant under a group $G$, then for $R\in G$ , $R\Psi$ has the same energy. Arfken

I've never taken quantum mechanics (only a class on modern physics) but I'm reading Arfken's section on Group theory. I don't understand what is trying to be proved here. It seems like a tautology to ...
0
votes
0answers
96 views

Spin 1/2 wavefunction transformation under inversion and mirror symmetry

I'm considering group-theory applications to condensed matter physics now. In particular I work with the following paper: http://journals.aps.org/pr/pdf/10.1103/PhysRev.100.580 and try to understand ...
3
votes
1answer
135 views

Difference between Cartesian product and tensor product on gauge groups

After a comment of John Baez from a question I asked about on MathOverflow I would like to ask what is the difference between, for example, $SU(3)\times SU(2) \times U(1) $ and $SU(3) \otimes SU(2) ...
1
vote
0answers
71 views

Bondi-Metzner-Sachs (BMS) related Question(s)

I started studying the BMS group in connection with the set of papers by A. Strominger et al., also related with the supposed solution of the "Black Hole Information Paradox" by S. W. Hawking ...
6
votes
1answer
78 views

Is there a systematic way to obtain all conserved quantities of a system?

I'd like to know whether, given a system, there's a way to obtain all the conserved quantities. For instance if the system consists of electric and magnetic fields, the fields must satisfy Maxwell's ...
0
votes
0answers
23 views

How to compute remnant discrete charges?

When a $U(1)$ gauge symmetry is broken by Higgs field with $U(1)$ charge $g$, we have a remnant $\mathbb{Z}_q$ symmetry. How can I compute the corresponding charges under this remnant group for ...
1
vote
1answer
70 views

Center group and Symmetry breaking

First, I will state the definition of center, $i.e$, the center of group $C_G$ is defined a sthe part of $G$ which commutes with all generators. I want to know the procedure of breaking gauge ...
2
votes
1answer
72 views

Why is the Mixed Faraday Tensor a matrix in the algebra so(1,3)?

The mixed Faraday tensor $F^\mu{}_\nu$ explicitly in natural units is: ...
0
votes
0answers
27 views

Dilations in momentum space

I don't quite understand what's going on here. Let's suppose I have a dilation in real space. The generator is $D=x^j \partial_j$, so an infinitesimal dilation is $\delta x^i = Dx^i = x^j \partial_j ...
2
votes
0answers
54 views

What does complexification mean for our particles in physics

As gauge group let's consider the popular $SO(10)$ group. The fundamental representation $\pi$ of the corresponding Lie algebra $\mathfrak{so}(10)$ is $10$ dimensional $$ \pi: \mathfrak{so}(10) ...
1
vote
1answer
146 views

How do simple two-component Fierz identities follow from a property of the Pauli matrices?

On page 51 Peskin and Schroeder are beginning to derive basic Fierz interchange relations using two-component right-handed spinors. They start by stating the trivial (but tedious) Pauli sigma identity ...
12
votes
1answer
194 views

Highest symmetric non-maximally symmetric spacetime

What is the highest number of symmetries (Killing vectors) that a (4-dimensional) spacetime can have without being maximally symmetric? From what I can see, it seems to be 7 (which includes the ...
-1
votes
1answer
95 views

Derivation of the Lorentz algebra explicity [closed]

I need the complete proof for commutation relation of the Lorentz group generators. The proof of Lorentz algebra using this commutation relation.
3
votes
1answer
169 views

Permutation symmetry - a continuous symmetry?

From quantum mechanics it is known that permutation between identical particles does not change the Hamiltonian. Assuming that the quantum system consists of a very high number of particles such that ...
1
vote
1answer
73 views

Regarding different representations of the Lorentz Group & its defining properties

Take $\Lambda$ to be a Lorentz matrix, it satisfying $\Lambda^T \eta \Lambda=\eta$. By writing $\Lambda=\exp[-\frac{i}{2}\omega_{\mu\nu}\mathcal J^{\mu\nu}]$, we find that the generators satisfy $$ ...
0
votes
0answers
30 views

trace of spherical components of an angular momentum multiplet

My basic mathematical question is whether or not there exist selection rules regarding the traces of the spherical tensor components of some operator acting on some subspace of definite total angular ...
1
vote
0answers
111 views

Quadratic Casimir Operator of $SO(5)$ [closed]

In the article A Four Dimensional Generalization of the Quantum Hall Effect, arXiv:cond-mat/0110572, by Zhang and Hu Quadratic Casimir operator for $SO(5)$ is given as $$p^2/2+q^2/2+2p+q .$$ When ...