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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4
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1answer
65 views

What is the $10$ in the $\mathbf{4}\otimes\mathbf{4}$ tensor product of $SO(6)$?

This is the question 22.D in Howard Georgi's Lie Algebras book, I thought about for a minute, but couldn't come up with a plausible answer. It's a fact that the SO(6) and SU(4) algebras are ...
0
votes
0answers
32 views

How do (special) groups such as U(1) and SU(2) do represent the electromagnetic and weak forces? [on hold]

I don't see how groups, such as the circle group somehow representing electromagnetism, represent the fundamental forces. Where is this connection between maths and theoretical physics? Also, as a ...
0
votes
0answers
11 views

How one can count how many phonon modes is there in the crystal?

I'm reading the review on phonon and Raman scattering in 2D transition metal dichalcogenides (2D TMDCs). At the beginning of Section 2.1 it is said that, since there are two X-M-X units in the ...
1
vote
1answer
69 views

Why is quark flavor just a SU(N) group?

In the standard model one has U(1) for electromagnetism, SU(2) for the weak sector and SU(3) for the color sector. One could say that in the quark part of the fermions, there are $$ ...
0
votes
0answers
49 views

What is the physical implication of a homomorphism between SU(2) and SO(3)?

It can be shown that there exists a homomorphism between $SU(2)$ and $SO(3)$. What is the physical implication of a homomorphism actually? I know that in physics $SU(2)$ acts on a space of spin 1/2 ...
13
votes
3answers
2k views

Hypercharge for $U(1)$ in $SU(2)\times U(1)$ model

I understand that the fundamental representation of $U(1)$ amounts to a multiplication by a phase factor, e.g. EM. I thought that when it is extended to higher dimensional representations, it would ...
0
votes
0answers
43 views

Homotopy group of the conformal group [migrated]

I would like to know which are the first three homotopy groups of the conformal group SO(4,2): $$ \pi_n(SO(4,2))=? \quad n=1,2,3 $$
0
votes
0answers
34 views

A Question about a $U(1)_{B-L}$

I know I can write the QCD lagrangian like this: $$ \mathcal{L} = (i\bar{q}_{R} \gamma_{\mu}\partial_{\mu} {q}_{R} + i\bar{q}_{L}\gamma_{\mu}\partial_{\mu} {q}_{L}) + \text{other terms} $$ When ...
1
vote
1answer
33 views

Enhanced D-Brane Gauge Symmetry and the string decoupling limit

Consider a system of $N_C$ $Dp$-branes (in Type IIA or Type IIB string theory, depending on whether $p$ is even or odd). In the Les Houches lectures on supersymmetric gauge theories by Berman and ...
12
votes
3answers
4k views

How to derive addition of velocities without the Lorentz transformation?

Lorentz contraction and time dilatation can be deduced without Lorentz transformation. Can you deduce also the theorem of addition of velocities $$w~=~\dfrac{u+v}{1+uv/c^2}$$ without Lorentz ...
1
vote
0answers
16 views
2
votes
1answer
109 views

Question on quotient groups and SLOCC [closed]

I have a math-physics question, which is based on an interest in SLOCC systems for black hole entanglement. The Cartan decomposition of a group $G$ such that $H = G/K$ is such that the derivation or ...
1
vote
2answers
46 views

What is the diagonal $U(1) \subset SU(2)$ and the diagonal $u(1) \subset su(2)$?

What is meant by the diagonal $U(1) \subset SU(2)$ and the diagonal $u(1)\subset su(2)$? I have read it above eqn. (10) in this paper http://arxiv.org/abs/0812.3572 but have also heard it mentioned in ...
2
votes
1answer
164 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 ...
1
vote
2answers
39 views

Rotation matrix for aligning x-axis in an arbitrary direction

I want to align the x-axis of my coordinate system, with an arbitrary direction in space $\hat{n}$. About which axis should I rotate? Ceratinty rotation about x-axis or $\hat{n}$-axis will not serve ...
1
vote
3answers
870 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 $[\sigma_2]_{ij}\text{ and }[\eta_1]_{xy}$ ...
3
votes
1answer
142 views

Dimension and Basis properties of $SU(N)$

$SU(N)$ is the group of special unitary matrices of dimension $N$, i.e., the set of all unitary ($U^{\dagger}U=I$) $N\times N$ matrices with $\det(U)=1$. For $N=2$, these matrices are spanned by the ...
3
votes
1answer
69 views

How many glueballs are there?

As I understand there are eight types of gluons (linear combinations of color/anticolor pairs with varying amplitudes) which can combine (for very short periods) to form glueballs. If there were no ...
0
votes
1answer
43 views

Notations for high symmetry points in the 1st Brillouin zone

I am trying to understand how I should interpret the letters like Г,K,M,T etc., that are usually there in the electronic band structure diagrams. So, let's assume we have graphene with its hexagonal ...
4
votes
1answer
162 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) ...
6
votes
2answers
197 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 ...
2
votes
1answer
116 views

Representation of U(1) on fock space

I am currently reading up on the use of group theory in physics using Peter Woit's book draft (available on his homepage). I do understand the mathematical concepts but have a bit of a problem making ...
0
votes
0answers
26 views

Applications of octonions in special relativity?

According to the Wikipedia article on octonions: Octonions [...] have applications in fields such as string theory, special relativity, and quantum logic. However, I couldn't find any ...
3
votes
2answers
177 views

Why do decompositons like $16 \otimes 16 = 10 \oplus 120 \oplus 126$ tell us which Higgs representations we can use?

EDIT: I found an answer, which I do not understand: In Gürsey - Symmetry breaking patterns in E6 he writes: " Because of Fermi-Dirac statistics of fermions they must occur in the symmetric part of ...
0
votes
0answers
19 views

Singularity of $\Gamma(g) - I$ [migrated]

Let $\Gamma$ be a non-trivial irreducible unitary representation(also suppose of finite degree $d$) of a nontrivial group $G$. Consider the matrices $\Gamma(g) - I$, where $g\in G$ and $I$ is the ...
1
vote
1answer
49 views

Magnetic monopoles gauge theories

I'm quoting 't Hooft: "[...] Locally stable field configurations may exist that have some topological twist in them [...].Careful analysis of the existing Lie groups and the way they may be ...
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 ...
3
votes
2answers
28 views

Standard-model flavor symmetry

If we consider the chiral Lagrangian after the spontaneous symmetry breaking, we have got fermion masses and Yukawa couplings to the physical Higgs boson. So it follows global symmetries in flavor ...
3
votes
1answer
63 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 ...
3
votes
1answer
105 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 ...
4
votes
1answer
279 views

Does the low-energy gauge structure depend on the choice of $SU(2)$ gauge freedom?

The starting point and notations used here are presented in Two puzzles on the Projective Symmetry Group(PSG)?. As we know, Invariant Gauge Group(IGG) is a normal subgroup of Projective Symmetry ...
3
votes
1answer
52 views

Using symmetry to determine a hydrogen electron's decay route from $|300\rangle$ to $|100\rangle$

Lets say we have an electron in state $|nlm\rangle = |300\rangle$ of the hydrogen atom. By selection rules, we know that it can only decay to ground state in 3 ways, namely through the $|21m\rangle$ ...
3
votes
1answer
79 views

Symmetry breaking to a special subalgebra?

This is a follow-up to my question here. For regular subalgebras of some group's Lie algebra the root system of the subalgebra is a subset of the root system of the original's group algebra. In ...
6
votes
1answer
78 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
103 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} ...
3
votes
0answers
79 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
53 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 $U(1)$", $$ ...
9
votes
1answer
753 views

Why do we identify symmetric 2nd rank tensors with spin-2 particles in string theory?

I am going through Tong's lecture notes on String Theory and came across the following irrep decomposition (Chap 2, p.43) of the bosonic string first excited states: $$\text{traceless symmetric} ...
0
votes
1answer
48 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
2answers
796 views

Orthochronous Lorentz transformations are time-preserving and $SL(2,\mathbb{R})$

Let's consider the pseudosphere/hyperboloid in $\mathbb{R}^{1,2}$ given by $$x^2+y^2-z^2=-R^2.$$ We know that the Lorentz group $$O(1,2)=\{ A \in Mat(3,\mathbb{R}): A^tGA=G \},$$ where ...
7
votes
2answers
115 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 ...
0
votes
1answer
51 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 ...
3
votes
2answers
77 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
27 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
23 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
79 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 ...
40
votes
16answers
14k views

Comprehensive book on group theory for physicists?

I am looking for a good source on group theory aimed at physicists. I'd prefer one with a good general introduction to group theory, not just focusing on Lie groups or crystal groups but one that ...
1
vote
0answers
46 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 ...
4
votes
0answers
85 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 ...
7
votes
1answer
167 views

Relations between diffeomorphism symmetry theories and invariant $SU(N), N \rightarrow \infty$ theories

Is it possible to have, an exhaustive panorama (as much as possible), about the relations between theories having a diffeomorphism symmetry, and theories having a $SU(N), N\rightarrow\infty$ ...