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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1answer
118 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^{(...
3
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3answers
87 views

What is an $\mathrm{SU}(2)$ Triplet?

Under $\mathrm{SU}(2)$ group, a doublet transforms like: $$\phi \rightarrow \exp\left(i\frac{\sigma_i}{2}\theta_i\right)\phi.$$ The doublet looks like $$\binom{a}{b} ,$$ which is easy to understand ...
5
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1answer
236 views

Difference between Cartesian product and tensor product on gauge groups

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

$(1/2, 0)$ representation of the Lorentz Group $SO(1,3)$

Let us consider the $(j, j') = \left(\frac{1}{2}, 0\right)$ representation of $SO(1, 3)\cong SU(2) \otimes SU(2)$. $j = \frac{1}{2}$ corresponds to $SU(2)$ generated by $$ \tag{1} N_i^+ = \frac{1}{...
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2answers
56 views

Module of $SU(2)$ [on hold]

I've read that "$SU(2)$ is the group of transformations in 2-dimensional complex space." What specifically are the two complex dimensions? Is one dimension the real axis and the other the ...
7
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2answers
226 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 ...
4
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1answer
131 views

Different ways of derivation of Gell-Mann-Okubo mass formula

Recently my teacher told me that there are many ways of deriving the Gell-Mann-Okubo mass formula by using group representation theory (by using dynamical group etc). Where can I read about these ways?...
7
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2answers
174 views

Normalising Generators of a Lie Algebra

Ok, so I'm asking this in physics because I'm currently working through part of Srednicki's text on QFT, even though it's really a maths question. In Srednicki's chapter on non-Abelian gauge theory, ...
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0answers
16 views

Moduli space of torus compactifications

I am trying to understand some general statements made in the lecture notes by Vafa entitled "Lectures on Strings and Dualities" concerning toroidal compactifications (arXiv:hep-th/9702201). Question ...
3
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0answers
33 views

Quantum groups and $q$-deformed $SU(2)$

I am looking for a self-contained introduction to quantum groups and $SU(2)_q$ in particular, which a person with background in relativity and particle physics would understand. It has to contain the ...
0
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1answer
165 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 ...
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3answers
63 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 ...
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1answer
55 views

Real irreducible representation of $SU(2)$ [closed]

Consider a real irreducible representation of $SU(2)$ group in $d$-dimensional space-time. How many components do the spinors (eigenvectors) have? For instance, a real irreducible spinor in 10-dim has ...
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17answers
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 ...
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1answer
181 views

How symmetry is related to the degeneracy?

I have several questions about symmetry in quantum mechanics. It is often said that the degeneracy is the dimension of irreducible representation. I can understand that if the Hamiltonian has a ...
3
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0answers
45 views

Relation between projective representations, connectivity of a group manifold and number of equivalence classes of paths

Projective representations of a multiply-connected group is defined as $$U(g_1)U(g_2)=c(g_1,g_2)U(g_1g_2)$$ where $c(g_1,g_2)$ is phase. Reading various articles, and this old post of mine, it appears ...
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1answer
173 views

Point group symmetries and unit cell

I was wondering if the unit cell (of a given lattice) had to have every point group symmetries of the lattice it defines ? I guess there is no unique way to define a unit cell and that it may not have ...
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3answers
2k views

Idea of Covering Group

$SU(2)$ is the covering group of $SO(3)$. What does it mean and does it have a physical consequence? I heard that this fact is related to the description of bosons and fermions. But how does it ...
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4answers
158 views

How can I tell that a Lagrangian has an $SU(2)\times SU(2)$ symmetry?

this is a very basic question and it probably has a very simple answer. I was reading through some handouts when I came over something that I did not understand. One considered the simple Lagrangian ...
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0answers
39 views

Why must the Vacuum Manifold contain the quotient group?

For a symmetry breaking pattern $G \rightarrow H$, the vacuum manifold must contain the quotient space $G/H$. In most cases, we take the vacuum manifold to be the quotient space. My question is, Is ...
2
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1answer
56 views

Trace of raman tensors

A raman tensor $\gamma_m$ is defined as the derivative of the polarizability tensor $\alpha$ with respect to a raman mode $Q_m$, so $$ \gamma_m =\frac{\partial\alpha}{\partial Q_m} $$ $\gamma_m$ will ...
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0answers
37 views

A question about character table of the group Td. Group theory

I am trying to get the character table of the group Td using the basis functions already defined. I have a problem with the basis function (2z^2-x^2-y^2, x^2-y^2) because I don´t get the character -1 ...
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1answer
56 views

Eigenvalues of spherical harmonics in $d$ dimensions

I'm working on the Schrodinger equation for a hydrogen atom in a $d$-dimensional space, so I'm interested in the possible eigenvalues of the angular momentum part of the $d$-dimensional Laplace ...
6
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1answer
330 views

Difference Between Algebra of Infinitesimal Conformal Transformations & Conformal Algebra

in Blumenhagen Book on conformal field theory, It is mentioned that the algebra of infinitesimal conformal transformation is different from the conformal algebra and on page 11, conformal algebra is ...
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1answer
47 views

$SU(3)$ structures and branching rules

I am reading this paper http://arxiv.org/abs/hep-th/0211102 and I would like to understand better about the branching rule $SO(6) \equiv SU(4) \rightarrow SU(3)$ used for eq. C.11 in the Appendix. I ...
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1answer
44 views

How to construct a space which is translation invariant but not rotation invariant

I am just confused by the following idea. Consider a 3-dimensional translation invariant space, we now have 3 translation generators. Then let us start with a point, the full 3-dimensional space ...
4
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2answers
85 views

Is there a relation between spin and the spin group?

In Quantum Mechanics spin appears as one type of angular momentum. Indeed, in Quantum Mechanics one angular momentum on the state space $\mathcal{E}$ is a triplet of observables $\mathbf{J}=(J_1,J_2,...
3
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1answer
64 views

Tensor product representation of $SO(3)$ in the Hilbert space of particle with spin $S$

For a particle with a spin $S$, the rotation operator is given by $$ e^{iJ_i\theta/\hbar} $$ where $J_i$ is the component of the total angular momentum along the direction of the rotation axis. The ...
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0answers
45 views

How to go from a Higgs which transforms in the adjoint representation to a 2x2 matrix? [closed]

I have a triplet transforming in the adjoint map of the lie albegra of su(2) but I don´t know how to include it in to a Lagrangian where I have two lepton doublets. It should be a 2x2 matrix but I don´...
4
votes
2answers
91 views

What does U(N)xU(N)/U(1) mean?

I am studying a model in which SSB occurs and the original symmetry group is U(N)xU(N)/U(1) it acts as: $M'=AMB^{-1}$ (M is an NxN matrix containing the fields) The groud state i have found is ...
3
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1answer
178 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 ...
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0answers
51 views

Derivation of Gell-Mann Okubo relation for mesons

In SU(3) quark model of hadronic structure one assumes that mass splitings between hadrons is due to difference between masses of $s$ quark and $u,d$. This is modeled by perturbation Hamiltonian $$ \...
8
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1answer
282 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 $$ X_{-1}^2+X_0^...
3
votes
1answer
66 views

$SU(N)$ Yang-Mills Theory

Yang-Mills theory is based on the gauge group $G$ which we take to be $SU(N)$. Consider an example; $$\mathcal{L}=-\frac{1}{4}F^a_{\mu\nu}F^{a\mu\nu}-\sum_{j=1}^N\bar{\psi}_j(i\gamma^\mu D_\mu-m)\...
5
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1answer
318 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 ...
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1answer
41 views

Size of the quotient of normalizer by the stabilizer in the Pauli group [closed]

Let $P_n$ be the Pauli group on the $n$-qubit and $S$ be a stabilizer subgroup of it. Let $N$ be the normalizer of $S$ in $P_n$. In p. 69 of Lidar & Brun, Quantum Error Correction, it mentioned ...
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1answer
55 views

Poincare Group (Wald, Chapter 4 Page 59)

In Wald's text on general relativity, he mentions that in special relativity, many different global inertial coordinate systems are possible and can be put into one-to-one correspondence with elements ...
3
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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 $SU(N)$?
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1answer
76 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 ...
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0answers
18 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 crystal'...
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1answer
80 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 $$ \underbrace{6}_\...
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0answers
51 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 ...
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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 ...
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0answers
36 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
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1answer
38 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 ...
13
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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 ...
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0answers
28 views
1
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1answer
132 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 ...
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2answers
53 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 ...
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1answer
177 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 ...