A vector space $\mathfrak{g}$ over some field $F$ and kitted with a bilinear, antisymmetric and Jacobi-identity-fulfilling product ("Lie Bracket" or "commutator"). In physics, most often arises as the Lie algebra (tangent space to the identity) of a Lie group; in gauge theories, basis vectors of the ...

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How to find the generators of a deformed boost?

I'm reading the paper arXiv:gr-qc/0012051 on doubly special relativity. In page 7, the author wants to find the generators of a deformed boost that preserves $$E^2 = p^2 + m^2 - l_p p^2 E$$ ($l_p$ is ...
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1answer
84 views

Angular momentum - proof for integer eigenvalues

I am confused about a proof my Quantum Mechanics textbook has left "as an exercise for the reader". So, we've got the angular momentum operator $\hat{L}$. We've also got the generalized angular ...
4
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1answer
116 views

Why $SU(3)$ has eight generators?

The generators of $SU(3)$ group are Gell-Mann matrices and one can construct these generators from Pauli spin matrices, basically expanding in 3d and rotating about each axis. Take $\sigma_3$, assume ...
4
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2answers
82 views

Spin, orbital angular momentum and total angular momentum

If I understand correctly, spin is an intrinsic property of particles, which follows the algebra of angular momentum, but has nothing to do with an "orbital angular momentum" in that the particle is ...
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1answer
52 views

Spin commutation relations

For orbital angular momentum defined as $L= r \times p $ we can prove, in quantum mechanics, the commutation relations. Also, we could prove these relationships through the study of rotations ...
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0answers
19 views

Covariant derivative of Noether current [closed]

I am working with a non-abelian gauge gauge theory that has one gauge field and a complex scalar field. I am supposed to prove that \begin{equation} (D_\mu j^\mu)^a=0, \end{equation} where ...
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1answer
28 views

Uniqueness of expression of a Lie group element

Just take the SU(2) group as an example. The three generators are $J_z$, $J_+$, and $J_-$. For an element $ g $, sometimes we want to express it as $$ g = e^{i a J_+} e^{i b J_z} e^{i c J_-} . $$ ...
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27 views

commutation relation of angular momentum operator in non cartesian coordinates

The angular momentum operator $J$ in quantum mechanics with the commutation relation \begin{equation*} [J_i,J_j]=i\hbar\epsilon_{ijk}J_k \end{equation*} has the structure of a Lie-algebra. It is ...
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2answers
122 views

$su(1,1) \cong su(2)$?

The three generators of $su(2)$ satisfy the commutation relations $$ [J_0 , J_\pm] = J_\pm , \quad [J_+, J_- ] = +2J_0 .$$ The three generators of $su(1,1)$ satisfy the commutation relations $$ ...
4
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1answer
103 views

Subgroup of Lorentz Group Generated by Boosts

It is common knowledge that a composition of boosts is not a boost, but involves a rotation. Further, in discussions of Thomas precession, it is often stated that boosting in $x$, then $y$, then back ...
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1answer
241 views

How to construct an isomorphism between the Complexified Special Linear Lie Group and the Special Unitary Group?

This may be an unenlightening question, but I'm just not sure about the result and hoping someone can help me varify it. $\\$ This question is related to these three questions. $\\$ I want to ...
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1answer
28 views

How to construct generators and Lie Algebra for Lorentz group?

I'm trying to figure out Lorentz group in 2+1. First of all, I'd like to think the special orthgonal group as a combination of rotation and boost in space. Then I construct it as below. First rotation ...
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1answer
60 views

How to use Clebsch-Gordan coefficients for 3 particles?

I have a Hamiltonian for 3 particles of spin 1 that I boiled down to: \begin{equation} k(\textbf{S}^2+\cdots), \end{equation} where: \begin{equation} \textbf{S}=\textbf{S}_1+\textbf{S}_2+\textbf{S}_3. ...
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32 views

Which representation do we start with in Grand Unified Theories?

The conventional approach in GUTs is to put all left-chiral fields $F_L$ of the standard model into one representation of the GUT group. For example, the 16 rep for $SO(10)$ GUT: $$ 16_L \rightarrow ...
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2answers
66 views

Why gauge fields are traceless Hermitian?

So I've had a read of this, and I'm still not convinced as to why gauge fields are traceless and Hermitian. I follow the article fine, it's just the section that says "don't worry about this ...
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1answer
47 views

Commutator of Gauge Covariant derivatives

What is the physical meaning of $$ [D_{\mu}, D_{\nu}] ~\propto~ F_{\mu, \nu}, $$ where $D_{\mu}$ is the gauge covariant derivative and $F_{\mu,\nu}$ is the field strength? Is it just a definition? ...
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382 views

Why do we need complex representations in Grand Unified Theories?

EDIT4: I think I was now able to track down where this dogma originally came from. Howard Georgi wrote in TOWARDS A GRAND UNIFIED THEORY OF FLAVOR There is a deeper reason to require ...
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1answer
54 views

Hamiltonian symmetry Lie algebra

What is the connection between complete set of commuting observables and generators of the Lie group? I have a Hamiltonian written down in second quantized formalism and I also checked that it ...
3
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1answer
68 views

Why is the gauge potential $A_{\mu}$ in the Lie algebra of the gauge group $G$?

If we have a general gauge group whose action is $$ \Phi(x) \rightarrow g(x)\Phi(x), $$ with $g\in G$. Then introducing the gauge covariant derivative $$ D_{\mu}\Phi(x) = ...
6
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1answer
167 views

Group representations as vectors and isomorphism between weights and matrix generators

This might be something basic, but it is unclear to me. So I am used to work with representations of groups as matrices. These matrices represent the structure of the Lie algebra by satisfying the ...
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3answers
163 views

Deriving cross product from angular momentum algebra

Is it possible to derive: \begin{equation} \hat{L}=\hat{r}\times \hat{p} \end{equation} from the angular momentum algebra: \begin{equation} [\hat{L}_i,\hat{L}_j]=i\ \hbar\ \epsilon_{ijk}\hat{L}_k\ ? ...
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2answers
53 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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2answers
103 views

What's the degree of freedom of this kind of matrix?

We first have a unitary matrix $$\{a_{ij}\}\quad(n\times n)$$ I know how to calculate its degree of freedom, which is $n^2$ if we consider a real variable as one degree of freedom. Now we have a ...
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1answer
32 views

Permissible combinations of colour states for gluons

My lecturer has said that there are 8 types of gluons (I'm assuming that the repetition of $r\bar{b}$ is a typo that is meant to be $r\bar{g}$) $$r\bar{b}, b\bar{r}, r\bar{g}, g\bar{r}, g\bar{b}, ...
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22 views

Recipe to compute dimension and decompose product of $SO(N)$ group representations [migrated]

As it is well known Young tableaux (YT) provide an efficient and very useful way to treat $SU(N)$ representation. This is principally based on these facts: There is a correspondence between irreps ...
2
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1answer
112 views

Irreducible Representations from Cartan Matrix

I need some help understanding how you can construct the irreducible representations of an algebra from knowing its roots or its Cartan matrix, which I am being told you most certainly can. ...
2
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2answers
103 views

Half-integer spin and infinitesimal rotations

On p. 692 of 'Quantum Mechanics' by Cohen-Tannoudji, he states that: Every finite rotation can be decomposed into an infinite number of infinitesimal rotations, since the angle of rotation can ...
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0answers
14 views

Classical limit and generalized coherent states

In quantum optics coherent states introduced by Glauber have a localized probability distribution in classical phase-space with maximum following classical equations of motions. This is not a ...
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20 views

Casimir operators of the Poincare Group

Say, $P^2$ and $W^2$ are the Casimir operators of the Poincare Group. Should the commutator of these Casimir operators be zero, because then they would be independent of each other and form a unique ...
12
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1answer
434 views

How to evaluate this sum of coupling coefficients?

I would like to evaluate the following summation of Clebsch-Gordan and Wigner 6-j symbols in closed form: $$\sum_{l,m} C_{l_2,m_2,l_1,m_1}^{l,m} C_{\lambda_2,\mu_2,\lambda_1,\mu_1}^{l,m} \left\{ ...
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2answers
255 views

Azimuthal quantum number $\ell$ and magnetic quantum number $m$ are from angular momentum?

Azimuthal quantum number $\ell$, and magnetic quantum number $m$ are defined when we do derivation for $L^2f=\ell(\ell+1){\hbar}^2f$ and $L_zf=\ell{\hbar}f$. This is my own conclusion after studying ...
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0
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40 views

The Lie algebra of the Lorentz group is $su(2) \oplus su(2)$. Is there a similar relation for the algebra of the Poincare group?

It can be shown easily, by introducing new generators from the usual ones that we can think of the Lie algebra of the Lorentz group as being built up by two copies of the $SU(2)$ Lie algebra: $$ ...
4
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1answer
66 views

Where does in GUT symmetry breaking $U(1)$ come from?

In GUTs one starts with some larger group, like $SU(5)$, which is then broken into smaller groups, for example $$SU(5) ~\longrightarrow~ SU(3) \times SU(2) \times U(1)$$ This can be seen, for ...
2
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1answer
217 views

Question on derivation of Ward identity

I'm currently reading these notes about the Ward identity (pages 259 - 261). I will repeat some of the steps to make the question self-contained. Let us consider a local transformation on the field ...
2
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1answer
30 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
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2answers
157 views

Unitary groups and infinitesimal transformations - Schwingers way of deriving Lie groups

In Schwinger's source theory book, he suggests if $G_a$ are the hermitian generators of the Unitary group, then we have an infinitesimal transformation is given by : $$ G = \sum_{a=1}^n ...
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0answers
26 views

How to expand free energy of Heisenberg spin chain?

In Dasgupta & Ma's 1979 paper "Low-temperature properties of the random Heisenberg antiferromagnetic chain", they give the free energy of a few interacting Heisenberg spins on a chain. I can't ...
2
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0answers
49 views

About generator of $SU(2)$ flavor symmetry group

I am reading the textbook, "weak interactions" by Howard Georgi which can be founded in his homepage. Here i am trying to solve the problem 1b-2. The problem is given as follows. Consider the ...
3
votes
1answer
93 views

What's the significance of the difference between the quantum numbers, $\ell$ and $m_{\ell}$?

I know that $m_{\ell}$ is associated with the projection of the angular momentum vector onto the $z$ axis and $\ell$ is associated with the length of the angular momentum vector. To me this implies ...
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1answer
51 views

How to derive the form of the invariant spinor inner product?

So we have gamma matrices that satisfy the spacetime algebra relations, $\{\gamma^\mu, \gamma^\nu\} = 2 \eta^{\mu\nu}$. We know that if we set $\sigma^{\mu\nu} = \frac{1}{4}[\gamma^\mu, \gamma^\nu]$ ...
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32 views

Name for the transformation into an accelerated frame?

A transformation into a frame that looks at an experiment from a rotated perspective is called a rotation. A transformation into a frame that moves with a different constant velocity is called a ...
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1answer
71 views

Why is orbital angular momentum quantized according to $I=\frac h{2 \pi} \sqrt{l(l+1)}$?

I simply have no idea how this result is found $$I=\frac h{2 \pi} \sqrt{l(l+1)}.$$ The result seems to just be dumped in textbooks rather than explained. I can get the result that ...
2
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1answer
40 views

Coset construction of Tricritical Ising CFT

In http://iopscience.iop.org/1742-5468/2008/03/P03010 the authors state that the Tricritical Ising Model (TIM) CFT can be obtained from a Wess Zumino Witten construction based in the coset ...
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3answers
532 views

The Asymmetry between Real and Imaginary in the three Pauli Spin Matrices

The Pauli spin matrices $$ \sigma_1 ~=~ (\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}), \qquad\qquad \sigma_2 ~=~ (\begin{smallmatrix} 0 & -i \\ i & 0 ...
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2answers
212 views

Why are the spin operators defined as they are?

$$\begin{align*}S_z &= \frac{\hbar}{2} \left(\left|+\right>\left<+\right| - \left|-\right>\left<-\right|\right)\\ S_y &= i\frac{\hbar}{2} \left(\left|-\right>\left<+\right| - ...
3
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1answer
90 views

About $SU(2)_L \times U(1)_L = U(2)_L $

In the many textbook of standard model, i encounter the relation \begin{align} SU(2)_L \times U(1)_L ~=~ U(2)_L. \end{align} Here $L$ means the left-handness. (It is a physical ...
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1answer
171 views

Symmetries in QM and QFT — operator transformation laws

In quantum mechanics, we implement transformations by operators $U$ that map the state $|\psi\rangle$ to the state $U|\psi\rangle$. Alternatively, we could transfer the action of $U$ onto our ...
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3answers
1k views

Why is the Symmetry Group for the Electroweak force $SU(2) \times U(1)$ and not $U(2)$?

Let me first say that I'm a layman who's trying to understand group theory and gauge theory, so excuse me if my question doesn't make sense. Before symmetry breaking, the Electroweak force has 4 ...
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1answer
48 views

Does invariance under infinite small transformation imply invariance to the finite one?

Let's say that I have finite chiral transform and I would like to show invariance of Dirac's Lagrangian when $m=0$ under it. The chiral transform is defined as: $$\psi(x) \rightarrow \psi'(x) =e^{i ...