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 ...

learn more… | top users | synonyms

0
votes
1answer
117 views

Is the fundamental representation of $SU(3)$ irreducible?

I want to check if the fundamental representation of $SU(3)$ is irreducible. The algebra is $$\mathbb{su}(3) = \{ m \in Mat(3,\mathbb{C} )\ |\ m = -m^+,\ Tr[m] = 0 \}$$ and I've found the generators. ...
1
vote
0answers
23 views
+50

Given a VEV how can I compute which generators remain unbroken using tensor methods?

This is a follow up to this question. A generator $T_a$ of a given gauge group $G$ remains unbroken after some Higgs field $\Phi$ gets a vev if $$ T_a \langle\Phi\rangle =0 $$ I'm trying to ...
2
votes
0answers
32 views

Thomas precession, Lie algebra of the Lorentz group and the conservation of energy

If you read this post Thomas Precession, you will see a very good answer by WetSavannaAnimal, on the subject of Thomas Precession, which I am currently working my through, in conjunction with some ...
1
vote
0answers
59 views

Quadratic Casimir Operator of $SO(5)$ [on hold]

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 ...
3
votes
2answers
80 views

Rotation matrix of Euler's equations of rotation relative to inertial reference frame

I was playing with simulation of Euler's equations of rotation in MATLAB, $$ I_1\dot{\omega}_1 + (I_3 - I_2)\omega_2\omega_3 = M_1, $$ $$ I_2\dot{\omega}_2 + (I_1 - I_3)\omega_3\omega_1 = M_2, $$ ...
1
vote
0answers
33 views

The derivation of the irreducible representations of the Lorentz group

I took the way of classification of Lorentz group representations from Sexl, Urbantke, Relativity, groups and particles (Germ. ed. 1975). But I don't understand it as I outline in the following: In ...
5
votes
2answers
515 views

Equivalent Rotation using Baker-Campbell-Hausdorff relation

Is there a way in which one can use the BCH relation to find the equivalent angle and the axis for two rotations? I am aware that one can do it in a precise way using Euler Angles but I was wondering ...
13
votes
1answer
474 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\{ ...
2
votes
1answer
105 views

How unique are the quantum numbers we commonly use?

We use the eigenvalues of the Cartan generators (=diagonal generators) of a given gauge group as quantum numbers in physics. Are these numbers somehow fixed and if not, what transformations are ...
1
vote
1answer
46 views

What is the definition of the duality group $E_{7(7)}$?

What is the definition of the duality group $E_{7(7)}$ that appears in ${\cal N}=8$ Supergravity and what are the basics properties? Moreover what is the relation with the Lie Algebra $E_7$? ...
0
votes
0answers
11 views

Anticommutative Sets of SU(N) Generators? Anticommutative Analogue to Cartan subalgebra?

I am currently studying SU(N) generators in order to find bases that may suit a problem at hand. I am especially interested in getting as large anticommuting sets within a basis as possible. In SU(2) ...
6
votes
1answer
268 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 ...
2
votes
1answer
118 views

Why is orbital angular momentum quantized according to $I= \hbar \sqrt{\ell(\ell+1)}$?

I simply have no idea how this result is found $$I=\hbar \sqrt{\ell(\ell+1)}.$$ The result seems to just be dumped in textbooks rather than explained. I can get the result that $I_z=\hbar m_j$. ...
0
votes
0answers
38 views

How can I compute the orbit of a Higgs field?

In many papers that deal with symmetry breaking a concept called orbit is introduced: It is worth noting that if the potential is a minimum $\phi_0$ at a value of the field, then from (3.13) it is ...
3
votes
1answer
137 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. ...
7
votes
2answers
129 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 ...
2
votes
1answer
33 views

Gluon have colour-anticolour; what about weak bosons?

Gluons can be red-antiblue, or green-antired, etc. What about weak interaction bosons? (Say before symmetry breaking, to make matters simpler.) Is there a similar "weak charge" structure of ...
6
votes
2answers
67 views

How are quadruple gluon vertices related to $SU(2)$ and $SU(3)$?

I once read that the non-commutativity of the Lie Groups $SU(2)$ and $SU(3)$ is the reason that the weak and strong interactions have Feynman diagrams with quadruple vertices, where four gauge bosons ...
1
vote
2answers
224 views

Pauli Matrices proof

The original question was to prove $$e^{i\sigma_z\phi} \sigma_y e^{-i\sigma_z\phi}=e^{2i\sigma_z\phi} \sigma_y.$$ Since left hand side is equal to $\sigma_y$, I thought in order for right hand ...
8
votes
1answer
191 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 ...
4
votes
0answers
107 views

Why the field strength tensor for locomotion at low Reynolds number may be written like that?

I've been studying locomotion at low Reynolds number for some time now and it has been a quite tough problem. I've already asked two questions about the problem here, and now there is this question ...
9
votes
2answers
709 views

Geometric/Visual Interpretation of Virasoro Algebra

I've been trying to gain some intuition about Virasoro Algebras, but have failed so far. The Mathematical Definition seems to be clear (as found in http://en.wikipedia.org/wiki/Virasoro_algebra). I ...
4
votes
1answer
379 views

How to get result $3 \otimes 3 = 6 \oplus \bar{3}$ for $SU(3)$ irreducible representations?

Let's have $SU(3)$ irreducible representations $3, \bar{3}$. How to get result that $$ 3\otimes 3 =6 \oplus \bar{3}~? $$ I'm interested in $\bar{3}$ part. It's clear that for $3 \otimes 3$ we can use ...
5
votes
1answer
231 views

Symplectic leaves, tori and Poisson manifolds

For classical systems we can define a configuration manifold, whose cotangent bundle is a momentum phase space equipped with a closed, non-degenerate 2-form. Upon the commutative algebra of smooth ...
13
votes
3answers
748 views

$\mathrm{SU(3)}$ decomposition of $\mathbf{3} \otimes \mathbf{\bar{3}} = \mathbf{8} \oplus \mathbf{1}$?

I have a question about the tensor decomposition of $\mathrm{SU(3)}$. According to Georgi (page 142 and 143), a tensor $T^i{}_j$ decomposes as: \begin{equation} \mathbf{3} \otimes \mathbf{\bar{3}} = ...
1
vote
1answer
76 views

Relationship between those two “exponentials”

Let $G$ be a Lie group and $L(G)$ it's Lie algebra. We know that every left-invariant vector field $X$ in $G$ is complete, and so one can consider the integral curve defined for all $t\in \mathbb{R}$ ...
2
votes
0answers
43 views

Traceless Tensors in $SU(3)$, Georgi's Lie Algebras

I'm doing a self-study through Georgi's Lie Algebra's in Particle Physics and there is a ''note without proof'' in the book that I have not managed to see through myself. In Section 10.3, Georgi ...
3
votes
3answers
68 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 $$ g = e^{i a J_+} e^{i b J_z} e^{i c J_-} . $$ ...
2
votes
2answers
115 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 ...
1
vote
1answer
77 views

How to prove that any rotation can be represented by 3 Euler angles

How can one prove that any rotation of a rigid object in 3-dimensional (3D) space can be represented by a sequence of three rotations around pre-fixed axes by 3 Euler angles? I see this statement in ...
1
vote
1answer
63 views

Representations of Lorentz algebra

It is well known that the Lorentz algebra can be written as two $SU(2)$ algebras. By defining $$N_i=\frac{1}{2}(J_i+iK_i), \qquad N^{\dagger}_i=\frac{1}{2}(J_i-iK_i)$$ we have ...
1
vote
0answers
28 views

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 ...
2
votes
1answer
93 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
votes
1answer
131 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
votes
2answers
119 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 ...
1
vote
1answer
127 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 ...
1
vote
0answers
29 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 ...
0
votes
0answers
39 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 ...
6
votes
2answers
148 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 $$ ...
0
votes
1answer
41 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 ...
3
votes
1answer
128 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. ...
1
vote
0answers
45 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 ...
1
vote
2answers
74 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 ...
1
vote
1answer
74 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? ...
17
votes
1answer
434 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 ...
1
vote
1answer
62 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
votes
1answer
79 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) = ...
2
votes
3answers
192 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\ ? ...
3
votes
2answers
67 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, ...
3
votes
1answer
52 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}, ...