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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3
votes
1answer
186 views

Quadratic Casimir operator of higher dimensional $\mathfrak{su}(3)$ representations

In higher dimensional representations of $\mathfrak{su(3)}$, what will be the quadratic Casimir operator? Is it same as in lower dimensions or different?
26
votes
2answers
712 views

Finding the vacuum which breaks a symmetry

I will start with an example. Consider a symmetry breaking pattern like $SU(4)\rightarrow Sp(4)$. We know that in $SU(4)$ there is the Standard Model (SM) symmetry $SU(2)_L\times U(1)_Y$ but depending ...
2
votes
0answers
49 views

Use Cartan subalgebra in spinor representation to find weights of vector representation

For $SO(2n)$ we can construct the lie algebra elements by using antisymmetric combinations of $\gamma_\mu$ which obey the Clifford algebra. Up to some prefactor the elements $ S_{\mu \nu} = \alpha ...
3
votes
1answer
535 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. ...
4
votes
1answer
111 views

Why complexify in order to construct Dirac representation?

Suppose we have a theory is covariant under the Spin group Spin(2n-1; 1). We consider the real vector space $V = R^{2n-1,1}$, which naturally comes with a Lorentzian inner product. On this vector ...
1
vote
0answers
70 views

Books on representation theory [duplicate]

Possible Duplicate: Best books for mathematical background? I'm looking for a textbook on the group/representation theory for a student-physicist. The main questions of interest are ...
0
votes
0answers
17 views

Why should an generator acts on an operator with the Lie bracket?

When we deal with ordinary symmetries which form a Lie group, we have an corresponding Lie algebra with a structure of Lie bracket $[,]$. A infinitesimal transformation can act on a state or an ...
1
vote
1answer
58 views

Decomposition of tensor product space into direct sum [closed]

Consider a tensor product space of two representations $j = \frac{3}{2}$ and $j = 1$. How to show $4 \otimes 3 = 6 \oplus 4 \oplus 2$?
2
votes
1answer
136 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 ...
16
votes
1answer
542 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\{ ...
9
votes
1answer
1k views

Generators of Poincare Groups

How can I determine the generators of the Poincare Group, $P(1,3)$ explicitly? Here $P(1,3)$ means a matrix Lie group.
3
votes
5answers
345 views

Eigenspaces of angular momentum operator and its square (Casimir operator)

The casimir operator $\textbf{L}^2$ commutates with the elements $L_i$ of the angular momentum operator $\textbf{L}$: $$ [\textbf{L}^2, L_i] = 0. $$ However, the $L_i$ do not commute among ...
0
votes
1answer
42 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 ...
7
votes
4answers
326 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
2answers
81 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} + ...
1
vote
1answer
41 views

Finding the proportionality constant of the Quantum Angular Momentum raising operator $T_{+}$ [closed]

This is a question about the mathematics of angular momentum operators in Quantum Mechanics- specifically a recursive relation from Robert Cahn's Semi-Simple Lie Algebras and their Representations ...
5
votes
1answer
182 views

Clever way to show a property of Lie transformation

Given the following Lie transformation: $$ \exp(\lbrace H, \cdot \rbrace):=\sum_{n=0}^{\infty} \frac{(\lbrace H, \cdot \rbrace)^n}{n!} $$ and apply it to a Poisson Bracket $\lbrace g_1, g_2 ...
10
votes
3answers
423 views

When do phase space functions' Poisson brackets inherit the Lie algebra structure of a symmetry?

I've seen several examples of phase space functions whose Poisson brackets (or Dirac brackets) have the same algebra as the Lie algebra of some symmetry. For example, for plain old particle motion in ...
0
votes
0answers
37 views

Angular momentum $J_z$, how do we get eigenvalue of $m\hbar$?

If we have angular orbital momentum for $z$-direction, we assume that for state $|j,m>$ that eigenvalue is $m\hbar$. Similarly for $J^2$, we assume $j(j+1)\hbar^2$ Can I get reference of ...
5
votes
0answers
55 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 $(\frac{1}{2},0)\bigoplus (\frac{1}{2},0)$ ? ...
2
votes
3answers
310 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
1answer
46 views

Broken symmetries realized nonlinearly

I'm trying to understand some concepts of spontaneous symmetry breaking, I'll write first the statement that I can't understand and later my questions. STATEMENT Consider a group $G$ and a subgroup ...
1
vote
0answers
48 views

$SU(3)$ Tensor Methods in a Tetraquark

I am trying to understand the Georgi chapter of tensor methods in $SU(3)$ representations, and I don't know how to resolve the tensor product of 2 matrices in a 2 heavy quark + 2 light antiquark ...
5
votes
1answer
71 views

Nonabelian global symmetries, $SO(N)$ charges in terms of creation and annihilation operators

Consider an $SO(N)$ symmetric theory of $N$ real scalar fields,$$\mathcal{L} = {1\over2} \partial_\mu \Phi^a \partial^\mu \Phi^a - {1\over2} m^2 \Phi^a \Phi^a - {1\over4} \lambda (\Phi^a ...
4
votes
0answers
80 views

How to obtain Thomas Precession from lie algebra of the Lorentz group? [closed]

it seems to be possible that you can get the Thomas Precession just through the commutation relations of the Lorentz group. With Thomas Precession i mean, that in general the product of two boosts is ...
4
votes
1answer
76 views

$SO(N)$ symmetric theory of $N$ real scalar fields, why do charges have correct commutation relations of generators?

Consider an $SO(N)$ symmetric theory of $N$ real scalar fields,$$\mathcal{L} = {1\over2} \partial_\mu \Phi^a \partial^\mu \Phi^a - {1\over2} m^2 \Phi^a \Phi^a - {1\over4} \lambda(\Phi^a ...
0
votes
1answer
45 views

Calculate mean number of particles of time evolution coherent state [closed]

I seem to be missing some identities. I know you need to calculate P_n = |<n|alpha_t>|^2 and mean number of particles is the infinite sum of nP_n. However I ...
1
vote
0answers
42 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 ...
1
vote
0answers
76 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 ...
5
votes
1answer
118 views

How to find symmetry transformations?

For a given Lagrangian $$ {\cal L} = - \frac{1}{4} F_{\mu \nu} F^{\mu\nu} + |D_{\mu} \phi|^2 -V (\phi) $$ with $\phi = \frac{1}{\sqrt{2}} (\phi^1 + i \phi^2)$, there are the infinitesimal local ...
16
votes
1answer
325 views

Why exactly do sometimes universal covers, and sometimes central extensions feature in the application of a symmetry group to quantum physics?

There seem to be two different things one must consider when representing a symmetry group in quantum mechanics: The universal cover: For instance, when representing the rotation group ...
0
votes
1answer
24 views

How proof the Lorentz algebra using the Poincaré algebra? [closed]

Show that $$[J_{i},J_{j}]=i\varepsilon_{ijk}J_{k},\quad [K_{i},K_{j}]=-i\varepsilon_{ijk}J_{k}, \quad [J_{i},K_{j}]=i\varepsilon_{ijk}K_{j},$$ using ...
4
votes
1answer
45 views

Spin $\frac{3}{2}$ representation in Georgi's book?

Georgi's book Lie Algebras in Particle Physics 2ed equation 3.32 lists the spin operators in the spin $\frac{3}{2}$ representation as: $$J_1=\left( \begin{array}{cccc} 0 & \sqrt{\frac{3}{2}} ...
2
votes
3answers
92 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 ...
1
vote
2answers
237 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. ...
0
votes
1answer
70 views

Pauli matrices identity with no repeating indices

I was just wondering if there is a proof of, or an example utilizing the following relation: ...
13
votes
3answers
933 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}} = ...
5
votes
1answer
635 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 ...
2
votes
2answers
140 views

Is there any $SU(\infty)$ gauge theory in quantum field theory?

The groups $U(N)$ and $SU(N)$ are the most important Lie groups in quantum field theory. The most popular are the $U(1),SU(2),SU(3)$ groups (these gauge groups form the Standard model). But is there ...
0
votes
0answers
26 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 ...
8
votes
2answers
176 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 ...
7
votes
1answer
346 views

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

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 ...
-1
votes
1answer
92 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.
5
votes
1answer
109 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
votes
1answer
179 views

What is the physical importance of the commutation relations of angular momentum?

What is the physical meaning of these commutation relations: $$[L_{z},L_{\pm}]=\pm\hbar L_{\pm}\tag{1}$$ and $$[L_{+},L_{-}]=2\hbar L_{z} ~?\tag{2}$$
6
votes
2answers
119 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, ...
0
votes
2answers
86 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 ...
1
vote
0answers
49 views

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
45 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 ...