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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27 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 ...
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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
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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 $4 \otimes 3 = 6 \oplus 4 \oplus 2$?
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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 ...
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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 ...
1
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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
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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 ...
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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
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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 $(\frac{1}{2},0)\bigoplus (\frac{1}{2},0)$ ? ...
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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} + ...
3
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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 ...
5
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1answer
70 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 ...
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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 ...
4
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1answer
72 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 ...
4
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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 ...
0
votes
1answer
44 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
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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 ...
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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
115 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 ...
0
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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
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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
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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 ...
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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: ...
16
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1answer
321 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 ...
2
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1answer
64 views

Why is the Mixed Faraday Tensor a matrix in the algebra so(1,3)?

The mixed Faraday tensor $F^\mu{}_\nu$ explicitly in natural units is: ...
2
votes
2answers
139 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 ...
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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 ...
4
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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 ...
3
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5answers
342 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 ...
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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.
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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 ...
2
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0answers
92 views

Quadratic Casimir Operator of $SO(5)$ [closed]

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 ...
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0answers
48 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 ...
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0answers
55 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 ...
3
votes
2answers
281 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, $$ I_1\dot{\omega}_1 + (I_3 - I_2)\omega_2\omega_3 = M_1, $$ $$ I_2\dot{\omega}_2 + (I_1 - I_3)\omega_3\omega_1 = M_2, $$ ...
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0answers
24 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) ...
2
votes
1answer
112 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 ...
2
votes
1answer
59 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
50 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 ...
2
votes
1answer
36 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
71 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 ...
2
votes
2answers
391 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 ...
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1answer
95 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
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0answers
60 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 ...
5
votes
1answer
243 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 ...
7
votes
4answers
324 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
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1answer
73 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 ...
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0answers
33 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
128 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
2answers
200 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 ...