Questions tagged [lie-algebra]

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 gauge group's Lie algebra correspond to Noether currents and conserved quantities.

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13
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1answer
531 views

What is “broken symmetry”?

For reference, I come from a mathematics background (mostly differential geometry). I have a very limited understanding of upper-level physics (I'm currently trying to fix this). This is my current ...
6
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0answers
256 views

Where in fundamental physics are Lie groups actually important (and not just Lie algebras)?

I was wondering where in fundamental physics the global structure of a Lie group actually makes a difference. Most of the time physicists are sloppy and don't distinguish groups and algebras ...
5
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1answer
230 views

How is the invariant speed of light encoded in $SL(2, \mathbb C)$?

In quantum field theory, we use the universal cover of the Lorentz group: $SL(2, \mathbb C)$, instead of $SO(3,1)$. The reason for this is, of course, that $SO(3,1)$ representations aren't able to ...
10
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2answers
469 views

Can Lie algebra $sl(2,\mathbb{C})$ be decomposed to direct sum of two $sl(2,\mathbb{R})$?

The number of generators of Lie algebra $sl(2,\mathbb{C})$ is 6, and $sl(2,\mathbb{R})$ has 3 generators, Can Lie algebra $sl(2,\mathbb{C})$ be decomposed to direct sum of two $sl(2,\mathbb{R})$? Say \...
4
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1answer
221 views

Relation between second cohomology and central extensions

In Blumenhagen's text on conformal field theory, after deriving the central extension of the Witt algebra, namely the Virasoro algebra, $$[L_m,L_n] = (m-n)L_{m+n} + \frac{c}{12}(m^3-m)\delta_{m+n,0}$...
4
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0answers
142 views

What Lie supergroup does the super-Poincare algebra generate?

Every Lie supergroup has an associated Lie superalgebra of generators (in general, some of which are bosonic and some fermionic). Which Lie supergroup(s) are generated by the Super-Poincare algebra ...
0
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1answer
307 views

Isospin of pions

Suppose that I have two pions with zero relative angular momentum. I want to find possible total isospin values. What I'm thinking is that their state should be symmetric since they are bosons. Each ...
3
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1answer
222 views

General analysis of internal symmetries in QFT

I am trying to understand as much as I can about internal symmetries in QFT, without using a Lagrangian or the canonical formalism (nor perturbation theory), but I am having a hard time to find good ...
6
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5answers
1k views

Spin operators in QM

In a text (Introduction to Quantum Mechanics by Griffiths) I am using it states without motivation that spin angular momentum has the same commutations relations as orbital angular momentum (these ...
1
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0answers
99 views

Why do we study the projective representations of SO(3) in the context of the theory of angular momentum? [duplicate]

I have been studying the group theoretic formalism of quantum mechanics and I have yet to find a satisfying explanation for the need for projective representations in the theory of angular momentum. I ...
0
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0answers
91 views

Understanding the Dirac spinor representation $(1/2,0) \bigoplus (0,1/2)$ of the Lorentz group?

I read that the generators of the Lie algebra in this representation are $$ J_k= \begin{pmatrix} \frac{1}{2}\sigma_k & 0 \\ 0 & \frac{1}{2}\sigma_k \end{pmatrix} $$ (Rotations) and $...
0
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0answers
105 views

Isospin symmetry as an $SU(2)$ symmetry

The generators for the isospin symmetry are given by $$ T_{+}=\left|\uparrow\rangle\langle\downarrow\right|, \qquad T_{-}=\left|\downarrow\rangle\langle\uparrow\right|, \qquad T_{3}=\frac{1}{2}\...
0
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1answer
97 views

Explicit calculation of $J_2$ induced rotation by $-\pi$ reversing the sign of $J_3$ operator

Let $\textbf{J}=(J_1,J_2,J_3)$ be the angular momentum QM operators in the x, y, z spacial directions respectively. These are also the generators of rotations around the x, y, z axes respectively. ...
1
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2answers
517 views

Why are 3 colors used in QCD?

The mapping of strong charge to RGB left me believing that there are only 3 conserved quantities in QCD. I recently came to the understanding that there are in fact 8 conserved quantities, as ...
2
votes
1answer
489 views

Generators of $SU(2)$ and $SU(3)$ Symmetry Groups

I've been reading about gauge field theories, and I keep coming across the generators of the $SU(3)$ and $SU(2)$ Symmetry Groups. I read that each generator corresponds to a gauge boson, but I'm ...
4
votes
1answer
337 views

Holstein-Primakoff Inconsistency?

I am trying to prove a simple relation involving the Holstein-Primakoff Transformation. Here is the transformation: $$S_+ = \sqrt{2S-b^{\dagger}b} b$$ $$S_- = b^{\dagger}\sqrt{2S-b^{\dagger}b}$$ $$S_z ...
1
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0answers
131 views

How to find generators of a transformation?

We have a 4-vector $V^\mu =(V,0,0,V)$ with sametime and $ z $ components. then we are asked to find the transformations that are not pure rotations and leave $ V $ invariant. Sone one such matrix is ...
3
votes
1answer
360 views

The adjoint representation and the gauge boson of $O(n)$

I'm learning "Gauge theory of Elementary Particle Physics Problems and Solutions" by Cheng and Li. In problem 8.4 "$O(n)$ gauge theory" on page 165, Under infinitesimal $O(n)$ representations a ...
3
votes
2answers
425 views

Clebsch-Gordan coefficents necessary and sufficent condition to be non-zero

I know that the Clebsch-Gordan coefficient, $$\left<J_1, m_1, J_2, m_2|J,M\right>,$$ is zero if the following conditions are not satisfied: $$|J_1-J_2| \le J \le J_1+J_2,$$ $$m_1+m_2=M,$$ $$|M| \...
7
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3answers
4k views

Is the adjoint representation of $SU(2)$ the same as the triplet representation?

Is the triplet representation of $SU(2)$ the same as its adjoint representation? Where the convention for the adjoint representation used is the one used in particle physics, where the structure ...
3
votes
0answers
211 views

How do gauge fields transform with an extra inhomogeneous term even though they are Lie Algebra valued in Non-Abelian gauge theories?

I am trying to work out Non-Abelian gauge theories but I couldn't get my head around the fact that gauge fields transform with an extra inhomogeneous term under the adjoint action of a group $G$, that ...
-1
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1answer
711 views

Lorentz group $SO(3,1)$ and $SU(2)\times SU(2)$

One way to classify Lorentz representations is to consider the Lie algebra isomorphic of Lorentz group to $SU(2)\times SU(2)$. So that we can classify it by two integers $(j_1,j_2)$. In this way I can ...
1
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1answer
553 views

Generalized Pauli matrix for spin larger than 1/2 [duplicate]

for Spin 1/2, we have Pauli matrix as in wiki. So what's the generalized 3-by-3 Pauli matrix for spin 1 or even larger spin? Is there a generalization method?
1
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2answers
1k views

Degree of Freedom of an $SU(n)$ Group

I've been thinking about the DOF of the $SU(n)$ group. I first consider the DOF of a unitary matrix. See if I get this right: Any unitary matrix can be written in the form of $e^{iH}$, where $H$ is a ...
4
votes
1answer
134 views

$E_{7(7)}$ symmetry in $(\mathcal{N}=8, d=4)$ Supergravity

$(\mathcal{N}=8, d=4)$ Supergravity has a hidden $E_{7(7)}$ symmtery, which acts on the scalar and vector fields of the theory. This $E_{7(7)}$, which is a 133-parameter group, can be decomposed as $...
9
votes
4answers
1k views

Matrix diagonalization in SU(2) and SO(3)

I'm currently using Nadri Jeevanjee's book on group theory for physicists to understand quantum mechanics. I came across these two pages that left me stuck: Example 4.19 $SU(2)$ and $SO(3)$ In ...
2
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1answer
1k views

Cross Products and Commutators

For non-commuting vector operators we have in general $$\mathbf{A}\times \mathbf{B}= - \mathbf{B}\times \mathbf{A} +\sum_{ijk} \epsilon_{ijk} \big[ A_i,\,B_j\big]\hat{\mathbf{k}} $$ For the case $\...
1
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0answers
213 views

Representation of Weyl-Heisenberg Lie Algebra [closed]

I'm reading the book: Coherent State in Quantum Physics, by Jean-Pierre Gazeau. In the page no. 35 of this book, the Weyl-Heisenberg Lie Algebra $\mathfrak{w}_m$ has been given that $$\mathfrak{w}_m=...
0
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0answers
158 views

Norm of Classical (Poissonian) Hamiltonian Operator

In the Poissonian formulation of classical mechanics, one finds that the time evolution of the phase space vector $\eta = (q_1,q_2\cdots q_n ; p_1, p_2\cdots p_n)^T$ can be put in terms of the ...
0
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1answer
136 views

N=4 Superconformal Algebra

The following is related to the article "Review of AdS/CFT Integrability, Chapter VI.1: Superconformal Symmetry" here The superconformal group is ${\rm PSU}(2,2|4)$ with corresponding Lie algebra $\...
1
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1answer
95 views

Continuous symmetry transformations Taylor expansion

Continuous symmetry transformations form a Lie group. The product of two such transformations is also a symmetry transformation: $T(\theta_1^a)T(\theta_2^a) = T(\theta_3^a)$ where $\theta_3^a=f^a(\...
0
votes
1answer
212 views

Difference Group Written in Capital Letters and in Lower Case Letters

I sometimes see that people write group names in capital letters and sometimes in lowercase letters. Like the 3 dimensional special orthogonal group is sometimes written as $SO(3)$ and sometimes as $...
3
votes
1answer
474 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}{...
4
votes
1answer
164 views

Lie Algebra for fermion fields

A key identity (e.g. when deriving BRST symmetry for gauge fields) is that: $$[c,d]_a =f_{abc}c_b d_c$$ where $c$ and $d$ are both Fermion Fields. How do I derive this from the lie algebra ...
2
votes
0answers
311 views

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

The projective unitary representations of a multiply-connected group $G$ 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 ...
1
vote
1answer
717 views

Generators of $SU(2)\times U(1)$

I'm currently reading about spontanous symmetry breaking. In particular about a Lagrangian that is invariant under $SU(2)\times U(1)$, in other words pretty standard QFT stuff. I know that the ...
1
vote
1answer
91 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 ...
1
vote
2answers
570 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 ...
2
votes
1answer
974 views

Virasoro Algebra vs Witt Algebra

I'm reading some notes on CFT, and there's a strange topic that I find quite confusing. We define the Witt algebra to be the generators of conformal transformations on the complex plane. $l_n = -z^{...
2
votes
1answer
224 views

Question on quotient groups and SLOCC

I have a math-physics question, which is based on an interest in stochastic local operator & classical communications (SLOCC) systems for black hole entanglement. The Cartan decomposition of a ...
2
votes
0answers
32 views

Equivariance Relation - Superconformal Hypermultiplets

I'm concerned with equation 2.24 of http://arxiv.org/abs/1601.00482 The superconformal hypermultiplets in this paper have a conic hyperkahler target manifold and the authors want to gauge some ...
4
votes
1answer
120 views

Integrating elements of a Lie group with respect to parameters of the corresponding Lie algebra

I am working with an operator $\textbf{M}$ that is represented by the Lie group SO(1,3), thus it can be written as, $$ \textbf{M} = \exp{\textbf{L}} $$ where, $$ \textbf{L} = \begin{bmatrix} 0&a&...
2
votes
1answer
327 views

Spontaneous symmetry breaking by two scalar multiplets

Consider a theory with two multiplets of real scalar fields $\phi_i$ and $\epsilon_i$, where $i$ runs from $1$ to $N$. The Lagrangian is given by: $$\mathcal L = \frac{1}{2} (\partial_{\mu} \phi_i) (\...
8
votes
2answers
226 views

Lie groups with same algebra

I had a problem when considering symmetry breaking in an SO(4) gauge theory: $\mathcal{L} = \left| D_\mu\phi \right|^2$ where $D_\mu$ is the SO(4) covariant derivative. Then assuming there is some ...
-2
votes
1answer
64 views

Doesn't modelling using Lie Groups assume spacetime is continuous?

Lie groups are used to some behaviors of quantum mechanics, as well as forming a basis for Kaluza-Klein, Yang-Mills, and String theory. But Lie groups are defined as involving a differentiable ...
0
votes
1answer
112 views

Decomposition of the adjoint representation of a spontaneously broken compact group

Let be $G$ a compact group, symmetry of the theory I am working with. $G$ is broken into one subgroup $H$. I define the generators of G as $T_A = \{T_a,T_\hat{a}\}$, where the first are the unbroken ...
7
votes
0answers
109 views

Why are generators defined oppositely in Weinberg's vs. Maggiore's QFT books?

I've been confused about the sign conventions used in Weinberg's QFT book for a long time. Here's my question: The generators $J^{\mu\nu}$ are defined in this book as $$U(1+\omega)=1+\frac{i}{2}\...
1
vote
1answer
283 views

ADHM construction and Momentum Map

while I was reading about ADHM construction I had some troubles with precise geometrical identification of the various quantities. My doubts is well manifest in these two Wikipedia pages 1) ADHM ...
1
vote
1answer
380 views

Does Operator Product Expansion form an algebra?

The operator product algebra in CFT is defined as $$\mathcal{O}_i(z,\bar{z})\mathcal{O}_j(\omega,\bar{\omega}) = \sum_{k} C^k_{ij}(z-\omega,\bar{z}-\bar{\omega})\mathcal{O}_k(\omega,\bar{\omega}).$$ ...
1
vote
1answer
157 views

Structure constant of the commutators of generators in broken symmetry

When I read a paper related to spontaneously global symmetry breaking, I cannot understand a statement: If we use the notation $T^i$ for the unbroken group generators in $H$ and $X^a$ the broken ...