The Stack Overflow podcast is back! Listen to an interview with our new CEO.

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.

Filter by
Sorted by
Tagged with
1
vote
0answers
89 views

How are symmetries defined mathematically? [duplicate]

I have started working on differential geometry very recently. I am little bit familiar with mathematical concepts such as manifolds, differential forms and associated concepts. As I was speeding ...
1
vote
0answers
109 views

Group of translations in two dimensions - A weird treatment

Again, as usual Schwinger leaves me startled as he writes, the Hermitian displacement operator in 2D is $$ G = p_1\delta x_1 +p_2 \delta x_2 $$ Now, we know clearly that this group is an Abelian ...
1
vote
0answers
138 views

Action of conformal generators on fields

I am calculating the action of the conformal generators on fields, to be more precise on wavefunctions. For now, I'm classical. I will just paste the part of my report on this to show what I am ...
1
vote
0answers
103 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 ...
1
vote
1answer
79 views

Irreducibility of $SU(N)$ rank-2 tensors [closed]

Given a rank-2 $\mathrm{SU}(N)$ tensor $X^{ab}$, it transforms as $X'^{ab} = U^a{}_c U^b{}_d X^{cd}$, where $U \in \mathrm{SU}(N)$. We can decompose it into a symmetric and an anti-symmetric part $$ X^...
0
votes
3answers
300 views

Commutator summation notation

I have the relation $ e^L M e^{-L}=\sum_{n=0}^\infty \frac 1{n!} [L,M]_{(n)}$ where $L$ and $M$ are operators. What does the subscript $n$ after the commutator bracket denote?
0
votes
2answers
59 views

How to prove $(α·σ)(β·σ) = α·β +iα×β·σ$ (where, $α$ and $β$ are 3 dimensional vectors and $σ$ represents Pauli matrices)?

I tried to evaluate the LHS first and obtained the first term of RHS easily. Then i tried to use the commutation relations of $\mathrm{SU}(2)$ group to proceed further to obtain the second term of the ...
0
votes
1answer
73 views

A doubt with Group Generators in group theory and their algebra

My doubt is this: I saw in a paper that the Lie Algebra is the relation between the commutator of the generators and the generators multiplied by structure constants. $$[S_{i},S_{j}]=c_{ij}^{k}S_{k}$...
0
votes
2answers
69 views

Incorrect proof that all gauge theories are abelian

Consider a gauge field $W_\mu = W_\mu^{a} \tau_a$ where $\tau_a$ are the generators of the Lie algebra and $W_\mu^{a}$ just numbers. Then: $$ W^2 = W_\mu W^\mu = W_\mu^a\tau_a W^{\mu b} \tau_b = W_\...
0
votes
2answers
163 views

Quark space tensor product Vs Angular momentum space tensor product

For two triplet angular momenta states, say $J=1$ and $I=1$, if we wanna look at it in the coupled basis $F=I+J$, we use the regular Angular Momentum rules: $$|I-J|\leq F\leq I+J,$$ and from that ...
0
votes
2answers
40 views

Eigenvalues of quadratic Casimirs of simple Lie groups

I want to find a generic formula for calculating eigenvalue of quadratic casimirs of Lie groups, in terms of Dynkin labels. For a simple example if we take $SU(2)$, with $[R]$ indicating the highest ...
0
votes
1answer
251 views

Why is the Galilean group not commutative?

As I understand it, the Galilean transformation is a matrix $$ \left[ {\begin{array}{ccccc} R_{11} & R_{12} & R_{13} & v_x & a_x\\ R_{21} & R_{22} & R_{23} & v_y ...
0
votes
2answers
371 views

${\bf su}(2)$ generators in 1, 2 and 3 dimensional matrix representations

The ${\bf su}(2)$ Lie algebra in a representation $\bf R$ is defined by $$[T^{a}_{\bf R},T^{b}_{\bf R}]=i\epsilon^{abc}T^{c}_{\bf R},$$ where $T^{a}_{\bf R}$ are the $3$ generators of the algebra. ...
0
votes
1answer
83 views

Show that when angular momentum $L_x$ and $L_y$ commute with operator $G$, then $L_z$ also commutes with $G$

I want to prove that if Angular momentum $L_x$ and $L_y$ commute with an operator $G$, angular momentum $L_z$ also commutes with $G$. if $[L_x , G] = [L_y, G] = 0$ then $[L_z , G] = 0$ I know that $...
0
votes
1answer
92 views

$\mathrm{SU}(2)$ as a representation of the rotation group

I have read in a book that the group $\mathrm{SU}(2)$ is one of the irreducible representations of the rotation group. The book begin saying that the rotation group has 3 generators $J_{1}, J_{2}$ and ...
0
votes
1answer
172 views

Does the sedenion algebra offer a grand unification theory?

Stephane Bronoff in The Standard Model of Particle Physics from Sedenions claims that studying the left-multiplication map of unit doubly-pure sedenions solves several mysteries of the standard model. ...
0
votes
1answer
36 views

Lie Subgroups of $SL(2,\mathbb{R})$

I'm wondering about the Lie subgroups of $SL(2,\mathbb{R})$. It's Lie algebra is the algebra of real traceless matrices and has basis elements $$L_0 = \left( \begin{matrix} -1 & 0 \\ 0 & 1 \...
0
votes
1answer
67 views

Generators of conformal transformations change of basis

I recently started going through Introduction to Conformal Field Theory by Blumenhagen and Plauschinn ( springer link ). On page 11, they glue together the generators of conformal transformations as ...
0
votes
1answer
73 views

Reference request for Lie algebras

My future adviser just published a beautiful paper, https://arxiv.org/abs/1904.08304, and I am looking for some references/textbooks to look into the following concepts: Lie algebra (central) ...
0
votes
1answer
50 views

Generator of 3D rotations in $\mathbb{C}^2 \otimes \mathbb{C}^2$

Let us consider a system of two spinors. The 3D rotation operator around the $\vec{n}$ axis in $\mathbb{C}^2$ is clearly $R(\theta) = \exp(i \frac{\theta}{2}\vec{n}\cdot\vec{\sigma})$. If I wish to ...
0
votes
1answer
89 views

Why can we write lagrangian for gauge theory without the traces?

I understand that trace is needed in order to preserve gauge invariance of the lagrangian equation by using the cycling property. But I fail to see why the following equation holds true: $$-\frac{1}{2}...
0
votes
1answer
71 views

How to prove commutation relation between charge and current in current algebra?

I am reading Gauge Theory of Elementary Particle Physics by Tapei Cheng and Lingfong Li. Proceeding equation 5.54, there is a statement which says Then from Lorentz covariance, we can include the ...
0
votes
1answer
63 views

Lie algebra valued potential vector [closed]

Maybe it is a simple question but I have some difficulty to understand the explicit matrix form of this usual relation: $$A_\mu=A^a_\mu \tau_a$$ where $A^a_\mu $ is the Lie algebra valued potential ...
0
votes
1answer
55 views

Lie algebra definition - Maggior'e book

I'm reading Maggiore's book about QFT, and I'm having a trouble understanding the notation in the part about Lie algebras (Section 2.1): The group generators are defined as $T^a_R=-i\frac{\partial ...
0
votes
1answer
58 views

Evaluating $n \otimes_A n^*$ in $SU(n)$

In "Quantum Field Theory in a Nutshell" pg424 the author (Zee) writes: $$(n\oplus n^*)\otimes_A(n \oplus n^*)\quad\cong\quad(n^2-1)\oplus 1 \oplus n(n-1)/2 \oplus ((n(n-1))/2)^*$$ From what I ...
0
votes
1answer
794 views

On the generators of the Lorentz group

We already know that the generators of Lorentz group are three boosts and three rotations. We have the relation $$ \left[ M_{\mu \nu},M_{\sigma \lambda} \right] = ig_{\mu \sigma} M_{\nu\lambda} - ...
0
votes
1answer
325 views

Why multiply the infinitesimal generator for a rotation $R$ by $i$ when constructing $U(R)$?

I'm sure this is a silly question, but I can't figure out the answer. Current I'm reading chapter 4 in Weinberg's Lectures on Quantum mechanics. Earlier in the book, he asserts that unitary operators ...
0
votes
1answer
340 views

Lorentz Group Representations

Consider for example the (trivial) spin-1/2 representation of the $SU(2)$ group. This representation has dimension two, which is clear from a quantum mechanical perspective since we need to specify ...
0
votes
1answer
175 views

Justification of gauge field transformations

I am trying to understand the gauge transformation of gauge fields in a gauge quantum field theory. As an example I considered this wikipedia article, section 'An example: Scalar $O(n)$ gauge theory'....
0
votes
1answer
238 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 $...
0
votes
1answer
278 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: $\sigma^i_{\alpha\beta}\sigma^j_{\gamma\delta}+\sigma^j_{\alpha\beta}\sigma^i_{\gamma\delta}+\delta^{ij}(\...
0
votes
4answers
176 views

Conceptualization and modelling of spin

I'm trying to get a decent understanding of the Bell inequality, and so am trying to understand spin both conceptually and mathematically. When I picture spin, I imagine a sphere rotating about its ...
0
votes
1answer
476 views

Eigenvalues of Angular Momentum in Quantum Mechanics

The eigenvalue equation of the $L^2$ operator is given by $$L^2f_l^m = \hbar ^2l(l+1)f_l^m$$ Side: So a determinate state for some observable $Q$ is a state where every measurement of $Q$ returns ...
0
votes
1answer
236 views

Derivation of the irreducible representations of SO(3)

Is there a way to derive the representations of $SO(3)$ without the usual method with the ladder operators which also gives the ones of $SU(2)$? The usual way to do these calculations is to start ...
0
votes
2answers
589 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 ...
0
votes
1answer
23 views

Center of $SU(3)$

I assumed a 3x3 matrix of the form $$A= \begin{pmatrix} a & b & c\\ d & e & f\\ k & l & m \end{pmatrix}$$ Then, since we know that the center is always an Abelian invariant ...
0
votes
1answer
50 views

Determining the manifold picture of a Lie group — and thus determining global properties

1) How do we determine a Lie group's global properties when the manifold that it represents is not immediately obvious? Allow me to give the definitions I am working with. A Lie group G is a ...
0
votes
1answer
46 views

A simple calculation in Peskin's and Schroeder's QFT book on page 608 chapter 18

I am trying to calculate the term: $$(t^a)_{ij} (t^a)_{kl}$$ In the book it's written that it equals to $$A\delta_{il}\delta_{kj}+B\delta_{ij}\delta_{kl}$$ and from using equation (18.40) $$tr[t^a](...
0
votes
1answer
25 views

Anticommutativity of an anticommutator of supercharges

In this paper, equation 38 gives the ${\cal N}=2$ Super-Poincare (extended with the central extension $\mathcal{Z}$). The anticommutation relation of the two different supercharges is given as: $$\{Q^...
0
votes
2answers
59 views

Lie algebra vs. position and momentum commutators

Most theoretical texts on high energy physics make statements like below: $$[A_i , A_j] = i C^k_{i,j} A_k $$ (I suppose $\hbar$ may or may not be needed) and of course they describe this as being ...
0
votes
1answer
96 views

Generators in Field Theory and Derivatives

Let's consider a representation of the multiplicative group $(0,\infty)$ on Minkowski space $\mathbb{R}^4$ by dilations. \begin{align} \rho:(0,\infty)&\rightarrow\text{GL}(\mathbb{R}^4)&\\ a ...
0
votes
1answer
121 views

Racah's Derivation of CG coefficients

I have what is hopefully a quick question. In Racah's 1942 paper Theory of Complex Spectra II, the author utilizes the action of $J_+$ on the Clebsch Gordan expression: $$ \begin{align*} &|j m\...
0
votes
1answer
298 views

Why we need $SU(2)$ symmetry? When we use it? [closed]

I am trying to learn Quantum mechanics and I am familiar with Pauli matrice but not with group theory. I want to understand SU2 symmetry in common language. When we talk about Pauli matrix x we simply ...
0
votes
1answer
139 views

Generators of conformal transformations

I'm currently reading about the Witt algebra, and I'm trying to understand in what sense the Witt algebra basis $L_n = -z^{n+1}\partial _z$ generates conformal maps in dimension $2$. From what I've ...
0
votes
1answer
58 views

Proposal of the Virasoro modes and algebra

Hi I am wondering what the first published paper on Virasoro modes was? And what about Virasoro algebra?
0
votes
1answer
169 views

How can the algebra of infinitesimal conformal transformations be infinite dimensional (in 2D)?

In Blumenhagen's book "Introduction to Conformal Field Theory", I found the statement The algebra of infinitesimal conformal transformations in an Euclidean 2-dimensional space is infinite ...
0
votes
1answer
41 views

A question about a commutation relation

Here $a$, $b$,$c$, $d$ $u$, $v$ range from 0 to 4 and the metric $g^{ab}=\text{diag}(-1,1,1,1)$. 16 $4 \times 4$ matrices $M^{ab}$ are defined as follows: \begin{equation*} (M^{ab})_{uv} = -i(\delta^...
0
votes
1answer
62 views

Problem calculating commutator with Casimir

I am trying to verify that the Casimir of the Lie group $SO(3)$ is actually $N^2=N_iN_i$, but I have problems, with indices surely, and I was wondering if someone could help me figuring out how to ...
0
votes
1answer
378 views

Gauge covariant derivative

I have seen distinct definitions of gauge covariant derivative (in Yang-Mills theory) $$ D_\mu \phi = (\partial_\mu + igA_\mu) \phi $$ vs $$ D_\mu \phi = \partial_\mu \phi + ig[A_\mu,\phi] .$$ I ...
0
votes
1answer
137 views

Representation of the Poincare algebra on the space of smooth functions

The following representation describes how a field $\varphi$ transforms under the Poincaré group $\mathcal{P}$. $$\mathsf{S} : \left\lbrace \begin{aligned} \mathcal{P} \times C^{\infty}(\mathcal{M})...