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
109 views

Relation between Lie algebra of conformal Killing vector fields and conformal algebra

I'm new to conformal transformations and I have a question. Following the book of Barrett O'Neill "Semi-Riemannian geometry with applications to relativity", there is a Lie anti-isomorphism between ...
5
votes
1answer
673 views

$\mathfrak{su}(3)$ structure constants

The $\mathfrak{su}(3)$ structure constants $f^{abc}$ are defined by $$[T^a,T^b] = i f^{abc} T^c,$$ with $T^a$ being the generators of the group $\mathrm{SU}(3)$. They are usually written out in a very ...
1
vote
1answer
72 views

How to interpret the extra indices of the generators of the Lorentz algebra in Peskin & Schroeder?

In Peskin & Schroeder p.39 they introduce the 4x4-matrices $$\left(\mathcal{J}^{\mu\nu}\right)_{\alpha\beta} = i \left(\delta^{\mu}_{\;\alpha} \delta^{\nu}_{\;\beta} - \delta^{\mu}_{\;\beta}\...
0
votes
1answer
66 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 ...
5
votes
1answer
216 views

Gauge transformations and Covariant derivatives commute

I would like to understand the statement "Gauge transformations and Covariant derivatives commute on fields on which the algebra is closed off-shell" which was taken from section 11.2.1 (page 223)...
3
votes
2answers
125 views

Global conformal group in 2D Euclidean space

This is a rather naive question, but I was just wondering. I know that the local conformal algebra of 2d Euclidean space is the direct sum \begin{equation} \cal{L}_0\oplus\overline{\cal{L}_0}, \end{...
0
votes
1answer
68 views

Construct components of tensor operator [closed]

I'm reading Georgi's textbook on Lie Algebras and have been struggling with this question for quite awhile. The entirety of Chapter 4 (Tensor Operators) has been much more difficult than anything I've ...
2
votes
1answer
96 views

Factorization of exponential of broken generators in parametrization of scalar multiplet in non-abelian SSB

Describing abelian symmetry breaking in his book on gauge theories, after favouring a vacuum (whose expectation value is $v$) from the symmetric continuum, Quigg parametrize the complex scalar as $$ ...
1
vote
2answers
137 views

How to prove the translation generator commutes with the spinors in SUSY algebra?

I was reading Modern Supersymmetry by John Terning, the book starts with SUSY algebra and says $$ \left[ P_{\mu} , Q_{\alpha} \right] = \left[ P_{\mu} , Q_{\alpha}^{\dagger} \right] = 0 $$ I am ...
2
votes
1answer
477 views

Trace of 4 Gell-Mann matrices

Does any one know what would be $tr(t^a t^b t^c t^d)$, where $t^a$ etc are Gell-Mann matrices? This came about when analyzing the color factor for the compton effect for QCD. So, must be pretty ...
2
votes
1answer
210 views

Unitary Representations in Conformal Field Theory

So I am currently studying conformal field theory from the perspective of the representation theory of Lie algebras. I am trying to understand exactly why we care about unitarizable Verma modules. For ...
0
votes
1answer
91 views

Link between dynamical algebra and symmetry group

I was wondering if there is a known link between dynamical algebra and symmetry group. In particular: Do all Hamiltonians belonging to certain dynamical algebra share the same symmetry group? Do ...
0
votes
1answer
129 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 ...
2
votes
0answers
39 views

Noether's theorem for fields and infinitesimal transformations [duplicate]

I'm starting to learn QFT by myself using many references, mostly (QFT for the Gifted Amateur, and Tong's lectures) and both present a proof of Noether's theorem using infinitesimal tranformations, ...
3
votes
2answers
189 views

Updating link variables in lattice $SU(N)$ gauge theory

I'm currently writing a basic program in python to simulate a 1 + 1 dimensional yang mills gauge theory with symmetry group $SU(2)$. On the lattice you work with link variables, which are $SU(N)$ ...
0
votes
1answer
67 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}$...
1
vote
1answer
99 views

False formula for Lie derivative

I have found, from this url, the link between Lie derivative and covariant derivative. It is said at the end of question that Lie derivative of of a vector field $\xi^{\alpha}$ with respect to a ...
-1
votes
2answers
209 views

Pauli matrices and Wikipedia

Wikipedia claims Pauli Matrices with an $i$: $i \sigma_1, i \sigma_2, i \sigma_3$ form a basis of $\mathfrak{su}(2)$. But what about the following relation?: $$[\frac{1}{2} \sigma_i, \frac{1}{2} \...
1
vote
1answer
84 views

How to prove a set of matrices form a representation of Lie algebra?

When reading Paul Langacker's The Standard Model and Beyond, I am quite confused on equation 3.29, which says with a set of fields $\Phi _a$, where $a$ goes from 1 to $n$, are chosen to be transformed ...
2
votes
1answer
26 views

For what angles (and why) does the equation for finite rotation fail to work?

I am learning rotations and group theory/representations and my lecturer's note mentioned that: "The group is considered connected, but not simply connected [...] As a result, the formula for a ...
1
vote
1answer
69 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^...
3
votes
2answers
279 views

A useful identity for Gell-Mann $su(3)$ matrices?

We have the following beautiful result for Pauli $su(2)$ matrices $$(\vec{\sigma}\cdot\vec{a})(\vec{\sigma}\cdot\vec{b}) = \mathbb{I} ~\vec{a}\cdot\vec{b} + i (\vec{a} \times \vec{b}) \cdot \vec{\...
5
votes
0answers
73 views

Modified Lie bracket

In a paper by Barnich, they use a different definition of the Lie bracket for vector fields at null infinity. Can somebody please give me the intuition behind using this Lie bracket in Equation 4.12: ...
0
votes
1answer
159 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. ...
1
vote
0answers
61 views

Time dependent Hamiltonian operator and $SU(1,1)$ generator method

In this screenshot of a paper I am reading, I have the following question: 1.What is a $SU(1,1)$ group and how do we find its generators? 2.From the expression for the Hamiltonian $\hat{H}$, how do ...
6
votes
1answer
368 views

Conformal field theory does not have… conformal symmetry?

This post is about 1+1d. It is often said that conformal field theory has an infinite-dimensional symmetry generated by the Virasoro algebra: $$ [L_n,L_m] = (n-m) L_{n+m} + \frac{c}{12} n (n^2-1) \...
1
vote
2answers
156 views

Going from full non-Abelian gauge transformation to its infinitesimal version in component notation

Let $A_\mu^a(x)$ be a non-Abelian gauge field, with $\mathrm{SU}(N)$ generators $T_a$. We can write the field as a Lie-algebra-valued object $$ \mathbf{A}_\mu \equiv A_\mu^a T_a.$$ The full local ...
7
votes
0answers
170 views

Complexifying Lie algebras confusion

I have been studying a course on Lie algebras in particle physics and I could never understand how complexifying helps us understand the original Lie algebra. For example, consider $\mathfrak{su}(2)$...
3
votes
0answers
71 views

Why is $\mathfrak{so}(3,1)_{\mathbb{C}}^\uparrow \cong \mathfrak{su}_\mathbb{C}(2) \oplus \mathfrak{su}_\mathbb{C}(2)$ [duplicate]

I am studying the orthochronous Lorentz algebra $\mathfrak{so}(3,1)^\uparrow $ and it reads $$ [X_i,X_j]=i \varepsilon_{ijk} X_k $$ $$ [X_i,Y_j]=i \varepsilon_{ijk} Y_k $$ $$ [Y_i,Y_j]=-i\varepsilon_{...
1
vote
0answers
47 views

Integrating over a symmetry vector field by exponential map

Take some constant-of-motion, $H$, and the Poisson bracket, $\{\cdot, \cdot\}$. Then, we recover the symmetry vector field of $H$ by \begin{equation} S_H = \{\cdot, H\} \end{equation} so take for ...
0
votes
1answer
56 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?
1
vote
1answer
54 views

Basis-free, non-power series definition of the exponential of linear operator?

Given an arbitrary linear operator $A$ (be it real, complex or whatever), how can the exponential of it ($e^A$) be defined naturally, without stuff like power series? The exponential for regular ...
1
vote
1answer
249 views

Angular momentum and Noether's theorem

Studying the lagrangian formulation of Noether's theorem and came upon how the invariance under rotations gives conservation of angular momentum. Whilst setting up the problem the notes state that if ...
1
vote
1answer
44 views

Decomposition of $E_6$ into $SO(p,q)$

I've seen the following decomposition of the fundamental representation 27 of $E_6$ into $$E_6 \rightarrow SU(2) \times SO(5,2) \times SO(1,1)$$ $$27 \rightarrow (1,1)(-4) + (1,7)(-2) + (2,8)(+1) + ...
3
votes
1answer
125 views

Conformal algebra

I am reading dr. Joshua Qualls lectures on conformal field theory. https://arxiv.org/abs/1511.04074 In section 2.4 Conformal group he defined the generators $$ \begin{aligned} J_{\mu,\nu}&=L_{\...
0
votes
1answer
62 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
154 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 ...
2
votes
1answer
369 views

Wigner Rotation

I'm trying to show that the composition of two Lorentz boosts produces a boost and a rotation using the generators from the Lorentz Group. If $\vec{K}$ denotes the Lorentz Boost generators and $\vec{S}...
4
votes
2answers
132 views

Construct an SO(3) rotation inside the two SU(2) fundamental rotations

We know that two SU(2) fundamentals have multiplication decompositions, such that $$ 2 \otimes 2= 1 \oplus 3.$$ In particular, the 3 has a vector representation of SO(3). While the 1 is the trivial ...
8
votes
1answer
398 views

Is there a generalised Wigner-Eckart theorem?

The Wigner-Eckart theorem gives you the matrix element of a tensor transforming according to a representation of $\mathfrak{su}(2)$, when sandwiched between vectors transforming according to another (...
3
votes
0answers
40 views

Is there any feature which distinguishes the Hamiltonian in the Poincare algebra?

The Poincare algebra is defined as \begin{align*} i[J^{\mu\nu},J^{\rho\sigma}]&=\eta^{\nu\rho}J^{\mu\sigma}-\eta^{\mu\rho}J^{\nu\sigma}+\eta^{\sigma\nu}J^{\rho\mu}-\eta^{\sigma\mu}J^{\rho\nu}\\ ...
0
votes
0answers
106 views

Eigenvalues of Cartan subalgebra and Casimir Operator

As I understand it, given a compact, semi-simple lie-algbera $\mathfrak{g}$, there exists a basis for $\mathfrak{g}$ such that the the components of the killing form $\kappa$ are $\kappa_{\alpha\...
2
votes
0answers
26 views

Finding the generators of the adjoint representation [duplicate]

Let $G$ be a lie group and $\mathfrak{g}$ it's associated lie algebra.I can quite easily show that if there is a Lie group representation $\rho_{G}$ from $G$ into $L(G)$: $$ \rho_{G}: G \rightarrow \...
1
vote
0answers
42 views

Representations of a symmetry group: what is actually being represented? [duplicate]

For definiteness, consider the group $SO(3)$. There is a Lie algebra $so(3)$ given by $$ [T_a, T_b] = if_{abc}T_c $$ The generators of this algebra can be exponentiated to form the elements of $SO(3)...
0
votes
0answers
20 views

Gaining intuition over central extension of algebras [duplicate]

Conformal algebra in 2 dimensions admits a central extension and while the procedure itself is explained in many textbooks quite well, I haven't found a source in which it is reasoned as to why we ...
3
votes
0answers
80 views

What is the physical meaning of Lie congruence classes?

Any weight $\lambda$ characterising a representation of $\mathfrak{su}(N)$ is an element of one of the $N$ congruence classes defined by (ref.1) $$ \lambda_1+2\lambda_2+\cdots+(N-1)\lambda_{N-1}\quad\...
4
votes
1answer
259 views

Composition of Lorentz transformations using generators and the Wigner rotation

I solved this problem by painful calculations of Lorentz matrices. However, I heard that there is a much easier solution using the generators of boosts and rotations and their commutation relations, ...
0
votes
1answer
40 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^...
1
vote
0answers
98 views

Explicit Quadratic Casimir for $sp(2N)$

We know that $so(3)$ has the explicit quadratic Casimir $$L^2=\sum L_{i}^2.$$ Are there analogs to this in other simple lie algebras? I know that for a simple lie algebra I can always use the ...
11
votes
3answers
1k views

Confusion about rotations of quantum states: $SO(3)$ versus $SU(2)$

I'm trying to understand the relationship between rotations in "real space" and in quantum state space. Let me explain with this example: Suppose I have a spin-1/2 particle, lets say an electron, ...