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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 ...

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44 views

Why is the $\mathrm{SU}(2)$ algebra taken over the complex field?

The lie algebra of $\mathrm{SU}(n)$ is composed by the $n \times n$ antihermitian matrix with null trace over the real field, but physicists prefer to use hermitian matrix. Does this mean taking the ...
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1answer
64 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 ...
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137 views

Notation question for exponential form of Lorentz transformations

So my question is, when we write the Lorentz transformation in the following form \begin{equation} \Lambda = e^{- \frac{\mathrm{i}}{2} \omega ^{\rho \sigma} M_{\rho \sigma}} \end{equation} Since the ...
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0answers
26 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 ...
4
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1answer
75 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 ...
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46 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}\...
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17 views

How to prove commutation relation between temporal and spatial current in current algebra?

I am reading Gauge Theory of Elementary Particle Physics by Tapei Cheng and Lingfong Li. Equation 5.56 says $$ \left[ J_0^a \left( \vec{x} , t \right) , J_i^b \left( \vec{y} , t \right) \right] = \...
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1answer
33 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 ...
4
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1answer
132 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
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2answers
65 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{...
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1answer
50 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 ...
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35 views

Meaning of a ladder operator being a scalar or a vector

I was reading an article [1] where the author is deducing the ladder operators for the three-dimensional isotropic harmonic oscillator. He does it two times one considering the said operators to be a ...
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2answers
70 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
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1answer
171 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
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1answer
92 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 ...
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1answer
70 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 ...
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0answers
44 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 ...
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0answers
31 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
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2answers
121 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 $SU(N)$ ...
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1answer
48 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}$...
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1answer
83 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 ...
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2answers
104 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} \...
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1answer
69 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
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1answer
20 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 ...
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0answers
25 views

Irreducibility of $SU(N)$ rank-2 tensors

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^...
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2answers
158 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{\...
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0answers
40 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: ...
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1answer
81 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. ...
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12 views

Compactification from non-orthogonal symmetries

For many theories like string theory, one extends the dimensions to a higher number D and then requires the space-time rotational symmetry group to be O(D-1,1). Then one compactifies the excess ...
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0answers
38 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 ...
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1answer
257 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) \...
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2answers
45 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 ...
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0answers
96 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 $\mathfrak{su}(2)$...
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0answers
54 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_{...
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0answers
36 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 ...
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1answer
45 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?
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1answer
52 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 ...
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1answer
61 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
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1answer
38 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) + ...
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1answer
49 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_{\...
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1answer
37 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 ...
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1answer
68 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
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1answer
178 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}...
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2answers
100 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 ...
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1answer
240 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 (...
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0answers
37 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}\\ ...
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0answers
48 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\...
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26 views

Why Casimir in super-Poincare is zero in massless case?

Trying to find the Casimir's of super Poincaré algebra, Calculating in the massless case: $P_{a} = (E, 0, 0, E).$ The Casimir: $C_{ab} = B{a}P{_b} - B_{b}P_{a} = Square[BP] - Square[B]Square[P]$ is ...
2
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0answers
23 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 \...
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40 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)...