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

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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\...
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1answer
2k views

Commutator of Lorentz boost generators : visual interpretation

I have always struggled to visualize the correctness of the commutation relation for the generators of the boost in the Lorentz group. We have $$[K_i,K_j] = i \epsilon_{ijk} L_k$$ I fail to picture ...
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3answers
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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 ...
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2answers
274 views

Gauge covariant derivative on form

Let $e$ be a one-form gauge field that belongs to the adjoint representation of the gauge group, that is SO(1,2). It is defined as \begin{equation} e = e_{\alpha}^{A}T_Adx^{\alpha}. \end{equation} ...
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1answer
61 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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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 ...
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4answers
320 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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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 ...
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1answer
2k views

The conjugate representation in $\mathfrak{su}(2)$

Cheng & Li gives the following problem: Let $\psi_1$ and $\psi_2$ be the bases for the spin-1/2 representation of $\mathfrak{su}(2)$ and that for the diagonal operator $T_3$, \begin{align} ...
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1answer
758 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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0answers
120 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 ...
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1answer
80 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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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 ...
5
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1answer
228 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)...
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4answers
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What do the Pauli matrices mean?

All the introductions I've found to Pauli matrices so far simply state them and then start using them. Accompanying descriptions of their meaning seem frustratingly incomplete; I, at least, can't ...
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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 ...
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2answers
204 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
227 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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2answers
146 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 ...
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1answer
503 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 ...
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1answer
98 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 ...
2
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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, ...
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1answer
2k views

Noether's Theorem: Lie groups vs. Lie algebras; finite vs. infinitesimal symmetries

I've had a brief look through similar threads on this topic to see if my question has already been answered, but I didn't find quite what I was looking for, perhaps it is because I'm finding it hard ...
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1answer
1k views

Index of representation of $SU(N)$ fundamental and adjoint

Im getting crazy trying to derive this simple expression. Say $f^{abc}$ are structure constants of a Lie algebra of $SU(N)$ with $[T^a, T^b]=i f^{abc}T^c$. Then chosing normalization such that $$\...
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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}$...
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1answer
104 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
218 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} \...
3
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2answers
293 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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1answer
85 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 ...
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1answer
28 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
78 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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2answers
371 views

$su(1,1) \cong su(2)$?

The three generators of $su(2)$ satisfy the commutation relations $$ [J_0 , J_\pm] = J_\pm , \quad [J_+, J_- ] = +2J_0 .$$ The three generators of $su(1,1)$ satisfy the commutation relations $$ [...
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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. ...
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0answers
65 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 ...
10
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1answer
369 views

Reference recommendation for Projective representation, group cohomology, Schur's multiplier and central extension

Recently I read the chapter 2 of Weinberg's QFT vol1. I learned that in QM we need to study the projective representation of symmetry group instead of representation. It says that a Lie group can have ...
7
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1answer
389 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) \...
3
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1answer
556 views

The states of the adjoint representation correspond to the generators

From section 2.4 of von Steinkirk's Introduction to Group Theory for Physicists [PDF] Defining a set of matrices $T_a$ as $$[T_a]_{bc} \equiv -if_{abc}$$ it is possible to recover (2.1.2): $$[...
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2answers
173 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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1answer
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Commutation relation of the generators of Lorentz Group

$$\left[J_i,J_j \right]=i\epsilon_{ijk}J_k$$ $$\left[J_i,M_j \right]=i\epsilon_{ijk}M_k$$ $$\left[M_i,M_j \right]=-i\epsilon_{ijk}J_k$$ where $J_i$ is the generator of rotation of Lorentz group, $M_i$ ...
3
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0answers
73 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_{...
52
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1answer
3k 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 $\mathrm{SO}(3)...
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0answers
48 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
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?
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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 ...
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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) + ...
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1answer
294 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 ...
3
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1answer
145 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
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
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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 ...
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1answer
128 views

Why is it that every locally conformal transformation can be extended to a global conformal transformation for $D>2$?

In $D=2$, we can have locally analytic transformations that cannot be globally well-defined. However, for CFTs in $D>2$, we have only the global group. Why is that? Also, is it a statement that ...