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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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3answers
119 views

Wrong sign in Conformal Casimir

The quadratic conformal Casimir in $d$-dimensional Euclidean space is given by \begin{equation} C = \frac{1}{2}L_{\mu \nu}L^{\mu \nu} - D^2 -\frac{1}{2}\left(P^\mu K_\mu + K^\mu P_\mu \right) \end{...
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1answer
107 views

On the homomorphism of the Lorentz algebra representation (1/2, 0)

I was reading this answer and I don't quite understand how the $\rho$ homomorphism works. The generators of the two copies of $\mathfrak{su}(2)$ in $\mathfrak{su}(2)\oplus\mathfrak{su}(2)$ are given ...
-2
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1answer
233 views

Show that the fundamental representation is a representation

I want to see that the fundamental representation is a representation. Suppose the structure constants $f^{abc}$ are given. We can assume there is at least one non-zero structure constant, otherwise ...
0
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0answers
112 views

Anti-commutator version of Zassenhaus formula

The Zassenhaus formula goes like $$ e^{t(X+Y)}= e^{tX}~ e^{tY} ~e^{-\frac{t^2}{2} [X,Y]} ~ e^{\frac{t^3}{6}(2[Y,[X,Y]]+ [X,[X,Y]] )} ~ e^{\frac{-t^4}{24}([[[X,Y],X],X] + 3[[[X,Y],X],Y] + 3[[[X,Y],Y],...
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1answer
117 views

How to derive an $E_8$ algebra?

What is the simplest way to derive an $E_8$ algebra? I am not interested in $E_8$ itself but what would compel one to think about it. I know for example why you would want to think about $SU(2)$ and ...
-2
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1answer
294 views

How to construct a supersymmetry algebra?

Starting with the general notion of supersymmetry: $$Q| boson \rangle = | fermion \rangle \\ Q| fermion \rangle = | boson \rangle$$ I want to construct the superalgebra relations. After applying $...
0
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1answer
104 views

What are the matrix representations of super Poincaré algebras?

I have seen that Lie superalgebras are classified by some algebras like $\mathfrak{osp}(m|2n)$, but I don't know how to fit super Poincar\'e algebras into this. Especially what are the fundamental ...
0
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1answer
149 views

Quantum spin, tensor product: a long time relationship [duplicate]

Anyone who has studied quantum mechanics know the following relation: $ 2 \otimes 2 = 3 \oplus 1 $ But how did a man woke up and said "Hell yeah, I'll use tensor product of two spin $1/2$ to simulate ...
2
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1answer
87 views

Infinitesimal generator flux of Lorentz trasformations in spacetime

I'm considering the following matrixs which I know that they form a flux of Lorentz trasformation in spacetime. I want to know how to calculate the infinitesimal generator of this flux. ...
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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 ...
5
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2answers
249 views

Lie group compactness from generators

Let us consider the Poincaré algebra, characterized by the following commutators: \begin{align} [H,P_i]&=0\\ [H,K_i]&=P_i\\ [P_i,P_j]&=0\\ [K_i,K_j]&=-\epsilon_{ijk}J_k\\ [P_i,K_j]&...
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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} - ...
3
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0answers
69 views

What are the relations between a ladder algebra and an $SU(2)$ algebra?

I am studying elementary models of the Quantum Hall Effect. I don't have a strong background in Lie algebras so I was hoping someone could elaborate on the following observation: For a free particle ...
1
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0answers
39 views

Representation of the Lorentz group from representation of its Lie algebra [duplicate]

In many books on particle physics, it is first shown that the Lie algebra of the Lorentz group is isomorphic to $$ su(2) \oplus su(2) , $$ then by this fact, it is implicitly assumed that for each ...
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0answers
134 views

Representation Theory in a Nutshell for Physicists? [duplicate]

Are there good references and introduction of Representation Theory as "Representation Theory in a Nutshell for Physicists"? For example, we hope that the book/ref contains the "introduction to ...
3
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0answers
93 views

How do WZW coset models contain perturbations?

I've been studying the coset construction. As far as I understand it, the Sugawara energy momentum tensor is a way of embedding the virasoro algebra inside the Lie algebra of your original WZW model. ...
1
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1answer
272 views

Symmetry of the Pauli group and its representations

The three traceless Pauli matrices $\sigma_x,\sigma_y,\sigma_z$ are arbitrary in the sense that any three operators with the appropriate commutation relations can be represented with those matrices. ...
7
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1answer
384 views

From irreducible representations of the Lorentz algebra to irreducible representations of the Lorentz group

My lecture notes state that we need to classify all finite-dimensional irreducible representations of the proper, orthochronous Lorentz group in order to formulate a QFT for particles with non-zero ...
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3answers
332 views

Finite lorentz transform for 4-vectors in terms of the generators

One or two sets of notes (one of them by Timo Weigand) on QFT that I have come across state explicitly that a finite lorentz transformation for 4-vectors can be written in terms of the generators $J^{\...
-1
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2answers
48 views

Failure or incomplete demostration about egeinvalues of $J^2$ and $J_z$ using lowering and raising operators

In many books show how find eigenvalues of $J^2$ and $J_z$ \begin{align} \hat{J}^2 |\ell,m\rangle & = \hbar^2 \ell(\ell+1) |\ell,m\rangle , \\ \hat{J_z} |\ell,m\rangle & = \hbar m |\ell,m \...
1
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1answer
77 views

Ambiguity with definitions of vector potential

In one of my books (the great Baez & Munian's "Gauge fields, knots and gravity"), the vector potential is defined as a $End(E)$ valued 1-form, with $End(E)$ endomorphisms of the fiber $E$. So, ...
1
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1answer
153 views

How does a wavefunction transform under an arbitrary boost?

How does a wavefunction $\psi$ transforms under an arbitrary boost? It's easy to find how it transforms under rotation or translation because the corresponding generators form a closed Lie algebra. ...
4
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1answer
727 views

Projector operator in Representation theory

I am reading some introductory stuff on Representation theory applied to physics and I am a bit confused about some things. The book I use is Lie Algebra in Particle Physics by Georgi (you can find it ...
3
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1answer
248 views

Quantization of angular momentum in $SO(3)$

When hermitian operators $L_1, L_2, L_3$ follow the commutation relations: $$ [L_1,L_2]=i\;L_3 \\ [L_2,L_3]=i\;L_1 \\ [L_3,L_1]=i\;L_2 $$ one can show that, assuming they are in finite number, their ...
1
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0answers
67 views

Gauge group properties VS Particle properties

Let us say that, for simplicity, we are studying gauge theory over 4-dimensional Minkowski spacetime, and that the gauge group is $SU(3)$, which is probably the simplest non-abelian gauge group after $...
1
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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 $$\...
2
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3answers
447 views

What is the relationship of Clebsch-Gordan decomposition with Young tableau?

Until recently, I had the impression that any representation $R_1 \otimes R_2$ for spins $J_1$ and $J_2$ is reducible, for example, into $(2 \min{(J_1,J_2)}+1)$ multiplets. $$ J_1 \otimes J_2 = (J_1 +...
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1answer
304 views

Infinitesimal Poincare transformations , Taylor expansion

Let $(\Lambda,a)\in\text{ ISO}_o(3,1)$ be a finite (proper) Poincare transformation and Let $U(\Lambda,b)$ be the corresponding unitary operator implementing this transformation on the Hilbert space ...
0
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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 ...
1
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1answer
166 views

Why must a fundamental particle's spin be a multiple of $\frac 1 2$? [duplicate]

A fermion is a particle whose spin is an odd multiple of $\frac 1 2$, and a boson is a particle whose spin is an integer. From what I've seen, these appear to be the only two possibilities; not only ...
3
votes
1answer
236 views

Why the Lorentz group has complex generators in QFT treatments? [duplicate]

In Schwartz' and Peskin's QFT books, when trying to deal with representations of the Lorentz group the authors study the representations of the Lie algebra of such group. By definition, if $SO(1,3)$ ...
2
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0answers
269 views

What was the first motive of physicists to interpret spin rotations as elements of $SU(2)$?

I have read several articles talking about the meaning of spins and spinor spaces. They only have a mathematical, or quantum mechanical identity. Thus one hardly finds a classical (geometrical) ...
0
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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
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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})...
3
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0answers
489 views

Cartan Killing metric and Casimir operators

I'm a little confused about Casimir operators and Cartan-Killing metric. The Lorentz group is a semi-simple group and its Cartan-Killing metric is non-degenerate, say $g_{ab}$; it is invertible and ...
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 ...
0
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2answers
339 views

Derivation of generators of Lorentz group for spinor representation

I want to prove $$S^{\mu \nu}=\frac{i}{4}[\gamma^\mu,\gamma^\nu].$$ I started from $$[\gamma^\mu,S^{\alpha\beta}]=(J^{\alpha\beta})^\mu_\nu \gamma^\nu$$ Putting the value of $(J^{\alpha\beta})^\mu_\...
0
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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 ...
2
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1answer
966 views

What is the difference between a Pauli spinor, a Weyl spinor, and a Cartan spinor?

I know that a spinor is a complex two components "vector", which is acted on by the $SU(2)$ group under a rotation. In the physics litterature, I often read "Weyl spinors", "Pauli spinors", "Cartan ...
3
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2answers
122 views

Do spins add when particles combine symmetrically?

Suppose I have three spin $s$ particles. What are the possible spins of a symmetric combination of these three particles? Will one of the states always have spin $3s$? Perhaps the above question is ...
2
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1answer
472 views

How to Derive Baryon Octet Matrix?

In every textbook on particle physics that I've read, I encounter the following matrix when reading about matrix of wave functions for the baryon octet: $$\left[\begin{array}{ccc} \frac{\Sigma^0}{\...
4
votes
1answer
144 views

How does one write a three dimensional de Sitter space as the quotient $SL(2, C)/SL(2, R)$?

In arXiv:hep-th/0110108 the $(2+1)-$dimensional de Sitter space is represented as a quotient space of $SL(2, C)/SL(2, \mathbb{R})$. I couldn't understand how, both mathematically and intuitively, is ...
6
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2answers
673 views

Interpretation of the field strength tensor in Yang-Mills Theory

In Yang-Mills theory the field strength tensor $F_{\mu \nu}$ can be calculated as \begin{equation} F_{\mu\nu} \equiv \frac{i}{g} [D_\mu,D_\nu] = \partial_\mu A_\nu - \partial_\nu A_\mu -ig[A_\mu,A_\nu]...
5
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0answers
391 views

Is there a Virasoro group?

On page 14 of the survey article Kac-Moody and Virasoro algebras in relation to quantum physics by Goddard and Olive, the authors show that smooth selfmaps of the circle form a Lie group corresponding ...
9
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1answer
420 views

Meaning of Lorentz Generators

I'm trying to understand infinitesimal Lorentz transformations in quantum field theory. I've studied some Lie theory from mathematicians, but I'm having trouble adjusting conceptually to how Lie ...
2
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1answer
328 views

Infinite dimensional representations of $\text{SO}(3)$

In the theory of angular momentum, we wish to study the projective representations of the rotation group $\text{SO}(3)$, for which we turn to the representation theory of the double cover $\text{SU}(2)...
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1answer
419 views

Representation and gauge transformation of Yang-Mills theory

Let us consider a classical field theory with gauge fields $A_{\mu}^{a}$ and a scalar $\phi^{a}$ such that the Lagrangian is gauge-invariant under the transformation of the gauge fields $A_{\mu}^{a}$ ...
1
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1answer
177 views

How to properly set up the relation between Lie algebras and observables in QM?

In quantum mechanics, given a Hermitian operator $A$, it gives rise to a symmetry/unitary operator by exponentiation $e^{i\lambda A}$, which can be properly defining using the eigenvector expansion, i....
2
votes
0answers
187 views

$SU(2)\times U(1) \rightarrow U(1)$ Symmetry Breaking

Consider a doublet of complex scalar fields $\mathbf{\Phi}$. The $SU(2)$ transformation is $\textrm{exp}\left(it^{i}\tau_{i}\right)$ with generators $\tau_{i}=\frac{\sigma_{i}}{2}$, whilst the $U(1)$ ...
6
votes
1answer
2k views

Difference between Lorentz group and Poincare group

I am currently studying the proper orthochronous Lorentz group $\text{SO}^+(1,3)$ and I have run into some confusion. I started by defining the Poincare group to be the set of all transformations ...