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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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0answers
141 views

Difficulty understanding Srednicki's derivation of the Lie Algebra Commutators of the Lorentz Group

I have just started reading through Srednicki's QFT in preparation for a few courses I am about to take. I have taken courses that have covered the basics of special relativity including ...
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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 ...
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0answers
54 views

Conformal Dimension and Highest Weight States of Coset CFT

I am trying to understand the vertex operator algebras of the following form: $$\frac{U(M|N)_{k_1}}{U(L)_{k_2}}$$ Where $U(M|N)$ is the unitary supergroup, $U(L)$ is the usual unitary group, and $...
2
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3answers
318 views

Solving the Lie algebra of generators: path from algebra to matrix representation

Given the Lie algebra, what is the systematic way to construct the matrix representation of the generators of the desired dimension? I ask this question here because it is the physicists for whom ...
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2answers
1k views

Why are the generators of rotation in the 4-dimensional Euclidean space correspond to rotations in a plane?

In three-dimensions, the rotation generators are represented by $J_1$, $J_2$ and $J_3$ where $1,2,3$ respectively stands for the generator of rotation about $x,y,z$ axes respectively. In general, in ...
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2answers
235 views

How do I show that a tensor product representation of $L(SU(2))\equiv su(2)$ is reducible?

So I have been reading about the irreducible representations of the Lie algebra $L(SU(2))$ and came across the Cartan-Weyl basis: $$ H = \sigma_3 $$ $$ E_+ = \frac{1}{2}(\sigma_1+i \sigma_2) $$ $$ E_-...
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1answer
32 views

What does “consistent at an infinitesimal level” mean?

I'm studying the canonical quantization of the real scalar field. I've managed to condense the Hamiltonian and momentum operators in the 4-momentum operator $P^{\mu}$ and have shown that its ...
4
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1answer
209 views

What's the geometric (or representation independent) definition of central charge of Lie algebra $\mathfrak{g}$?

There is a common way(Weinberg QFT Vol.1 P83) to introduce the central charge which I can't understand. Given a unitary projective representation $U(g)$ of Lie group $G$. $$U(g_1)U(g_2)=e^{i \phi(g_1,...
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1answer
156 views

Operators - how to motivate they must be linear ? Is this comment a hint? [duplicate]

Is there a way to motivate, retrospectively, that observables must be representable by linear operators on a Hilbert space? Specifically, there seems to be a hint to something in the accepted ...
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1answer
169 views

Building $\mathfrak{so}(1,3)$ reps using $\mathfrak{so}(1,3)\cong \mathfrak{su}(2)\oplus \mathfrak{su}(2)$

I'm going through the representation theory of $\mathfrak{so}(1,3)$, building Dirac/Weyl spinors and vectors, and I'm a bit confused on the mathematical definitions involved. We have $\mathfrak{so}(1,...
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1answer
560 views

Infinitesimal transformation

I came across this statement in the book "Quantum Field theory and the Standard Model" by Schwartz. "We would now like to find all the representations of the Lorentz group. The Lorentz group ...
3
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3answers
125 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 ...
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1answer
237 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 ...
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0answers
114 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 ...
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1answer
300 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 $...
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1answer
106 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 ...
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1answer
150 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 ...
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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
63 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
823 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} - ...
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70 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 ...
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0answers
40 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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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
281 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
392 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
341 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^{\...
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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, ...
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1answer
156 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
763 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
253 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 ...
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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
468 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
310 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 ...
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1answer
255 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 ...
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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
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1answer
239 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) ...
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1answer
383 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 ...
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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
499 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
374 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 ...
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2answers
346 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_\...
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1answer
338 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
995 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 ...