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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7
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1answer
257 views

Topological/Geometrical justification for $\text{CFT}_2$ being special

It is known as a fact that conformal maps on $\mathbb{R}^n \rightarrow \mathbb{R}^n$ for $n>2$ are rotations, dilations, translations, and special transformations while conformal maps for $n=2$ are ...
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0answers
44 views

How to prove that given operators form an algebra? [closed]

I am new to Group theory and representations and I'm having trouble with this problem in an exercise: Given the two oscillator algebra $$[a, a^†] = 1$$ $$[b, b^†] = 1$$ $$[a, b] = 0$$ show that ...
3
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2answers
245 views

Adjoint representation in Liouville-von Neumann equation

I am having trouble understanding the adjoint representation of a Lie algebra in the scope of a very specific example, so I thought physics.SE would be the best place to ask. Background: A $N \times ...
2
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4answers
493 views

Why does the Lie algebra corresponding to the unitary group contain Hermitian operators?

I saw an awesome derivation of Schrodinger's equation on Wikipedia. Part of it relies on: We also know that when $t' = t$, we must have the unitary time evolution operator $U(t, t) = 1$. Therefore, ...
1
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2answers
651 views

Casimir Operators and the Poincare Group

Following along in QFT (Kaku) he introduces the Casimir Operators (Momentum squared and Pauli-Lubanski) and claims that the eigenvalues of the operators characterize the irreducible representations of ...
2
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1answer
109 views

axial anomaly for adjoint fermion v.s. fundamental fermion

It is known that the axial anomaly (chiral anomaly, the left L- right R) shows that $U(1)_A$-axial symmetry is not a global symmetry at quantum level. In particular, one can consider the (1) ...
1
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1answer
148 views

“Color charge” of the adjoint fermion?

What kind of "color charge" does the adjoint fermion carry? Let us consider the SU(N) gauge theory. The gauge field is in the adjoint representation (rep). Well-Konwn: If the fermion is in SU(N) ...
2
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1answer
93 views

What is the physical meaning of the Gell-Mann matrices generating SU(3)?

I understand on a surface level that there are these matrices that generate the group SU(3). However, when reading books on gauge theory they appear to make the jump from SU(3) having 8 generators to ...
5
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2answers
251 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} ...
8
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2answers
976 views

Relation between the Dirac Algebra and the Lorentz group

In their book Introduction to Quantum Field Theory, Peskin and Schroeder talk about a trick to form the generators for the Lorentz group from the commutators of the gamma matrices, using their anti-...
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1answer
49 views

Lie algebra definition - Maggior'e book

I'm reading Maggiore's book about QFT, and I'm having a trouble understanding the notation in the part about Lie algebras (Section 2.1): The group generators are defined as $T^a_R=-i\frac{\partial ...
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0answers
125 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 ...
0
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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
46 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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2answers
246 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 ...
9
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2answers
923 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 ...
2
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2answers
190 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
200 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,...
4
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1answer
149 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 ...
3
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1answer
163 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,...
2
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1answer
473 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
109 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
104 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
220 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
102 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],...
1
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1answer
116 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
283 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
102 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
131 views

Quantum spin, tensor product: a long time relationship

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
84 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. ...
0
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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
votes
2answers
239 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]&...
0
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1answer
722 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
63 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
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
124 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
86 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
254 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
365 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 ...
1
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3answers
300 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
76 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
146 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
votes
1answer
654 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
votes
1answer
229 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
990 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 $$\...
3
votes
2answers
403 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 +...
1
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1answer
268 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 ...