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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4
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2answers
143 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 ...
2
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0answers
320 views

Relation between projective representations, connectivity of a group manifold and number of equivalence classes of paths

The projective unitary representations of a multiply-connected group $G$ is defined as $$U(g_1)U(g_2)=c(g_1,g_2)U(g_1g_2)$$ where $c(g_1,g_2)$ is phase. Reading various articles, and this old post of ...
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0answers
40 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
113 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\...
2
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0answers
26 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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44 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)...
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0answers
26 views

Gaining intuition over central extension of algebras [duplicate]

Conformal algebra in 2 dimensions admits a central extension and while the procedure itself is explained in many textbooks quite well, I haven't found a source in which it is reasoned as to why we ...
4
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1answer
724 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 ...
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0answers
82 views

What is the physical meaning of Lie congruence classes?

Any weight $\lambda$ characterising a representation of $\mathfrak{su}(N)$ is an element of one of the $N$ congruence classes defined by (ref.1) $$ \lambda_1+2\lambda_2+\cdots+(N-1)\lambda_{N-1}\quad\...
6
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2answers
523 views

Inonu-Wigner Group Contraction

I am trying to understand how one obtains the Galilean algebra from the Poincare algebra, specifically through the method of central extension. I'm doing this by imposing that the generators of the ...
11
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3answers
1k views

Confusion about rotations of quantum states: $SO(3)$ versus $SU(2)$

I'm trying to understand the relationship between rotations in "real space" and in quantum state space. Let me explain with this example: Suppose I have a spin-1/2 particle, lets say an electron, ...
5
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2answers
1k views

Equivalent Rotation using Baker-Campbell-Hausdorff relation

Is there a way in which one can use the BCH relation to find the equivalent angle and the axis for two rotations? I am aware that one can do it in a precise way using Euler Angles but I was wondering ...
0
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1answer
41 views

A question about a commutation relation

Here $a$, $b$,$c$, $d$ $u$, $v$ range from 0 to 4 and the metric $g^{ab}=\text{diag}(-1,1,1,1)$. 16 $4 \times 4$ matrices $M^{ab}$ are defined as follows: \begin{equation*} (M^{ab})_{uv} = -i(\delta^...
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0answers
105 views

Explicit Quadratic Casimir for $sp(2N)$

We know that $so(3)$ has the explicit quadratic Casimir $$L^2=\sum L_{i}^2.$$ Are there analogs to this in other simple lie algebras? I know that for a simple lie algebra I can always use the ...
1
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1answer
86 views

Vector spaces in second quantization

Studying about fermionic commutation relations, the convention I'm following is to consider a set of creation (destruction) operators $\hat{a}_{i}^{\dagger}\left(\hat{a}_{i}\right)$ with $i=1,...,n$ ...
-1
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1answer
50 views

How to find all the roots of a Lie Algebra starting from Cartan marix, following Georgi's book?

I am reading chapter 8 of Georgi's book, in particular sections 8.8-8.11, where he shows a diagrammatic procedure to find all the roots starting from simple roots. I understand that at each step, we ...
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2answers
79 views

What do quantum spin hamiltonians describe?

I've learned all particles are either fermions or bosons, obeying their respective operator algebras, and then I've seen Hamiltonians describing models carrying one of these two types of particles. So ...
0
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1answer
106 views

How to show that two operators in the forms of vectors commute?

This is an exercise from Peskin&Schroeder's book. The exercise requires to show that $\textbf{J+}$ and $\textbf{J-}$ commute with each other. What is the exact meaning of commutation between ...
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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
253 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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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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2answers
780 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 ...
16
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2answers
589 views

When are there enough Casimirs?

I know that a Casimir for a Lie algebra $\mathfrak{g}$ is a central element of the universal enveloping algebra. For example in $\mathfrak{so}(3)$ the generators are the angular momentum operators $...
2
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1answer
112 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) ...
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1answer
163 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
105 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 ...
2
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1answer
962 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 ...
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2answers
1k 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-...
13
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1answer
538 views

What is “broken symmetry”?

For reference, I come from a mathematics background (mostly differential geometry). I have a very limited understanding of upper-level physics (I'm currently trying to fix this). This is my current ...
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0answers
135 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
55 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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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
52 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
279 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 ...
10
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2answers
995 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
213 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_-...
2
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2answers
309 views

Multiplicity of eigenvalues of angular momentum

Reading Dirac's Principles of Quantum Mechanics, I encounter in § 36 (Properties of angular momentum) this fragment: This is for a dynamical system with two angular momenta $\mathbf{m}_1$ and $\...
4
votes
1answer
207 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
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 ...
1
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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
154 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
votes
1answer
166 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
529 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 ...
8
votes
1answer
700 views

Why is the gauge potential $A_{\mu}$ in the Lie algebra of the gauge group $G$?

If we have a general gauge group whose action is $$ \Phi(x) \rightarrow g(x)\Phi(x), $$ with $g\in G$. Then introducing the gauge covariant derivative $$ D_{\mu}\Phi(x) = (\partial_{\mu}+A_{\mu})\...
3
votes
3answers
118 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{...
-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],...
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 ...
1
vote
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 ...
1
vote
2answers
565 views

The Lie algebra of the Lorentz group is $su(2) \oplus su(2)$. Is there a similar relation for the algebra of the Poincare group?

It can be shown easily, by introducing new generators from the usual ones that we can think of the Lie algebra of the Lorentz group as being built up by two copies of the $SU(2)$ Lie algebra: $$ \...