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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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7 views

Isomorphism between the conformal group and SO

In CFT, one constructs the generators of various confromal transformations, and re-expresses them in terms of $J_{ab}$'s that manifestly satisfy commutation relations of $so(d+1,1)$ (taking Euclidean ...
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What does the classification of finite dimensional simple Lie algebras tell in physics? [on hold]

A simple Lie algebra is a non-abelian Lie algebra whose only ideals are zero and itself. We know finite-dimensional simple Lie algebras are classified into four families $A_n,B_n,C_n,D_n$, and five ...
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A basic question about how to apply the gauge covariant derivative in Yang-Mills theory

I am sorry if this question is too stupid... We know that Yang-Mills equation (without source) can be written as $$D^\mu F_{\mu\nu}=0,\tag{1}$$ where $$D^{\mu}=\partial^\mu-ig A^{\mu}$$ and $$A^\mu=A^...
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118 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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Basis of the Lie Algebra of a Group [migrated]

It is known that the Lie algebra of a group is a vector space. The question i have is this: Is there a way to find a basis of the Lie algebra of the group? Also, if i have a set of matrices that ...
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How does complexifying a Lie algebra $\mathfrak{g}$ to $\mathfrak{g}_\mathbb{C}$ help me discover representations of $\mathfrak{g}$?

I have been studying a course on Lie algebras in particle physics and I could never understand how complexifying helps us understand the original Lie algebra. For example, consider $\mathfrak{su}(2)$...
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3answers
2k views

Proof that $(1/2,1/2)$ Lorentz group representation is a 4-vector

Taken from Quantum Field Theory in a Nutshell by Zee, problem II.3.1: Show by explicit computation that $(\frac{1}{2},\frac{1}{2})$ is indeed the Lorentz vector. This has been asked here: How do I ...
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47 views

Generators of $su(2) \oplus su(2)$ in the $(A, B)$ representation

Let us call the generators of $su(2)$ in the spin $A$ or spin $B$ representation $J^A_i$ and $J^B_i$ respectively. What are the generators of $su(2) \oplus su(2)$ in the $(A, B)$ representation ? ...
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Highest and Lowest $SU(3)_F$ states

For the finite dimensional $(p,q)$-irreducible representation of $SU(3)_F$, we can label the states as $\mid T_3,Y\rangle$. Where $T_3$ is the third component of isospin and $Y$ is the hypercharge. ...
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317 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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Lie Groups and group extensions?

Is $U(1)\times SU(2) \times SU(3)$ a vector space over a field? I saw an article here that seemed to me that a similar concept to a field extension was being used. In QFT, is each particle ...
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What are the generators of spherical symmetry?

The title says it all. I think this should be a pretty simple question but I just couldn't find the answer. Ok -- I'll give a bit more context to my question. I'm encountering this in the context of ...
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How do you subtract colors and divide them by irrational numbers? (Gluons) [closed]

There is a gluon that is $$\frac{1}{\sqrt{3}} (red \cdot\overline{red} + blue\cdot\overline{blue} - 2\cdot green \cdot\overline{green})$$ This confuses me because I do not understand how adding and ...
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19 views

Evaluation of Commutation Relations in Ballentine's book

I'm reading chapter 3 of Leslie Ballentine's book Quantum Mechanics : A Modern Development but there are a few derivations I don't understand. Question 1 : In the middle of page 74, it says ...
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29 views

$SU(3)$ $U$ and $V$ spin normalization

As we know the $SU(3)$ algebra can be written in terms of $3$ dependent $SU(2)$ algebras viz $(T_+,T_-,T_3),(U_+,U_-,U_3)$ and $(V_+,V_-,V_3)$. There action on a standard state is $\mid T_3,Y\rangle$ ...
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Inconsistency in the expression for $d\alpha= d\alpha_k \wedge d\xi^k$ contracted with pairs of vector fields

Classical Dynamics: A Contemporary Approach, Jorge v. Jose and Eugene J. Saletan,(1998) states in page 227 that Globally $d\alpha$ is defined by its contraction with pairs of vector fields: $$...
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28 views

Center of $SU(3)$

I assumed a 3x3 matrix of the form $$A= \begin{pmatrix} a & b & c\\ d & e & f\\ k & l & m \end{pmatrix}$$ Then, since we know that the center is always an Abelian invariant ...
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Inner product on group theoretic coherent states and anti-commutator of Lie algebra generators

This question is related to group theoretic coherent states (Gilmore, Perelomov etc.). I consider a semi-simple Lie group $G$ with Lie algebra $\mathfrak{g}$ and a unitary representation $U(g)$ acting ...
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1answer
129 views

Induced representation in Zee's Group Theory

I am trying to understand the topic of Induced representation of the euclidean Group E(2) in A. Zee's Group theory in a Nutshell in Chapter IV.i3. The Lie algebra of E(2) has three elements $P_1, P_2,...
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Why does the Weyl vector $\rho = \sum_{\lambda_i\in\Lambda_W} \lambda_i = \dfrac 12 \sum_{\alpha_i \in\Lambda_R^+}\alpha_i$ represents vacuum?

I was reading https://arxiv.org/pdf/1003.2861.pdf, and in p.5, just below equation (3.1), it was written that The state $|\rho\rangle$ associated with the Weyl vector $\rho$ corresponds to the ...
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55 views

Determining the manifold picture of a Lie group — and thus determining global properties

1) How do we determine a Lie group's global properties when the manifold that it represents is not immediately obvious? Allow me to give the definitions I am working with. A Lie group G is a ...
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1answer
51 views

(Physics version of) Taylor expansion. In the the context of deriving a Lie groups generators (a Lie algebra from a Lie group)

Statement which I'm confused about: "Consider some n-dimensional Lie group whose elements depend on a set of parameters $\alpha = (\alpha_1 ... \alpha_n)$, such that $g(0) = e$ with e as the identity,...
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66 views

Matrix Expression of the Maurer-Cartan Form

I'm looking for clarification re: the 'classical' matrix expression for the Maurer-Cartan form $$g^{-1} dg$$ (I have seen the related posts, they don't answer my specific question.) Specifically I ...
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Lorentz algebra and group question with regards to operator representaion of $M^{\mu\nu}$

1) Likely an Einstein summation confusion. Consider Lorentz transformation's defined in the following matter: I aim to consider the product $L^0{}_0(\Lambda_1\Lambda_2).$ Consider the following ...
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Modern form of Brown-Henneaux formula

Almost every paper mentioning Brown and Henneaux's matching of asymptotic symmetries of AdS$_3$ with the Virasoro algebra of a $1{+}1$-dimensional CFT summarizes their results in the formula $$c=\frac{...
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Finding commutators without cyclic permutations

I've been trying to solve Problem 2.4 in Srednicki's Quantum Field Theory textbook. This involves proving the identity $$[J_i,J_j]=i\hbar\epsilon_{ijk}J_k$$ where $J$ is the angular momentum operator, ...
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3answers
465 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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100 views

Do total derivatives have anything to do with central extensions?

I recently got interested in the Galilean group and its central extension and found a paper "Quantization on a Lie group: Higher-order Polarizations" by Aldaya, Guerrero and Marmo. Before asking my ...
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42 views

Representation of $SU(2)$, i.e., spin

Let \begin{equation} X= \begin{bmatrix} 0 & 1\\ 0 & 0\\ \end{bmatrix}, \qquad Y= \begin{bmatrix} 0 & 0\\ 1 & 0\\ \end{bmatrix}, \qquad H= \begin{bmatrix} 1 & 0\\ 0 & -1\\ \end{...
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Can we parameterise $SU(3)$ in such a way that there are clearly 2 parameters corresponding to the cartan torus?

We can parameterise the lie algebra of $SU(3)$ using the Gell-Mann matrices, so that a general element of LA is $\theta_i T_i$, where $T_i=\lambda_i/2$ and $\lambda_i$ are the Gell-Mann matrices. ...
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Conformal weight of a coset model, and a specific case

Given a coset model $(G\times SO(2d))/H$, what is the expression for its conformal weight (in terms of its central charge or, alternatively, in terms of the highest weights of irreducible ...
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Weight $h$ as a function of the central charge $c$ in a Unitary Highest Weight Irrep of the Virasoro algebra

A highest weight irreducible representation of the Virasoro algebra can be labelled uniquely by $(c,h)$, with $c$ the central charge and $h$ the "Witt weight" (the weight of the Witt algebra part of ...
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1answer
58 views

Casimir operator and particle worldline

I'm studying Killing vectors in 2d Minkowski space-time, with signature $(+,-)$, the usual metric given by $ds^2=dt^2-dx^2$. I have found these Killing vectors: $\xi^{(1)}=(1,0)=\partial_t\equiv ...
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3answers
635 views

Different definition of SL(2,R) algebra?

I'm looking into $SL(2,\mathbb{R})$ group and it's algebra. I found on line that the $sl(2,\mathbb{R})$ algebra is given by the two by two real matrices of trace zero. This Lie algebra has dimension ...
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Product of structure constants of $U(N)$

In $SU(N)$, one can derive the following identity: $$f^{abe}f^{cde} = \frac{2}{N} \left(\delta_{ac}\delta_{bd} - \delta_{ad}\delta_{bc} \right) + d_{ace}d_{bde} - d_{bce}d_{ade}\tag{1}$$ with $f^{...
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330 views

Hamiltonian symmetry Lie algebra

What is the connection between complete set of commuting observables and generators of the Lie group? I have a Hamiltonian written down in second quantized formalism and I also checked that it ...
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36 views

Vanishing Poisson bracket with non-vanishing Moyal bracket

Let $M=\mathbb{R}^{2n}$ be the phase space with standard Poisson bracket on smooth functions on $M$. Fix a classical hamiltonian $h$ (function on $M$) and function $f$ generating symmetry of $h$ i.e. $...
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576 views

Commutator relationships and the exponential

I am currently trying to prove that the two following commutator relationships are equivalent (for an operator $\hat{A}(s)$ that depends on a continuous parameter $s$), so if one holds the other one ...
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1answer
156 views

Gauge Field Transformation Properties

I'm a bit confused about the gauge transformation properties of non-abelian gauge fields, and I just wanted some clarification. I keep seeing the statement that "gauge fields transform in the adjoint ...
3
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1answer
265 views

Levi-Civita tensor and the Lorentz group generators in the vector representation

In the vector representation of the Lorentz group its generators are given by - $$(J^{\mu\nu})_{\alpha\beta} = i(\delta^\mu_\alpha\delta^\nu_\beta-\delta^\mu_\beta\delta^\nu_\alpha)$$ It can be ...
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2answers
135 views

Anticommutator of spin-1 matrices

We know that in the spin-1/2 representation the anticommutation relation of the Pauli matrices is $\{\sigma_{a},\sigma_{b}\}=2\delta_{ab}I$. Does a similar relation hold for the spin-1 representation?
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1answer
50 views

Which matrices represent unitary projective representations of ${\rm SO(3)}$?

I was reading this post which triggered the following question. The group ${\rm SO(3)}$ is real orthogonal. However, it is possible to consider representations of ${\rm SO(3)}$ on a complex vector ...
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2answers
64 views

Single sequence of angular momentum ladder in quantum mechanics? — Why there is only a

How do you prove that there is only one sequence of angular momentum eigenstates connected by the ladder operator, within the subspace where the squared modulus of the angular momentum has a given ...
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1answer
111 views

$SU(2)$ and $SO(3)$ WZW models

It seems that the $SU(2)_1$ and $SO(3)_1$ Wess-Zumino-Witten models are quite different despite the Lie algebras being identical. The $SO(3)_1$ model has central charge 3/2 and is equivalent to 3 ...
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1answer
189 views

How much information about a quantum operator is determined by its Poisson bracket Lie algebra?

Hamiltonian quantum mechanics is often built using many ideas from Hamiltonian classical mechanics like the Poisson bracket to determine the commutator between quantum operators, which is appropriate ...
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2answers
140 views

Hypercharge normalization for $SU(5)$ GUT

Reading about $SU(5)$ unification, texts says that they use the renormalization factor $\sqrt{3/5}$ for weak hypercharges in order to embed SM into a $SU(5)$ group. This implies a new $U(1)_Y$ ...
61
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20answers
31k views

Comprehensive book on group theory for physicists?

I am looking for a good source on group theory aimed at physicists. I'd prefer one with a good general introduction to group theory, not just focusing on Lie groups or crystal groups but one that ...
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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 ...
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30 views

There is a classification of matrix representations of Euclidean group $\mathrm{E}(n)$ (also called $\mathrm{ISO}(n)$)?

I'm wondering if someone has already done a classification of matrix representations of the Euclidean group $\mathrm{E}(n)$, more specifically I'm trying to classify $\mathrm{E}(2)$. I watched a ...
3
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1answer
58 views

Fierz identity for symplectic group

For the fundamental representation of $SU(N)$, there is a Fierz identity: $$ \sum_iT^i_{ab}T^i_{cd}=\frac{1}{2}\left(\delta_{ad}\delta_{bc}-\frac{1}{N}\delta_{ab}\delta_{cd}\right) $$ where $T^i$ is ...