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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488 views

Question about angular momentum operator

To show that the eigenvalue to $L^2$ is proportional to $\hbar^2$ is shown from $L_z=xP_y-yP_x$ $p_y=-i\hbar\frac{\partial}{\partial y}$ $p_x=-i\hbar\frac{\partial}{\partial x}$ $L_z=-i\hbar(x\...
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303 views

Spin operator eigenstate in Fock space

I am creating an operator group from representation of spin 1 operators $$J_{x} = \frac{1}{\sqrt{2}}\left(\begin{array}{ccc} 0 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0 \end{array} \...
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186 views

splitting spin representation when reducible

I'm looking at the spin representation of the orthogonal algebra $so(2m,C)$. This representation is $2^m$ dimensional and an explicit construction with gamma matrices is well known (see for example ...
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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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41 views

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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29 views

Generators of 2D global conformal group in terms of differential operators?

I'm looking for a reference that lists generators of two dimensional global conformal group on a complex sphere in terms of differential operators that may act on quasi primary fields $\phi(z,\bar z)$....
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32 views

Torsion form and exterior covariant derivative

The torsion form can be defined as the exterior covariant derivative of a solder form, $\Theta=d_\omega\theta$. This derivative is always in the fundamental representation of the algebra $\mathfrak g$ ...
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42 views

Regarding notation used for infintesimal parameters of the Lorentz algebra and generators of the Lorentz group

I have a confusion regarding the notation that is used for infintesimal Lorentz transformations and the parameters that define the Lorentz transformation (used in various books such as Srednicki's and ...
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20 views

Physics-y resource request on Killing-Cartan forms

Most books that treat nonabelian gauge theory do not contain detailed discussion on Killing-Cartan forms, they'll usually just say that in $\text{SU}(N)$ Yang-Mills theory, one can choose generators $...
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27 views

Calculating the Commutation Relation of the Generators of $SO(n)$ [duplicate]

I'm working through problems in the book Einstein Gravity in a Nutshell by Zee, and I'm stuck on one of the harder problems. The problem is Calculate $[J_{(mn)}, J_{(pq)}]$. We are given that $[J_{(...
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Virasoro algebra commutation (part 2)

This was a sub-question in my previous post that I ask separately now. In Introduction to Conformal Field Theory by Blumenhagen and Plauschinn (springer link) the Virasoro algebra is introduced the ...
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Angular velocity is $\dot{g}$ carried to the identity element of the group

I was reading the example below from Arnolds book I can't really understand why the angular velocity is $\dot{g}$ carried to the identity element of the group. I would appreciate if someone who ...
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27 views

Legal values of spin-1 field can take: $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, ..?

For the spin-1/ boson field $A_\mu$, we may choose it to be a vector which needs to be real $\mathbb{R}$ usually for photon field. The field strength $F= dA$ is also real. Same for the nonabelian ...
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35 views

Lie Algebra in Particle Physics

In his book " Lie Algebra in Particle Physics" Georgie directly put the relation $$(1-P)D(g)(1-P)=D(g)(1-P)...(1)$$ This came from the two previous relations: $$PD(g)P=D(g)P$$ $$PD(g)P=PD(g).$$ where ...
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47 views

Simultaneous shifted diagonalization of bunch of operators

I have the following Lie algebra which is generated by $\{L_n|n\geq 0\}.$ It satisfies the following commutation rule $$ \Big[ L_i ,L_j \Big]= (i-j)(L_{i+j}-\frac14 L_{i+j-1}).....*$$ My question is ...
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54 views

Commutator of Lie Group Generators

This is from Maggiore's "A Modern Introduction to Field Theory", Page 15. I have a Lie group with matrix generators $$ T^{a}_{R}$$ Where $a$ takes values from 1 to the dimension of the Lie group. ...
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Measuring the lorentz transform generators $J$, $K$, and providing evidence that photons have no internal continuous d.o.f

I am reading Weinberg's first QFT book. We looked for (and I suppose found) unitary representations of the Lorentz group: $$U(\Lambda) = 1 - i (\vec{\theta}\cdot\vec{J}-\vec{\eta}\cdot \vec{K})$$ ...
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100 views

Products of Lie-Groups versus Lie-Group Extensions in Physics

The Standard Model of elementary particle physics is a gauge theory based on the Lie group $U(1) \times SU(2) \times SU(3)$. From the mathematical perspective I read that: Simple Lie groups have ...
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16 views

Covariantly constant Lie algebra-valued field with Dirichlet boundary condition

I have a question about a statement in Witten's paper 'Analytic Continuation of Chern-Simons Theory' (https://arxiv.org/abs/1001.2933). On page 66, below equation 4.13, he discusses a Lie algebra-...
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11 views

Calculating adjoint representation of Lie group/algebra [duplicate]

How do I calculate adjoint representation of Lie group and Lie algebra? I would be thankful if anyone can give good example or general formula on calculating adjoint of any Lie group
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53 views

Decomposition of the complex conjugate of the fundamental representation of $SU(5)$ in $SU(3)\times SU(2)\times U(1)$

I know I can decompose the fundamental representation (denoted as $5$) of $SU(5)$ as: $$ (3,1)_{-2c/3} \oplus (1,2)_{c} $$ But how do I get the decomposition of the complex conjugate of this ...
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40 views

Weights of $SU\left(5\right)$ representation

Consider the representation $\Lambda^2V$ of $su\left(5\right)$ where $V$ is the fundamental representation. How can I work out the Dynkin labels of its weights? Are these the correct Dynkin labels ...
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68 views

Construct components of tensor operator [closed]

I'm reading Georgi's textbook on Lie Algebras and have been struggling with this question for quite awhile. The entirety of Chapter 4 (Tensor Operators) has been much more difficult than anything I've ...
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1answer
95 views

Link between dynamical algebra and symmetry group

I was wondering if there is a known link between dynamical algebra and symmetry group. In particular: Do all Hamiltonians belonging to certain dynamical algebra share the same symmetry group? Do ...
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112 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\...
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22 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 ...
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1answer
101 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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108 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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103 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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147 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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35 views

On what kind groups of Lie groups are Yang-Mills theories based? [duplicate]

According to https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory: Yang–Mills theory is a gauge theory based on [...] any compact, semi-simple Lie group. But the symmetry group of Standard ...
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93 views

Understanding the Dirac spinor representation $(1/2,0) \bigoplus (0,1/2)$ of the Lorentz group?

I read that the generators of the Lie algebra in this representation are $$ J_k= \begin{pmatrix} \frac{1}{2}\sigma_k & 0 \\ 0 & \frac{1}{2}\sigma_k \end{pmatrix} $$ (Rotations) and $...
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105 views

Isospin symmetry as an $SU(2)$ symmetry

The generators for the isospin symmetry are given by $$ T_{+}=\left|\uparrow\rangle\langle\downarrow\right|, \qquad T_{-}=\left|\downarrow\rangle\langle\uparrow\right|, \qquad T_{3}=\frac{1}{2}\...
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161 views

Norm of Classical (Poissonian) Hamiltonian Operator

In the Poissonian formulation of classical mechanics, one finds that the time evolution of the phase space vector $\eta = (q_1,q_2\cdots q_n ; p_1, p_2\cdots p_n)^T$ can be put in terms of the ...
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98 views

Which representation do we start with in Grand Unified Theories?

The conventional approach in GUTs is to put all left-chiral fields $F_L$ of the standard model into one representation of the GUT group. For example, the 16 rep for $SO(10)$ GUT: $$ 16_L \rightarrow ...
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1answer
153 views

Does invariance under infinite small transformation imply invariance to the finite one?

Let's say that I have finite chiral transform and I would like to show invariance of Dirac's Lagrangian when $m=0$ under it. The chiral transform is defined as: $$\psi(x) \rightarrow \psi'(x) =e^{i \...
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116 views

What is ``thermal" about a thermal quotient of EdS and EAds?

This is in continuation of my previous question and is in reference to this paper. I guess that the authors are interested in $S^n$ and $\mathbb{H}^n$ since these are the Euclideanized versions of $...
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2answers
214 views

Pauli matrices and Wikipedia

Wikipedia claims Pauli Matrices with an $i$: $i \sigma_1, i \sigma_2, i \sigma_3$ form a basis of $\mathfrak{su}(2)$. But what about the following relation?: $$[\frac{1}{2} \sigma_i, \frac{1}{2} \...
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1answer
293 views

Derivation of the Lorentz algebra explicity [closed]

I need the complete proof for commutation relation of the Lorentz group generators. The proof of Lorentz algebra using this commutation relation.
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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 \...
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1answer
409 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 $A_{\mu}^{a}$ ...
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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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1answer
718 views

Lorentz group $SO(3,1)$ and $SU(2)\times SU(2)$

One way to classify Lorentz representations is to consider the Lie algebra isomorphic of Lorentz group to $SU(2)\times SU(2)$. So that we can classify it by two integers $(j_1,j_2)$. In this way I can ...
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1answer
292 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
66 views

Doesn't modelling using Lie Groups assume spacetime is continuous?

Lie groups are used to some behaviors of quantum mechanics, as well as forming a basis for Kaluza-Klein, Yang-Mills, and String theory. But Lie groups are defined as involving a differentiable ...
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1answer
229 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 ...