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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The Abelian versus the non-Abelian commutator of covariant derivatives in field theory
In the case of Abelian symmetry, the covariant derivative is defined as $D_\mu\equiv \partial_\mu + ieA_\mu$, where $e$ is an arbitrary constant and the vector field, $A_\mu$ is a called a gauge field....
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What implements finite conformal transformations in two dimensions?
In a two dimensional conformal field theory I have two sets of generators giving a representation of the Virasoro algebra
$$L_n, \bar{L}_n, n \in \mathbb{Z}$$
$$[L_n,L_n] = (m-n) L_{m+n} + c\frac{m(m^...
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Construction of the Lorentz group in higher or arbitrary dimensions [closed]
I'm interested in generalizations of the Lorentz group and its representations.
Suppose a pseudo-riemannian manifold with signature $ (p,q) $. since the conformal group is
$$ \operatorname{Conf}(p,q) \...
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Extracting the dimension of an operator from algebra
I may misinterpret the question. In the lecture note of conformal field theory, arXiv:2207.09474, it says the following
where for $P^\mu=i\partial_\mu$ and $D=ix^\mu \partial_\mu$. I am confused ...
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How can I calculate action of $\mathfrak{su}(3)$ or other simple algebra ladder operators on "states" from the algebra commutators?
I wanted a way to "derive" Gell-Mann matrices for $\mathfrak{su}(3)$ and generalise this to other semi-simple algebras $\mathfrak{g}$. The way I wanted to approach this is start from the ...
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Multiplying two $SO(3)$ representations
In Group Theory by Zee in Chapter IV.2, he discusses the multiplication of two $SO(3)$ representations on p. 207. Suppose you have a symmetric traceless tensor $S^{ij}$ which furnishes a $5$-...
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Where does the "arbitrary constant" in the $L_{0}$ Virasoro operator come from?
In the 2007 "String Theory and M-Theory" textbook by Becker, Becker, Schwartz there is the following claim about the canonical first quantization of a bosonic string: the quantization of the ...
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Can $W^\pm$ bosons act by changing the weak isospin state of particles?
From what I understand, the weak isospin part of the symmetry-unbroken electroweak interaction consists of the $\mathfrak{su}(2)$ valued gauge field $\hat W_\mu = \sum_i^3W^i_\mu \hat T^i$, where $\{\...
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Why is $\epsilon$ at most quadratic in CFT with $d\geq 3$? [duplicate]
I am trying to read through these notes on CFT, and author reaches a point in chapter $2$ saying:
$$\Big(\eta_{\mu\nu}\square + (d-2)\partial_{\mu}\partial_{\nu}\Big)(\partial\cdot\epsilon) = 0\tag{2....
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A question about the generators of proper orthochronous Lorentz transformations
In Special Relativity, we are introduced that any proper orthochronous Lorentz transformation can be given by
\begin{align}
L=e^{u\cdot K+\theta\cdot J}
\end{align}
where $u\cdot K=u_1K_1+u_2K_2+...
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On the $0$ representation of massive multiplets
So my doubt involves the massive multiplet of $\mathcal{N}=2$. I am not being able to deduce what particles does the states represents. For example,
The $\mathcal{N}=2$ massive short hypermultiplet $s=...
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Classifying projective representations
Blagoje Oblak in their thesis "BMS particles in three dimensions” says that
"Given a group $G,$ suppose we wish to find all its projective unitary representations. The above considerations [...
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Can the generators of a Lie group furnish its adjoint representation?
For generators of the Lie group under an arbitrary representation: $[T^a,T^b]=if^{abc}T^c$
$[T_A^c]^{ab}=-if^{cab}$ is the generator of the adjoint representation.
Is $\ \ e^{i\theta^d T^d}T^ae^{-i\...
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Are the SHO's ladder operators induced from a Lie group action?
Consider a quantum system with a hamiltonian $\hat{H}$, which is invariant under the action of a lie group $G$, meaning we have a unitary representation of $G$, $\hat{U}(g)$, in Hilbert space, and $\...
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What kind of combinations of field components are equal under $SO(9)$ symmetry?
My question is a bit long and chaotic since I haven't learnt group theory systematically.
I am looking at the Banks-Fischler-Shenker-Susskind (BFSS) matrix model. It consists of 9 bosonic matrices $...
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$SU(3)$ adjoint representation and irreducibility
Consider the Gell-Mann matrices, with
$$
\lambda_3 = \operatorname{diag}(1, -1,0), \quad \lambda_8 = \frac{1}{\sqrt{3}}\operatorname{diag}(1, 1, -2), \quad, ... \ ,
$$
they span the Lie algebra $\...
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Infinitesimal transformation of the Yang-Mills field
I am trying to obtain the infinitesimal transformation for the Yang-Mills field $A_{\mu}$. I want to show that
$$ A^{\prime a}_\mu=A_\mu^a-\partial_\mu \theta^a-g_s f^{a b c} \theta^b A_\mu^c $$
For ...
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Conformal symmetry and group in arbitrary dimensions [duplicate]
As far as i understand, the full symmetry of relativity is conformal symmetry.
This is represented by the conformal group $ \operatorname{Conf}(1, 3) $
Of Minkowski spacetime which is $ \mathbb{R}^{1, ...
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Finite dimensional irreducible projective representations of $SO^+(1,3)$
Conventions:
Take the Minkowski metric tensor to have signature $(+, -, -, -)$. Use Hermitian (instead of skew-symmetric) generators of rotations $J_i$ and anti-Hermitian (instead of symmetric) ...
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Lie group symmetry in Weinberg's QFT book
In Weinberg's QFT volume 1, section 2.2 and appendix 2.B discuss the Lie group symmetry in quantum mechanics and projective representation. In particular, it's shown in the appendix 2.B how a ...
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How do Maxwell's equations follow from the action of Lorentz generators on field strength?
Following Warren Siegel's book on Field theory (pg. 223), one might derive the action of Lorentz generators $S_{ab}$ on an antisymmetric 2-tensor field strength $F_{cd}$ which arises for example in ...
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Does the gauge transformation rule on the gauge fields satisfy the definition of group action?
According to definition group action, it is required that action of a group $G$ on a set $X$ must satisfy the compatibility condition:
\begin{equation}
g \cdot (h \cdot x) = (gh) \cdot x \text{ for ...
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Physical motivation behind the relationship between spin, $SO(3)$, and angular momentum
I am only focusing on non-relativistic quantum mechanics in this post. My current understanding is that a particle of spin $\ell$ can be described as an element of the tensor product $L^2(\mathbb{R^3})...
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Degrees of freedom in $(A,B)$ representation of the Lorentz group
Matthew D. Schwartz in his QFT and the Standard Model subsection 10.1.2, shows that $SO(1,3) \cong SU(2) \times SU(2)$ which according to this PhysicsSE post is actually untrue. I was confused about ...
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Why are there only three independent rotations and three independent boosts?
In what sense, there are only three independent rotations (i.e., rotations $R_x, R_y$, and $R_z$ about $x$, $y$, and $z$ axes, respectively)? Is it because any infinitesimal rotation about an ...
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Rotational states in higher dimensions: multiple magnetic quantum numbers
In 4 dimensions, arbitrary rotations are usually double rotations (rotations which can be understood as happening independently on two different planes with different rotation angles). It certainly ...
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Lie algebra and Lorentz transformation
Denote the matrĩ $\eta=$ diag$(-1,1,1,1)$. The group $O(1,3)$, called Lorentz group, is the group of all matrix $L\in M_4(\mathbb R)$ such that
\begin{align}
L^\top\eta\,L\,=\,\eta.\tag1
\end{align}
...
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$SU(5)$ Gauge Field Theory, symmetry breaking
If I start with an $SU(5)$ gauge group and discover that the vacuum is preserved only by matrices of the form $G$
$$\begin{bmatrix}
A & 0 \\
0 & B \\
\end{bmatrix}$$
where ...
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Lie algebra basis and generators
In Supergravity by Freedman (specifically sections 1.2.1 & 1.2.3), he states that given an element $R$ of the group $SO(n)$, the infinitesimal form of $R$ is given by,
\begin{equation}
R^i_j = \...
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How to raise and lower indices of the Poincarè group generators?
I'd like to clarify how raising and lowering indices works when working with Poincarè group generators.
For example, suppose that we want to compute the first component of the Pauli-Lubanski operator:
...
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How to obtain the explicit form of Lorentz transformation matrix using Lie algebra?
Consider the Minkowski space $\mathbb R^4$ with the Minkowski metric tensor
\begin{align}
\langle,\rangle:\ \mathbb R^4\times\mathbb R^4&\longrightarrow\mathbb R
\\ (u,v)&\longmapsto\langle u,...
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Commutator of raising operator in angular momentum with partial derivative wrt z
While fiddling around with certain commutation relations, i noticed the following relation while using spherical coordinates.
What could this relation mean intuitively? Let me know if any information ...
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Weights and roots of $SU(3)$
I am self-studying group theory from Lie Algebras in Particle Physics by H. Georgi and I am having trouble following some of his arguments. In section 7.2 titled Weights and roots of $SU(3)$ he starts ...
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Bloch-Messiah decomposition for 2-mode Gaussian unitary
The Bloch-Messiah decomposition states that any Gaussian unitary can be decomposed as a sequence of passive elements and single-mode squeezers (i.e. beamsplitters, rotations, single-mode squeezing). ...
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Product of Dirac $\gamma^0$ and $\gamma^\mu$ generate a representation of some algebra?
I need your help with an issue about Dirac gamma matrices. Precisely, I need to know if $\gamma^0\gamma^\mu$ generates an irreducible representation of some algebra. This problem has come out in the ...
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Help deriving integral representation of derivative of exponential map for Lie algebras/groups using specifically using Riemann sum [duplicate]
I'm trying to derive the following equation using a Riemann sum formulation only
$$\frac{d}{d t}e^{A(t)} = \int_0^1 ds \quad e^{sA(t)}(\frac{d A(t)}{dt})e^{(1-s)A(t)}$$
What I've done so far is make ...
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Help deriving the integral representation of the derivative of the exponential map [duplicate]
I'm trying to derive the following equation using a Riemann sum formulation only $$ \frac{d}{d t}e^{A(t)} = \int_0^1 ds \,\,\,\,e^{sA(t)}(\frac{d A(t)}{dt})e^{(1-s)A(t)}. $$
The book is using einstein ...
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Simple definition for the generator of an infinitesimal transformation
Studying quantum mechanics, or QFT, the concept of generator $G$ of an infinitesimal transformation $T$ keeps showing up. My problem is that I don't have in mind a solid (dare I say "rigorous&...
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Quadratic form of $SO(2,1)$ from the actions of its generators
I have 3 operators which are generators of $SO(2,1)$. I don't know the expressions of these operators as differential operators, but I know their action on a basis $\left|l',m\right>$:
$$J_{\mu\nu}\...
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Inverse of anti-symmetric rank 4 tensors?
I am trying to find an inverse of a tensor of the form
$$M_{\mu\nu\rho\sigma}$$ such that $M$ is anti-symmetric in the $(\mu, \nu)$ exchange and $(\rho, \sigma)$ exchange. The inverse should be such ...
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What justifies the statement that a Dirac spinor can be written as two Weyl spinors?
I've cross listed this post on math SE in case it is more appropiate there. That post can be found here: https://math.stackexchange.com/q/4833722/.
I am approaching this from a Clifford algebra point ...
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Are $s > 1/2$ spin matrices unitary and $\det = 1$?
A fact we learn in upper undergraduate/early graduate quantum mechanics is that the set of Pauli matrices (i.e spin-$\frac{1}{2}$ matrices) are generators of the group $SU(2)$. A common definition (...
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$SO(3,1)$ is locally $SU(2)\times SU(2)$, what does *locally* mean here?
I am learning Lie group and Lie algebra. I saw in a YouTube video "Supersymmetry lecture 02" from OpenCourseWare (OCW) at University of Cambridge at 11:17 that
$SO(3,1)$ is locally $SU(2) \...
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What do we mean by "generator" in general relativity?
Killing vector field is often referred as "generator" of infinitesimal isometry. However from what I understood this would mean that the exponential map of these generators would be the ...
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What goes wrong when we quantise a classical system without using $[X,P]=i\hbar$?
Let's say we have a classical system with a Poisson bracket. We quantise this system to get a quantum theory where we choose some variable to operator replacement : $x\rightarrow X, p\rightarrow P$, ...
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2
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Lorentz group representation of spin 2 particles
I am beginner in particle physics and I have the following understanding of Lorentz group. Correct me if I am wrong. We know that Lorentz algebra is characterised by the eigenvalues $m(m + 1)$ and $n(...
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Killing vectors on the unit sphere
I am asked to find the Killing vector fields on $S^2$ where the line element is given by $ds^2=d\theta\otimes d\theta +\sin^2\theta d\phi\otimes d\phi$.
I know how to solve this problem by considering ...
0
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0
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Georgi's "Lie Algebras in Particle Physics" Theorem 1.2 proof
In Georgi's "Lie Algebras in Particle Physics", Theorem 1.2 reads
Every representation of a finite group is completely reducible.
The proof that follows contains the following lines
If it ...
1
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0
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64
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CFT algebra calculation [closed]
Hi I'm reading https://arxiv.org/abs/2006.13280 and following its calculation but I'm stuck in page16, eqn 4.4a.
I got different result than the one in the paper, but I can't find the point I missed.
...
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1
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73
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(Anti-)Fundamental Representation of $SU(5)$ GUT
In many places, it has been mentioned that the sum of electrical charges of the particles present in $\overline{5}$ of $SU(5)$ is zero since the trace of $SU(5)$ generators is zero. I do not ...