Questions tagged [group-theory]
Group theory is a branch of abstract algebra. A group is a set of objects, together with a binary operation, that satisfies four axioms. The set must be closed under the operation and contain an identity object. Every object in the set must have an inverse, and the operation must be associative. Groups are used in physics to describe symmetry operations of physical systems.
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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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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:
$$\Big(\eta_{\mu\nu}\square + (d-2)\partial_{\mu}\partial_{\nu}\Big)(\partial\cdot\epsilon) = 0\tag{2....
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Generator of two-qubit quantum gate
I would like to know how to derive the explicit form of the GENERATOR of a general two-qubit gate (also here), e.g., controlled-rotation Y.
From the definition: $$\exp(-i\theta G) \ ,$$
I see it is: $$...
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Have all the symmetries of the standard model of particle physics been found?
Background
The standard model of particle physics is entirely determined by writing down its Lagrangian or, equivalently, writing down the corresponding system of PDEs.
Every set of PDEs has a ...
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(Time-)Orientability in the Language of Fiber Bundles
I'm currently studying spin geometry through Hamilton's book Mathematical Gauge Theory. At a given point, Hamilton considers a pseudo-Riemannian manifold, which I'll take to be Lorentzian in $d=3+1$ ...
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How to find the full global symmetry group of a Lagrangian of $N$ complex scalars?
I have the Lagrangian
$$ \mathcal{L} = \frac{1}{2}D_\mu \Phi^\dagger D^\mu \Phi - \frac{m^2}{2} \Phi^\dagger \Phi - \frac{\lambda}{4}(\Phi^\dagger \Phi)^2 - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} $$
where $\...
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Simplification of $\sum_{a=1}^3\sigma^a_{\alpha\beta} S^a_{\gamma\delta}$ where $\sigma^a,S^a$ are representations of $SU(2)$?
As the question asks, I am dealing with a problem where I'd like to simplify
$$\sum_{a=1}^3 \sigma^a_{\alpha\beta} S^a_{mn}$$
where $\sigma^a$ are the spin-1/2 Paulis and $S^a$ are some higher-spin ...
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In QFT when performing path integral, why don’t we divide it by the volume of Poincaré group, as what we did for gauge group?
When performing path integral in gauge theory, we naively want to compute
$$
Z = \int DA \exp(iS[A])
$$
But we noticed, that because the action is the same for gauge equivalent conditions, we should ...
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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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QFT visual interpretation of $U(1)$
Anyone has (even a "pictorial") way of visualize what the group $U(1)$ does on the fields in the QFT framework?
I know that $U(1)$ can be seen as a circle and the operation of the groups is ...
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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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On the derivation of Wess-Zumino term
$G$-$\text{WZW}$ model on a Riemann surface $\Sigma$ at the level $k$ is defined as
$${\displaystyle S_{k}(\gamma )=-{\frac {k}{8\pi }}\int _{\Sigma }d^{2}x\,{\mathcal {K}}\left(\gamma ^{-1}\partial ^{...
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Weyl spinors under the Lorentz transformation
I am reading An Modern Introduction to Quantum Field Theory by Maggiore. On page 28, it says
Using the property of the Pauli matrices $\sigma^2 \sigma^i \sigma^2 = -\sigma^{i*}$ and the explicit form ...
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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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