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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Are projective representiations of a Lie group a representation of the semi-direct product of the group with $U(1)$ if the norm is preserved?
Let's say we have a function $f(x_{\mu},t)$ that transforms under the action of an $N$-parameter group $G(a_{\nu})$. Then a projective representation of $G(a_\nu)$ in the $f(x_\mu,t)$ basis would ...
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Global properties of the gauge group
In this very good P.E. answer, it is explained precisely what it means for a quantum system/theory to have a symmetry group $G$ (where $G$ is a Lie group): going back to first principles, it means ...
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Proving $so(3,1)\simeq sl(2,\mathbb{C})$ by redefining generators
First of all, I am a pedestrian in group theory. I have a general question and two particular ones.
General question: I am trying to show that $so(3,1)\simeq sl(2,\mathbb{C})$ by redefining its ...
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Using Galilean covariance to find conditions on physical observables
Let's suppose that coordinates have to transform accoring to the Inhomogenous Galilean Group. Then
$$ x' = x + a + v(t+b) $$
$$ t' = t + b $$
Let's use a funtion $\psi(x,t)$ of $x$ and $t$ as the ...
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Why semi-simple and compact Gauge Group in YM Theory? [duplicate]
I'm studying the Yang-Mills theory, with the Action:
$$
S=-\frac{1}{2}\int\mathrm{tr}_{\rho}(\mathcal{F}\wedge\star\mathcal{F})
$$
where $\mathcal{F}:=\mathrm{d} \mathcal{A}+\frac{1}{2}[\mathcal{A},\...
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Wilson loop is not an element of $\mathrm{SU}(3)$ in color deconfinement
The center symmetry in QCD comes from the
$$a\ \mathcal{P}\mathrm{exp}\left(ig_s \int_C dx^\mu \ A_\mu(x)\right) a^{-1} = \mathcal{P}\mathrm{exp}\left(ig_s \int_C dx^\mu \ A_\mu(x)\right),$$
where $C$ ...
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Can the composition law of a group be defined only when considering a representation or realisation of the Group?
When we talk about, lets say, the Lorentz group, we define the action of the Lorentz transformation $\varLambda$ on
\begin{alignat}{1}
x^{\mu} & \in\mathbb{R}^{1,3},\\
x^{\mu} & \rightarrow x'^...
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Confusion about tensors in $SU(3)$
I have some confusion regarding the notion of tensors in $SU(3)$ (or some other matrix Lie group, but let's keep the discussion to $SU(3)$).
For concreteness, I will refer to Peskin and Schroeder's ...
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Can we make a Bloch sphere for Weyl spinors?
If spinors are the "square root" of 3-vectors [$\mathrm{SU}(2)$ double cover of $\mathrm{SO}(3)$], Weyl spinors can be thought of as the "square root" of 4-vectors [$\mathrm{SL}(2,\...
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Why are there triclinic and monoclinic lattices, but biclinic is never mentioned?
When classifying the Bravais lattices we have the
triclinic (point group ${\rm C_i}$) and the monoclinic $({\rm C_{2h}})$ cases, but we do not see the "biclinic" case listed. Why not?
It ...
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Custodial symmetry of the standard model symmetry group $SU(2)_L \times SU(2)_R$
I am studying the standard model including the Higgs sector and electroweak interactions. Here, all of my terms have their usual meanings. Therefore my symmetry group is $SU(2)_L \times SU(2)_R \times ...
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How can a scalar field have components and how do I interpret these components?
From lecture notes$^\zeta$ I've been reading that:
Consider a real three-component scalar field
$$\phi=\begin{pmatrix}\phi_1 \\\ \phi_2 \\\ \phi_3\end{pmatrix}\tag{a}$$
with Lagrangian
$$\mathcal{L}=\...
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How to rotate an electron mathematically?
Im a mathematics student who just learned about the fact that if you rotate an electron by $2 \pi$ its spin state changes but if you turn it by $4 \pi$ it stays the same.
I understand all the ...
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How can I construct a projective representation when the group is not simply connected?
S. Weinberg, in his book "The quantum theory of fields", states this theorem (page 83): The phase of any projective representation $U(T)$ of a given group can be chosen so that $\phi =0$ if ...
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Conserved current transforming under adjoint
If we have a Lagrangian with a global internal symmetry $G$. Why do the conserved currents transform under the adjoint representation of $G$? Is it a general statement (if this is the case, how can we ...
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