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There is a subtle difference between saying $(2,2)$ and $2\otimes 2$. In the latter case we are thinking of both reps as transforming under the same element of the group $SU(2)$. In the former case we are thinking of $(2,2)$ as transforming under the Lorentz group, which contains two distinct copies of $SU(2)$. Call one copy the $L$ copy and the other the ...

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We must distinguish between the gauge group $G$, stereotypically the Lie group $\mathrm{SU}(N)$, and the group of gauge transformations $\mathcal{G}$, which are all $G$-valued smooth functions of spacetime. There is no issue if you only write down quantities that transform in proper representations of the group of gauge transformations $\mathcal{G}$. The ...

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A gauge field transforms in the adjoint of the gauge group, but not in the adjoint (or any other) representation of the group of gauge transformations. In detail: Let $G$ be the gauge group, and $\mathcal{G} = \{g : \mathcal{M} \to \mathcal{G} \vert g \text{ smooth}\}$ the group of all gauge transformations. A gauge field $A$ is a connection form on a ...

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Usually the first step in deriving the reps of Poincaire is to go to the rest frame of the particle. This amounts to choosing a basis where $P^0$ acting on the state is nonzero, and where the eigenvalue a of $P^i$ are zero. We can do this if the momentum is timelike, that is if the eigenvalue of $P_\mu P^\mu$ is negative (in -+++ signature). Furthermore the ...

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First off the Standard Model (SM) is chrial, so left and right handed fermions are in different representations of the gauge group. The rep of $SU(3)$ is determined by the color charge. Gluons are in the adjoint of $SU(3)$, which is the 8 of $SU(3)$. Both left and right handed quarks are in the fundamental rep, which is a $3$ (or a $\bar{3}$) (for example, ...

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$$\boldsymbol{3}\boldsymbol{\otimes}\boldsymbol{3}\boldsymbol{\otimes}\boldsymbol{3}= \boldsymbol{1}\boldsymbol{\oplus}\boldsymbol{10}\boldsymbol{\oplus} \boldsymbol{8}^{\boldsymbol{\prime}}\boldsymbol{\oplus}\boldsymbol{8}$$ We talk about this because it explains the structure of a number of baryons in Particle Physics made ...

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As the above only deals with the gauge boson side of things This is a wrong assumption. The format represents the total knowledge from innumerable data of particle physics that have been fitted with SU(3)xSU(2)xU(1) . The particles are slotted into representations of the groups and there are rules of how the interactions happen within the structure of ...

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Let me attempt to answer your question, since your question is about SO(10) GUT model, so I will assume that you have the knowledge of simpler version of GUT namely SU(5) GUT model and also little of group theory. You have 4 different questions >>> 01. Isn't this term ($\psi^{T} C \psi$) already invariant under SO(10)? 02. Doesn't this term ($\psi^{T} C ... 2 The spin of a quantum field is related to the representation of the Lorentz group they transform under: scalar fields transform under the trivial representation, spinors transform under the spinorial representation, gauge bosons under the vectorial representation, gravitons (if they exist) under the second-rank tensorial representation... If you restrict to ... 2 Comments to the question (v3): Recall that the restricted Lorentz group $$\tag{1} SO^+(3,1)~\cong~ SL(2,\mathbb{C})/\mathbb{Z}_2$$ is locally isomorphic to the Lie group of complex$2\times 2$matrices with unit determinant, cf. e.g. my Phys.SE answer here. The Lie group of 3D rotations $$\tag{2} SO(3)~\cong~ SU(2)/\mathbb{Z}_2$$ is a subgroup thereof. The ... 2 SECTION A : What remains invariant for a complex$\:3\times 3\:$tensor depends upon its transformation law under$\:U \in SU(3)\:$CASE 1 :$\:\boldsymbol{3}\boldsymbol{\otimes}\boldsymbol{3}=\boldsymbol{6}\boldsymbol{\oplus}\overline{\boldsymbol{3}}\:$The transformation law for the complex$\:3\times 3\:$tensor$\:\mathrm{X}\:$in this case is ... 9 Section A : The connection of the transformations of complex$\:3\times 3\:$antisymmetric tensors and their representative complex$\:3$-vectors. Let$\:U\:$be a special unitary transformation in$\:SU(3)\:$represented by the$\:3\times 3\:\$ complex matrix U= \begin{bmatrix} u_{11} & u_{12} & u_{13} \\ u_{21} & u_{22} ...

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