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Ref. 1 does not seem to mention a symmetry-breaking $U(1)$, which must belong to the part of $SU(5)$ which is not in the standard model. In this answer, we will assume that OP is really asking about the weak hypercharge $U(1)$ gauge factor of the standard model. At the Lie algebra level, recall that the Lie algebra $su(n)$ consists of Hermitian traceless ...

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How can we see that the group $N$ generated by $$g = (e^{2\pi i/3} I, -I, e^{i\pi /3}) \in SU(3)\times SU(2)\times U(1)$$ acts trivially on all fields in the Standard Model? First of all, note that $g$ is in the center of $SU(3)\times SU(2)\times U(1)$. Therefore its representative in the adjoint representation is the identity. Since gauge bosons ...

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The main point is that if one has a consistent gauge theory with matter with gauge group $$G:=SU(3)\times SU(2)\times U(1),$$ if one divides $G$ with a normal subgroup $N$, the matter representations of the matter fields could potentially become multi-valued. However, it is possible to choose $N=\mathbb{Z}_6$ in such a way that the standard model matter ...

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Baez actually has another paper (with Huerta) that goes into more detail about this. In particular, Sec. 3.1 is where it's explained, along with some nice examples. The upshot is that the hypercharges of known particles work out just right so that the action of that generator is trivial. Specifically, we have Left-handed quark Y = even integer + 1/3 ...

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Comments to the question (v2): Note that the Lie group $SU(2)$ is not a vector space; only a manifold. But it is a subset of the vector space ${\rm Mat}_{2\times 2}(\mathbb{C})$. So $\{{\rm 1}_{2\times 2},\sigma_1,\sigma_2,\sigma_3\}$ is formally speaking a (complex) basis for ${\rm Mat}_{2\times 2}(\mathbb{C})$; not $SU(2)$. The lecture notes refer to the ...

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This notation is typical of the terrible habit of high energy theorists to label irreps by their dimension, and some educated guess is required to figure out what is $50$ and what is $50^*$. I will label representations of su(5) by their highest weight (or Dynkin labels), i.e. by the 4-dimensional vector of non-negative integers ...

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The confusion here arises because we are not fully analogous to non-relativistic QM here. Given a (quantum or classical) field $\phi$, we usually specify whether it is a "scalar", "spinor", "tensor", whatever field. This refers to a finite-dimensional representation $\rho_\text{fin}$ of the Lorentz group the field transforms in as an element: $$\phi ... 1 I may add a few recent works that use coadjoint orbits to better understand the space of solutions of 2+1 gravity. With a cosmological constant you get Virasoro group coadjoint orbits, and without a cosmological constant you get BMS_3 coadjoint orbits. These are the symmetries of the spaces of solutions of the corresponding gravitational theories. The ... 2 I) The main point is that the half-angle \frac{\theta}{2} doubles when we go from the ket$$\tag{1} |\psi\rangle~=~\begin{bmatrix}\cos\frac{\theta}{2} \cr e^{i\phi}\sin\frac{\theta}{2}\end{bmatrix}, \qquad ||\psi||~=~1, $$to the density matrix/operator$$\tag{2}\rho~=~| \psi\rangle \langle\psi | ~=~\frac{1}{2}\left({\bf 1}_{2\times 2}+ \vec{r}\cdot ...

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From the way it is defined $\left| \Psi \right\rangle$ is not a vector on the sphere, but rather a vector along the z-axis between $-\hat{z}$ and $\hat{z}$, because it is a linear combination of $\left|0\right\rangle$ and $\left|1\right\rangle$ which are both vectors along the z-axis. Now we want $\left|\Psi(\theta = 0 , \phi =0)\right\rangle = ... 2 The definition suggested by joshphysics and clarified by Qmechanic already exists in the literature under then name of representation operator. This is discussed in, e.g., Sternberg's Group Theory and Physics, as well as the somewhat more elementary text An Introduction to Tensors and Group Theory for Physicists by Jeevanjee. 0 On a vector space$V$with metric$g$- be that euclidean, lorentzian or whatever - the Orthogonal group$O(V,g)\subset GL(V)$is defined to be the group of (linear) isometries on$V$. More precisely, for an element$\Lambda\in O(V,g)$, $$g(\Lambda v,\Lambda u)=g(u,v)$$ holds for all$u,v\in V$. Orthorgonal trafos preserve lengths and angles. Expanding ... 1$g$denotes the metric. For Euclidean space the metric is just the unit matrix$I$. For Minkowksi space, which is of interest when talking about the Lorentz group it's the Minkowski metric$\eta_{\mu \nu}$. The lower right matrix inside the Minkowski metric is the 3-dimensional unit matrix and therefore for the space-like components of the Minkowski metric ... 1 Tricritical Ising model belongs to the family of minimal models ($M(5,4)$). There are several different coset constructions that represent them, one of them is the following:$M(m+1,m)=SU(2)_{m-2} \times SU(2)_1/SU(2)_{m-1}$2$SU(N)$is the$N$-fold cover of$PSU(N)$. They share the same Lie algebra, so the Yang-Mills action would look identical locally. The center of$SU(N)$is just$Z_N$. At the level of representations, the fundamental representation of$SU(N)$is a projective representation of$PU(N)$, and only the adjoint ones are linear representations of$PU(N)\$. If the ...

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