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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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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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with usual notations l is the orbital quantum numbers associated with the orbital motion of the quantum particle and m(with suffix l) is the magnetic quantum number associated with the space quantization of l(ie projection of l along the z-axis directed magnetic field.These quantum numbers along with spin quantum numbers (s) generates the whole angular ...

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If we choose the signature of the metric $\eta$ to be $(1,-1,-1,-1)$ and choose the gamma matrices $\gamma^{\mu}$ to be unitary (as they can be because they form a representation of a finite group), then it would follow from the commutation relations $\{\gamma^{\mu},\gamma^{\nu}\}=2\eta^{\mu\nu}$ that $\gamma^{0}$ would be Hermitian and ...

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The value $m_{l}$ is the eigenvalue of the operator $L_{z}$ determined by seeing the action of this operator on the eigenstate $|~l,m_{l}>$ or in other words $$L_{z} |~l,m_{l}> = m_{l} |~l,m_{l}>$$ While $l$ is related to the total angular momentum operator $L$ and it acts on the same eigenstate giving you $$L^2 |~l,m_{l}> = l(l+1) ... 2 This comes from the representation theory of the rotation group \mathrm{SO}(3). Quantum mechanics takes place in a vector space, and observables are operators on this space. The total amount of angular momentum is obtained from the angular momenta L_x,L_y,L_z about the three axes of space as$$ L = \sqrt{L_x^2+L_y^2+L_z^2}  since that is the length of ...

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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: $M(m+1,m)=SU(2)_{m-2} \times SU(2)_1/SU(2)_{m-1}$

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