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6

This is what happens when physicists try to do group theory but don't bother introducing the proper mathematical notions. There is no isomorphism between $\mathrm{SO}(1,3)$ and $\mathrm{SU}(2)\times\mathrm{SU}(2)$, the former is non-compact, the latter is compact. More around this apparently confusing topic can be found in this answer. Furthermore, using ...

4

You already got your answer, all right, several times over, but I will emphasize the central puzzle of your question which you only got indirect answers for, connected to the peculiar special structure of SO(4). Any self-respecting text introducing the standard model more or less has it. I'll skip all superfluous issues like lagrangian terms, the U(1)s, etc....

3

The group elements are in principle abstract objects defined by the way they act on some structure. For example, the rotation group in three dimensions is formed by elements that rotate coordinate systems in some appropriate way. In order to make things easier to understand and visualize we assign linear representations, i.e. matrices to the elements of ...

2

This is a general aspect of representation theory. The polarizability tensor $\alpha$ is rank (1,1), and is acted on by a group of transformations $G$. The class of all possible polarization tensors forms a vector space, that decomposes into mutually orthogonal representations of $G$. One of these representations is the 'trivial' representation, invariant ...

2

One reason there are more possible eigenvalues of the Casimir operator of the rotations than appear in the spherical harmonics is that the spherical harmonics are proper representations of $\mathrm{SO}(n)$ while the possible values for the Casimir operator classify the possible irreducible representations of $\mathfrak{so}(n)$. By general Lie theoretic ...

2

The $SU(2)$ triplet results from the Adjoint Representation $\mathrm{Ad}: SU(2)\to SO(3)$ of $SU(2)$, whereby $SU(2)$ acts on its own Lie algebra. As a $2\times2$ matrix, an element of the Lie algebra $\mathfrak{su}(2)$ can be written: X=\left(\begin{array}{cc}i\,z&i\,x - y\\i\,x + y&-i\,z\end{array}\right)=i\,(x\,\sigma_x+y\,\sigma_y + z\,\...

2

In quantum mechanics, operators $\{J_x,J_y,J_z\}$ measuring the angular momentum of a state are required to obey the commutation relations $$[J_i,J_j]=i \sum_k \epsilon_{ijk} J_k.$$ If we only care about the spin of a particle, which does not know about the wavefunction, the state of a particle becomes a length $n$ vector (we do ...

1

A 2-dimensional vector space requires 2 basis vectors $v_1$ and $v_2$ to span it. These vectors should not be thought of as either real or complex per se. Instead, for a real vector space arbitrary combinations $a v_1 + b v_2$, $a$ and $b$ real, are also in the space, while for a complex vector space $a$ and $b$ can be complex. An illuminating example is ...

1

For $SU(2)$ the spinor representation has dimension 2. Your questions is not clear, but for rotation groups (or more precisely, their associated spin groups), we have: For $so(2n)$ (with $n\ge 2$), there are two spinor representations of dimension is $2^{n-1}$. E.g., for $so(2\times 5)$, the spinor is 16. For odd dimensions, $so(2n+1)$, the spinors have ...

1

Answer of this question is quite subtle. First let us consider the most general Higgs potential which is renormalizable and invariant under $SU(2)_{L}\otimes U(1)_{Y}$ gauge transformations, which has the form $$V = \lambda(\phi^{\dagger}\phi-\mu^{2})^{2}$$ Where \phi = \frac{1}{\sqrt{2}}\begin{pmatrix} \phi_{1}+...

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