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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 \begin{equation} [J_i,J_j]=i \sum_k \epsilon_{ijk} J_k. \end{equation} 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 ...


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 ...


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 ...


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 ...


0

$j=\frac{1}{2}$ representation is the fundamental representation of the group $SU(2)$. $SU(2)$ is the group of $2\times 2$ Unitary matrices with determinant $+1$. The group $SU(2)$ does act on a two dimensional complex space that you have described. $SU(2)$ has an algebra $su(2)$ whose representations may act on vector spaces of different dimensions. ...


0

One place you could look for a rather neat derivation (that I haven't really found elsewhere) is Lecture 38 and 39 from the series that Sidney Coleman gave at Harvard in 1976. The series is available online at the Harvard physics website. He says he learnt that method himself from Smorodinsky (Russian mathematician) at the Dubna conference (probably in the ...


-1

There is a very quick and clean way of doing this, which is presented in Building an Orthonormal Basis from a 3D Unit Vector Without Normalization. JR Frisvad. J. Graphics Tools 16 no. 3, 151 (2012). Suppose you have a normalized vector $\vec n=(n_x,n_y,n_z)^T$, and you want a rotation matrix that will take the $z$ axis into $\vec n$. (Here it's ...


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 ...


0

I suggest Group Theory in a Nutshell for Physicists by A. Zee


0

Before going into the details, let me describe pictorially how the Hamiltonian, the Symmetry group, and the Dynamical group look in a basis in which the Hamiltonian is diagonal. Hamiltonian $$ H = \begin{bmatrix} \begin{bmatrix} \lambda_1 \mathbf{1} \end{bmatrix} & & & \\ & \begin{bmatrix} \lambda_2 \mathbf{1} \end{...


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....


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 \begin{equation} V = \lambda(\phi^{\dagger}\phi-\mu^{2})^{2} \end{equation} Where \begin{equation} \phi = \frac{1}{\sqrt{2}}\begin{pmatrix} \phi_{1}+...


0

After a bit of discussion I believe there is actually a $SU(2)\times SU(2)$ symmetry in a sense. So in principle there is a $U(2)$ symmetry if $\phi=(\phi_1,\phi_2)^T$, $\phi^\dagger=(\phi_1^*,\phi_2^*)$ and the lagrangian $$\mathscr{L}=\partial_\mu \phi^\dagger\partial^\mu \phi-m\phi^\dagger\phi-\lambda(\phi^\dagger\phi)^2,$$ simply sent $\phi\to U\phi$, ...


-1

If the field is a simple complex scalar field, than the symmetry is just $U(1)$. For a higher symmetry, $\phi$ need to be higher dimensional too, for instance you can add a vector index $\phi_i$ with $i=1,2$ for simplicity, which means that you add an additional complex field. If these two fields interact, you can have two cases now: Each field has a $U(1)$ ...


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 ...



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