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4

We interprete OP's question (v3) as essentially asking Is $SU(3) \times SU(2)\times U(1)$ a normal subgroup of $SU(5)$? Or in terms of the corresponding Lie algebras, Is $su(3) \oplus su(2)\oplus u(1)$ an ideal of $su(5)$? Here we identify $su(5)$ with antihermitian $5\times 5$ matrices; $su(3)$ with antihermitian $3\times 3$ block matrices in ...


0

They do not lie in $\mathfrak{so}(3)$ but they lie in its complexification, which would be $A_1$ in the usual mathematical classification. Much of Lie representation theory is set up this way: you work at the level of the complexification then go back to the real form. For compact groups it's not a big deal; for non-compact groups extra care is needed. So ...


1

How to understand non-associative composition of velocities in STR? Special relativity introduces a weirdness about how your axes can be related to other observers' axes: if your axes are aligned with observer A's axes and theirs are aligned with observer B then special relativity (i.e. the Lorentz transformations) say that B's axes will be rotated with ...


3

Are Lorentz transformations more adequate representations of motion, than the more intuitive velocities? Yes. The non-associativity that bothers you simply arises because there is no group of three dimensional boosts. Confined to one dimension, boosts form a rather lovely one parameter subgroup of the Lorentz group $SO^+(1,3)$. So everything "works", ...


3

why it is such a prevalent idea. In elementary particle physics and nuclear physics groups and their representations have played a very crucial role in developing the standard models. The elementary particles in the table in the link above have a lot of quantum numbers. These quantum numbers have lead to observed symmetries, that can be described by ...


9

It's an enormous subject, but briefly: Lie groups are smooth groups. Technically, Lie groups are sets that are both a smooth manifold, like a sphere for instance, and also have a group structure (multiplication operator, inverses, and an identity). The group multiplication and inverse must be smooth (differentiable) functions on the manifold. As you ...


2

I guess what you are missing is the following: given a representation $\rho(g)$ of $g\in$SU(2) acting on some vector space $V$. We define the representation $\rho_\otimes$ of SU(2) (not of SU(2)$\times$SU(2)) on $V\otimes V$ as $$\rho_\otimes(g) (v_1 \otimes v_2) = \rho(g) v_1 \otimes \rho(g) v_2.$$ So in fact we are defining the tensor product of two ...


0

So this explanation (my first post on stackexchange!) is based on H. Georgi's "Lie algebras in particle physics", chapter 4. Since $Q_{ij}$ is symmetric, real and traceless, it has 5 independent degrees of freedom. So it's possible to express $Q_{ij}$ into a 'spherical' basis $Q^s_l$, where $s=2$ in this case and $l$ takes on values -2, -1, 0, 1, 2. (For ...


0

I find things are clearer using the dotted and undotted spinor notation. The L-spinors $\chi_{L}$ are dotted vectors $\chi^{\dot{A}}$ and the R-spinors $\xi_{R}$ are undotted vectors $\xi^{A}$ with index $A=1,2$. The parity operator has to be a tensor $P^{\dot{A}}_{B}$ and another tensor $P^{A}_{\dot{B}}$ in order to change the way each type of spinor ...


0

You are looking for a unitary representation of partity on spinors. That it should be unitary can be seen from the fact, that partity commutes with the Hamiltonian. Compare this to time-reversal and charge conjugation, which anticommute with $P^0$ and hence need be antiunitary and antilinear. They involve complex conjugation. As demonstrated parity ...


1

I assume to deal with an autonomous, first order (at least $C^1$ or smooth) system of ordinary differential equations and that the hypotheses sufficient for existence and uniqueness of maximal solutions are satisfied. You may always reduce to the case of a first order system by adding auxiliary variables, $\dot{x}$, to the initial system of differential ...


8

The point is that the symmetries in QM (bijective operations sending states to states preserving the transition probability) can be represented by either unitary or antiunitary operators. This is the statement of a famous theorem due to Wigner. It is possible to prove also that, if the Hamiltonian of a system is bounded below, time reversal must be ...


1

It is indeed possible to break $ SU(3) $ to $ SU(2) \times U(1) $. To see that we need to check that $ SU(2) $ and $ U(1) $ are subgroups of $ SU(3) $. Its easy to see that $ SU(2) $ is a subgroup since the first three Gell-mann matrices are given by, \begin{equation} \lambda _i = \left( \begin{array}{cc} \sigma _i & 0 \\ 0 & 0 \end{array} ...


1

D. Hilbert derived the (same as Einstein's) equations of general relativity by demanding the invariance (form of symmetry) of the Einstein-Hilbert action under general differentiable coordinate transformations, i.e diffeomorphisms So this is the symmetry associated with General Relativity, also refered to as general covariance. UPDATE: Note that all ...



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