The SU(2) Group

This is the group, non abelian, of of $2\times2$ complex matrices $U$ which verify

$$\begin{array}{rll} U^{\dagger} U & = U U^{\dagger} = 1 &\;;\;[unitary] \\ \det U & = 1 &\;;\;[unimodular] \end{array}$$ (179)

This group depends on three continuous parameters $(\alpha_{1},\alpha_{2} ,\alpha_{3})$. The elements of the group can be written

$$U = exp [i \sum_{j}I_{j} \alpha_{j}] = e^{i I_{j} \alpha_{j}}$$ (180)

with

$$I_{j}^{\dagger} = I_{j}$$ (181)

and

$$Tr\{I_{j}\} = 0$$ (182)

A possible election of $I_{j}$ is

$$I_{j} = \frac{1}{2} \tau_{j}$$ (183)

where $\tau_{j}$ are the Pauli matrices which verify the commutation rules

$$[\frac{1}{2} \tau_{i},\frac{1}{2} \tau_{j}]= i \epsilon_{ijk} \frac{1}{2} \tau_{k}$$ (184)

Writing an infinitesimal transformation

$$U(\alpha_{1},\alpha_{2},\alpha_{3}) \simeq 1 + i I_{j} \alpha_{j}$$ (185)

it is clear that the $I_{j} ; (j=1,2,3)$ are the generators of the group.

The Lie theorem, which shows that the commutation rules are valid independently of the representation of the generators, allows to write

$$[I_{i},I_{j}]= i \epsilon_{ijk} I_{k}$$ (186)

and therefore $\epsilon_{ijk}$ are the structure constants of the non abelian Lie algebra of SU(2).

There exists an operator which commutates with the three generators

$$I^{2} = I_{1}^{2} + I_{2}^{2} + I_{3}^{2}$$ (187)

which is the Casimir operator of the group. As a consequence, the irreducible representations of $SU(2)$ are characterized by the eigenvalues of $I^{2}$ . Resulting

$$I^{2} \mid j,m> = j (j+1) \mid j,m>$$ (188)

$$I_{3} \mid j,m> = m \mid j,m>$$ (189)

with $-j\leq m \leq j$. The states $\mid j,m>$ belong to an irreducible basis of dimension $d=2 j + 1$.

There exists also an empirical relation of Gell-Mann and Nishijima for the electric charge of a particle, namely

$$Q = I_{3} + \frac{1}{2} Y = I_{3} + \frac{B+S}{2}$$ (190)

That makes clear then, from the commutation relations, that

$$[Q,I_{j}]\neq 0 \;;\; for\;j=1,2$$ (191)

that is to say, the electric charge breakes the conservation of isospin as it should be expected.