introduction-weak-interaction-volume-one

of T 3 the Pauli spin matrix (1 .7 .18) :
T 3
0 - 1
(5 .1 .4 )
Often it is useful to have an operator which transforms the proton into th e
neutron and vice-versa . Thus we defin e
T1 9, 1_ (5 .1 .5 )
Tl 9- _ 9+ . (5 .1 .6 )
Once again, we use the Pauli matrix representation, so tha t
T1
T0 1
L1
0
We also introduce, in complete analogy to real spin ,
T 2 = -j T 3 Tl = LO -j
Li
0
A further two useful operators are defined
T'
so that
T+
0 11
0 0J
T 2 )
(5 .1 .7 )
(5 .1 .10 )
0 0
T
1 0
(5 .1 .11 )
Thu s
T I- n = p (5 .1 .12 )
T+ p = 0 (5 .1 .13 )
T - n = 0 (5 .1 .14 )
T +p = n . (5 .1 .15)
Since our representation and formalism for isospin is identical to that fo r
real spin, we may now define a vector I such that
I =
[1 0
-{ T 1 , T 2 , T3 ) , (5 .1 .16 )
so that (see Appendix B )
(5 .1 .8 )
(5 .1 .9 )
[ Tk , l' ll jTm .
(k, 1, m cyclic) (5 .1 .17 )
The total isotopic spin I defined by (5 .1 .16) must be a conserved quantity .
However, since it was based only upon the characteristics of the strong interaction ,
there is no reason to suppose that it is conserved in anything except the stron g