introduction-weak-interaction-volume-one

and thus, from (1 .7 .10) and (1 .7 .26), we hav e
J ( a / a.r) ( ' yr T' ) = i (m Zy - m'TIJ) = 0 , (1 .8 .3 )
and
P =
(Ili) S4 - i4 Y4 ) = r-lj ,
(1 .8 .4 )
which is the conventional quantum mechanical expression for probability density ,
(1 .4 .9) .
The Dirac equation may be proved to describe the wave functions o f
massive particles of spin or total angular momentum-, such as the electron o r
proton, and this proof is given in Nuirhead : Elementary Particle Physics ,
Pergamon 1971, pp . 50-52 . We now examine a few of the simpler solutions o f
the Dirac equation, which represent possible wave functions for spin 2 particles .
Obviously, since it must be compatible with the
f matrix in (1 .7 .10), any
solution must be a four-component wave function, known as a spinor . We writ e
the plane wave solution a s
(r~ t)
_
uJ eJPixj
where u is a spinor. Now we set
(J = 1 , 4) , (1 .8 .5 )
2 p 3 , (1 .8 .6 )
and thus we obtain, from (1 .7 .10) ,
(-E + m)ul + pu3 0 (1 .8 .7 )
(-E -+ m)u2 - pu4 0 (1 .8 .8 )
( E + m)u3 - pul
( E + m )u4 -f.
put
We know that
0 (1 .8 .9 )
0 . (1 .8 .10 )
p 2 = E2 - m 2
(1 .8 .11 )
and
that
2
= 1 , (1 .8 .12 )
since u is normalized . Hence,from (1 .8 .7)
u3iu1
= P/( E - m )
and writing
ul = 1 ,
we
deduce that
u 3 = PMI's m ) ,
(1 .8 .13 )
(1 .8 .14 )
(1 .8 .15 )
u 2 = u 4 = 0 . (1 .8 .16)