introduction-weak-interaction-volume-one

CHAPTER TbO : FIELD THEORY .
2 .1 The Operators PCT .
The parity or space reflection operator, P, reverses the sign of the
x-co-ordinates of a wave function, which is equivalent to reflecting it in th e
plan e
x _ 0 . (2 .1 .1 )
Most physical systems can be described by wave functions which are eigenfunction s
of the P operator . For these, we may write
P ).T(z) (-R) = E, 4r(r) . (2 .1 .2 )
From geometrical considerations, it is obvious that
P2 Ni(r.) _ (=) (2 .1 .3 )
and henc e
EZ = 1, C _ } 1 (2 .1 .4 )
Thus, for systems with only one linearly independent eigenfunction correspondin g
to a particular eigenvalue, the wave function has a definite intrinsic parity, P ,
which may be even (P = 1) or odd (P = -1) .
We assume that we may write a Hamiltonian in terms of the three-vector
x and that
(2 .1 .5 )
i .e . the Hamiltonian is invariant under the P operator . The time-dependen t
Schrddinger equation (1 .3 .7) written in Hamiltonian form i s
H ( x ) 1)-(x, t ) _ jY(ox1t)
(2 .1 .6 )
in natural units ( = c =1) . Replacing x by -x in (2.1 .6) and defining
' (x, t ) _ t ), (2 .1 .7 )
we have, using (2 .1 .5) ,
H(z) ~ '(x, _ t) _ J {x't )
(2 .1 .8)
a t
Thus the function Zf'(x, t) is a solution to the same differential equatio n
as 1Y(x, t) .
For the Dirac equation, we write the equation of motion of a particl e