introduction-weak-interaction-volume-one

4 .4 Currents in Leptonic Reactions .
Recalling our Hamiltonian for muon decay (4 .2 .1), it seems possible
that this might initially be constructed from simpler units such a s
~fe rr(1 +
)r5)1v e
(4 .4 .1 )
d
ant,, 1r(1 + Y5 )1J . (4 .4 .2 )
(4 .4 .1) and (4 .4 .2) are strongly reminiscent of the electromagnetic curren t
Sr (x) _ e'~j e (x) o e(x) . (4 .4 .3 )
This is known as a local operator, since it is dependent on a single point x
in space—time . The current (4 .4 .3) obeys the causality commutation relation
LSr(x), S s(x') I — 0 (4 .4 .4 )
where x and x' are two epacelike separated points (see 2 .6) . Bohr and Rosenfel d
(15) showed that when a system of elementary charges in a state A may b e
treated macrocosmically by the classical approximation, the matrix element
(4 .4 .5 )
is simply the classical current density . The electromagnetic interaction
Hamiltonian has the form
HL = Sr (x) Ar( x ) . (4 .4 .6 )
This is identical to the Hamiltonian in classical electrodynamics . From
(4 .4 .3), we should be able to obtain the electromagnetic charge density b y
integrating over all space :
P A(t) = J d 3x . (4 .4 .7 )
(4 .4 .7) is turned into a conserved quantity by the classical equation
(a/ xr ) Sr(x) = 0 .
(4 .4 .8 )
Returning to muon decay, we see that this could be considered as th e
interaction of the two currents (4 .4 .1) and (4 .4 .2), represented a s
e e ) (4 .4 .9 )
an d
(irk.) (4 .4 .10)
We may now ask whether the complete weak interaction Hamiltonian for leptoni c
reactions written in current form contains any other leptonic current—curren t