Lecture 28.
Handouts:
Midsemester Grade Summary.
Today we showed that the Dirac Lagrangian is Lorentz-invariant, and rewrote both it and the Dirac equation in a manifestly Lorentz-covariant form.
We recalled how 
transforms under rotations and boosts in the
Weyl basis, with the only two bilinear products invariant under both
being 
and 
. We then showed that the mass term 
gave the sum of precisely those two Lorentz-invariant
terms.
We defined the Dirac adjoint 
as a dual to ; that is, we defined 
so that the
product 
always gives a scalar. This is just as we
defined 4-covectors to absorb the matrix 
necessary to
obtain Lorentz-invariant inner products between 4-vectors.
This feature of transforming exactly opposite to the the way that
transforms is so useful that we rewrite our Lagrangian to
involve only and . We then considered the
gradient term in the Lagrangian, and rewrote it as 
, where
We then showed that the
bilinear 
transforms under
Lorentz transformations as a spacetime 4-vector: that the Lorentz
transformation matrices acting on and in
spin space induce an effective Lorentz transformation of the
spacetime indices 
of 
. Thus the gradient term in
the Lagrangian, , transforms as a Lorentz scalar, with
transforming in the
opposite direction as 
in spacetime.
Finally we introduced the Feynman slash notation, writing the Dirac
equation in its compact, manifestly Lorentz-covariant form 
.
-- KB