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 \psi transforms under rotations and boosts in the Weyl
basis, with the only two bilinear products invariant under both being
\psi_R^\adjoint \psi_L and \psi_L^\adjoint \psi_R. We then showed that
the mass term \psi^\adjoint \beta \psi gave the sum of precisely those
two Lorentz-invariant terms.

We defined the Dirac adjoint \psibar = \psi^\adjoint \beta as a dual
to \psi; that is, we defined \psibar so that the product \psibar \psi
always gives a scalar. This is just as we defined 4-covectors to
absorb the matrix g_{\mu\nu} necessary to obtain Lorentz-invariant
inner products between 4-vectors.

This feature of transforming exactly opposite to the the way that \psi
transforms is so useful that we rewrite our Lagrangian to involve only
\psi and \psibar. We then considered the gradient term in the
Lagrangian, and rewrote it as i\hbar \psibar \gamma^\mu \del_\mu
\psi, where \gamma^\mu = (\beta, \beta \vec{\alpha}). We then showed
that the bilinear \psibar \gamma^\mu \psi transforms under Lorentz
transformations as a spacetime 4-vector: that the Lorentz
transformation matrices acting on \psi and \psibar in *spin* space
induce an effective Lorentz transformation of the *spacetime* indices
\mu of \gamma^\mu. Thus the gradient term in the Lagrangian, i\hbar
\psibar \gamma^\mu \del_\mu \psi, transforms as a Lorentz scalar,
with \psibar \gamma^\mu \psi transforming in the opposite direction as
\del_\mu in spacetime.

Finally we introduced the Feynman slash notation, writing the Dirac
equation in its compact, manifestly Lorentz-covariant form (i\hbar
\delslash - m) \psi = 0.

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KB