Lecture 11.

We discussed general scalar field Lagrangians, derived their
Euler-Lagrange equations, and stated Noether's theorem relating
conserved currents to symmetries.

We had already derived the Lagrangian and field equations for a scalar
field \phi, which gives the displacement of a continuous medium from
its equilibrium position, in some single chosen direction. We showed
that these field equations enforce Hamilton's principle that classical
motions extremize the action. We now consider more general scalar
field theories, whose interaction forces have no pre-existing analog
in Newtonian mechanics.

We wrote the action as an integral over spacetime of a Lagrangian
density, defined locally at each single point in spacetime. That
Lagrangian density (which we will sloppily call the Lagrangian)
depends on the field \phi, its four-gradient \partial_\mu \phi, and
(for an interacting system) the spacetime coordinate x^\mu. It's form
is (usually) the free four-gradient contribution we have already
derived, minus an arbitrary potential V(\phi).

We derived the Euler-Lagrange equations, by considering all possible
paths from an initial field configuration to a final field
configuration, *restricted to paths that do not differ at spatial
infinity*. We calculated the first order variation of the action about
such a path, setting it to zero for an extremal path. This gave the
Euler-Lagrange equations. We checked that the Euler-Lagrange equations
gave a wave equation for our displacement field \phi, then derived the
Klein-Gordon equation for a massive scalar field \phi.

Finally, we introduced Noether's theorem, pointing out that the
locality of field theory leads not just to a conserved total charge,
but to local charge conservation, where at any given point the charge
density changes only by locally flowing in or out. This implies a
conserved current, \partial_\mu j^\mu = 0.

We stated Noether's theorem: Every continuous symmetry determines a
conserved current.  Here a continuous symmetry is some transformation
of the fields --- a rotation, translation, etc --- that can be done in
a continuous way through some distance (or angle) \lambda, and that
leaves the field equations invariant. We define \lambda D\phi as the
first order change in \phi due to the transformation, for small
\lambda. The equations of motion are invariant if the Lagrangian
changes only by a four-divergence: we showed that, in integrating the
Lagrangian over four-space to get the action, the spatial derivatives
give only surface terms, which vanish for field variations that go to
zero at spatial infinity. The time derivative, as in classical
particle mechanics, contributes to the action only a term which is
constant for all varied paths. Thus the four-divergence can't change
which path is extremal.  Noether's theorem states that, for a symmetry
tranformation that changes the field by \lambda D\phi and the
Lagrangian by \lambda \partial_\mu F^\mu (and F^\mu needn't be a
Lorentz four-vector, only an array of functions), the current j^\mu =
\pi^\mu D\phi - F^\mu is conserved; that is, \partial_\mu j^\mu = 0.
Here \pi^\mu is just shorthand for the partial derivative of the
Lagrangian with respect to the gradient \partial_\mu \phi.

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KB