Lecture 12.

We proved Noether's theorem for scalar field theory; examined the
conserved currents for spacetime translation; and generalized our
coverage of classical field theories to theories with multiple fields
and with Lorentz covariant fields.

A transformation is a symmetry if it only changes the Lagrangian
(density) by a total 4-divergence. We calculated explicitly the change
in the Lagrangian, using the chain rule, and found that it is the
four-divergence of \pi^\mu D\phi. Equating this to the defining
expression for a symmetry, we found the conserved current j^\mu =
\pi^\u D\phi - F^\mu (where F^\mu is the array of functions whose
4-divergence gives the change in Lagrangian).

We noted that the conserved current j^\mu is ambiguous: adding some
array of functions \Lambda^\mu to j^\mu doesn't change the continuity
equation, as long as its four-divergence vanishes. This gives us a lot
of freedom in choosing a nice form for the conserved current j^\mu.

Finally, we worked out the example of spacetime translations, Taylor
expanding the fields and Lagrangian about x^\mu to find the first
order change in each when they are evaluated at the translated
spacetime position x^\mu + \lambda e^\mu. We found four conserved
currents j^\mu (one for each axis of spacetime), and wrote them in
terms of the stress-energy tensor T^{\mu\nu}. We calculated T^{00},
the zero-component of the current associated with time translation
(i.e., with translation in the e^\mu = (1,0,0,0) direction). We showed
that T^{00} was equal to the Hamiltonian density; specifically, to the
Hamiltonian density we would have found by discretizing the system to
field values \phi_i at grid points i, calculating the canonical
momenta p_i = \partial L /\partial \dot{\phi_i}, and calculating the
Hamiltonian ( \sum_i p_i \dot{\phi_i} ) - L. Thus T^{\mu 0}, the
conserved current associated with time translation, corresponds to the
local density of the momentum 4-vector (E, \vec{p}). The total charge,
which is the integral over all space of the density (time) component,
is just the Hamiltonian, or total energy. This is constant, since
current conservation relates it to the current outflow at spatial
infinity, which vanishes because the field and its gradients go to
zero there.  For a scalar field with no derivative interactions, the
stress-energy tensor is symmetric under the reversal of index order;
we showed that this implies that spatial translation invariance gives
conserved momentum, with the conserved current describing how momentum
flows about locally within the system.

Finally, we generalized our coverage of classical field theory to
include theories with many fields (indexed by a), or Lorentz covariant
fields with many components \mu. We claimed that:

1) the Lagrangian depends on all fields and components a and \mu; the
contributions due to each are summed.

2) there is one Euler-Lagrange equation for each a and \mu, depending
only on the Lagrangian's partial derivatives with respect to that
specific A^\mu_a, and its 4-gradient \partial_\nu A^\mu_a .

3) there is *one* conserved current for each continuous symmetry,
which is given by the sum of contributions of the standard form due to
each field and component a and \mu.

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KB