Lecture 35.

Handouts:
26. Exam 2 solutions.
27. Writing Assignment 2.
28. Final Project Requirements.

We reviewed the propagator, its relationship to general scattering
amplitudes, and its expression as the Feynman path integral.  We also
reviewed its perturbative expansion, in terms of free propagators and
insertions of point interactions at intermediate vertices.

We discussed how terms in the perturbative expansion can be
represented pictorially, as Feynman diagrams.  With each propagator,
we associate a line segment; with each interaction vertex, a point
i, which means we must insert the quantity -i/\hbar\ \int\ dx_i\
V(x_i). We explained how to combine all factors to find the diagram's
associated contribution to the full propagator; and how to find its
contribution to the scattering amplitude from the diagram's initial
state to its final state.

We also discussed Fourier transformed propagators and scattering
amplitudes, which describe scattering between momentum eigenstates.
We wrote the Feynman rules associating lines with momentum-space
propagators (which conserve 4-momentum as they go along) and vertices
with insertions of the Fourier-transformed potential.

We noted that the momentum-space picture is most useful for field
theory, where we observe scattering from some initial set of particles
of fixed momentum to some final set of particles of fixed
momentum. The perturbative expansion of the full propagator is the
same: we write it as a free propagator, plus a term with one
intermediate insertion of the potential, plus a term with two
intermediate insertions of the potential, etc, etc. However, in
momentum space, the potential we are inserting has terms which are
products of creation and annihilation operators, for various
momenta. These act to create and annihilate particles at the
intermediate point. We explained that spacetime symmetry implies that
the Fourier transformed potential we obtain always involves
products of creation and annihilation operators that conserve
momentum: that is, that create and destroy particles without changing
the total momentum.

We discussed how to write down an interaction vertex for a product of
fields, at which each field either creates or annihilates a single
particle or antiparticle. (Which is created and annihilated depends on
which creation and annihilation operators appear in the Fourier
expansion of the field: a complex field \phi has particle
annihilation operators and antiparticle creation operators, while
\phi^\adjoint has particle creation operators and antiparticle
annihilation operators. A real field's particles are their own
antiparticles, so the field has both particle creation and particle
annihilation operators.)

With this particle creation/annihilation interpretation of the
interaction, the Feynman diagram becomes more than a shorthand for an
integral in a perturbative expansion. It becomes a story of how one
process that goes from an initial state to a final state can occur,
with incoming particles propagating, then getting annihilated at
various interaction vertices; virtual particles being created at one
interaction vertex, then propagating until being annihilated at a
later interaction vertex; and outgoing particles being created at an
interaction vertex then propagating to the final state. As the Feynman
path integral, or full propagator, is the sum over all processes going
from an initial state to a final state, a single Feynman diagram (or
term in the perturbative expansion) describes one process that
contributes to that sum.

We described how, while all four components of p_\mu are conserved
at interaction vertices, virtual particles can have --- for brief
amounts of time --- an energy p_0 that does not obey the classical
dispersion relation E^2 = |p|^2 + m^2. Such particles are called ``off
mass shell''. This is where we see the quantum nature of the
propagator: all processes contribute, though classical processes ---
where all particles are on mass shell --- make the dominant
contributions. (Note that our propagator has a pole, which is dominant
when the classical dispersion relation relating energy to momentum is
obeyed.) 

We noted that in principle we have to sum over all Feynman diagrams to
predict scattering amplitudes for any physical process.  In practice,
each interaction vertex comes in with a factor of a coupling constant,
which is small. Thus amplitudes are dominated by processes with small
numbers of intermediate interactions, and we in practice truncate the
perturbative expansion. We also noted the formality that many of the
integrals often turn out to be divergent. For many well-defined
theories, these infinities can be absorbed by careful redefinition of
the coupling constants, order by order in perturbation theory. Such
theories are called ``renormalizable,'' and are the only kinds of
theories we will consider here.


--- KB