Lecture 35.
We introduced the ideas of the propagator, the Feynman path integral and Feynman diagrams, in the context of quantum mechanics.
First we introduced the propagator 
as the
answer to the following physical question in quantum mechanics: if we
have a particle at position 
at time 
, what is the
probability amplitude to find it at position x at some later time
? We showed how the propagator can be calculated by evolving
the initial state 
forward a time t, then taking its
projection onto the position eigenstate 
.
We noted that the propagator contains all the information about the
time evolution of a quantum system. We showed that we can obtain any
time-evolved wave function 
from an initial 
, by integrating the initial wavefunction over x', times
the propagator from x' to x, all over the spatial variable x'.
(We showed this simply, in Dirac notation, by inserting the complete
set of position eigenstates x'.) That is, the propagator is the
Green's function for Schrodinger's equation: it gives us an object to
integrate an initial wavefunction against to convert it into a final
wavefunction. Physically, the propagator implements Huygens'
principle: the amplitude to find the particle at x at time
is the sum over amplitudes for the particle to be at all points x'
at which propagate to x in time t.
We considered splitting the particle's propagation from to x
into two steps, by inserting a complete set of position eigenstates
at time 
. We found that the propagator, or amplitude
to go from to x, was the sum of the amplitudes for all
two-step paths from to x, including paths that passed through
all intermediate points 
. We broke the propagation into
more and more intermediate steps, ending with a limit in which the
number of steps is infinite, and the paths are continuous curves. Even
in that case, the propagator from to x is the sum of the
amplitudes for all trajectories (covering all intermediate points)
that go from to x in time t.
We noted that Feynman showed that the propagator in this limit could
be written as a functional integral over all trajectories x(t), with
each trajectory weighted by the phase factor 
, where S
is the classical action of the path. We noted why, in the classical
limit where 
goes to zero, this results in the classical
trajectory, for which the action is extremal -- thus making many
contributions to the path integral which are all in phase, and
which reinforce each other instead of cancelling due to a rapidly changing
phase.
Finally, we discussed the perturbative calculation of the path
integral. Since we can usually solve for the propagator in the case
of a free theory (no interactions), we write a perturbative expansion
of the full propagator by Taylor expanding the contribution 
to the path integral due to interactions. This
expansion told us that the full propagator from to x is the
sum of the following terms: at zeroth order, the free propagator from
to x; at first order, an integral over a single intermediate
point , of the free propagator from to , times the
potential 
, times the free propagator from to x; at
second order, an integral over two intermediate points, of products of
propagators between all points, and potentials evaluated at all
intermediate points; and similarly for higher order terms j with j
intermediate points. We mentioned how we could associate with each
term a ``Feynman diagram,'' which associates line segments with
propagators, and points with potential insertions.
We will continue with this pertubative expansion, next time.
-- KB