Chapter 49

Further Topics

There is a long list of other topics that belong in a more complete discussion of
the general subject of this volume. Some of these are standard topics which are
well-covered in many physics or mathematics textbooks. In this chapter we’ll
just give a short list of some of the most important things that have been left
out.
Several of these have to do with quantum field theory:

• Lower dimensional quantum field theories. The simplest examples
  of quantum field theories are those with just one dimension of space (and
  one dimension of time, so often described as “1 + 1” dimensional). While
  it would have been pedagogically a good idea to first examine in detail this
  case, keeping the length of this volume under control led to the decision to
  not take the time to do this, but to directly go to the physical case of “3
  +1” dimensional theories. The case of two spatial dimensions is another
  lower dimensional case with simpler behavior than the physical case.


• Topological quantum field theories. One can also formulate quantum
  field theories on arbitrary manifolds. An important class of such quantum
  field theories has HamiltonianH= 0 and observables that only depend on
  the topology of the manifold. The observables of such “topological quan-
  tum field theories” provide new sorts of topological invariants of manifolds
  and such theories are actively studied by mathematicians and physicists.
  We have already mentioned a simple supersymmetrical quantum mechan-
  ical model of this kind in section 33.3.

49.1 Connecting quantum theories to experimen-

tal results

Our emphasis has been on the fundamental mathematical structures that occur
in quantum theory, but using these to derive results that can be compared to real
world experiments requires additional techniques. Such techniques are the main