topics of typical standard physics textbooks dealing with quantum mechanics
and quantum field theory. Among the most important are:
- Scattering theory. In the usual single-particle quantum mechanics, one
can study solutions to the Schr ̈odinger equation that in the far past and
future correspond to free particle solutions, while interacting with a poten-
tial at some intermediate finite times. This corresponds to the situation
analyzed experimentally through the study of scattering processes.
In quantum field theory one generalizes this to the case of “inelastic scat-
tering”, where particles are being produced as well as scattered. Such
calculations are of central importance in high energy physics, where most
experimental results come from colliding accelerated particles and study-
ing these sorts of scattering and particle production processes. - Perturbation methods. Rarely can one find exact solutions to quantum
mechanical problems, so one needs to have at hand an array of approxima-
tion techniques. The most important is perturbation theory, the study of
how to construct series expansions about exact solutions. This technique
can be applied to a wide variety of situations, as long as the system in
question is not too dramatically of a different nature than one for which
an exact solution exists. In practice this means that one studies in this
manner Hamiltonians that consist of a quadratic term (and thus exactly
solvable by the methods we have discussed) plus a higher-order term mul-
tiplied by a small parameterλ. Various methods are available to compute
the terms in a power series solution of the theory aboutλ= 0, and such
calculational methods are an important topic of most quantum mechanics
and quantum field theory textbooks.
49.2 Other important mathematical physics top-
ics
There are quite a few important mathematical topics which go beyond those
discussed here, but which have significant connections to fundamental physical
theories. These include:
- Higher rank simple Lie groups. The representation theory of groups
likeSU(3) has many applications in physics, and is also a standard topic
in the graduate-level mathematics curriculum, part of the general theory
of finite dimensional representations of semi-simple Lie groups and Lie
algebras. This theory uses various techniques to reduce the problem to the
cases ofSU(2) andU(1) that we have studied. Historically, the recognition
of the approximateSU(3) symmetry of the strong interactions (because
of the relatively light masses of the up, down and strange quarks) led
to the first widespread use of more sophisticated representation theory
techniques in the physics community.