- Euclidean methods. Quantum field theories, especially in the path
integral formalism, are analytically best-behaved in Euclidean rather than
Minkowski space-time signature. Analytic continuation methods can then
be used to extract the Minkowski space behavior from the Euclidean space
formulation of the theory. Such analytic continuation methods, using a
complexification of the Lorentz group, can be used to understand some
very general properties of relativistic quantum field theories, including the
spin-statistics and CPT theorems. - Conformal geometry and the conformal group. For theories of
massless particles it is useful to study the groupSU(2,2) that acts on
Minkowski space by conformal transformations, with the Poincar ́e group
as a subgroup. The complexification of this is the groupSL(4,C). The
complexification of (conformally compactified) Minkowski space turns out
to be a well known mathematical object, the Grassmannian manifold of
complex two dimensional subspaces ofC^4. The theory of twistors exploits
this sort of geometry ofC^4 , with spinor fields appearing in a “tautological”
manner: a point of space-time is aC^2 ⊂C^4 , and the spinor field takes
values in thatC^2. - Infinite dimensional groups. We have seen that infinite dimensional
gauge groups play an important role in physics, but unfortunately the rep-
resentation theory of such groups is poorly understood. Much is known if
one takes space to be one dimensional. For periodic boundary conditions
such one dimensional gauge groups are loop groups, groups of maps from
the circle to a finite dimensional Lie groupG. The Lie algebras of such
groups are called affine Lie algebras and their representation theory can be
studied by a combination of relatively conventional mathematical meth-
ods and quantum field theory methods, with the anomaly phenomenon
playing a crucial role. The infinite dimensional group of diffeomorphisms
of the circle and its Lie algebra (the Virasoro algebra) also play a role in
this context. From the two dimensional space-time point of view, many
such theories have an infinite dimensional group action corresponding to
conformal transformations of the space-time. The study of such confor-
mal field theories is an important topic in mathematical physics, with
representation theory methods a central part of that subject.
lily
(lily)
#1