Quantum theory may be formulated using Hilbert spaces over any of the three
associative normed division algebras: the real numbers, the complex numbers and
the quaternions. Indeed, these three choices appear naturally in a number of
axiomatic approaches. However, there are internal problems with real or
quaternionic quantum theory. Here we argue that these problems can be resolved
if we treat real, complex and quaternionic quantum theory as part of a unified
structure. Dyson called this structure the "three-fold way". It is perhaps
easiest to see it in the study of irreducible unitary representations of groups
on complex Hilbert spaces. These representations come in three kinds: those
that are not isomorphic to their own dual (the truly "complex"
representations), those that are self-dual thanks to a symmetric bilinear
pairing (which are "real", in that they are the complexifications of
representations on real Hilbert spaces), and those that are self-dual thanks to
an antisymmetric bilinear pairing (which are "quaternionic", in that they are
the underlying complex representations of representations on quaternionic
Hilbert spaces). This three-fold classification sheds light on the physics of
time reversal symmetry, and it already plays an important role in particle
physics. More generally, Hilbert spaces of any one of the three kinds - real,
complex and quaternionic - can be seen as Hilbert spaces of the other kinds,
equipped with extra structure.Comment: 30 pages, 3 encapsulated Postscript figure
