Division Algebras and Quantum Theory |
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Authors: | John C Baez |
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Institution: | 1.Centre for Quantum Technologies,National University of Singapore,Singapore,Singapore;2.Department of Mathematics,University of California,Riverside,USA |
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Abstract: | 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. |
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