# Quantum theory in quaternionic Hilbert space: How PoincarĂ© symmetry reduces the theory to the standard complex one

September 26, 2017

We extend some results of group representation theory and von Neumann
algebras to the quaternionic Hilbert space case, proving the double commutant
theorem (whose quaternionic proof requires a different procedure) and extend to
the quaternionic case a result concerning the classification of irreducible von
Neumann algebras. Secondly, we consider elementary relativistic systems in
Wigner's view defined as a locally-faithful irreducible strongly-continuous
unitary representation of Poincar\'e group in quaternionic Hilbert space. We
prove that, if the squared-mass operator is non-negative, the system admits a
natural, Poincar\'e invariant and unique up to sign, complex structure
commuting with the observables of the system leading to a physically equivalent
reformulation in complex Hilbert space. Differently from the quaternionic
formulation, all selfadjoint operators are now observables, Noether theorem
holds and composite systems may be given in terms of tensor product. In the
third part, we use a physically more accurate notion of relativistic elementary
system: irreducibility regards the algebra of observables only, symmetries are
automorphisms of the restricted lattice of elementary propositions and we adopt
a notion of continuity referred to the states viewed as probability measures on
the elementary propositions. We prove that again there exists a unique (up to
sign) Poincar\'e invariant complex structure making the theory complex.
Relativistic elementary systems are naturally and better described in complex
Hilbert spaces even if starting from a real or quaternionic Hilbert space
formulations and this complex description is uniquely fixed by physics.

Keywords:

quaternionic quantum mechanics, spectral theory, representation theory, von Neumann algebras