Quantum theory in real Hilbert space: How the complex Hilbert space structure emerges from Poincar\'e symmetry

Reviews in Mathematical Physics, 29, 1750021 (2017) As established by Sol\`er, Quantum Theories may be formulated in real, complex or quaternionic Hilbert spaces only. St\"uckelberg provided physical reasons for ruling out real Hilbert spaces relying on Heisenberg principle. Focusing on this issue from another viewpoint, we argue that there is a fundamental reason why elementary quantum systems are not described in real Hilbert spaces: their symmetry group. We consider an elementary relativistic system within Wigner's approach defined as a locally-faithful irreducible continuous unitary representation of the Poincar\'e group in a real 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 which commutes with the whole algebra of observables generated by the representation. All that leads to a physically equivalent formulation in a complex Hilbert space. Differently from what happens in the real picture, here all selfadjoint operators are observables in accordance with Sol\`er's thesis, and the standard quantum version of Noether theorem holds. We next focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them and making our model physically more general. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions and we adopt a notion of continuity referred to the states. Also in this case, the final result proves that there exist a unique (up to sign) Poincar\'e invariant complex structure making the theory complex and completely fitting into Sol\`er's picture. This complex structure reveals a nice interplay of Poincar\'e symmetry and the classification of the commutant of irreducible real von Neumann algebras.