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

摘要

As established by Sol\`er, Quantum Theories may be formulated in real,complex or quaternionic Hilbert spaces only. St\"uckelberg provided physicalreasons for ruling out real Hilbert spaces relying on Heisenberg principle.Focusing on this issue from another viewpoint, we argue that there is afundamental reason why elementary quantum systems are not described in realHilbert spaces: their symmetry group. We consider an elementary relativisticsystem within Wigner's approach defined as a locally-faithful irreduciblecontinuous unitary representation of the Poincar\'e group in a real Hilbertspace. We prove that, if the squared-mass operator is non-negative, the systemadmits a natural, Poincar\'e invariant and unique up to sign, complex structurewhich commutes with the whole algebra of observables generated by therepresentation. All that leads to a physically equivalent formulation in acomplex Hilbert space. Differently from what happens in the real picture, hereall selfadjoint operators are observables in accordance with Sol\`er's thesis,and the standard quantum version of Noether theorem holds. We next focus on thephysical hypotheses adopted to define a quantum elementary relativistic systemrelaxing them and making our model physically more general. We use a physicallymore accurate notion of irreducibility regarding the algebra of observablesonly, we describe the symmetries in terms of automorphisms of the restrictedlattice of elementary propositions and we adopt a notion of continuity referredto the states. Also in this case, the final result proves that there exist aunique (up to sign) Poincar\'e invariant complex structure making the theorycomplex and completely fitting into Sol\`er's picture. This complex structurereveals a nice interplay of Poincar\'e symmetry and the classification of thecommutant of irreducible real von Neumann algebras.