# Quantum theory in real Hilbert space: How the complex Hilbert space structure emerges from PoincarĂ© symmetry

November 28, 2016

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 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.

Keywords:

relativistic quantum theory, real spectral theory, representation theory