Quantum theory in quaternionic Hilbert space: How Poincar\'e symmetry reduces the theory to the standard complex one

TL;DR

This paper demonstrates that quaternionic Hilbert space formulations of relativistic quantum systems naturally reduce to the standard complex Hilbert space framework, clarifying the foundational role of complex structures in quantum theory.

Contribution

The authors extend group representation theory and von Neumann algebra results to quaternionic Hilbert spaces and show that relativistic systems inherently admit a unique complex structure, simplifying their description.

Findings

Quaternionic proofs of the double commutant theorem are established.

Relativistic systems with non-negative squared mass admit a unique Poincaré invariant complex structure.

Quantum systems are more naturally and accurately described in complex Hilbert spaces, even if initially formulated in quaternionic spaces.

Abstract

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.