Quantum theory in real Hilbert space: How the complex Hilbert space structure emerges from Poincaré symmetry

Abstract

As earlier conjectured by several authors and much later established by Solèr (relying on partial results by Piron, Maeda-Maeda and other authors), from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. Stückelberg provided some physical, but not mathematically rigorous, reasons for ruling out the real Hilbert space formulation, assuming that any formulation should encompass a statement of Heisenberg principle. Focusing on this issue from another -in our opinion deeperviewpoint, we argue that there is a general fundamental reason why elementary quantum systems are not described in real Hilbert spaces. It is their basic symmetry group. In the first part of the paper, we consider an elementary relativistic system within Wigner's approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a real Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the real one, all selfadjoint operators represent observables in accordance with Solèr's thesis, and the standard quantum version of Noether theorem may be formulated. In the second part of this work we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. 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 of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exist a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr's picture. This complex structure reveals a nice interplay of Poincaré symmetry and the classification of the commutant of irreducible real von Neumann algebras. In memory of Rudolf Haag√ −c . With this definition we find J ∈ B(H), J * = −J and J * J = −I, i.e., J is a complex structure as wanted, and A = aI + bJ for a, b ∈ R.Remark 2.17 The result in (i) can be made stronger with the help of Theorem 5.3 we shall prove later, observing that if A ∈ B(H) commutes with the unitary representation U it also commute with the von Neumann algebra generated by U. Square root and polar decomposition in real (and complex) Hilbert spacesAnother technical tool, which will be very useful in this work, is the polar decomposition theorem demonstrated in a version which is valid f...

Quantum theory in quaternionic Hilbert space: How Poincaré symmetry reduces the theory to the standard complex one

“…The main goal and result of this work is the proof that, also in the case of a quaternionic formulation with Poincaré symmetry, the theory can be re-formulated into a standard complex Hilbert space picture, where all self-adjoint operators represent observables, the standard version of Noether relation between continuous symmetries and conserved observables is restored and the notion of composite system can be implemented by the standard tensor product. The complex structure is, as in [23], uniquely imposed by Poicaré symmetry and is Poincaré invariant. We shall establish this result into a pair of distinct theorems.…”

“…On the other hand no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex-Hilbert space framework. This complex description is unique and more precise than the real one as for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group.…”

“…Physically speaking this is due to the existence of incompatible elementary propositions (e.g., see [4,22]). The quantum lattice L enjoys a set of peculiar properties that can be phenomenologically motivated (e.g., see [4]) even if some non-trivial interpretative problems remain [7]: (i) orthomodularity, (ii) σ-completeness, (iii) atomicity, (iii)' atomisticity, (iv) covering property, (v) separability, (vi) irreducibility (see the appendix of [23] for a brief illustration of these definitions).…”

“…(2) Observables are the Projector-Valued Measures (PVMs) over the real Borel sets (see [23] for the real and complex case and [13] for the quaternionic case), taking values in…”

The Correct Formulation of Gleason’s Theorem in Quaternionic Hilbert Spaces

“…We shall not address here the problem of the apparent absence of physical systems described in real Hilbert spaces [St60,StGu61,MoOp17] and the possibility (or impossibility) of quaternionic formulations [FJSS62,Ad95,Gan17,MoOp17b]. Instead, we restrict attention to a celebrated result, provided by Gleason's theorem [Gl57], regarding the notion of quantum state.…”