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 Theories can be formulated in real, complex or quaternionic Hilbert spaces as established in Solér's theorem. Quantum states are here pictured in terms of σ-additive probability measures over the non-Boolean lattice of orthogonal projectors of the considered Hilbert space. Gleason's theorem proves that, if the Hilbert space is either real or complex and some technical hypothes are true, then these measures are one-to-one with standard density matrices used by physicists recovering and motivating the familiar notion of state. The extension of this result to quaternionic Hilbert spaces was obtained by Varadarajan in 1968. Unfortunately, the formulation of this extension [Va68] is partially mathematically incorrect due to some peculiarities of the notion of trace in quaternionic Hilbert spaces. A minor issue also affects Varadarajan's statement for real Hilbert space formulation. This paper is devoted to present Gleason-Varadarajan's theorem into a technically correct and physically meaningful form valid for the three types of Hilbert spaces. In particular, we prove that only the real part of the trace enters the formalism of quantum theories (also dealing with unbounded observables and symmetries) and it can be safely used to formulate and prove a common statement of Gleason's theorem.

After reviewing the theory of "causal fermion systems" (CFS theory) and the "Events, Trees, and Histories Approach" to quantum theory (ETH approach), we compare some of the mathematical structures underlying these two general frameworks and discuss similarities and differences. For causal fermion systems, we introduce future algebras based on causal relations inherent to a causal fermion system. These algebras are analogous to the algebras previously introduced in the ETH approach. We then show that the spacetime points of a causal fermion system have properties similar to those of "events", as defined in the ETH approach. Our discussion is underpinned by a survey of results on causal fermion systems describing Minkowski space that show that an operator representing a spacetime point commutes with the algebra in its causal future, up to tiny corrections that depend on a regularization length.

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