==This is an early document. The insights of the author have refined. See the Wikiversity Hilbert Book Model Project for a more recent interpretation==
Quantum physics applies Hilbert spaces as the realm in which quantum physical research is done. However, the Hilbert spaces contain nothing that prevents universe from turning into complete chaos. Quantum physics requires extra mechanisms that ensure sufficient coherence.
Reality has built-in principles. If you understand these built-in principles, then these principles teach a lesson.
The foundation of reality already supports the built-in principles. A foundation must have a simple structure and that structure must be easily comprehensible. It must install restrictions such that extension of the foundation runs according predetermined lines that preserve sufficient coherence, such that the installed principles are keeping their validity. This makes the discovery of the foundation a complicated affair, because not every simple structure will provide these requirements. Still a sensible candidate for such foundation was discovered eighty years ago. It is a relational structure and it discovery was reported in 1936. The structure implements a law of reality. That law cannot be phrased in the form of a formula, because the relational structure only contains unnamed elements and it defines tolerated relations between these elements. Thus the relational structure does not contain numbers that could be used as variables in the formula. Instead the most fundamental law of reality can be stated in the form of a commandment. That commandment runs:
"Thou shalt construct in a modular way".
Look around you and you will see the implications of this commandment. All discrete objects are modules or modular systems. Apart from that continuums exist.
In the first part of the evolution of the universe its creation used stochastic modular design. After that period the modular design process achieved the generation of intelligent species. From that moment on these species can actively participate in the modular design process. That was the introduction of intelligent modular design. The lesson that the intelligent designers get from the fundamental commandment is:
“Economize your environment and protect your resources!”
Preserve your own species and care about the creatures that you depend on.
The foundation that was discovered by the duo Birkhoff and von Neumann does not contain numbers. In their introductory paper they showed that the set of closed subspaces of a separable Hilbert space has exactly this orthomodular relational structure. The orthomodular lattice only knows relations and elements that are connected by these relations. It is an atomic lattice. This means that multiple elements exist that are not themselves a result of a relation. In the Hilbert space, these atoms are represented by subspaces that cannot be split into other subspaces and therefore they are spanned by a single Hilbert vector. A special operator connects every atomic Hilbert vector with a quaternion that acts as its eigenvalue. In this way, each orthomodular atom corresponds with a matching quaternion. Quaternions consist of a real scalar and a three dimensional vector. The scalar can represent a progression value and the three dimensional vector can represent a spatial location. This shows that the selected foundation indirectly emerges into notions of progression and geometric location. However, this interpretation couples every atom to a single progression instant and a single spatial location. This is a static and not a dynamic geometrical location.
The discoverers of the orthomodular lattice saw this structure as a logical system. They saw the atoms as logical statements and not as Hilbert vectors and also not as quaternions that might represent dynamic locations. The question now is what the atomic elements of the lattice will be if they do not represent logical statements and also do not represent dynamic locations. After all, a dynamic location only makes sense if at other progression instants it may take a different location value. However, that different location would then belong as eigenvalue to a different Hilbert vector as the eigenvector. This dilemma can be solved when a somewhat broader interpretation is given to the representation of an orthomodular atom. The dilemma is cured if we allow the representation to possess more persistence. We allow the elementary object that represents the orthomodular atom to cover more progression instants and more corresponding geometric locations. This means that on other progression moments the elementary object exists on other locations. After reordering of the progression instants the elementary object appears to hop along a hopping path. After a large number of hops, the landing locations form a location swarm. Both the hopping path and the location swarm now represent the elementary object. Without further measures, nothing prevents the elementary object to use a completely arbitrary hopping path and a chaotic location swarm. In this way, the orthomodular lattice cannot ensure the relatively coherent behavior that we know from the reality that surrounds us. Something must exist that ensures the coherence of the hopping path and the corresponding location swarm. We therefore postulate a mechanism that establishes this coherence by ensuring that the swarm gets a coherent shape and a location density distribution that can be characterized by a continuous function. We go one step further by postulating that this distribution owns a Fourier transform. This requirement corresponds to the condition that the swarm owns a displacement generator. This means that in first approximation the swarm itself moves as one unit. The Fourier transform of the location density distribution is the characteristic function of the elementary object. The location density distribution corresponds to the squared modulus of the wave function of the elementary object. This indicates that we are on the right track. However, in this model the wave function is replaced by the characteristic function of the stochastic process that defines the landing locations. This goes a lot deeper than the concept of the wave function.
The most important aspect of the foregoing is that the existence of the Hilbert space automatically follows from the existence of the underlying orthomodular lattice. So if this orthomodular lattice structure is indeed the foundation of physical reality, then physical reality also contains the structure of the Hilbert space with everything that goes with it and that's a lot. The mechanisms that ensure coherence are not part of the Hilbert space. They form an addition to the model and that addition does not emerge from the selected foundation.
More is explained in: http://vixra.org/abs/1606.0028 ; “Mechanisms that keep reality coherent”