==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”
