The Hilbert Book Model contains a base model that is constructed from a quaternionic infinite dimensional separable Hilbert space and its unique non-separable companion that embeds its separable partner. The quaternionic number system exists in many versions that differ in the way that they are ordered. Cartesian and polar coordinate systems can define these orderings and let these versions act as parameter spaces. These parameter spaces can be represented by eigenspaces of special reference operators that reside in the separable Hilbert space. The operators connect the countable eigenvalues with an orthonormal base of eigenvectors. This procedure only applies the rational members of the number system. The version of the number system that defines the inner products of pairs of Hilbert vectors defines the background parameter space of the Hilbert space. An interesting subspace is spanned by the eigenvectors of the background reference operator for which the real part of the eigenvalue corresponds with a selected real value that we will call the progression value. As function of the selected progression value, the resulting subspace scans over the complete base model.

In this scanning subspace, the elementary modules are represented by a ray, which is a one-dimensional subspace. In this ray, a normalized vector is eigenvector of a special operator that represents the elementary module and applies a quaternionic eigenvalue that contains a time-stamp that equals the selected progression value and a three-dimensional spatial location. This spatial location is supplied by a mechanism that locates outside of the base model and that applies a stochastic process, which owns a characteristic function. As a result, the elementary module hops around in a stochastic hopping path and after a while the hop landing locations have formed a hop landing location swarm. The characteristic function ensures that this swarm represents a dense and coherent object that in first approximation moves as a single unit. The reason of this result is that the characteristic function acts as a displacement generator. The coherence of the swarm enables its description by a location density distribution and the action as a displacement generator means that the characteristic function is the Fourier transform of this location density distribution. The squared modulus of the wavefunction of the elementary module equals the location density distribution of the hop landing locations warm. 

The relation between location density distribution and displacement generator maybe fairly obvious for continuous functions, but it is far from comprehensible for the discrete stochastic process. It is not immediately clear why the hopping path does not run into complete chaos instead of resulting in a coherent and dense swarm that moves as a coherent object. 

The hopping path occurs inside the platform that belongs to the considered elementary module. The platform owns a private parameter space. A version of the quaternionic number system constitutes that parameter space. The platform with everything on it, floats over the background platform. The stochastic process takes the locations from this platform. This selection passes the properties of the platform to the generated swarm.

I have developed a multimix path algorithm that enables the investigation of the effect of the characteristic function on the generated hop landing location swarm. The multimix algorithm applies the hopping path together with the displacement generator. To get an idea what the influence of the displacement generator can be, we start with a location q i in the stochastic hopping path {j} and generate the next hop. Instead of taking the spatial hop, we take a U turn into Fourier space. The inner product of the eigenvector |q i, which belongs to the current location q i and the eigenvector | p nwhich belongs to the current value p n of the displacement generator describes this step. Next, we implement the action of the displacement generator. Subsequently we step back to configuration space. The three steps involve multiplications rather than additions. However, the middle factors are unitary and the other factors compensate in the neighbor hops.

The inner product q i|p n⟩ represents the step from configuration space location q i to Fourier space location p n.

The factor exp(⟨p n,q i+1-q i⟩) represents the influence of the displacement operator for configuration space step q i+1-q i.

The inner product n|q i+1⟩ represents the step from Fourier space location p n to configuration space location q i+1.

Together the three factors result in the product ⟨q i|p n⟩ exp(⟨p n,q i+1-q i⟩)⟨p n|q i+1

If we combine two subsequent terms, then we get

q i|p n⟩ exp(⟨p n,q i+1-q i⟩)⟨p n|q i+1⟩⟨q i+1|p n⟩ exp(⟨p n,q i+2-q i+1⟩)⟨p n|q i+2

Now we reduce this by using: ⟨p n|q i+1⟩⟨q i+1|p n⟩=1 and exp(⟨p n,q i+1-q i⟩) exp(⟨p n,q i+2-q i+1⟩)= exp(⟨p n,q i+2-q i⟩)

Accounting the result for the next step results in ⟨q i|p n⟩ exp(⟨p n,q i+2-q i⟩)⟨p n|q i+2

Accounting for all hopping path steps results in ⟨q 0|p n⟩ exp(⟨p n,q N-q 0⟩)⟨p n|q N

p n,∑ i=0..N ( q i+1-q i)⟩=⟨p n,q N-q 0

This enables the approximation

exp(p n,∑ i=0..N( q i+1-q i)⟩)⇔ exp(∫L dτ)

p n,∑ i=0..N( q i+1-q i)⟩ ⇔ ∫L dτ

L=⟨p, dq/dτ⟩

L is known as the Lagrangian


∂L/∂vk=pk; vk=dqk/dτ


H is known as the Hamiltonian



In these equations,we used proper time τ rather than coordinate time t.

The result of the“multi-mix algorithm” is expected. The “step” of the swarm equals the sum of the steps of the hops. After a complete swarm regeneration cycle, the center of mass of the swarm will coincide with the geometric center of the platform. This may require an adjustment of the geometric center of the platform. This represents a change in kinetic energy of the platform. 

The “multi-mix algorithm” is introduced to show the similarity with the “path integral.” The “path integral” is taken over all possible paths. The multi-mix algorithm only takes the actual hopping path. Starting from the Lagrangian introduces the “path integral” algorithm approach. Here we started the “multi-mix algorithm” from the hopping path and the“multi-mix algorithm” “results” in the Lagrangian.

The essence of the mulimix path algorithm is that hop q i+1-q in configuration space results in a unitary factor exp(p n,q i+1-q i⟩) in Fourier space. 

The extra kinetic energy might be provided by absorption or emission of the engergy that is carried by warps.