The Hilbert Book Model impersonates a creator (HBM). At the instant of the creation, the HBM stores all dynamic geometric data of his creatures in a read-only repository that consists of a combination of an infinite dimensional separable quaternionic Hilbert space and its unique non-separable companion that embeds its separable partner. The storage applies quaternionic eigenvalues of operators. The quaternionic containers hold a scalar real number valued time stamp and a three-dimensional spatial location vector that represents the imaginary part of the quaternion. Mechanisms that apply stochastic processes generate these data. The stochastic processes own a characteristic function. This attribute ensures the coherence of the generated data. In this way, quaternionic differential calculus can describe the dynamic relations between the stored data. The characteristic function acts as a displacement generator. Consequently, at first approximation, the generated location swarm moves as a single coherent unit. The generated locations act as artefacts for the embedding continuum. This continuum is eigenspace of an operator that resides in the non-separable Hilbert space.

Partial quaternionic differential equations that apply the quaternionic nabla ∇ describe the interaction between a field and a point-like artifact.

∇ ≡ {∂/∂τ, ∂/∂x, ∂/∂y, ∂/∂z}

∇ ≡ {∂/∂x, ∂/∂y, ∂/∂z}

∇ᵣ ≡ ∂/∂τ

τ is progression or proper time.

In the quaternionic differential calculus, differentiation with the quaternionic nabla is a quaternionic multiplication operation:

c = cᵣ + c= ab ≡ (aᵣ + a) (bᵣ + b) = aᵣbᵣ − 〈a,b〉 + abᵣ + aᵣb ± a×b

Here the real part gets subscript ᵣ and the imaginary part is written in bold face.

The right side covers five different terms.

〈a,b〉 is the inner product.

a×b is the external product.

± indicates the choice between right and left handedness.

Now the partial differential equation that describes the first order behavior of a continuum is given by:

Φ = ϕᵣ + Φ = ∇ψ ≡ (∇ᵣ +∇) (ψᵣ + ψ) = ∇ᵣψᵣ − 〈∇, ψ 〉 + ∇ψᵣ + ∇ᵣ ψ ± ∇× ψ

ϕᵣ = ∇ᵣψᵣ − 〈∇, ψ 〉

Φ =∇ψᵣ + ∇ᵣ ψ ± ∇× ψ

〈∇, ψ 〉 is the divergence of ψ

∇ψᵣ is the gradient of ψᵣ

∇× ψ is the curl of ψ

In physics some of the terms get new symbols.

E=−∇ψᵣ−∇ᵣ ψ

B=∇× ψ

Double differentiation leads to the second order partial differential equation:

ρ = ∇*ϕ = (∇ᵣ−∇) (∇ᵣ+∇) (ψᵣ+ ψ) = (∇ᵣ∇ᵣ+〈∇, ∇〉) (ψᵣ+ ψ)=ρᵣ+J

This equation splits into two first order partial differential equations Φ = ∇ψ and ρ = ∇*ϕ.

ρᵣ=〈∇,E〉

J =∇× B −∇ᵣE

∇ᵣ B =−∇×E

Two quite similar second order partial differential operators exist. The first is described above.

(∇ᵣ∇ᵣ + 〈∇, ∇〉) ψ = ρ

This is still a nameless equation.

The second is the quaternionic equivalent of d’Alembert’s operator (∇ᵣ∇ᵣ − 〈∇, ∇〉). It defines the quaternionic equivalent of the well-known wave equation.

(∇ᵣ∇ᵣ − 〈∇, ∇〉) ψ = φ

Both second order partial differential operators are Hermitian differential operators.

These equations are pure mathematical equations and hold for all fields!

Apart from waves, the solutions of the homogeneous versions of these second order partial differential equations describe the super-tiny objects that were subject of a previous blog post. Warps constitute photons and clamps give elementary particles their mass.

The simple differential equations describe what happens with the stored dynamic geometric data. They do not describe the information that observers perceive. The data are stored in the Euclidean quaternionic storage format. The elementary particles do not own limbs that touch other particles. Instead observers perceive via a continuum that transfer the information about the observed event via deformations and vibrations of a selected continuum that embeds both the observed event and the observer. Observers perceive in space-time format. The Lorentz transform converts the Euclidean storage format into the perceived space-time format. This includes the necessary time dilation and length contraction. The Lorentz transform is a hyperbolic transform.

If the locations {x,y,z} and {x',y',z'} move with uniform relative speed v, then

c t'=c t cosh(ω)-xsinh(ω)

x'=x cosh(ω)-c t sinh(ω)

cosh(ω) = ½(exp(ω)+exp(-ω)) = c/√(c²-v²)

sinh(ω) = ½(exp(ω)-exp(-ω)) = v/√(c²-v²)

cosh(ω)²-sinh(ω)²=1

This is a hyperbolic transformation that relates two coordinate systems.

The Lorentz transform describes the coordinate transform correctly when the continuum that transfers the information is flat. However, the massive elementary particles that hop around in their hopping path deform the continuum.

Thus, the observers perceive extra changes since the path of information transfer is no longer a straight line. Instead, the information travels along geodesics that bend with the deformation of the continuum.

In summary, quaternionic partial differential equations describe the dynamics of the geometric data, which the creator archived in the read-only repository. These equations describe the interaction between artifacts and embedding continuums. Observers travel with the scanning subspace and can only retrieve data that have a historical time-stamp. They receive their information via a continuum that transfers this information via deformations and vibrations. That is why observers can only perceive in space-time format. The deformation of the continuum by massive objects, bends the path of information transfer. This also affects the information transfer.

The mentioned partial differential equations do not contain physical units and they hold for all basic fields. This includes the field that represents our living space and embeds all massive objects. It also includes the symmetry-related field that has symmetry-related charges as its sources/drains. These fields differ in their start and boundary conditions and are affected by different artifacts. These two basic fields are coupled via the platforms on which the elementary particles reside.