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.
