Abstract

In the
eighteenth century, scientists discovered the ingredients of basic quantum field
theory. In those times quantum physics played no role. In the twentieth century,
these ingredients were forgotten and stayed ignored.

This blog post introduces two categories of super-tiny dark objects that represent the most
basic field quanta. Warps represent a tiny bit of energy. Clamps represent a
tiny bit of mass. Observers cannot perceive
these objects as individual items. The objects are the tiny dark objects that science is still missing. The LHC
and its successors will never be able to detect them.

Introduction

Quantum
field theory requires a continuum that can be deformed or vibrated and
actuators that cause this deformation or vibration. Next the activity of these
actuators must be quantized. Thus, the strength of the deformation or vibration
occurs in a set of fixed values. Deformation or vibration can be temporarily or
persistent.

Field dynamics

A function for which both the
parameter space and the target space are multi-dimensional describes a
continuum. Dynamics requests a progression parameter,
and the spatial part requests a multi-dimensional spatial parameter.
Quaternions have the advantage that they combine storage for the progression part, and
the spatial part and quaternionic calculus defines a multiplication procedure
for the combination of the two. Quaternions can also store the scalar part and
the vector part of the target value of a quaternionic function. Quaternions can
describe the behavior of dynamic fields via quaternionic differential calculus.
Partial second order differential equations describe the interaction between
point-like artifacts and quaternionic continuums.

The combination of a
quaternionic infinite dimensional separable Hilbert space and its unique
non-separable companion Hilbert space that embeds its separable partner offers
the playground where this interaction can take place. This playground stores
separate quaternions in eigenspaces of operators that reside in the separable
Hilbert space and can store quaternionic continuums as eigenspaces of operators
that reside in the non-separable Hilbert space. Quaternionic functions define these continuums.

A subspace that scans this
base model as a function of a selected
progression value represents the static status quo of the model and splits it between a historical part, the current
static status quo, and a future part.

The embedding maps the
discrete quaternions onto an embedding continuum. The embedding process occurs
inside the scanning subspace.

Solutions of the
differential equations

The dynamic
solutions of the homogeneous second order partial differential equations do not
occur spontaneously. They are generated by actuators that determine what kind
of solution is generated. For example, a periodic harmonic actuator causes
wave solutions of a homogeneous second order partial differential equation, which is
therefore known as the wave equation. Also, another, quite similar homogeneous
second order partial differential equation exist that does not offer waves as
its solutions. This equation splits into two first order partial differential
equations. Both homogeneous second order partial differential equations offer
solutions that are triggered by one-shot actuators that generate shock fronts.
These solutions occur in two versions. Warps are one-dimensional shock
fronts that during travel keep their amplitude. Clamps are spherical
shock fronts that quickly fade away because their amplitude diminishes as 1/r
with distance r from the trigger location. In the meantime, clamps integrate
into the Green’s function of the carrier field. This means that they
temporarily deform the carrier. Warps carry a standard bit of energy and clamps
carry a standard bit of mass. This makes them the most basic quanta of the
carrier field.

Super-tiny dark objects

Warps and
clamps form two categories of super-tiny objects that in separation cannot be
perceived. Only organized in huge collections these objects become observable.
For example. If emitted at equidistant instants, the warp strings become a
frequency, and if these strings obey the Einstein-Planck relation, then the
strings implement the functionality of photons.

If
recurrently regenerated by dense and coherent swarms of hop landing location
triggers, the clamps become noticeable as elementary particles. Less coherent
assemblies of warps can create a noticeable amount of dark energy. Less
coherent assemblies of clamps can create a noticeable amount of dark mass.

Elementary
particles are elementary modules. Together these elementary modules generate
all other modules and the modules construct modular systems.

Ensuring coherence

A private mechanism
that applies a stochastic process, which owns a characteristic function
generates a hopping path and a hop landing location swarm. The characteristic function
acts as a displacement generator and ensures that the process generates a
coherent swarm, which moves as a single unit. The location density distribution
of the swarm is the Fourier transform of the characteristic function and equals
the squared modulus of the wavefunction of the object that the swarm
represents.

The generated swarms represent
elementary modules. They show both particle and wave behavior. The
characteristic function of the stochastic process explains the wave behavior.

Elementary modules reside on
private platforms that own a private parameter space that a version of the quaternionic
number system generates. The platforms float over a background parameter space,
which the version of the quaternionic number system
that the Hilbert spaces use to define their inner product generates. The differences in ordering symmetry
between parameter spaces give rise to symmetry-related charges. These charges
locate at the geometrical centers of the platforms and produce symmetry-related
fields.

The elementary modules inherit
the properties of the platforms on which they reside. In this way, a range of
different elementary modules exist.

Modules and spectral
binding

Together the elementary
modules constitute all other modules and the modules constitute modular
systems.

Also here, stochastic
processes that own a characteristic function generate the footprints of modules.
Therefore, the modules also move as a single unit.

The characteristic function of
the module equals the superposition of the characteristic functions of the
components of the module. The superposition coefficients determine the internal
locations of the components. These coefficients may oscillate.

The superposition installs a
very strong kind of spectral binding.

Gravity and attractive
symmetry-related charges may add to the effect of spectral binding.

History

The
solutions of the wave equation are known for more than two and a half centuries
[1]. In those times physicists where not aware of the quantization of space,
but some awareness was growing about the quantization of wave packages. The
shock fronts are not waves. They do not feature a frequency. Wave packages
disperse when they move. Shock fronts do not disperse. It is strange that
during the development of quantum physics the shock fronts escaped the
attention of the early quantum physicists. Otherwise, quantum field theory
would have become a straight forward part of quantum theory.

Mathematics

Partial quaternionic differential equations that apply
the quaternionic nabla ∇ describe the interaction between a field and a
point-like artifact [2].

∇ ≡ {∂/∂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

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.

Solutions

Waves

f (τ, x) = a exp
(i ω(cτ-|x-x' |));  c=±1

solves ∇ᵣ∇ᵣ f = 〈∇, ∇〉 f = −ω² f

Warps

ψ = g(x i±τ)

Clamps

ψ = g(r i±τ)/r

References

[1] https://en.wikipedia.org/wiki/Wave_equation#General_solution

[2] https://en.wikiversity.org/wiki/Hilbert_Book_Model_Project/Quaternionic…