In quaternionic physics one equation plays a major role. It is a mixture of a differential equation, a continuity equation and a coupling equation.

Φ = ∇ ψ = m φ

Here Φ, ψ and φ are continuous quaternionic distributions or more specially they are quaternionic probability amplitude distributions. ψ and φ are normalized. m acts as a coupling factor. ∇ is the quaternionic nabla. The real part of it takes the differential with respect to the progression parameter.

m follows from

m² = ∫ |Φ|² dV

Here we focus on

Φ = ∇ ψ

It represents not only the definition of the differential, it also specifies a differential continuity equation. This becomes clear when a continuous quaternionic distribution is interpreted as the combination of a scalar field and a 3D vector field.

The integral form of this equation runs

∫ Φ dV = ∫ ∇ ψ dV

The right part can be split into a volume integral that confines to the variation with respect to progression and a second surface integral that can be written as a surface integral. See the above picture.

The corresponding balance equation reads:

Total change
within *V* = flow into *V* + production inside *V*

A zero surface integral renders the surface into an inversion surface.

The inversion surfaces divide universe into compartments. These universe pockets contain matter. If there is enough matter in the pocket it will form a black hole. In that case the rest of the pocket is will be cleared from its mass content. At the same time the size of the pocket may increase. This represents the expansion of the universe. Inside the pocket the holographic principle governs. The black hole represents the densest packaging mode of entropy. This indicates that in this region nature returns to its natal state.

The pockets may merge. Thus at last a very large part of the universe may return to its natal state, which is a state of densest packaging of entropy. The mass in the rest of the universe, which is positioned at a huge distance will enforce a uniform attraction. This uniform attraction will install an isotropic extension of the considered package. This will disturb the densest packaging quality of that package. The motor behind this is formed by the combination of the attraction through distant massive particles, which installs an isotropic expansion and the influence of the small scale random re-localization which is present even in the state of densest packaging. This re-localization is described by the quantum state functions of the packed particles.

The rupture of the package and the departure of its densest packaging state
represent a new episode in the history of the compartment. This describes an
eternal process that takes place in and between the pockets of an affine space.

The story described here is part of a much larger project with the name Hilbert Book Model. The Hilbert Book Model (HBM) is a simple model of fundamental physics that is strictly based on traditional quantum logic. This results in a quaternionic approach of quantum physics. On its turn it shows that quantum physics is a kind of fluid dynamics. The project runs since 2009 and its latest full description is treated in

http://vixra.org/abs/1211.0120

and is continuously updated and extended at

http://www.scitech.nl/English/Science/OnTheHierarchyOfObjects.pdf

The supporting mathematics are compiled in

http://vixra.org/abs/1210.0111

The site http://www.e-physics.eu is devoted to this project.

The pictures are taken from a slide presentation: http://www.e-physics.eu/HBM_Slides .

I am aware that the HBM is unconventional and in many aspects controversial.

The HBM departs in many aspects from contemporary physics. I found good reasons for doing this.

Its main target is the exploration of the undercrofts of physics.

I do my best to keep the HBM well founded and self-consistent.

I am open to proper (preferably well founded) criticism.

If you read "On the Hierarchy of Objects" then do not forget to read the conclusion.

In your theories, what are quaternions geometrically?

For example, how do they transform under arbitrary coordinate systems?

In what sense is the piece you refer to as a "scalar field" actually a scalar value?

Note:

Your last line is not displaying correctly for me. It ends abruptly after an integral sign. Also, the phrase "and a second surface integral that can be written as a surface integral" conveys no information.