Quaternionic Fields

Quaternionic Field Equations

This page is currently being adapted by User:HansVanLeunen

Introduction

Maxwell equations apply the three-dimensional nabla operator in combination with a time derivative that applies coordinate time. The Maxwell equations derive from results of experiments. For that reason, those equations contain physical units.

In this treatment, the quaternionic partial differential equations apply the quaternionic nabla. The equations do not derive from the results of experiments. Instead, the formulas apply the fact that the quaternionic nabla behaves as a quaternionic multiplying operator. The corresponding formulas do not contain physical units. This approach generates essential differences between Maxwell field equations and quaternionic partial differential equations.

The quaternionic partial differential equations form a complete and self-consistent set. They use the properties of the three-dimensional spatial nabla. The corresponding formulas are taken from Bo Thidé's EMTF book., section Appendix F4. Another online resource is Vector_calculus_identities.

The quaternionic partial differential equations do not change the data format. The format of the information that the field transmits to observers, which the field embeds is affected by the information transfer. Instead of the Euclidean storage format, which governs at the location of the observed event, the observers perceive a spacetime format, which features a Minkowski signature. The Lorentz transform describes the format conversion.

Maxwell equations use coordinate time, where quaternionic differential equations use proper time.

Regarding quaternions, the norm of the quaternion plays the role of coordinate time. These time values apply not in their absolute versions. Thus, only time intervals apply.

Hilbert spaces can only cope with number systems that are division rings. In a division ring, all non-zero members own a unique inverse.

Only three suitable division rings exist. These are the real numbers, the complex numbers, and the quaternions.

Thus dynamic geometric data that are characterized by a Minkowski signature must first be dismantled into real numbers before they can serve in a Hilbert space.

Quaternions can store and retrieve without dismantling.

Quantum physicists use Hilbert spaces for the modeling of their theory. However, most quantum physicists apply complex number based Hilbert spaces.

Quaternionic quantum mechanics appears to represent a natural choice.

Format conversion

A read-only repository in the form of the combination of a quaternionic infinite dimensional separable Hilbert space and its non-separable companion stores the dynamic geometric data that constitute the observed event in a Euclidean format in the form of combinations of a timestamp and a three-dimensional spatial location. Quaternions act as storage containers. A private timestamp and a central spatial location characterize the observer. The observer can only access storage locations whose timestamp predates his own timestamp. A continuum transfers this information to the observer. The speed of information transfer of the continuum is fixed. Therefore, the information transfer affects the format of the information that the observer perceives. A non-zero speed difference between observed event and observer will contract observed lengths will dilate durations. The Lorentz transform is a hyperbolic transform that describes the format conversion.

Quaternionic differential calculus describes the interaction between discrete objects and the continuum at the location where events occur. Converting the results of this calculus by the Lorentz transform will describe the information that the observers perceive. Observers perceive in spacetime format. This format features a Minkowski signature. The Lorentz transform converts from the Euclidean storage format at the situation of the observed event to the perceived spacetime format.

Storage model

In this model, the instant of storage of the event data is irrelevant as long as it precedes the stored time stamp. Thus the model can store all data at an instant, which precedes all stored timestamp values. This impersonates the model as a creator of the universe in which the observable events and the observers exist.

The repository merges Hilbert space operator technology with quaternionic function theory and quaternionic differential and integral calculus. The separable Hilbert space typically stores the discrete quaternionic data. These can occur as spurious data, as coherent swarms or as ordered distributions. Coordinate systems can order dense coherent swarms, which then become ordered distributions. Location density distributions can describe these ordered swarms. The non-separable Hilbert space embeds the separable Hilbert space, and in this way, the data sets become part of the non-separable Hilbert space. The non-separable Hilbert space stores continuums. In the non-separable Hilbert space, quaternionic functions describe continuums. The coherent swarms can embed in a continuum. The embedding process involves a convolution of the location density distribution of the coherent swarm with the Green's function of the continuum. Differential equations describe the behavior of the continuums. In this page, we only consider continuums that mostly continuous quaternionic functions can describe.

Fields

In the Hilbert Book Model fields are eigenspaces of operators that reside in the non-separable Hilbert space. Continuous or mostly continuous functions define these operators and apart from some discrepant regions their eigenspaces are continuums. These regions might reduce to single discrepant pointlike artefacts. The parameter spaces of these functions are constituted by quaternionic number systems. Consequently the real number valued coefficients of these parameters are mutually independent and the differential change can be expressed in terms of a linear combination of partial differentials. Now the total differential change $df$ of field $f$ equals

$df={\frac {\partial {f}}{\partial {\tau }}}d\tau +{\frac {\partial {f}}{\partial {x}}}{\vec {i}}dx+{\frac {\partial {f}}{\partial {y}}}{\vec {j}}dy+{\frac {\partial {f}}{\partial {z}}}{\vec {k}}dz$

(1)

In this equation, the partial differentials ${\frac {\partial {f}}{\partial {\tau }}},{\frac {\partial {f}}{\partial {x}}},{\frac {\partial {f}}{\partial {y}}},{\frac {\partial {f}}{\partial {z}}}$ are quaternions. The quaternionic nabla $\nabla$ assumes the special condition that partial differentials direct along the axes of the Cartesian coordinate system. Thus

$\nabla =\sum _{i=1}^{n}{\vec {e}}_{i}{\partial \over \partial x_{i}}={\frac {\partial {}}{\partial {\tau }}}+{\vec {i}}{\frac {\partial {}}{\partial {x}}}+{\vec {j}}{\frac {\partial {}}{\partial {y}}}+{\vec {k}}{\frac {\partial {}}{\partial {z}}}$

(2)