In the first decades of the twentieth century Minkowski introduced the field strength tensor. This tensor raises several false suggestions. In order to elucidate this criticism, it is necessary to compare quaternionic differential calculus with Maxwell based differential calculus.

BASIC FIELDS

Basic fields exist whose differentials give existence to secondary fields. Basic fields can be affected by artifacts, which cause local discontinuities into the basic field. The reason of existence of these artifacts is not treated here. The electric field and the magnetic field are examples of secondary fields. The electric scalar potential and the electric vector potential are examples of basic fields. In fact they are components of the same basic electro field. Electric charges form the artifacts that deform the basic electro field. The field that represents our deformable living space is also an example of a basic field. Massive elementary objects form the artifacts that deform our living space.

Scalar basic fields and corresponding basic vector fields can be combined in a single basic quaternionic field. Apart from a countable set of localized discontinuities these basic fields are continuous. By selecting appropriate parameter spaces, the basic fields can be described by multidimensional functions. Different parameter spaces correspond to different functions.

Quaternionic basic fields can be stored in a structured way into the eigenspaces of corresponding operators that reside in a non-separable quaternionic Hilbert space. There the fields exist independent of the parameter spaces that connect them to the functions. However, quaternionic number systems provide natural parameter spaces. The problem with such natural parameter spaces is that due to the four dimensions of quaternions several versions of quaternionic number systems exist that differ in the way that their elements are ordered.

DIFFERENTIAL EQUATIONS

In this text, two sets of differential equations will be given that correspond to two different choices of parameter spaces. One set holds for a parameter space that corresponds to a Cartesian coordinate system. Once the axes are selected, still eight choices for the ordering on these axes can be defined. One of these choices is taken. Further, the real part of the quaternions are taken as part of the parameters in the first set of differential equations. In the second set instead local quaternionic distance is taken as part of the parameter space. Both sets describe the behavior of the same basic field! Great similarities, but also significant differences between these two sets of equations exist. The differences culminate in the two second order partial differential equations.

Tensor calculus is applied to relate for a given field the differential equations and the selected parameter spaces.

QUATERNIONIC VERSUS MAXWELL DIFFERENTIAL CALCULUS

QUATERNIONIC DIFFERENTIAL CALCULUS

In quaternionic differential calculus the differential operator acts as a multiplier.

Further, the real part acts as the scalar part of the parameter space. The imaginary part acts as the vector part of the parameter space. The real part is marked with subscript ₀.

▽ is the quaternionic nabla operator. ∇₀ is its real part and ⩢ is its vector part.

We will use ⊛ as a sign switch. ⊛ equals 1 for quaternionic equations. ⊛ equals −1 for Maxwell equations.

ϕ ≡ ϕ₀+Φ = ▽φ ≡ (∇₀+⩢)(φ₀+φ) = ∇₀φ₀−〈⩢,φ〉+⩢φ₀+∇₀φ ± ⩢×φ

ϕ₀ = ∇₀φ₀−⊛ 〈⩢,φ〉 ; ⊛=1

Φ = ⩢φ₀+∇₀φ ± ⩢×φ

The next equations define terms of the vector part of the differential

ℰ ≡ −⩢φ₀−∇₀φ

ℬ ≡ ⩢×φ

∇₀ℬ = ⩢× ℰ

〈⩢,ℬ〉 = 0

∇₀ℰ = −∇₀⩢φ₀−∇₀∇₀φ

〈⩢,ℰ〉 = 〈⩢,⩢〉φ₀+∇₀〈⩢,φ〉

∇₀ϕ₀ =∇₀∇₀φ₀−⊛ ∇₀〈⩢,φ〉

⩢ϕ₀ =⩢∇₀φ₀−⊛ ⩢〈⩢,φ〉=⩢∇₀φ₀−⊛ ⩢×⩢×φ+⊛ 〈⩢,⩢〉φ

This results in the quaternionic second order partial differential equations:

γ ≡ γ₀+Υ = ▽▽ ̽φ ≡ (∇₀+⩢)(∇₀−⩢) (φ₀+φ) = (∇₀∇₀+〈⩢,⩢〉)φ

∇₀∇₀φ₀+⊛ 〈⩢,⩢〉φ₀ =∇₀ϕ₀+⊛ 〈⩢,ℰ〉

∇₀∇₀φ+⊛ 〈⩢,⩢〉φ =−∇₀ℰ−∇₀⩢φ₀−⩢ϕ₀+⩢∇₀φ₀+⊛ ⩢×⩢×φ =−∇₀ℰ−⩢ϕ₀+⊛ ⩢×ℬ

γ₀=∇₀ϕ₀+⊛ 〈⩢,ℰ〉

Υ=−∇₀ℰ−⩢ϕ₀−⊛ ⩢× ℬ

The quaternionic second order partial differential equations are not wave equations! The homogeneous form of these equations do not offer waves as solutions. Instead these homogeneous second order partial differential equations offer shape keeping fronts as solutions.

MAXWELL BASED DIFFERENTIAL CALCULUS

In Maxwell differential calculus the sum of a scalar and a vector is not supported. Thus, the differential operator cannot be used as a multiplier. Further, the local four dimensional distance acts as the scalar part of the parameter space. The spatial part acts as the vector part of the parameter space.

ϕ ↔ {ϕ₀,Φ} ↔ {∇₀,⩢}{φ₀,φ}

ϕ₀ =∇₀φ₀−〈⩢,φ〉 is not a genuine Maxwell equation. Instead we use the gauge

ϰ = ∇₀φ₀−⊛ 〈⩢,φ〉 ; ⊛=−1

or we use ϰ as the scalar part of the differential. In that case it acts as an additional differential equation. Here the sign switch signals a different sign!

Apart from the different sign ⊛=−1, all other equations are the same.

This results in the WAVE equations:

∇₀∇₀φ₀+⊛ 〈⩢,⩢〉φ₀ = ∇₀ϰ−⊛ 〈⩢,ℰ〉

∇₀∇₀φ+ ⊛ 〈⩢,⩢〉φ = −∇₀ℰ−⩢ϰ−⊛ ⩢×ℬ

γ₀=∇₀ϕ₀ ⊛ 〈⩢,ℰ〉+∇₀ϰ

Υ=−∇₀ℰ−⊛ ⩢× ℬ−⩢ϰ

Also here ϰ generates extra terms!

Thus, most of the Maxwell based equations are similar to the equivalent quaternionic equations. In some cases a sign difference exists. This is indicated by the sign switch ⊛.

TENSOR CALCULUS

Tensor calculus has provided means to relate the parameter space selections with the describing functions. This especially holds for the differentials of the fields.

The definition of the field strength tensor exists already for about a century and during its existence this tensor has raised the false suggestion that the underlying fields are asymmetric by nature. It also hides the potential existence of some naturally symmetric secondary fields. It hides the fact that besides the secondary vector fields another field exists that presents the scalar part of the differential of the basic field. That field has not yet got a name. Since the scalar potential and the vector potential exist, will also the derived symmetric secondary fields exist. The vector parts of the differential of this basic electro field deliver the electric field and the magnetic field. The same occurs for other basic fields!

Next the field strength tensor is also called the electromagnetic tensor. This second name surpasses the fact that the field that is treated by the tensor has a far more general nature than the electromagnetic field. The underlying differential equations hold as well for other fields, such as the field that represents our deformable living space.

FIELD STRENGTH TENSOR

The field strength tensor is defined as

Fᵤᵥ=∂ᵤAᵥ−∂ᵥAᵤ

This is a sensible choice for the ℬ field. Its components are naturally asymmetric functions of partial differentials. However, especially in quaternionic differential calculus, the members of the ℰ field do not naturally correspond to

ℰᵥ=∂₀Aᵥ−∂ᵥA₀

That is only the case when mixed differentials are treated. The choice is arbitrary.

∇₀φ+⩢φ₀↔∇₀A−⩢A₀ or ∇₀φ−⩢φ₀↔∇₀A+⩢A₀

http://vixra.org/abs/1511.0007