Possibly related scalar field ideas:

There is a more advanced paper that follows on from that of the last post. (K.J. van Vlaenderen’s, Generalised Classical Electrodynamics for the prediction of scalar field effects)

Electrodynamics with the scalar field

This paper goes into more of the mathematical backing through the use of quaternion and biquaternion form.

Through the use of quaternion form it defines the scalar field/seventh field component S as;


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There is another paper that is similar to the above; it discusses a new field component being revealed, and it also uses quaternion form to derive this field.

Physical Space as a Quaternion Structure, I:
Maxwell Equations. A Brief Note.

Here the scalar field T is defined as;


There are clear similarities between the theories; not only from the definition of the scalar field, but also the prediction of “thermoscalar” effects.

I’m continuing to look into these theories and familiarising myself with the mathematics involved.

Wireless Transmission of Power and related fields

This project has been inspired by Nikola Tesla and his various inventions. A quick search of the Internet will provide a vast amount of information on his work and possible related fields, a lot of this information is clouded with inaccuracies mainly caused by speculation and those passionate about conspiracy theories. In my approach to this topic I will try to be objective and pragmatic, this will lead me to be sceptical of “fact” from both the classical and alternative schools of thought, yet to remain open and analytical of both approaches. This will lead me to not only analyse the mathematics behind theories but also the conceptual substance that they are based; from this theories with sound conceptual properties that may lack mathematical completeness will still be considered (if not a little more sceptically).

As my investigation has progressed somewhat now, I’ve decided to post my findings thus far:

This is an extract from one of Tesla’s patents that I think is of significance:
N. Tesla: Apparatus for the Transmission of Electrical Energy
U.S. Patent #: 649,621 Patented May 15, 1900

“It is to be noted that the phenomenon here involved in the transmission of electrical energy is one of true conduction and is not to be confounded with the phenomenon of the electrical radiation which from the very nature and mode of propagation would render practically impossible the transmission of any appreciable amount of energy to such distances as are of practical importance”

Here he seems to be explicitly stating that he does not use regular EM radiation. Here I shall pose a question, could he mean something like displacement current?
Displacement Current


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Here is an article I've found on the Internet, it has not been peer reviewed, but its contents are very intriguing, as it has many examples of observational effects to back the theory, including an explanation of what Tesla may have been referring to.
I am currently looking into this paper and have contacted the author:

Generalised Classical Electrodynamics
for the prediction of scalar field effects

In this paper it refers to gauge invariance and conditions in relation to the vector (magnetic) and scalar (electric) potentials in classical electromagnetics, these conditions (Lorentz gauge for example) are assumptions made to simplify the mathematics.

Electromagnetism by Pollack & Stump:
p406:
“In contrast, the potentials are introduced as a mathematical convenience; at least that is true in classical electromagnetism. *”
At bottom of page “* The Bohm-Aharanov effect in quantum electrodynamics is a physical effect that is directly related to the vector potential A.”

Aharonov-Bohm effect

The Aharonov-Bohm effect is very interesting as it detects effects that are non-existent in classical electromagnetism, and make the potentials more than just a “mathematical convenience”.

p561:
“Also, it is important in radiation theory to impose upon A(x,t) and V(x,t) the Lorentz gauge condition,”


Here I will draw your attention to the phrase “important in radiation theory”, this may be neglected if in fact we are not dealing with "radiation" of the usual sense.

I've also worked through the first 12 equations from this paper, they do seem to be in accordance to classical EM, with the exception of the extra 7th field component.
Equations 1-12