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Revisiting the Klein-Gordon equation

With the Klein-Gordon equation (short: KGE; please also read the article on Wikipedia-GB, extern link), we get a new possible application for the intrinsic method.

This opportunity does not appear immediately. It is merely the result of an inverse logical analysis of my seminal work concerning the non-trivial decompositions of deformed cross products; see [a].

The first part of the demonstration in [a] ends with a so-called “initial theorem.” That theorem says that: “Any non-trivial decomposition is irremediably associated with a proper polynomial of degree two depending on the three components of the first argument involved in that product (i. e.: the projectile in my semantic).”

 

This theorem suggests the existence of an opposite logical way. Namely: if we dispose of such a polynomial, then we may suspect the eventual presence of a deformed cross product which has been decomposed in a non-trivial manner. I improve this way of thinking in the document titled: “The Klein-Gordon Equation revisited.”

In fact, in that document, I go a little bit further and do not only apply the results obtained in [a]. I cross them also with the consequences arising from the extrinsic method, [b], and get a new exciting insight on the problematic. I also note that this further information suggests a possibility to represent the spinors in M(3, C). They also indirectly indicate that the polarization tensor can be written in that set, instead than in SU(2) -extern link: Wikipedia GB.

 

In that document, I do not emit any judgment on the utility of the Klein-Gordon equation. Neither do I try to indicate a way to circumvent the real difficulties concerning it (negative probability, etc.).

This work should be read as an exercise of application for [a] and [b].

Anyway, I give indications for particular applications in that sense that calculations prove that any KG plane wave is theoretically Lense-Thirring effect (See article on Wikipedia GB, extern link) sensitive. Since KGE are describing massive waves, this fact is not surprising.

 

Personal contributions

[a] https://cordes-cosmiques.fr/mathematical-methods/intrinsic-method/

[b] https://cordes-cosmiques.fr/mathematical-methods/extrinsic-method/

 

© Thierry PERIAT, 20 March 2019.