The Klein-Gordon equation for free fields in a four dimensional context. The scientific document has been updated – 02 May 2019.

The dissertation is now reaching the domain in which mathematics and physics are meeting together.

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I already have examined the KGE in [a]. I did it in involving a 3 + 1 approach and the intrinsic method exposed in [b]. In that new document, following a new strategy, I shall apply the extrinsic method in a full four-dimensional context.


The Klein-Gordon equation for free fields is recalled. The dispersion relation in vacuum is re-obtained. It describes two families of particles: (i) massless particles (m = 0), in which case the impulse-energy vector p is isotropic; (ii) non-massless particles (m ¹ 0), in which case that dispersion relation represents a unit sphere. The normalized speed V = u/c can be parametrized with the Euler-Rodrigues formula. The particle can then be described with the help of a matrix in M(3, C) depending on the spatial part of the normalized speed and on the ratio between its energy, E, and its energy at rest: m.c2.


The discussion is questioning if that dispersion relation is signing the presence of a generic ÄA(, p) deformed tensor product where neither the cube A nor the projectile are known in the initial formulation of the theory. That generic product describes a whole set of deformed tensor products which, in peculiar, is including the gravitational term ÄG(u, p) appearing in the Lorentz-Einstein law (LEL).


Answers to the questioning are progressively gained with the help of the extrinsic method which I expose extensively again for the pedagogy at the beginning of the document. The answers are then organized as systematically as possible in considering all logical possible categories of situations. Beside a meaningless impulse-energy vector (configuration 1), there are two main categories: configuration 2, the deformed tensor product is decomposed exactly: configuration 3, the deformed tensor product is not decomposed exactly, and the error is the particle itself.


Personal works

[a] PERIAT, T.: Revisiting the Klein-Gordon Equation (in a three-dimensional context); ISBN 978-2-36923-079-3, 20 March 2019.

[b] PERIAT, T.: Decompositions of deformed Lie products (in a three-dimensional context); ISBN 978-2-36923-084-7, v1, 31 January 2016.


© Thierry PERIAT, 25 April 2019.



The document below is written in English, has 24 pages and 19 references.