The Helmholtzian Operator and Maxwell-Cassano Equations of an Electromagnetic-nuclear Field

Affiliation: Santa Clara, California, USA * Corresponding author: Claude Michael Cassano, Santa Clara, California, USA, Tel: 1-650-603-1264, E-mail: cloudmichael@hotmail.com Citation: Cassano CM. The helmholtzian operator and maxwell-cassano equations of an electromagnetic-nuclear field (2019) Edelweiss Appli Sci Tech 3: 8-18 Received: Dec 10, 2018 Accepted: Jan 29, 2019 Published: Feb 12, 2019 Copyright: © 2019 Cassano CM. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Abstract This article demonstrates that when acting on a four-vector doublet, the Helmholtzian may be factored into two 4×4/8×8 differential matrices, resulting in a four-vector doublet Klein-Gordon equation with source. This factorization enables yielding of a mass-generalized set of Maxwells equations.


Introduction
The He1mholtz differential equation [1][2][3] is a linear second order differential equation, generalization of the wave equation.A Helmholtzian operator is a linear second order differential operator, typically in four independant doublet-variables; a generalization of the d'Alembertian operator [4] where it's additional constant vanishes.When the time-independant version (in three independent (space) variables acting on a function or vector vanishes, the resulting equation is called the Helmholtz's equation.In four dimensions this equation is referred to as the Klein-Gordon equation [5][6][7][8][9] (with imaginary constant).
(Because the Klein-Gordon equation is the four-dimensional generalization of the three-dimensional Helmholtz equation; and each, of course may be generalized to higher dimensions, I have chosen to denote the operator with the Helmholtz designation).This article demonstrates that when acting on a four-vector doublet, the Helmholtzian may be factored into two 4×4/8 × 8 differentials matrices in two distinct ways, as follows: Citation: Cassano CM.The helmholtzian operator and maxwell-cassano equations of an electromagneticnuclear field (2019) Edelweiss Appli Sci Tech 3: 8-18 9 Where: ( ), ( ), ,

Discussion
It doesn't take much more than a cursory look to see that this Helmholtzian operator and factorization is a generalization of the d'Alembertian operator and its factorization [4].
The four-vector-doublet Klein-Gordon equation may be written as a matrix product.This, thus, when operated on a four-vector-doublet, gives a matrix product definition for the Helmholtzian operator.(The scalar Klein-Gordon equation is a special case of this four-vectordoublet version, where there are three restrictions on the four component-doublets of the potential A , leaving the single independent function-pair.)(Note that the Dirac equation [5][6][7][8][9] is a set of equations on a scalar doublet; so there is, in this way, Helmholtzian factorization consistency.)(And,there is a deeply intimate relationship with the Dirac equation).
Another great thing about using this description of the Helmholtzian is that if the matrix is applied to the column vector, the result may be expressed in terms of generalized E and B vector components associated with generalized electromagnetic field (with the appropriate definitions of A0 and x0 )(just as with the d'Alembertian operator).And, when the final matrix is applied, the result is mass-generalized.
and similar for the other factorization.
Citation: Cassano CM.The helmholtzian operator and maxwell-cassano equations of an electromagneticnuclear field (2019) Edelweiss Appli Sci Tech 3: 8-18 13 Additionally, the above process may be generalized for any dimension power of two, thus generalizing the Helmholtzian (and, thus the d'Alembertian) as well as the mass-generalized Maxwell's equations to any dimension power of two (using the weighted matrix product to construct suitable algebras and doing the same analysis on it).
Comparing to the above [12] d'Alembertian factorization, a massgeneralization of Maxwell's equations is a clear result.However, they are in a form beyond standard notation; so expanding them is in order.
  Citation: Cassano CM.The helmholtzian operator and maxwell-cassano equations of an electromagneticnuclear field (2019) Edelweiss Appli Sci Tech 3: 8-18 15 Citation: Cassano CM.The helmholtzian operator and maxwell-cassano equations of an electromagneticnuclear field (2019) Edelweiss Appli Sci Tech 3: 8-18 16 Since the mass constituents/components of anti-fermions are the negative of their corresponding fermion, these may be simply written: Equations ( 15) may be subtracted to yield: Citation: Cassano CM.The helmholtzian operator and maxwell-cassano equations of an electromagneticnuclear field (2019) Edelweiss Appli Sci Tech 3: 8-18 17 20)Citation: Cassano CM.The helmholtzian operator and maxwell-cassano equations of an electromagneticnuclear field (2019) Edelweiss Appli Sci Tech 3: 8 the mass constituents/components of anti-fermions are the negative of their corresponding fermion, these may be simply written: generalized Maxwell's equations may be simply written: