Research Article :
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 Maxwell s equations. The Helmholtz differential equation [1-3] 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 dAlembertian
operator [4] where its 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 Helmholtzs equation. In four
dimensions this equation is referred to as the Klein-Gordon equation [5-9]
(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: It doesnt take much
more than a cursory look to see that this Helmholtzian operator and
factorization is a generalization of the dAlembertian 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-vector-doublet 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 [5-9] 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 dAlembertian operator). And, when the final matrix is
applied, the result is mass-generalized. Maxwells inhomogeneous field equations [10-15] (with the
gauge fixing term)(because the homogeneous field equations are identities,
which actually appear as such in the final computation by all those terms
canceling each other out). So, this is an incredibly compact way of writing
both the Klein-Gordon equation and mass-generalized Maxwells equations of an
electromagnetic-nuclear field [10-15]. Additionally, the above process may be generalized for any
dimension power of two, thus generalizing the Helmholtzian (and, thus the dAlembertian)
as well as the mass-generalized Maxwells 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] dAlembertian factorization, a
mass-generalization of Maxwells equations is a clear result. 1.
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Formulas, Graphs and Mathematical Tables (1964) Dover Publications, New
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https://en.wikipedia.org/wiki/Hermann_von_Helmholtz 4.
Cassano CM. The dAlembertian operator and Maxwells equations
(2018) J Mod Appl Phys 2: 26-28. 5.
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https://en.wikipedia.org/wiki/Klein%E2%80%93Gordon_equation 10. Kovetz
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G. The Cambridge Handbook of Physics
Formulas (2000) Cambridge University Press, New York, USA. 12. Feynman
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https://en.wikipedia.org/wiki/Maxwell%27s_equations
The Helmholtzian Operator and Maxwell-Cassano Equations of an Electromagnetic-nuclear Field
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