Helmholtz-Hodge Theorems: Unification of Integration and Decomposition Perspectives

Jose G. Vargas, PST Associates, United States

We develop a Helmholtz-like theorem for differential forms in Euclidean space $E_n$ using a uniqueness theorem similar to the one for vector fields. We then apply it to Riemannian manifolds, $R_n$, which, by virtue of the Schlafli-Janet-Cartan theorem of embedding, are here considered as hypersurfaces in $E_N$ with $N$ greater than or equal to $n(n+1)/2$. We obtain a Hodge decomposition theorem that includes and goes beyond the original one, since it specifies the terms of the decomposition. We then view the same issue from a perspective of integrability of the system $d\alpha = \mu$, $\delta \alpha = \nu$ relating boundary conditions to solutions of $d\alpha=0$, $\delta \alpha = 0$, [$\delta$ is what goes by the names of divergence and co-derivative, inappropriate for the Kaehler calculus, which has been used to obtain the foregoing].

$U(1) \times SU(2) \times SU(3)$ from the Tangent Bundle

Jose G. Vargas, PST Associates, United States

In this paper, we revisit E. Kaehler's treatment of Poincare symmetry as it reflects itself in members of the left ideals —in the Clifford algebra of scalar-valued differential forms— generated by primitive idempotents for time translation and third component of angular momentum symmetry (read rest mass and spin). We replace that algebra with a Clifford algebra of Clifford-valued differential forms that makes the unit imaginary unnecessary. It has a commutative subalgebra of “mirror elements”, i.e. where the valuedness is dual to the differential form and where even the idempotents for components of angular momentum commute. In Kaehler's treatment, total angular momentum is a differential $2$-form operator (not a vector operator!). One need not first square it before using it. Of course, one reproduces standard results from repeated application of the operator. Additional subtleties and enhanced computational power follow from reproducing his treatmen in the commutative subalgebra. The so enriched theory of angular momentum and of primitive idempotents becomes the foundation for an interplay of all three components of angular momentum and time and space translations. The expansion of the set of primitive idempotents uncovers an algebraic representation of quarks, which does not mean, however, similar behavior for the new members of that set. Quarks are "different type particles". An algebraic stoichiometry of fundamental particles emerges in that commutative algebra. It includes neutron decay and pair creation-annihilation. The geometric nature of the algebras involved makes obvious why there are only three generations with two flavors each, only three colors and why the triple of electron, muon and tau are respectively associated with the three generations.