9.2 Second Order Euler-Lagrange Equations

One of the Claims of Geometric Unity is that we have been unsuccessful in Unifying the four basic equations for Gravity, Non-Gravitational force, Matter and Higgs phenomena because they belong to a Dirac Pair. That is, we believe that the Einstein and Dirac equations belong to a unifying equation which is in the sense of Dirac something of a square root of a different equation or Lagrangian related to the Yang-Mills-Maxwell equation and the Higgs version of the Klein-Gordon equation. Thus we should seek to unifying our equations and Lagrangians much the way Dirac unified first and second order equations with his masterstroke To this end we focus on second Lagrangian of 2nd order:

9.3 The Fermionic Sector

In the case of the Fermionic content we can at the classical level take fields ν, ¯ ¯ζ, ν, ζ on Y to be four distinct fields when ultimately we will wish to integrate out the Dirac like operator to take a Berezinian ‘integral’ in the quantum theory.

noting that other versions of the theory exist including one with a non-trivial map in the lower right quadrant of the operator. This two can be made to look closer to the Dirac Theory of Spinorial Fermions:

where χ contains three generations of observed Fermions as well as LookingGlass matter, dark Spinorial Matter, Rarita-Schwinger matter and more while D subsumes the Dirac Operators, and the various subfields of ω accomodate the functionings of the CKM matrix, the Higgs-Like soft mass fields, the Yukawa couplings, Gauge Potentials and the like.

Let us compile the Bosonic and Fermionic variations of the Spinorial Lagrangian terms in a single term:

as a kind of 1-form on some SuperSpace-like structure over A:

which can be combined with the variation of either a first or second order purely Bosonic Lagrangian so as to form:

At this point, we wish to take this mixed spinorial-tensorial Υω and ask whether we are attempting to penalize this expression in our extremization because it is actually the obstruction term for a cohomology theory.

That is, we choose to view ‘wedging’ with Υω as the application of a zeroth order operator and ask if it possesses a non-trivial square root the in form of a first order differential operator δ ω so that:

to get a Lagrangian Cohomology theory of at least two steps:

of a geometrically meaningful cohomology complex.