1.8.7 Summary

Jason Glowney

Relativistic mechanics when presented as the $e^{\mathrm{A} + i\Omega}$ generator product reveals it fundamentally to be a consequence of the hyperbolic geometry inherent to the Unified Field. This hyperbolic geometry is a direct result of the real-valued portion $\alpha$ of the Lie algebra representation of the form and the anti-form

\begin{array}{r} \mathfrak{f}\left( \mathcal{z} \right)= \mathfrak{f}(\alpha + i\omega), \end{array}

(1.8.453)

\begin{array}{r} \overline{\mathfrak{f}(z)}= \mathfrak{f}( - \alpha - i\omega), \end{array}

(1.8.454)

which, as algebras, both lie on the boundary of the unit singularity. The form and the anti-form group generators are the result of the matrix exponentiation of these Lie algebras, and all commutative groups that are the result of the matrix exponential of their respective Lie algebra will take the form of our example

\begin{array}{r} e^{\mathrm{A}}e^{i\Omega} = e^{\mathrm{A} + i\Omega} = \begin{pmatrix} e^{\alpha} & 0 & 0 \\ 0 & \cosh(\alpha) & \sinh(\alpha) \\ 0 & \sinh(\alpha) & \cosh(\alpha) \end{pmatrix}\begin{pmatrix} e^{i\omega} & 0 & 0 \\ 0 & \cos(\omega) & {isin}(\omega) \\ 0 & {isin}(\omega) & \cos(\omega) \end{pmatrix}, \end{array}

(1.8.455)

revealing every group is a balance between the degree of its hyperbolic and spherical geometry. In the above, when $\mathcal{a} = 0$ , the hyperbolic matrix reduces to the identity matrix. This case will yield a purely spherical result in geometry and is the equivalent case of a massless object system not bound by relativistic mechanics. In the case when $\beta = 0$ , the result is a system purely hyperbolic in geometry and equal to $E = mc^{2}.$

For generators with variables $\alpha,\omega \neq 0$ , cohesive mathematics shows us the interplay between hyperbolic and spherical geometries inherent in the $e^{\mathrm{A} + i\Omega}$ generator products underlie the wave-particle duality concept in quantum mechanics. Innate in the $e^{i\Omega}$ the portion of the generator is spherical geometry, which confers wavelike properties to the object system, while the hyperbolic geometry of the $e^{\mathrm{A}}$ portion gives the object system its real-valued and observable properties in spacetime. The $e^{\mathrm{A} + i\Omega}$ generator acts on a coordinate vector, a zero-dimensional point in the Unified Field's geometric space, and this generator-coordinate vector action is the basis for excitations in the field. These excitations in geometric space are coordinate-based and account for the observed particle-like properties of the object system.

The form/anti-form generator products are closely related to symplectic geometry, defined by a smooth and differentiable even-dimensional manifold, and it traces its origins to the Hamilton classical mechanics formulation. Symplectic geometry functions similarly to that of the metric tensor in Riemann geometry. It defines the phase space of a system, which is central to dynamical systems, representing all the possible states of a system. The phase space of a mechanical system will typically use position and momentum variables to represent all its possible states.

1.8.7 Summary

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