Gravity and electromagnetism on conic sedenions
Introduction
In order to qualify and sufficiently support a quantum theory of gravitation on genuine conic sedenion arithmetic (from C. Musès’ hypernumbers program), it was first shown that the Dirac equation in physics is expressible on hyperbolic octonion arithmetic [1]. Rotation in the (1, i0) plane yields a counterpart on circular (Euclidean) geometry, which exhibits certain primitive properties of quantum gravity [2]. For large-body (non-quantum) physics, an alignment program was developed from an invariant modulus condition [3], and shown to be consistent with the General Relativity formalism for gravity.
This paper will conclude definition of the computational framework for a proposed quantum theory of gravitation and electromagnetism on hypernumbers, by supplying a physical force field to the formulation. An electromagnetic field will be added to the hyperbolic subalgebra and shown to be equivalent to current description in physics, supporting a conjecture by analogy for the circular subalgebra to describe quantum gravitational interaction.
Certain mixing effects become apparent at extreme energies, which cannot be separated into the individual constituent forces of gravity and electromagnetism anymore. This makes conic sedenion arithmetic a needed tool for further investigation into such effects.
Referring to the power orbit concept, it will be remarked that hypernumbers offer additional mathematical versatility to satisfy a generalized invariant hypernumber modulus theorem, beyond the description of gravity and electromagnetism herein.
Section snippets
Electromagnetism
The Dirac equation with electromagnetic field is a fundamental building block in describing dynamic interaction of spin 1/2 particles (like electrons or protons). In a simple form it can be written as(from [4] equation 32.1) and contains an operator and certain implicit summations. In order to map this relation to conic sedenion arithmetic as in [1], it will now be written explicitely
Gravity
In an identical procedure to electromagnetism above, the circular octonionic Dirac equation proposed for the description of gravity in [2] (relation 9 there) can be extended by a field eA0 using the definitionstoThis is obtained by demanding invariance of under the same transformation as (7) above,(with identical definition of
Mixing gravity and electromagnetism
The transformation leaves the modulus ∣ΨGr,EM∣ unchanged and introduces a physical force field Aμ which acts on the particle’s electric charge e. Depending on the mixing angle α, the effect of this force is either traditional electromagnetism (α = π/2), proposed quantum gravity (α = 0), or a combination thereof. Since a change in α generally changes ∣ΨGr,EM∣, α must remain constant under space-time coordinate transformation. Therefore, there are three constant properties of a spin
Conclusion and outlook
Conic sedenion arithmetic has been used to describe both the classical hyperbolic Dirac equation with electromagnetic field and a counterpart on circular geometry proposed for quantum gravity. The same field Aμ from electromagnetism is also generator of gravitational interaction in this description. The observed difference between the two forces is quantified through a mixing angle α, which becomes a third particle property next to its electrical charge e and mass at rest m. At experimentally
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