Gravity and electromagnetism on conic sedenions

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Abstract

Conic sedenions from C. Musès’ hypernumbers program are able to express the Dirac equation in physics through their hyperbolic subalgebra, together with a counterpart on circular geometry that has earlier been proposed for description of gravity. An electromagnetic field will now be added to this formulation and shown to be equivalent to current description in physics. With use of an invariant hypernumber modulus condition, a description of quantum gravity with field will be derived. The resulting geometry reduces in very good approximation to relations expressible through customary tensor algebra. However, deviations are apparent at extreme energy levels, as shortly after the Big Bang, that require genuine conic sedenion arithmetic for their correct description. This is offered as method for exploration into bound quantum states, which are not directly observable in the experiment at this time. Extendibility of the invariant modulus condition to higher hypernumber levels promises mathematical flexibility beyond gravity and electromagnetism.

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 asγpˆ-eA-mΨ=0,(from [4] equation 32.1) and contains an operator pˆ 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 definitionscon16Gr(0,0,0,0,0,3,-2,1,0,eA0,0,0,0,eA3,-eA2,eA1),Ψcon16Gr(ψ0r,ψ0i,ψ1r,ψ1i,-ψ2r,ψ2i,ψ3r,ψ3i,0,0,0,0,0,0,0,0),to(con16Gr-m)Ψcon16Gr=0.This is obtained by demanding invariance of Ψcon16Gr under the same transformation as (7) above,Ψcon16GrΨcon16Grei0χ,(with identical definition of

Mixing gravity and electromagnetism

The transformation ΨGr,EMΨGr,EMei0χ 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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