Electron spin, nutation, and wave-particle duality

We present a diagonal spacetime 4-manifold to model the wave-particle duality with two derivations of the energy ratio of wave to particle to be 1 to 3, by which we calculate the least time for an electron nutation as based on its quadrupole spin states in a 720-degree rotation of its associated electromagnetic wave in a model of Zitterbewegung .


Introduction and motivation
The pursuit of ever faster computing speed in juxtaposition with an increasingly acute environmental concern has motivated the search for more efficient uses of electricity, such as through a better manipulation of the electron spin, as the information (0,1)-bits correspond to the magnetic dipoles, so that the higher the speed of electron nutation, the quicker the information processing (cf. [1,2]). Meanwhile, nano science with its engagement in electromagnetic waves has been more and more entwined with quantum mechanics, resulting in a mixed field of quantum nanoscience, e.g., [3,4]. Yet herein lies a geometric incongruence: whereas spin polarization as practiced by the former targets the physical 3-space, e.g., [5,6], the construct of spin as originated from Dirac's spinor of period 4π 3 4π turn (cf. [1,2,10] for electric quadrupole): that is, relative to a fixed reference frame, its associated wave begins with a semi-circular rotation from ( )  where the first term corresponds to the particle of m and the second term, the wave of m; assuming the energy ratio of particle to wave being 3:1(to be derived below), then one has This shows that an electron by way of Breit-Wheeler pair production may be described as an electromagnetic wave of length λ spinning along a pair of perpendicular semicircles, stopping at the two intersection points (for altering the angular momentum, cf. Zitterbewegung [12]), whence slowing down the otherwise speed c to 0.66c [9]. Therefore, at either stopping point the duration t ∆ is 34 M as wave M as particle ≡ dictates an energy entity M that has dual representations, hence the wave-particle duality, and either has a Euclidean ambient spacetime. In the following we shall derive the energy distribution of M .

Derivations for wave-particle energy ratio
We begin with an evaluation of gravitational effect of the waves in B :  The above geometry arises from a departure from Dirac's square-root approach to the mass-shell equation by taking instead the complex conjugates as in [14]; in doing so, the abstract Clifford algebra as involved in the spinor solution underlying the Standard Model (SM) is avoided (see [15] for the comment on quantum mechanics being 'very ungeometric') and the construct of intrinsic spin is replaced with a pair of perpendicular semi-circular rotations in the Euclidean space (cf. [14]). Here, amid many re-examinations of SM, it is noteworthy that the working equations for the strong force in particular are reckoned as 'QCD inspired' rather than derived [16], which has great significance however: it means that all the experimentally verified algebraic results continue to be of engineering value, as the above introduced spacetime provides only a geometric background. In the same vein, gauge invariance specializes to frame invariance for this Euclidean model. In this connection, the proposed geometry respects the CPT invariance, where a left-handed positron is identified with a right-handed electron via a spatial transformation, hence providing an explanation of the anti-matter asymmetry and thereof a potentially more efficient way to produce and store anti-particles, benefiting medical technology for example [9], and otherwise it furnishes the quantum vacuum a physical space of wave energies, with their energy densities identified as probability densities via their formal resemblance.

Summary Remark
In this note we presented two derivations of the energy ratio of particle to wave to be 3:1. By this relationship, we calculated the least nutation time for an electron, with a view to increasing the nutation frequency for more efficient use of electricity. We note that our model here is of a geometric nature, to serve as a supplement to SM. Yet this topic of wave-particle duality is so fundamental that it has been addressed since the early days of quantum mechanics such as [18][19][20]. Later treatments can be found in, e.g., [21][22][23][24][25]. Most of these analyses have been a reconfirmation of the principle of complementarity that no experiments are possible to measure both the wave and the particle attributes at the same time notwithstanding they have made contributions on the quantitative conditions for 'which way (welcher-weg)' the duality will assume [26]. In closing, we identify our model here with a diagonal spacetime manifold.
'Diagonal manifold,' artificial as it may resonate, nevertheless leads to a definition of the topological construct of the Euler characteristic.
Conflict of interest statement --As the sole author of this paper, I declare that there is no conflict of interest.
Data availability statement --The arguments contained in this paper go by mathematical derivations, hence entailing no data.