On Geometric Structure of Point Particles

Abstract

I propose, as geometric structure in an internal space, a helical field that is responsible for intrinsic properties of point particles, particularly, electron. For the novel theoretical development, plasma astrophysical analogy is made extensively. Transition between our conventional space and the infinitesimal space is considered in an operational manner. It is shown that rotational eigenvalue equation satisfied by the vector field equivalent to Gromeka-Beltrami flow provides a coordinate-rotor that captures complex orthogonality between the internal coordinate space and isotopic, angular momentum space. Self-consistent normalization of rotational coordinate owing to the rotor is compared to renormalization of electric charge. It is also found that chiral asymmetry of the helical eigenflows can be reflected in electroweak symmetry breaking. The theoretical prototype suggests possible geometrical features of a fundamental framework ruling matters and forces with higher dimensions.

Figures (8)

• Figure 1: Schematics of the wavenumber transformation $k\to k_{2}$, which comes about when light (wavy solid curves) propagates from plasma region I to II across the boundary (dashed lines): a surface of discontinuity of the density $n$ (solid lines). Provided the angular frequency of the light $\omega$ is invariant for the transformation, the cases of $\omega_{p}<\omega_{p1}<\omega$ (a) and $\omega_{p1}<\omega<\omega_{p}$ (b) are shown, where $\omega_{p1}$ and $\omega_{p}$ denote the plasma frequency of the region I and II, respectively.
• Figure 2: Plots of $\kappa_{\mathrm R}^{(+)}(\tilde{\xi})$ [equation \ref{['eq:4.9']}] (a) and $\kappa_{\mathrm R}^{(-)}(\tilde{\xi})$ [equation \ref{['eq:4.10']}] (b). The functions positive in the discrete domains are shown by the solid curves. Both the figures have a common horizontal axis. In (b), the point at which $\kappa_{\mathrm R}^{(-)}$ takes the minimum is indicated by the open circle.
• Figure 3: Plots of $\kappa_{\mathrm L}^{(+)}(\tilde{\xi})$ [equation \ref{['eq:4.13']}] (a) and $\kappa_{\mathrm L}^{(-)}(\tilde{\xi})$ (b). The functions positive in the discrete domains are shown by the solid curves. Both the figures have a common horizontal axis. In (a), the point at which $\kappa_{\mathrm L}^{(+)}$ takes the minimum is indicated by the filled circle.
• Figure 4: Plots of $\kappa_{\mathrm R}^{(-)}(\tilde{\xi})$ [equation \ref{['eq:4.16']}] (a) and $\kappa_{\mathrm L}^{(+)}(\tilde{\xi}) =-\kappa_{\mathrm R}^{(-)}(\tilde{\xi})$ (b). These functions having, respectively, the domain of $j_{0,n}<\tilde{\xi} <j_{1,n}$ for odd and even number of $n\,(=1,2,3,\dots)$ are shown by the solid curves. Both the figures have a common horizontal axis, on which positions of $j_{0,n}$ are indicated, in place of scale of the linear axis. The points at which $\kappa_{\mathrm R}^{(-)}$ and $\kappa_{\mathrm L}^{(+)}$ take the minima in their domains are indicated by the open and the filled circle, respectively. The broken curves in (a,b) correspond to $\kappa_{\mathrm L}^{(-)}$ and $\kappa_{\mathrm R}^{(+)}$ for $j_{1,n}<\tilde{\xi} <j_{2,n}$ with $n$ odd and even, respectively (see text).
• Figure 5: Cyclotron motion of a single electron having the spin ${\bf S}$ and its longitudinally polarized state (a), and the corresponding rotation of unit radial vector $\hat{\bf r}={\bf R}/\|{\bf R}\|$ and change of the polarization for the one orbital turn: $\bm{\sigma}\to\bm{\sigma}^{\prime}$(b). In (a), the angular momentum coupling is depicted.