States of the Tunneling Stylus

Electron conduction in metals involves the Drude model, the Heisenberg chain model, and various types of band structures that describe the energetic landscape and dispersion relation of charged particles as they travel like waves in lattices in various dimensions. This is really complicated and best described in modern physics by the Schrödinger equation, whose wave recursion relation describes the spatial curvature of a particle as its kinetic energy and the potential energy from the field source, usually some sort of Coulomb field. What is interesting is thinking of more novel multi-valued solutions to the wave equation in quantum mechanics, the fractal, the self-similar curve, the expansion of the state past Huygens to the reality of Hilbert's space filling curve. Of course, this requires new mathematics, something that has been worked out in Interaction Quantum, a book soon to be published which describes in detail the waves beyond waves and fractal nature of states. Weiss et al. have developed an exciting test bed for fractal wave analysis, which should be combined with Fourier analysis and mappings to generate what will likely be Riemann zeta function descriptions of nodal behavior - isomorphic to the zeroes of the zeta are the bifurcations of the fractal. Naturally, fractals are multi-dimensional and the audience is only allowed a two dimensional representation. What reality contains is a multi-valued fractional dimensional curve eikonal that exists in the many dimensions of reality. Yang-Mills theory suggests a SU(n) end state that is not infinite due to IR and UV divergence. Summations are truncated and infinities represent the inflationary bounds linking quantum to CMB to Hubble constant to Planck's constant. What we should see is a modification of the Standard Model. This is a bit of a digression and it is better to return to the tip. The states are modeled by the lattice potential, the field generated by the Schrödinger equation, the novel solutions using a periodic potential, and a regime where the tunneling tip state combines with the surface state beyond the forbidden zone.

Let's think for a moment about what this implies. Waves are not simple sinusoidal functions. What there is now becomes increased bandwidth in signal analysis, new phased array sensor development, things like POPOP in signal transforms and waveform capture with phase difference calculation, and a quantum picture beyond traditional postulates. The self-similarity will show that the kernel of waves is related to the Riemann zeta prime nodal zero spacing, inducing pseudo-randomness, or at least a better bound on the uncertainty relation.

Field emission and states at the tunneling tip enter into the forbidden region, superluminal travel through a potential barrier. It is spooky-action-at-a-distance. The results are measurable and profound. Tunneling provides the scientist with a testbed not only for quantum behavior, but for particle wave constitution and fractional dimensional analysis.