A Cross-Disciplinary, Quantum-Gravity Approach to Everything from Consciousness and Gravity to the Riemann Hypothesis and Topological Insulators

Earlier in this article, it was suggested how everything in space-time could be formed from binary digits, Mobius strips, figure-8 Klein bottles, interacting gravitational and electromagnetic waves acting in synchrony with Wick Rotation. Referring to Figure 2 - The Riemann hypothesis, proposed in 1859 by the German mathematician Georg Friedrich Bernhard Riemann, is fascinating. The Riemann hypothesis doesn’t just apply to the distribution of prime numbers but can also apply to the fundamental structure of the mathematical universe’s space-time, as the following shows. In mapping the distribution of prime numbers, the Riemann hypothesis is concerned with the locations of “nontrivial zeros” on the “critical line”, and says these zeros must lie on the vertical line of the Complex Number Plane i.e. on the y-axis in Figure 2 (this circular placement may reflect future spacetime warping greatly magnifying General Relativity’s concept of curved space-time). Besides having a real part, zeros in the critical line (the y-axis) have an imaginary part. This is reflected in the real +1 and -1 of the x-axis in Figure 2, as well as by the imaginary +i and -i of the y-axis. In the upper half-plane of Figure 2, a quarter rotation plus a quarter rotation equals a half – both quadrants begin with positive values and ¼ + ¼ = ½. (The Riemann hypothesis states that the real part of every nontrivial zero must be 1/2.) While in the lower half-plane, a quarter rotation plus a negative quarter rotation equals zero: 1/4 + (-1/4) = 0. In the Riemann zeta function, there may be infinitely many zeros on the critical line. This suggests the y-axis and the universe (space + its partner time) described by Wick rotation are literally infinite. By this paragraph’s extension, the Riemann hypothesis would also describe the universe as infinite in space and infinite in time (eternal). * Relativity’s curvature would need to be exaggerated by future space-time warping so that the future could directly feed back on the past in a circular loop. Life (possibly multicellular and intelligent) and the genetic code could then possibly come from humans acquiring knowledge of these things over the centuries, then applying that knowledge – via terraforming, accumulation of raw materials like amino acids and nucleic acids, performance of genetic engineering - to a time in the past when life didn’t exist. From that origin, life could evolve through innumerable mutations and adaptations, with humans once again acquiring knowledge of it in cyclic (nonlinear) time.

* The zeros are on the potentially infinite y-axis of the Complex Number Plane (Wick Rotation). Wick rotation and the Riemann hypothesis both describe space-time. If space and time truly go on forever without any beginning (in linear time) or end, then the zeros could mean infinity equals zero). In that case - after space-time has undergone warping, there’d be absolutely no distance to any other galaxy, or to the past and the future. This is how warping might be done - Cosmology’s holographic principle suggests the 3rd dimension results from information in the 2nd dimension. The 2nd D might be the Mobius strips comprising particles and the 3rd D might be capable of being deleted by programming the binary digits (used in electronics) which act as Hidden Variables that are compatible with quantum mechanics (not with known probabilistic quantum mechanics but with quantum certainty, for they give precise calculations). Of course, zero would also equal infinity. There would be null distance between the Mobius strips comprising particles and the Wick rotation embedded within the strips (Wick can be regarded as a programmed subroutine). This nil distance can be converted into infinite distance, producing a timelike multiverse (limitless multitude of universes) representing each infinitesimal fraction of a second in the lifetime of the one physical cosmos. The multiverse could be reunified with the universe using infinity = 0, meaning the multiverse is actually part of the universe.

The General Theory of Relativity will be useful in this article. Specifically – the analogy of the theory’s curvature of space-time to a rubber sheet. A small body like the Earth is said to warp space-time only a little and create a dimple in the sheet. A larger body such as the

Sun curves space-time much more and forms a deep valley in the rubber. And a black hole is often pictured as warping space-time so much that it tears a hole through the rubber fabric. In 2004, U.S.A. physicist Charles Kane noticed something strange in his computer simulations of electrons flowing through different materials: an insulator whose quantum state had the equivalent of a hole. Kane had not found the first quantum black hole but had discovered the first topological insulator – a then theoretical material that could conduct electricity on its surface but not within its interior. (In 2007, American physicist M. Zahid Hasan led the team that made the first 3D topological insulator.)

In 1929, while experimenting with the equations of quantum physics, German physicist Hermann Weyl showed that a massless and charged particle (now called the Weyl fermion) could theoretically exist. The Majorana fermion was predicted in 1937 by Italian physicist Ettore Majorana playing with the same quantum math that had intrigued Weyl. Like a Weyl fermion, a Majorana fermion has no mass. It also has no charge, despite being made of a bunch of negatively charged electrons. The Weyl fermion can be related to Topological Insulators, * the Majorana fermion can be related to quantum computers’ Topological Superconductors. Topological insulators and topological superconductors may be regarded as the (Mobius dependent) inverse of each other, with the properties of surfaces and holes being interchangeable as a result of the twisting in their Mobius-strip / figure-8-Klein-bottle composition.

* A topological insulator is a material that behaves as an insulator in its interior but whose surface contains conducting states. However, the conducting surface is not the unique character of topological insulators, since the ordinary band insulators can also support conductive surface states. What is special is that the surface states of topological insulators are symmetry protected. Symmetry Protected Topological (SPT) Order is a kind of order in topological insulators where, if symmetry is preserved during the deformation undergone in topology, a phase transition from one state of matter to another must occur (in this case, between insulation and conduction). In other words, if the shape of a Möbius strip (or the union of two strips into a Klein bottle) is preserved, phase transition must occur just as orientation-reversing curves occur in the Möbius and Klein. In three-dimensional topological superconductors, it’s more common to have multiple surfaces. But if subatomic particles making up topological superconductors are composed of the topological Mobius strip, they can theoretically only have one surface. Topological insulators can also be composed of Mobius strips. The key aspect is that the topologically protected states are robust against certain types of perturbations, regardless of the number of surfaces.

Referring to Figure 1, Side DC of parallelogram = Vector 1 electrons and Side DA of parallelogram = Vector 2 electrons.

The two vectors (two groups of charged electrons) interact to form the resultant diagonal DB (the electrons travel ADB and CDB, coming together to behave like a single charged particle called a Weyl fermion). Tensor calculus converts the points on the sides and diagonal into a single scalar point on a nominated side (say, in the centre of the diagonal). And the mass of the vector 1 electrons minus the mass of the vector 2 electrons [(x MeV/c^2) - (x MeV/c^2)] equals zero, and the massless Weyl. If the electrons flow in the reverse direction, they go in the BD direction, then split and follow the paths DA and DC.This preserves information if one pathway is interfered with, giving robustness against perturbations. They produce the chargeless Majorana because the negative vector-1 electrons minus the negative vector-2 electrons equal (-y) - (-y) = 0. The Majorana’s lack of mass is attributed to the same process by which the Weyl particle becomes massless.

Why is subtraction essential? This appears to be a consequence of matter, and the Higgs boson, both emerging from photon-graviton interaction. Two adjoining sides of a parallelogram represent the vectors of the photon’s spin 1 and the graviton’s spin 2. The resultant diagonal represents the interaction of the sides/vectors (1÷2 = the spin ½ of every matter particle: and division is merely repeated subtraction e.g. 4 subtracted from 20 five times equals zero, therefore 20 ÷ 4 = 5). (By the way - in calculus, the quotient of two vectors is called a quaternion.) Speaking of the Higgs which resides on the diagonal in Figure 1 and has spin 0: zero can be arrived at through (1 - 2) + 1 which uses both subtraction and the experimental data of a photon existing in two places simultaneously (it uses the graviton’s spin 2 being taken away from the photon’s spin 1, and the spin motion of 1 being in more than one place at the same time).

According to their representation by vector-tensor-scalar geometry, the chargeless Majorana’s negative vector-1 electrons minus the negative vector-2 electrons equal (-y) - (-y) = 0. This can be expressed as y + (-y) = 0, which clearly highlights its similarity to the Riemann hypothesis’ ¼ + (-1/4) = 0 (in terms of the paragraph’s first equation, this is : -1/4 – (-1/4) = 0. The last pair of expressions resolve conflict regarding whether the first term in the lower half-plane of Riemann’s hypothesis is positive or negative (either can be used).