The Standard Model in the framework of the combined spacetime 4-manifold of particles and waves-a heuristic introduction

Abstract

This paper represents a culmination of our previous work. We started with the hypothesis of a pair of independent Einstein Field Equations responsible for two spacetime 4-manifolds M[1] and M[2], each with its own gravitational constant G[1] and G[2]. We combined M[1] and M[2] into M[3], with G[3] equal to the observed value. We then derived the values of G[1] and G[2]. We found that G[2] is so large that a cosmic black hole B could easily be formed in M[2] and the center of B by its infinite energy density opened up M[1], the Big Bang, transforming the hitherto pure electromagnetic waves contained in B of M[2] into particle-waves. By Feynman's analysis of the electromagnetic mass, we distributed energy E[3] = E[1] + E[2] = (3/4)E[3] + (1/4)E[3]. We cast Pauli matrices in M[3] and derived the motion of a free electron-wave via the Dirac equation to be a uniform circular motion around two perpendicular connected semicircles ; at their intersection virtual photons are emitted for electromagnetic interactions. We derived the unique wave length that was the diameter of M[1] at the Big Bang to be the Planck length and showed how the union of two spinning circles of electromagnetic wave energy within this wave length could be decomposed into a pair of electron-wave and positron-wave. In this paper we go one step further to show how the above decomposition at the Big Bang could also result in all the observed fermions and bosons with their associated antiparticles, thus essentially concluding our theory.

The standard model in the combined four-manifold

“…This paper solidifies our previous result of the particle-wave spinning motions of all fermions and bosons by a direct derivation from the "mass-shell" equation, E 2 = (m 0 c 2 ) 2 + (pc) 2 , instead of using the Dirac equation as we did previously (see [8]). In doing so, we eliminate the existence of negative energy as is present in the Dirac equation.…”

“…Remark 1 For an elaboration of the above proposition, see [8]. As an added supporting argument, we cite from Feynman ([1] , III-10-14) about how the three Pauli matrices emerge from a pair of differential equations for solving for the probability amplitudes, C 1 and C 2 (≡ ψ 1 and ψ 2 in our notation), of the two alternative spinning states of an electron with magnetic moment µ in a given magnetic field B = (B x , B y , B z ) :…”

“…In this paper we have factored the mass-shell equation into a pair of complex conjugates that (1) lends support to the geometry of our combined spacetime 4-manifold {(t + ti, x + yi, y + zi, z + xi)}, (2) eliminates the problem of negative energies, (3) explains pair creation of matter and anti-matter and at the same time the scarcity of anti-matter, and (4) re-asserts our particle-wave motions that explain the spontaneous symmetry breaking and the nuclear interactions of the Standard Model (for the strong force, see [8]). We have also examined the internal logic associated with the Dirac equation and found that (1) a formal square of the Dirac operator is mathematically invalid, (2) in order to square the Dirac equation to the mass-shell equation one must identify anti-matter with matter, thus reducing the dimension of the spinor space from four to two, (3) the two spin states represent a change of reference frames by our linear transformation R in R 3 , thus reducing the dimension of the spinor space from two to one, and (4) the remaining single equation of E = m 0 c 2 + pc, which we share, is a manifestation of the Morse lemma.…”

“…We clarify that E = ω = hν > 0 with ω = 2πν > 0 can be cast in four alternative settings of using a real line = {t} to cover S 1 = {it}, that is, (±R) ×(clockwise, counterclockwise), and therefore anti-matter can readily take the guise of matter. In one of our earlier papers ( [9]), we made a revision of the Planck energy formula to be E = hν + h s 2 ν , where h s 2 ν = the Heisenberg uncertainty energy due to the uncertainty of the phase at which the particle-wave emerges from our cosmic black hole B ⊂ M [2] (the wave universe) into the particle universe M [1] (see [8]). With the uncertainty energy included in E, the massshell equation accounts for all entities of energy.…”