If we put aside the absurd interpretations of quantum mechanics, we have a successful model of two partial differential equations: the Schroedinger and Dirac’s equation. Now assume that the universe is and the physical universe is a hypersurface. Then there can be alternative explanations for the following Dirac’s equation: take the wave equation
where is the Dirac operator on and restrict this to a hypersurface our physical universe. We should get a version of the Dirac equation for the physical universe.
The Dirac operator restricted to has the form (see Hijazi eq (8))
So the Dirac equation for the hypersurface
This is formally in the Dirac equation form with potential . Let’s consider the original Dirac equation for comparison:
The analogy suggests the correspondence between mass and the mean curvature of the hypersurface. This simple picture gives some indication regarding the identity of the zero-point energy introduced by Max Planck.
On the issue of ‘wave-particle duality’ it is useful to consider what the eigenfunctions of the Dirac operator looks like in say. Now these have been computed in terms of Jacobi polynomials by Camporesi and Higuchi in 1996 (Camporesi-Higuchi-EigenfunctionDiracOnSpheres). The separation of variables produces a scalar part:
So consider say and . We have something like this with axis $latex[0,2\pi]$. In an S4 universe, the eigenspinors of the Dirac operator are going to be highly localized (‘point’) objects. Note that the compactness and the geometry of the universe is crucial to these things. Big Bang theory is a significant blunder of science which is blocking us from obtaining the correct laws governing physics.
Now it is tempting indeed to interpret the Dirac equation in terms of a movement in the ‘normal direction’ to the physical universe. If the physical universe were the great sphere $latex $^3(1/h) \subset S^4(1/h)$ and is an eigenspinor of the Dirac operator representing some ‘particle’ then is a representation of an evolution of the physical universe in the normal direction.