Here is a simple geometric explanation of the redshift in a static Einstein (1917) model which is a scaled S^3.  The issue turns on the fact that the obvious frequency-wavelength relation f=c/\lambda that holds for wave propagation on the plane does not hold for wave propagation on a sphere.  Now Einstein’s static universe model is a scaled S^3.  The wave equation on an n-sphere,

(1/c^2 \partial_t^2 - \Delta_{S^n})\psi = 0

can be solved explicitly by very standard methods in physics, separation of variables and using spherical harmonics Y_{\ell,m}(x) (see for example orthoganalpolnomialsinddimesnions for a treatment of spherical harmonics).  The frequency of the waves on S^3(R) are

\omega_{\ell} = (c/R)\sqrt{\ell(\ell+2)}

This is in sharp contrast to the frequency of a wave on a circle of radius R which is \omega = c\ell/R.  It is this discrepancy that explains the observed redshift.

An intrinsic curvature for a closed universe has a positive cosmological constant without any need for a dark energy, and of course the universe is static and will not have any expansion since redshift is a geometric phenomenon.  De Sitter relativity is one approach that can apply, but more interesting is the fact that quantization of energy can be explained easily also as a geometric feature.

Of course I hold the firm conviction that not only is our actual universe a compact three dimensional hypersurface of a fixed four-sphere of fixed radius (which is responsible for all quantum phenomena) but having been convinced after some years that unless I connect up my views with the mainstream this will not be accepted, I am slowly working on studying standard material.  In fact the geometry of the situation I am fairly sure is true in actual reality can be considered through the lens of topological quantum field theories, where one considers a three-manifold Y that is a submanifold of a four-manifold X.  Dijkgraaf and Witten had a nice paper for which they consider in what way the Chern-Simons action on a three manifold is related to the Yang-Mills action on the bounding four-manifold.  This is an interesting thing to do for our situation.  A useful analysis of theirs is the following:  if you take a principal G-bundle on Y and want to extend this bundle to X the four-manifold, this can be accomplished for a compact Lie group case essentially because the odd dimensional homology groups H_{k+1}(BG,\mathbf{Z})=H_{k}(G,\mathbf{Z})=0.  So you get this from the fact that the short exact sequence G \rightarrow EG \rightarrow BG has a contractible space in the middle (by definition of classifying space) and then taking the long exact sequence in homology to get an isomorphism of the homology of BG and $G$ in the appropriate indices, then use Poincare duality and the fact that the cohomology of compact simple groups are known — Milnor’s Morse Theory has a proof that the loop space over G not only has the homotopy type of a CW complex but has only even dimensional skeleta and then there is an isomorphism of loop group cohomology and that of the Lie group.  The conclusion is that $H_3(BG,\mathbf{Z})=0$ for compact Lie groups such as SU(n).   The Witten Dijkgraaf (dijkgraaf-witten) procedure is to consider the classifying map associated to a bundle E\rightarrow Y, say \gamma:Y\rightarrow BG and consider whether the image is the boundary of some four-manifold in BG.  The image \gamma_*[Y] has no boundary and is three-dimensional therefore is represented in $H_3(BG)$ and since this is trivial for compact Lie groups of interest we can extend our classifying map over X and then pull back the universal bundle to X.  This is a very nice method.  Now in our case we have an ambient four-sphere and we actually know the instantons over the four-sphere by the wonderful results of Atiyah-Drinfeld-Hitchin-Manin who produced explicit solutions for instantons for a family of principal bundles on S^4.

Gromov came up with a very interesting set of concepts such as ‘filling radius’ and ‘filling volume’ — see here (filling-riemannian-manifolds.  Filling radius is essentially the radius of the largest ball that fits inside the manifold if it is isometrically embedded and filling volume is defined similarly.  Interesting bounds involving these are FillRad(Y)^n \le FillVol(Y) \le C_n Vol(Y)^{(n+1)/n} and FillRad(M) \ge InjRad(M)/2(dim M + 2).  So we can obtain an upper bound on the injectivity radius of our universe and a bound for the volume as well by following geometric analysis of this type.



Kontsevich had a spectacular solution of the Witten conjecture in 1991, that the generating function of the intersection numbers of genus g surfaces with n marked points satisfies the KdV equations which came from equating two different models of two dimensional gravity:  kontsevichthesis.  So the question is whether there is a nontrivial generalization for gravity in three dimensional hypersurfaces of a four-sphere of fixed radius.  I don’t know whether the physicists have already produced the generalization but this is probably what needs to be done to connect the S4 ideas with established direction in physics …

Geometric quantization for three-dimensional configuration spaces M can be described in the following way when it embeds into S^4(1/h).  First, the tangent bundle and its dual cotangent bundle are both topologically trivial in three dimensions isomorphic to M\times\mathbf{R}^3 and so are its powers which describe N-particle phase spaces in classical mechanics.  Now prequantization requires a line bundle L\rightarrow T^*M=M\times\mathbf{R}^3.  The line bundle can be identified with a circle bundle topologically.  We can map the fibers L_x of the circle bundle choosing a fixed base section to $M\subset S^4(1/h)$.  This allows us to map the fibers to the normal circle at every point from the embedding M\subset S^4(1/h).  Now we have a surjective map L\rightarrow S^4(1/h)\times\mathbf{R}^3 that factors through the bundle projection L\rightarrow M\times\mathbf{R}^3 and the embedding map M\rightarrow S^4(1/h).  Now a connection on the circle-bundle L is equivalent to a 1-form defined on the total space.  We should be able to produce a 1-form on S^4(1/h) which pulls back to L.  Geometric quantization prescribes a connection on L whose curvature is i\omega where \omega is the symplectic form.

In geometric quantization one picks a polarization, a maximal distribution T on T^*M such that \omega(v,v')=0 and the quantum algebra consists of functions f such that [X_f,T]\subset T.

Geometric quantization maps functions f\in C^{\infty}(T^*M) to the operators with values in the prequantization line bundle $f\cdot + i\hbar^{1/2} \mathcal{L}_{X_f}$.  So we can assume a geometric quantization and use the map L\rightarrow S^4(1/h) to make a trivial change where the values are in Slatex S^4(1/h)$.  But this is most likely the tip of the iceberg for most likely the entire geometric quantization programme is hiding an actual structure of the actual universe which is being approached by geometric quantization in a roundabout manner.


This is a giant field that has a complicated history on which I am not an expert.  I studied pute mathematics and not physics in college.  The idea that the universe is S4(1/h) came out of some inspiration I had in 2008 in Brooklyn.  The idea occurred from thinking back to 1900 what would have resolved the problem of the blackbody problem.  If all that is necessary is quantization of energy, this can be explained by a spherical universe.  The problem is how many dimensions and the obvious answer is four dimensions, macroscopic.  If one then takes the leap of faith of accepting a fourth macroscopic dimension, then it’s easy to see that something like the gravitational field equations occur when taking the Ricci curvature of any hypersurface whatsoever with a cosmological constant.  This is purely geometric intuition.  So it seemed right and I still have not managed to educate myself on quantum gravity in physics because I was too sure that gravity would just make sense in this model but these are nontrivial issues.  So I wasted 2008-16, eight years on this wild goose chase.  Well better physicists have spent their lives on their own wild goose chases so this is not the end of the world, but yes, quantum gravity, B. de Witt has a great deal more to say (dewitt-quantumgravity).  The folks at Classical and Quantum Gravity don’t respect me, which is reasonable since I was suggesting that quantum gravity is a fairy tale because gravity and quantization are some simple mathematical consequences of a four-sphere physics.  Emersonian adherence to sticking to intuition before it is proved seemed appropriate.  I am sorry but mathematical complexity does not equate to laws of nature.  Renormalization issues are too ugly to be right.

My model for our actual universe is a hypersurface of a round four-sphere.  In general, for generic metrics on closed spin three-manifolds, there are no harmonic spinors by anghel-genericvanishingtwisteddiracharmonicspinorsammann-dahl-humbert-surgeryharmonicspinors.  Thus for generic hypersurfaces of the four-sphere, the kernel of the Dirac operator is zero, i.e. there are no massless fermions.  So I think that the recent discovery of apparently massless Weyl fermions is probably a repetition of the error when neutrinos were considered massless on discovery.  I don’t believe that there can be massless fermions in the actual universe.


Here is Chen-Kumagai two sided bound for jump diffusion with some assumption on the jump kernel similar to that of \alpha-stable jump 1/|x-y|^{-n-\alpha}.  We use \sim to denote two-sided bound.  Suppose J(x,y) \sim \frac{1}{|x-y|^n \varphi(|x-y|)} is the jump kernel where \varphi(0)=0 and \varphi(1)=1.  Then

p(t,x,y) \sim (t^{-n/2} \wedge \varphi^{-1}(t)^{-n}) \wedge (p^c(t,c(|x-y|))+p^j(t,|x-y|)


p^c(t,r) = t^{-n/2} \exp(-r^2/t)

p^j(t,r) = \varphi^{-1}(t)^{-n}\wedge \frac{t}{r^n\varphi(r)}