Archive for April, 2014

The standard mass gap problem is to show whether there is a mass gap for a Yang-Mills theory in flat Minkowski space.  I do not claim to solve this problem.  I solve a mass gap problem for the model of the universe that is a scaled 4-sphere of numerical radius 1/h.  I have provided empirical evidence that the actual universe itself must be a scaled 4-sphere.  With this model, a physical universe can only occur as a smooth hypersurface of S^4(1/h).  The S4 mass gap problem is to show that no smooth hypersurface M can carry a nontrivial harmonic spinor.  Nontrivial harmonic spinors are the same as massless fermions:  the time-dependent Dirac equation can be solved just as in the case of wave equations multiplying e^{i \omega t} with the eigenspinors.

Now it is well-known that no spin manifold with positive scalar curvature can carry harmonic spinors by the Lichnerowicz formula for D^2 = \nabla^*\nabla + S/4 where S is the scalar curvature which applies to the ambient S^4 as well as minimal submanifolds such as the equator and the Clifford tori.

Suppose M is a physical hypersurface of S^4 that only contains massless fermions and no massive fermions.  Mass corresponds to geometry of M by increasing its mean curvature.  If we accepts the correspondence, then a physical universe with only massless fermions will be a minimal hypersurface which have rigidity in S^4: the possible scalar curvatures are discrete and positive by the Chern conjecture and its resolution.  Since all the minimal hypersurfaces have positive scalar curvature, no massless fermions are possible.  We can then conclude that there is a mass gap for these hypersurfaces.

The correspondence between mass and mean curvature is elucidated by the work of LawnRothIsometricImmersionS4Spinors who show how two ‘antiparticle’ massive fermion solutions of the Dirac operator:
D\phi_1 = 3/2(H+k)\phi_1 and D\phi_2 = -3/2(H+k)\phi_2

where H is a function can be used to embed the manifold into a sphere whose radius is a simple function of k for which H appears as the mean curvature of the embedding.  Their work shows that the correspondence between mass and mean curvature can be made precise.



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A physical universe in the S4 universe is a hypersurface of S^4.  We know that the equatorial S^3 has positive scalar curvature.  If we can deform our hypersurface smoothly through embeddings to the equatorial S^3 we can then use the Gromov-Lawson theorem that says that positive scalar curvature condition is preserved by this ‘spin cobordism’ and therefore M has positive scalar curvature and hence no harmonic spinors by the Lichnerowicz theorem.  This is another way to show that our actual universe cannot have massless fermions.

There is a problem with the argument above unfortunately, which is that the Gromov-Lawson spin cobordism theorem assumes that the dimensions of the manifolds are at least 5.  Therefore we have to analyze the situation further.


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Our model of the universe is a scaled 4-sphere where the physical universe is an evolving hypersurface.  The fact that the normal geodesics from M all have the same length suggests that this is the circle (of length 1/\hbar) is confused with a circle bundle that defines electromagnetism in the standard account.  One possible answer is that electromagnetism is simply defined by a 1-form on the 4-sphere similarly to how the connection form is defined on the total space of the circle bundle.  Indeed, one can embed the one-form into the Clifford bundle and consider electromagnetism to be defined by a potential.  The fact that there exists a Dirac equation for the photon implies that it might be reasonable to consider electromagnetism to be defined by a single spinor field potential.

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Studying the scalar curvature of a hypersurface M of S^4 is one way of trying to show nonexistence of harmonic spinors (positive scalar curvature would do).  Another approach is to assume that a nontrivial harmonic spinor exists on M and try to extend it to a harmonic spinor on S^4 by solving an appropriate boundary value problem.  If there is a canonical procedure for doing this, then we can use the positive scalar curvature of S^4 to show that the extension must be zero.  The physical problem of interest is the existence of massless fermions.

The right boundary value problem was given by Atiyah-Patodi-Singer.

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There are no harmonic spinors on a hypersurface M of S^4 if the scalar curvature is positive by the Lichnerowicz theorem.  Therefore it is useful to look at the formula for the scalar curvature.  It is (see ):

n(n-1)r = n(n-1) + n^2 H^2 - S

generally for S^{n+1} where H is the mean curvature and S is the squared length of the second fundamental form.  This will be negative obviously when S is large compared to the mean curvature term.  The issue of when the scalar curvature will be positive is delicate but we can get some sense by considering the minimal surfaces with H=0 to consider what happens: a small number of possibilities arise all of which have positive scalar curvature.  Thus a simple conjecture that there should be rare or impossible to have negative scalar curvature; then we can conclude that massive fermions cannot exist in the physical universe.


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Our model of the physical universe is a hypersurface of a scaled 4-sphere and so it is valuable to consider the spectrum of the Dirac operator (with time frozen).  Corollary 4.3 of BarBoundsDirac reads:

Let M be an n-dimensional hypersurface of S^{n+1}.  Let M carry the induced spin structure.  Let H denote the mean curvature of the embedding.  Then the classical Dirac operator on D_M has at least 2^{[n/2]} eigenvalues satisfying
\lambda^2 \le \frac{n^2}{4} + \frac{n^2}{4 vol(M)} \int_M H^2

This result gives us two eigenspinors of the Dirac operator controlled essentially by the integral of the mean curvature.  There is also a lower bound of the first nonnegative eigenvalue but in terms of pointwise infimum of the mean curvature.  Thus we have our basic link between the ‘lightest possible mass’ and mean curvature of the physical universe.

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An intuitive identification of mass would be to consider it as the possible non-negative eigenvalues of the Dirac operator in the physical universe which we assume is a hypersurface of S4(1/h).  The analysis of Dirac operators for hypersurfaces performed by Hijazi et. al. HijaziDiracHypersurfaces show that the first nonnegative eigenvalue of the Dirac operator has the lower bound \lambda_1 \ge \inf_M H where H is the mean curvature of the hypersurface M.  Thus geometrically speaking, it is reasonable to develop the rule of thumb that mass is mean curvature.  This gives us an intuitive picture of what happens geometrically in a physical hypersurface of S4(1/h): matter represented by eigenspinors of the Dirac operator gets heavier with mean curvature of the hypersurface geometry.  The Einsteinian picture of gravity in terms of a metric of the three-manifold is sharpened here: the mean curvature of the metric increases with heavier objects such as Dirac eigenspinors with high eigenvalues.  This intuitive picture is marred by my current lack of understanding regarding why Dirac eigenspinors should have strong localization properties to serve as models for particles but if one accepts the identification then the relationship between hypersurface geometry and mass of a fermion (Dirac eigenspinor) is clearer.




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