Calculus without limits # Is there physics for the connection Laplacian?

The classical potential $V(x,y) = 1/|x-y|$ has infinite range which violently clashes with relativity. Solving this problem had required a completely new theory: GR. It remains also a fundamental problem still in general relativity: a Gedanken experiment in which the particles in the sun suddenly transition to particles without mass shows this. [This is forbidden by energy conservation but energy conservation is not a theorem which follows from GR. It is a rule we know from empirical observation. Indeed, in quantum field theory, on a fundamental level, there is no energy conservation. There are virtual particles for example. There is a tiny probability (of course ridiculously small but not zero, maybe $10^{-10^{1000}}$ or so) in which the event happens that the sun disappears. What happens with the Pseudo-Riemannian tensor g in that case? ] This is a case at the boundary of the theory

The mathematical problem is to look at solutions of the Einstein equations if the energy-momentum mass tensor has compact support in space time. In part of the space, when the sun is present, it should look like a Schwarzschild solution, at a later time like a flat space time. General relativity is a model which works well in certain ranges and the “switching off the sun Gedanken experiment” might appear as silly as the Schrödinger cat experiment. But such musings have their value: they show where the boundaries of the theory are. It is a basic principle of physics that it consists of models which work in some range. And every model has its limitations. The switching off of the sun is close to a real enigma in GR. What happens if a black hole evaporates completely? What happens at the moment when it disappears? t’Hooft exlaims for this problem that a healthy theory should allow us to compute there. We obviously have no such theory as no theory gives a prediction what happens then. An other problem is even if somebody could come up with an explanation, we have no idea how one would test it.

The transition Newton->Gauss (ODE to PDE) for gravity is followed by Gauss->Einstein which is mathematically much more difficult as linear partial differential equations become now non-linear partial differential equations in which also boundary values like asymptotic behavior plays a role. While Newton -> Gauss was a transition from forces to potentials, the transition Gauss -> Einstein moves from potentials to curvature. In the discrete we see that this has already happened as the potential $V(x) = \sum_y g(x,y)$ obtained by adding up the potential energies over all simplices $y$ (including $x$) on $x$ can be interpreted as a curvature $k(x)$ for which Gauss-Bonnet holds. This was the basic insight to prove the energy theorem. When we take that allegory serious, then the Euler characteristic should play the role of the Hilbert functional. There were other indications that this is reasonable as for a four dimensional complex $G$, the Euler characteristic can be seen as an expectation value over all sectional curvatures defined algebraically. As in differential geometry, the scalar curvature is an expectation over all possible sectional curvatures, this is a nice analogy.

Could the connection Laplacian L be a Laplacian for gravity in the sense of Gauss? One has not to be too hopeful as speculation is not that hard. (Thousands of papers speculating in some or the other direction prove how easy it is to come up with ideas.) Coming up with a “good theory” which allows predictions of not-yet observed events or quantitative verifications of actual measurements is what is hard.

[Added December 10, 2017: If we assume that L is a relevant Laplacian, then the fact that the total potential theoretical energy defined by it is the Euler characteristic, and that this energy is also the difference between the positive and negative eigenvalue count, then we have the picture that positive energy states of L contribute positively to curvature and negative energy states contribute negatively. In any way, the fact that a quantum mechanical energy values are linked to geometry in such a way that the total potential theoretical energy matches an eigenvalue count and so relate to an integrated density of states is intriguing. As in solid state physics, spectral quantity is linked with a geometric property. ]

In order to validate a theory of gravity, one can use the “Kepler test”: can the theory compute the Kepler two body problem in any scale? In the Macro scale it should give something like general relativity, in the Meso scale something like classical gravity as well as in the Nano scale it should give something like the quantum mechanics of the Hydrogen atom. The theory should be elegant, not a patch work from existing theories and do things with on a comparable complexity than the existing theories. Of course, we can model anything with finite models as when we build a mathematical model and implement it in the computer, we do that. But these models are not pretty. Numerical schemes almost always lack elegance. Indeed, elegance and efficiency appear to be reciprocal to each other. Already the 2-body problem in GR is a mess. It requires Newtonian approximations and hacks to merge general relativity with gravitational radiation.

On the Meso scale, having a potential $V(x,y)$ and a path $C$ in $G$, one can look at the action $\sum_{(x,y) \in C} V(x,y)$. Given two points $x,y \in G$ we can now try to describe paths from $x$ to $y$ which minimize the action. One can now hope that for a geometry modeling a central mass point the motions are close to a Kepler problem. On the Micro scale, one could simply look at the unitary motion $e^{it L}$ on the Hilbert space. One can then hope that the analogue of the Ehrenfest correspondence theorem holds and that the center of mass of a wave moves close to the minima given by the action principle. The Maco scale, which is classically still in the dark is also not clear here. It requires to have a natural time deformation of the complex. Obviously any discrete deformation of the geometry is a hack. But why discrete time. But we know that every simplex in $G$ can be associated to a state in the Hilbert space. Any quantum mechanics defining an unitary path $U(t)$ now defines also a deformation of the geometry but this happens in a more general space as the original geometry given by an orthogonal frame in the Hilbert space is deformed into orthogonal frames for which there is no classical geometry analog. Especially, the total gravitational energy $\sum_{x,y} L_t^{-1}(x,y)$ will change with time if we don’t take $L$ as a Hamiltonian. It does not change for $e^{i t L}$ or $e^{i t L^2}$ although. Since we can associate eigenvectors of L with simplices (the correspondence is given once a CW complex structure is defined on G). So, we have a time evolution of space itself.

The hope is now that if we start with a geometry modelling a Kepler problem with two masses, then the motion of the two bodies is classically given by Newtonian dynamics. It is by nature a quantum evolution but we would like to have eigenvectors matched with the eigenvectors in the Hydrogen atom. In the macroscopic case, the dynamics of the binary system should produce measurable gravitational radiation further away. We don’t have to worry about the Einstein equations as the Einstein ewquations “define” the mass distribution. What we have to worry about is whether the time evolution of the masses follows geodesics. We expect this to happen as exp(i L t) is a wave evolution by d’Alembert and the Huygens principle assures that waves move on geodesics. If we look at the d’Alembert solution however, it makes more sense to see L as a “Dirac operator” and look at $e^{i t L^2}$ instead, mirroring the classical Hodge case, where the Hodge Laplacian $H = D^2$ is a square of the Dirac operator $D=d+d^*$. What does it mean that unlike D, which comes from an exterior derivative, there is no cohomology to $L$. Actually, while in the Hodge case super symmetry gives a bijective correspondence between the positive and negative eigenvalues of H, this correspondence is broken for L. We have proven that the number of positive eigenvalues minus the number of negative eigenvalues is the Euler characteristic of G. Super symmetry for L is not just broken in an arbitrary way. It is broken in a beautiful way. The discrepancy is Euler characteristic.