math-ph — Mathematical Physics

Cancellation of all ultraviolet divergences of quantum gravity by a scale-invariant gas of branes

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya

A frozen Poisson gas of branes with no preferred size, each brane a layer of independent cells that shift the proper time of a crossing graviton, multiplies the graviton kinetic operator by an entire function without zeros, $\exp[\tfrac n2\Ein(\Box^2/M^4)]$ times a random factor that tends to a constant at high momentum, in almost every realization. We prove that for $n\ge4$ the theory of gravity built on it has no ultraviolet divergence at any order of perturbation theory: every graph with two or more loops converges, the one-loop poles are those of a local theory and are cancelled by three operators quartic in curvature with finite coefficients, Newton's constant and the cosmological constant do not run, the zero-point energy of the graviton is $3nM^4/(64\pi^2)$ for every regulator, and the physical $S$ matrix is unitary at every loop order. The propagator keeps the graviton pole alone, and the potential of a point mass is finite at the origin. A one-loop graph in closed form confirms the cutting rule and grows faster than any exponential of the energy, so the loop expansion is an expansion in a small parameter only below an energy between $1.4M$ and $1.9M$. The free Euclidean field is a Gaussian measure with finite curvature that is not reflection positive, which states what a construction beyond perturbation theory would have to supply.