Abstract
We prove the equivalence between non-local gravity with an arbitrary form factor and a non-local gravitational system with an extra rank-2 symmetric tensor. Thanks to this reformulation, we use the diffusion-equation method to transform the dynamics of renormalizable non-local gravity with exponential operators into a higher-dimensional system local in spacetime coordinates. This method, first illustrated with a scalar field theory and then applied to gravity, allows one to solve the Cauchy problem and count the number of initial conditions and of non-perturbative degrees of freedom, which is finite. In particular, the non-local scalar and gravitational theories with exponential operators are both characterized by four initial conditions in any dimension and, respectively, by one and eight degrees of freedom in four dimensions. The fully covariant equations of motion are written in a form convenient to find analytic non-perturbative solutions.
Highlights
There is cumulative evidence that theories with exponential non-local operators of the form e−( /M 2)n (1.1)have interesting renormalization properties
We prove the equivalence between non-local gravity with an arbitrary form factor and a non-local gravitational system with an extra rank-2 symmetric tensor
We use the diffusion-equation method to transform the dynamics of renormalizable non-local gravity with exponential operators into a higher-dimensional system local in spacetime coordinates
Summary
There is cumulative evidence that theories with exponential non-local operators of the form e−( /M 2)n (1.1). Calling R(r, x) the curvature invariants of a putative localized theory, since the diffusion equation ( − ∂r)R(r, x) = 0 would be non-linear in the metric gμν, one would have [ (g), ∂r]R(g) = 0 ,. We impose the diffusion equation on φμν: the linearity problem is immediately solved and one can proceed to localize the non-local system, count the initial conditions and identify the degrees of freedom, which are finite in number. Counting non-local degrees of freedom is a subject surrounded by a certain halo of mystery and confusion in the literature To make it hopefully clearer, we will make a long due comparison of the counting procedure and of its outcome in the methods proposed to date: the one based on the diffusion equation and the delocalization approach by Tomboulis [49]
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