Abstract
Anti-Newtonian expansions are introduced for scalar quantum field theories and classical gravity. They expand around a limiting theory that evolves only in time while the spatial points are dynamically decoupled. Higher orders of the expansion re-introduce spatial interactions and produce overlapping lightcones from the limiting isolated world line evolution. In scalar quantum field theories, the limiting system consists of copies of a self-interacting quantum mechanical system. In a spatially discretized setting, a nonlinear “graph transform” arises that produces an in principle exact solution of the Functional Renormalization Group for the Legendre effective action. The quantum mechanical input data can be prepared from its 1 + 0 dimensional counterpart. In Einstein gravity, the anti-Newtonian limit has no dynamical spatial gradients, yet remains fully diffeomorphism invariant and propagates the original number of degrees of freedom. A canonical transformation (trivialization map) is constructed, in powers of a fractional inverse of Newton’s constant, that maps the ADM action into its anti-Newtonian limit. We outline the prospects of an associated trivializing flow in the quantum theory.
Highlights
Anti-Newtonian expansions are introduced for scalar quantum field theories and classical gravity
Most computations are based on Euclidean signature, including those in a foliated setting. This is unproblematic in flat space quantum field theories but not so in curved spacetimes, where a satisfactory notion of “Wick rotation” remains elusive [3]. This is compounded with the conformal factor instability. (ii) The Legendre effective action Γ is a highly nonlocal functional of the mean fields and no structural characterization of the terms that can occur is known
The truncation ansätze employed in solving the Functional Renormalization Group (FRG) lack a clear ordering principle
Summary
Non-perturbative techniques come in many guises (as do non-elephants). Non-perturbative series expansions often use an artificial control parameter (number of flavors) and are not applicable to pure gravity or single flavor systems. The functional integral based on Equation (1) comes with an ab-initio ultraviolet regularization and is well-defined whenever the quantum mechanical one is Correlation functions or their generating functionals can be computed by expanding in powers of the hopping term. Should be such that, for the spatially homogeneous truncation φ(t, x ) = φ(t), the pair a(t), φ(t) solves the coupled ordinary differential equations entailed by the Einstein field equations This can be built in, essentially by parameterizing the potential U (φ) in terms of a superpotential u(φ), which in turn arises as the logarithmic derivative of the scale factor a = a( φ) viewed as a function of φ. The successful implementation of Equation (6) turns out to imply the counterpart of Equation (3) for the Hamiltonian fields equations, but without troubles from constraint non-propagation
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