Quantum field theory, gravity and cosmology in a fractal universe

Abstract

We propose a model for a power-counting renormalizable field theory living in a fractal spacetime. The action is Lorentz covariant and equipped with a Stieltjes measure. The system flows, even in a classical sense, from an ultraviolet regime where spacetime has Hausdorff dimension 2 to an infrared limit coinciding with a standard D-dimensional field theory. We discuss the properties of a scalar field model at classical and quantum level. Classically, the field lives on a fractal which exchanges energy-momentum with the bulk of integer topological dimension D. Although an observer experiences dissipation, the total energy-momentum is conserved. The field spectrum is a continuum of massive modes. The gravitational sector and Einstein equations are discussed in detail, also on cosmological backgrounds. We find ultraviolet cosmological solutions and comment on their implications for the early universe.

Standard Model in multiscale theories and observational constraints

“…(59). Since the system in the fractional picture (43) is the same as the one in the integer picture (46), there is no nontrivial information in (43). A similar exact equivalence between the fractional and the integer picture was shown in [40] for electrodynamics.…”

“…These have been proposed either as stand-alone models of exotic geometry [31,32,40,43] or as an effective means to study, in a controlled manner, the change of dimensionality with the probed scale (known as dimensional flow 1 of these models and of their status). The model with ordinary derivatives was the first to be proposed [4,46,47] but it cannot be a fundamental theory due to some issues regarding its momentum space and quantization. The theory with fractional derivatives is most promising especially as far as renormalization is concerned but, apart from a general power-counting argument [32], its physical properties have not been studied yet; this will be done in the near future.…”

Geometry of fractional spaces

“…In the attempt to encode dimensional flow in a physically intuitive framework, in [23,39,40] a field theory living in a fractal spacetime was proposed. The key point is to replace the standard Lebesgue measure in the action with a Lebesgue-Stieltjes measure with anomalous scaling,…”

“…for some scalar v. In particular, the Poincaré algebra of these models is deformed [39] and at sufficiently small scales an observer should see deviations from Lorentz invariance. At the quantum level, (1.2) helps field theories with Lebesgue-Stieltjes measure to be power-counting renormalizable, gravity included.…”

Complex evolution of a multi-particle system

“…We are considering multi element systems going from huge down to micro scales, such as cosmological [16][17][18][19][20][21][22][23][24][25][26][27] and evo-lutionary [28][29][30][31] phenomena. A meta-level comprehensive formulation of the laws governing and encompassing all sys-tems is far from being tackled.…”