Geometry and field theory in multi-fractional spacetime

Abstract

We construct a theory of fields living on continuous geometries with fractional Hausdorff and spectral dimensions, focussing on a flat background analogous to Minkowski spacetime. After reviewing the properties of fractional spaces with fixed dimension, presented in a companion paper, we generalize to a multi-fractional scenario inspired by multi-fractal geometry, where the dimension changes with the scale. This is related to the renormalization group properties of fractional field theories, illustrated by the example of a scalar field. Depending on the symmetries of the Lagrangian, one can define two models. In one of them, the effective dimension flows from 2 in the ultraviolet (UV) and geometry constrains the infrared limit to be four-dimensional. At the UV critical value, the model is rendered power-counting renormalizable. However, this is not the most fundamental regime. Compelling arguments of fractal geometry require an extension of the fractional action measure to complex order. In doing so, we obtain a hierarchy of scales characterizing different geometric regimes. At very small scales, discrete symmetries emerge and the notion of a continuous spacetime begins to blur, until one reaches a fundamental scale and an ultra-microscopic fractal structure. This fine hierarchy of geometries has implications for non-commutative theories and discrete quantum gravity. In the latter case, the present model can be viewed as a top-down realization of a quantum-discrete to classical-continuum transition.

Standard Model in multiscale theories and observational constraints

“…(A) Multiscale spacetimes were originally proposed, with quantum gravity in mind, as a class of theories where the renormalization properties of perturbative quantum field theory could be improved, including in the gravity sector [4,32]. Later on, it was shown that the two theories considered in the present paper do not have improved renormalizability [41], while the arguments of [32] still apply to the theory with fractional derivatives.…”

“…In this context, we focus on theories of multiscale spacetimes [4,[30][31][32][33][34][35][36][37][38][39][40][41][42][43][44]. 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).…”

“…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 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. The scenario with weighted derivatives and the one with q-derivatives, on both of which we focus here, have been analyzed in greater detail both in quantum field theory [32,38,40,41] and cosmology [43] and a wealth of new phenomenology has begun to emerge.…”

Convexity and the Euclidean Metric of Space-Time

“…Since the theory does not possess any "memory" in its formulation, how then one can make any prediction based on the behavior of geodesics which largely encode the underlying geometry, if the theory has branching geodesics? If, for instance, the action S or the resulting kinematic equations possessed some form "memory" as in the case of systems being modelled by fractional derivatives [72][73][74], then the use of geometric structures with branching geodesics might not pose a serious problem to predictability. Knowing this, one may wish to stay as far away as possible from using metrics that may allow for the possibility of branching geodesics.…”