Nonlocality in string theory

Abstract

We discuss an aspect of string theory which has been tackled from many different perspectives, but incompletely: the role of nonlocality in the theory and its relation to the geometric shape of the string. In particular, we will describe in quantitative terms how one can zoom out from an extended object such as the string in such a way that, at sufficiently large scales, it appears structureless. Since there are no free parameters in free-string theory, the notion of large scales will be unambiguously determined. In other words, we will be able to answer the question: How and at which scale can the string be seen as a particle? In doing so, we will employ the concept of spectral dimension in a new way with respect to its usual applications in quantum gravity. The operational notions of worldsheet and target spacetime dimension in string theory are also clarified and found to be in mutual agreement.

Diffusion in multiscale spacetimes

“…In general, one of the indicators characterizing quantum geometry, the spectral dimension d S of spacetime, changes with the scale, running from d S 2 (or exactly d S = 2) in the ultraviolet (UV) to the usual, classical value d S ∼ 4 in the infrared (IR). Numerical and analytic examples can be found in causal dynamical triangulations (CDT) [4,5], random combs [6,7] and random multigraphs [8,9] (both sharing some properties with CDT), quantum Einstein gravity (QEG, also called asymptotic safety) [10,11], spin foams [12][13][14][15], Hořava-Lifshitz gravity [16,17], noncommutative geometry at the fundamental [18,19] and effective [20][21][22] levels, field theory on multifractal spacetimes [23][24][25] (in particular, in the realization within multifractional geometry [26][27][28][29][30][31]), and nonlocal super-renormalizable quantum gravity [32][33][34].…”

“…Due to the young age of the proposal, these tools are largely unexplored and the purpose of this paper is to continue the investigation initiated in Refs. [26][27][28][29], in the meanwhile improving the understanding of dimensional flow as a general phenomenon of quantum geometry.…”

“…[27]. Systems with anomalous dimension are described by fractional versions of the standard diffusion * calcagni@iem.cfmac.csic.es; Present address: Instituto de Estructura de la Materia, CSIC, Serrano 121, 28006 Madrid, Spain.…”

“…After the extension of fractional measures to bilateral worlds [29], the calculation of the spectral dimension of Ref. [27], with further comments on the diffusion probability, can be generalized to these cases and to fractional Laplacians. In Secs.…”

“…III B-III D we exploit the recent construction of a momentum transform in fractional spaces [29] and include a class of fractional Laplacians not considered in Refs. [27,28]. The classification of different types of diffusion is heavily based upon known results in the literature on probability theory, stochastic processes, and diffusion, although here we give the subject a different spin by identifying the diffusion medium with spacetime itself (Sec.…”