English

Quantum spectral dimension in quantum field theory

High Energy Physics - Theory 2016-04-13 v3 Mathematical Physics math.MP

Abstract

We reinterpret the spectral dimension of spacetimes as the scaling of an effective self-energy transition amplitude in quantum field theory (QFT), when the system is probed at a given resolution. This picture has four main advantages: (a) it dispenses with the usual interpretation (unsatisfactory in covariant approaches) where, instead of a transition amplitude, one has a probability density solving a nonrelativistic diffusion equation in an abstract diffusion time; (b) it solves the problem of negative probabilities known for higher-order and nonlocal dispersion relations in classical and quantum gravity; (c) it clarifies the concept of quantum spectral dimension as opposed to the classical one. We then consider a class of logarithmic dispersion relations associated with quantum particles and show that the spectral dimension dsd_s of spacetime as felt by these quantum probes can deviate from its classical value, equal to the topological dimension DD. In particular, in the presence of higher momentum powers it changes with the scale, dropping from DD in the infrared (IR) to a value dsUVDd_s^{\rm UV}\leq D in the ultraviolet (UV). We apply this general result to Stelle theory of renormalizable gravity, which attains the universal value dsUV=2d_s^{\rm UV}=2 for any dimension DD.

Keywords

Cite

@article{arxiv.1408.0199,
  title  = {Quantum spectral dimension in quantum field theory},
  author = {Gianluca Calcagni and Leonardo Modesto and Giuseppe Nardelli},
  journal= {arXiv preprint arXiv:1408.0199},
  year   = {2016}
}

Comments

26 pages, 3 figures; v2: discussion clarified and improved at several points, typos corrected, results unchanged; v3: some material confined to an appendix, discussion streamlined, results unchanged

R2 v1 2026-06-22T05:18:30.856Z