Interpretation of Quantum Field Theories with a Minimal Length Scale

Quantum gravity is probably the most challenging and fascinating problem of physics in the 21st century. The most impressive indicator is the number of people working on it, even though so far there is no experimental evidence that might guide us from mathematics to physical reality. During the last years, the priority in the field has undergone a shift towards the phenomenology and possible predictions [1–12]. The phenomenology of quantum gravity has been condensed into effective models which incorporate one of the most important and general features: a minimal invariant length scale that acts as a regulator in the ultraviolet. Such a minimal length scale leads to a generalized uncertainty relation and it requires a deformation of Lorentz-invariance which becomes important at high boost parameters. The construction of a quantum field theory that selfconsistently allows such a minimal length makes it necessary to carefully retrace all steps of the standard quantization scheme. So far, there are various approaches how to construct a quantum field theory that incorporates a minimal length scale and the accompanying deformed special relativity (DSR), generalized uncertainty principle (GUP) and modified dispersion relation (MDR). Most notably, there are approaches which start from the DSR [13–18], the κ-Poincaré Hopf algebra [19–24] and those which start with the GUP [25–29]. Besides this, there exists the possibility to examine specific effects like reaction thresholds or radiation spectra starting from the MDR without aiming to derive a full quantum theory in the first place [30–32]. Relations between several approaches have been investigated in [33]. In this paper we aim to closely examine the ansatz starting with the GUP by paying special attention to the interpretation of the effective theory. Since this starting point is conceptually different from the DSRmotivated one, it does not suffer from some of the prob-