6 Summary

We have seen in this review that there are many motivations for the existence of a minimal length scale. Various thought experiments suggest there are limits to how well we can resolve structures. String theory and LQG, presently the two most widely pursued approaches to quantum gravity, both bring with them a minimal length scale, if in very different realizations. It has been argued that a minimal length scale also exists in the scenario of ASG, and that non-commutative geometries have a minimal length scale built in already.

With that extensive motivation, many models have been proposed that aim at incorporating a minimal length scale into the quantum field theories of the standard model, rather than waiting for a theory of quantum gravity to be developed and eventually connected to the standard model. We have discussed some of these approaches, and also identified some key open problems. While a lot of work has been done directly studying the implications of modified dispersion relations, deformations of special relativity and a GUP, the underlying framework is not yet entirely understood. Most importantly, there is the question of how to construct physically-meaningful observables. One possibility to address this and some other open questions is to develop an axiomatic approach based on the geometry of phase space.

Exploring the consequences of a minimal length scale is one of the best motivated avenues to make contact with the phenomenology of quantum gravity, and to gain insights about the fundamental structure of space and time.

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