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"Minimal Length Scale Scenarios for Quantum Gravity"

Sabine Hossenfelder

	Abstract
1	Introduction
2	A Minimal History
3	Motivations
	3.1	Thought experiments
	3.2	String theory
	3.3	Loop Quantum Gravity and Loop Quantum Cosmology
	3.4	Quantized conformal fluctuations
	3.5	Asymptotically Safe Gravity
	3.6	Non-commutative geometry
	3.7	Miscellaneous
	3.8	Summary of motivations
4	Models and Applications
	4.1	Interpretation of a minimal length scale
	4.2	Modified commutation relations
	4.3	Quantum mechanics with a minimal length scale
	4.4	Quantum field theory with a minimal length scale
	4.5	Deformed Special Relativity
	4.6	Composite systems and statistical mechanics
	4.7	Path-integral duality
	4.8	Direct applications of the uncertainty principle
	4.9	Miscellaneous
5	Discussion
	5.1	Interrelations
	5.2	Observable consequences
	5.3	Is it possible that there is no minimal length?
6	Summary
	Acknowledgements
	Index
	References
	Footnotes
	Figures
	Tables

	Figure 1: Heisenberg’s microscope. A photon moving along the $x$ -axis scatters off a probe within an interaction region of radius $R$ and is detected by a microscope (indicated by a lens and screen) with opening angle $𝜖$ .
	Figure 2: The length of a string is not the same as its average extension. The lengths of strings in the groundstate were studied in [173].
	Figure 3: Spacetime uncertainty (red, solid) vs uncertainty from spherical black holes (blue, dotted) in $D = 10$ dimensions, for $g < 1 s$ (left) and $g > 1 s$ (right). After [318], Figure 1. Below the bound from spacetime uncertainty yet above the black-hole bound that hides short-distance physics (shaded region), the concept of classical geometry becomes meaningless.
	Figure 4: Cross section for scattering of two scalar particles by graviton exchange with and without running Planck mass, in units of the low-energy Planck mass $√ --- 1∕ G0$ . The dot-dashed (purple) line depicts the case without asymptotic safety; the continuous (blue) and dashed (grey) line take into account the running of the Planck mass, for two different values of the fixed point, $∘ --- 1∕ &tidle;G∗ = 0.024$ and $0.1$ respectively. Figure from [261]; reproduced with permission from IOP.