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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

According to the Hellinger–Toeplitz theorem, an everywhere-defined symmetric operator on a Hilbert space is necessarily bounded. Since some operators in quantum mechanics are unbounded, one is required to deal with wave functions that are not square integrable. The same consideration applies here.