1. Tatsu Takeuchi Virginia Tech @Osaka University, January 10, 2012

2. Work in collaboration with: Zack Lewis Djordje Minic Lay Nam Chang Also contributions from: Naotoshi Okamura (Yamanashi Univ.) Sandor Benczik (quit physics) Saif Rayyan (MIT, physics education)

3. The Minimal Length Uncertainty Relation Suggested by Quantum Gravity. Observed in perturbative String Theory.

4. Deformed Commutation Relation

5. Harmonic Oscillator

6. Multi Dimensional Case

7. Isotropic Harmonic Oscillator in D dimensions

8. 2D Case

9. Some Details (1D case) :

10. Solution:

11. Wave-functions: n=0 n=1 n=2

12. Uncertainties of the Harmonic Oscillator: β=0

13. Uncertainties of the Harmonic Oscillator: β≠0 How can we get onto the Δx~Δp branch?

14. Harmonic Oscillator with negative mass:

16. Uncertainties of the Harmonic Oscillator: β≠0, m<0

17. Classical Limit:

18. Classical Harmonic Oscillator:

19. m>0 case:

20. m  ∞  limit:

21. m<0 case:

22. Compactification:

23. Classical Probabilities:

24. Comparison with Quantum Probabilities:

25. Work in progress: Discreate energy eigenstates have been found for the negative mass case. Uncertainties are difficult to calculate. What do we mean by x=0 ?

26. The minimal length uncertainty relation allows discreate energy “bound” states for “inverted” potentials. In the classical limit, these “bound” states can be understood to be due to the finite time the particle spends near the phase-space origin. Particles move at arbitrary large velocites. Do the non- relativistic negative mass states correspond to relativistic imaginary mass tachyons? Conclusions: