1. Peng Wang Sichuan University
Implications of Minimal Length Scale on Quantum Mechanics and Quantum Field Theory
Peng Wang
Sichuan University
弦论、粒子物理和宇宙学201​2研讨会
2012/10/27

2. Why Minimal Length
Suggested by Quantum Gravity. Observed in perturbative String Theory.
Consider a particle with energy E which is not a black hole, Its radius r must satisfy
where 1/E Compton Length and E come from Hoop conjecture proposed by Kip Thorne.

3. Higher Energy needed to probe shorter distance, but if energy is too high what happens?

4. Effective Model Generalized Uncertainty Principles (GUP)
For the Usual QM
For GUP modified QM, is model dependent. For example

5. Minimal Length Uncertainty Relation
Assume
Start with
This leads to

6. GUP Modified Schrödinger Equation
Start with
Representation of x and p
In “coordinate” space
The Hamiltonian

7. Phys.Lett.B678: ,2009, arXiv:
arXiv:

8. Finite Square Potential
We only consider bound state（0＜E＜V0）
Stationary Schrödinger Equations：
To order of
To order of
Impose symmetry

9. Solutions to order of
Boundary Conditions Symmetry

10. Solutions to order of
Boundary Conditions Symmetry

11. Punch Line
Sharp Potentials are not appropriate in GUP and hence one may resort to smooth potentials.
Future work:
Consider the potential .
It is interesting to see what happens as

12. What is the Schwinger Mechanism?
The Schwinger mechanism refers to the production of charged fermion-antifermion pairs out of the vacuum by a static external electric field.
Non-perturbative particle production in a strong classical electric field.
It has motivated analysis in many parts of particle physics and quantum field theory:
Models of string breaking in QCD
Insights into Hawking radiation near black holes
Schematics of semi-classical tunneling

13. Derivation of Schwinger Formula
Using path integral, the vacuum to vacuum amplitude
where L is a Lagrangian for a charged spinor
and are eigenvalues of
The rate per unit volume per unit time for pair production in a constant electrical field is

14. GUP Modified QFT Model (a)
where they consider modified commutators
and simply replace for a gauge field theory
Finally, one has for a charged fermion

15. GUP Modified QFT Model (b)
where they define frequency and wave vector through
and make replacement for a gauge field theory
Finally, one has for a charged fermion

16. GUP Modified Schwinger Formula
Model (a)
Model (b)
Up to

17. Uncertainty Principles and Hawking Temperature
Ordinary QM
GUP

18. Schwinger Mechanism and Hawking Temperature (Naive) Ordinary

19. Schwinger Mechanism and Hawking Temperature (Naive) GUP

20. Conclusion
Finite potential square is considered in GUP framework. We claim that sharp potential is not appropriate in GUP.
Pair production rates in a constant electrical field are calculated in two models of GUP modified QFT. Through hand waving arguments, we can obtain expressions for Hawking temperature from the pair production rates.

21. Welcome to Chengdu
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