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2012研讨会 2012/10/27
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.
Higher Energy needed to probe shorter distance, but if energy is too high what happens?
Effective Model Generalized Uncertainty Principles (GUP) For the Usual QM For GUP modified QM, is model dependent. For example
Minimal Length Uncertainty Relation Assume Start with This leads to
GUP Modified Schrödinger Equation Start with Representation of x and p In “coordinate” space The Hamiltonian
Phys.Lett.B678:497-499,2009, arXiv:0906.5396 arXiv:1107.3164
Finite Square Potential We only consider bound state（0＜E＜V0） Stationary Schrödinger Equations： To order of To order of Impose symmetry
Solutions to order of Boundary Conditions Symmetry
Solutions to order of Boundary Conditions Symmetry
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 .
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
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
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
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
GUP Modified Schwinger Formula Model (a) Model (b) Up to
Uncertainty Principles and Hawking Temperature Ordinary QM GUP
Schwinger Mechanism and Hawking Temperature (Naive) Ordinary
Schwinger Mechanism and Hawking Temperature (Naive) GUP
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.
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