1. D-Branes and Noncommutative Geometry in Sting Theory Pichet Vanichchapongjaroen 3 rd March 2010

2. Introduction The Need For a New Model Noncommutative Geometry in String Theory Quantum Mechanics in Noncommutative Phase Space INTRODUCTIONINTRODUCTION

3. The Need For a New Model General Relativity (GR)  highly gravitating objects Quantum Mechanics (QM)  small objects What about But GR+QM does not work. GR requires smooth spacetime String Theory  noncommutative geometry (NCG) Inside Black HoleTime around Big Bang Need new model of spacetime THENEEDFORANEWMODELTHENEEDFORANEWMODEL Pictures from:http://commons.wikimedia.org/wiki/File:Black_Hole_in_the_universe.jpg http://en.wikipedia.org/wiki/File:Universe_expansion2.png

4. Strings STRINGSSTRINGS Quantise Particles and Fields

5. Commutation Relations Fields: NO Background: Flat String: Neutral Boundary Conditions Neumann Dirichlet D-BRANESD-BRANES D-Branes

6. Boundary Conditions Neumann Dirichlet NONCOMMUTATIVED-BRANENONCOMMUTATIVED-BRANE Commutation Relations Fields: constant NS-NS B-field Background: Flat String: Charged Noncommutative D-Brane

7. Topics in Quantum Field Theory in Noncommutative Spacetime UV/IR mixing Morita Equivalence etc. NONCOMMUTATIVEQFTNONCOMMUTATIVEQFT

8. Commutation Relations Boundary Conditions Neumann Dirichlet Fields: constant NS-NS B-field Background: pp-wave String: Charged D-Brane in pp-wave Background PP-WAVEBACKGROUNDPP-WAVEBACKGROUND D-BRANEIND-BRANEIN

9. To Study Physics in Noncommutative Phase Space Goal: Quantum Field Theory Quantum Field Theory  Lots of Simple Harmonic Oscillators Problem: Coordinate and Momentum Space Representation no longer works Need to view phase space as a whole Study Phase Space Quantisation NONCOMMUTATIVEPHASESPACENONCOMMUTATIVEPHASESPACE

10. Two Dimensional Simple Harmonic Oscillator 2DSHO2DSHO

11. Two Dimensional Simple Harmonic Oscillator in Noncommutative Phase Space 2DSHOINNCPHASESPACE2DSHOINNCPHASESPACE

12. 2DSHOINNCPHASESPACE2DSHOINNCPHASESPACE

13. 2DSHOINNCPHASESPACE2DSHOINNCPHASESPACE

14. 2DSHOINNCPHASESPACE2DSHOINNCPHASESPACE

15. 2DSHOINNCPHASESPACE2DSHOINNCPHASESPACE

16. Conclusion The need of a new model D-brane becomes noncommutative in some situations Noncommutative Phase Space: Use Phase Space Quantisation to study Simple Harmonic Oscillator  hope to get starting point for QFT Energy level of Noncommutative SHO is generally nondegenerate Sign of degenerate vacuum and vanished vacuum  further investigation CONCLUSIONCONCLUSION

17. References F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer. Deformation theory and quantization. II. Physical applications. Annals of Physics, 111:111–151, Mar. 1978. C.-S. Chu, P.-M. Ho, Noncommutative Open String and D- brane, Nucl. Phys. B550:151-168, 1999. C.-S. Chu and P.-M. Ho. Noncommutative D-brane and open string in pp-wave background with B-field. Nucl. Phys., B636:141–158, 2002. REFERENCESREFERENCES