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D- Branes and Noncommutative Geometry in Sting Theory

D- Branes and Noncommutative Geometry in Sting Theory. Pichet Vanichchapongjaroen 3 rd March 2010. Introduction. The Need F or a New Model. General Relativity (GR)  highly gravitating objects Quantum Mechanics (QM)  small objects What about But GR+QM does not work.

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D- Branes and Noncommutative Geometry in Sting Theory

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  1. D-Branes and Noncommutative Geometry in Sting Theory PichetVanichchapongjaroen 3rdMarch 2010

  2. Introduction

  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 Hole Time around Big Bang Need new model of spacetime 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 • Strings • Particles and Fields Quantise NS-NS B-Field

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

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

  7. Topics in Quantum Field Theory in NoncommutativeSpacetime • UV/IR mixing • Morita Equivalence etc.

  8. D - B R A N E I N P P - W A V E B A C K G R O U N D D-Brane in pp-wave Background Boundary Conditions Fields: constant NS-NS B-field Background: pp-wave String: Charged • Neumann • Dirichlet Commutation Relations

  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

  10. Two Dimensional Simple Harmonic Oscillator • Hamiltonian • Commutation Relations • Spectrum • Degeneracies 1 state 2 states 3 states

  11. Two Dimensional Simple Harmonic Oscillator in Noncommutative Phase Space • Hamiltonian • Commutation Relations • Spectrum • This analysis valid for

  12. Two Dimensional Simple Harmonic Oscillator in Noncommutative Phase Space • Small

  13. Two Dimensional Simple Harmonic Oscillator in Noncommutative Phase Space • (noncommutativespacetime)

  14. Two Dimensional Simple Harmonic Oscillator in Noncommutative Phase Space • General

  15. 2 D S H O I N N C P H A S E S P A C E Two Dimensional Simple Harmonic Oscillator in Noncommutative Phase Space • Assume continues to work for • Degenerate vacuum with • No vacuum as

  16. C O N C L U S I O N 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

  17. R E F E R E N C E S 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.

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