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

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

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:

Strings STRINGSSTRINGS Quantise Particles and Fields

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

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

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

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

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

Two Dimensional Simple Harmonic Oscillator 2DSHO2DSHO

Two Dimensional Simple Harmonic Oscillator in Noncommutative Phase Space 2DSHOINNCPHASESPACE2DSHOINNCPHASESPACE

2DSHOINNCPHASESPACE2DSHOINNCPHASESPACE

2DSHOINNCPHASESPACE2DSHOINNCPHASESPACE

2DSHOINNCPHASESPACE2DSHOINNCPHASESPACE

2DSHOINNCPHASESPACE2DSHOINNCPHASESPACE

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

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