D-Branes and Noncommutative Geometry

in Sting Theory

Pichet Vanichchapongjaroen3rd March 2010

Introduction

The Need For a New Model

Noncommutative Geometry in String Theory

Quantum Mechanics in Noncommutative Phase Space

INTRODUCTION

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 modelof spacetime

THE

NEED

FOR

A

NEW

MODEL

Pictures from: http://commons.wikimedia.org/wiki/File:Black_Hole_in_the_universe.jpghttp://en.wikipedia.org/wiki/File:Universe_expansion2.png

Strings

STRINGS

Strings

Quantise

Particles and Fields

NS-NS B-Field

[ x̂ i , p̂ j ]=iℏδ❑ij ,❑

❑ [ x̂ i , x̂ j ]=[ p̂i , p̂ j ]=0Commutation Relations

Fields: NOBackground: FlatString: Neutral

Boundary Conditions

•Neumann•Dirichlet

D-BRANES

D-Branes

Boundary Conditions

•Neumann•Dirichlet

NONCOMMUTATIVE

D-BRANE

[ x̂ i , p̂ j ]=iℏ𝛿❑ij ,❑

❑ [ x̂ i , x̂ j ]=i𝜃ij ,❑❑ [ p̂i , p̂ j ]=0Commutation Relations

Fields: constant NS-NS B-fieldBackground: FlatString: Charged

Noncommutative D-Brane

Topics in Quantum Field Theory in Noncommutative

Spacetime

•UV/IR mixing•Morita Equivalence

etc.

NONCOMMUTATIVE

QFT

[ x̂ i , p̂ j ]=iℏ𝛿❑ij ,❑

❑ [ x̂ i , x̂ j ]=i𝜃ij ,❑❑ [ p̂i , p̂ j ]=i ϕij

Commutation Relations

Boundary Conditions

•Neumann•Dirichlet

Fields: constant NS-NS B-fieldBackground: pp-waveString: Charged

D-Brane in pp-wave BackgroundPP-

WAVE

BACKGROUND

D-BRANE

IN

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

NONCOMMUTATIVE

PHASE

SPACE

¿ x⃗ ⟩¿ p⃗⟩

Two Dimensional Simple Harmonic Oscillator

• Hamiltonian

• Commutation Relations

• Spectrum

• Degeneracies1 state2 states3 states

2D

SHO

Two Dimensional Simple Harmonic Oscillator in Noncommutative Phase Space

2D

SHO

IN

NC

PHASE

SPACE

• Hamiltonian

• Commutation Relations

• Spectrum

• This analysis valid for

Two Dimensional Simple Harmonic Oscillator in Noncommutative Phase Space

2D

SHO

IN

NC

PHASE

SPACE

• Small 𝜃𝜙<1

Two Dimensional Simple Harmonic Oscillator in Noncommutative Phase Space

2D

SHO

IN

NC

PHASE

SPACE

• (noncommutative spacetime) 𝜃𝜙<1

Two Dimensional Simple Harmonic Oscillator in Noncommutative Phase Space

2D

SHO

IN

NC

PHASE

SPACE

• General

𝜙=1

𝜃𝜙<1

Two Dimensional Simple Harmonic Oscillator in Noncommutative Phase Space

2D

SHO

IN

NC

PHASE

SPACE

• Assume

continues to work for • Degenerate vacuum with

• No vacuum as

𝜃𝜙≥1

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

CONCLUSION

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.

REFERENCES