D-Branes and Noncommutative Geometry in Sting Theory Pichet Vanichchapongjaroen 3 rd March 2010
Mar 23, 2016
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