Download D-Branes and Noncommutative Geometry in Sting Theory

D-Branes and
Noncommutative Geometry
in Sting Theory
Pichet Vanichchapongjaroen
3rd March 2010
Introduction
The Need For a New Model
Noncommutative Geometry in String
Theory
Quantum Mechanics in
Noncommutative Phase Space
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The Need For a New Model
• General Relativity (GR)  highly gravitating objects
• Quantum Mechanics (QM)  small objects
• What about
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Inside Black Hole
Time around Big Bang
• But GR+QM does not work.
Need new model
• GR requires smooth spacetime
of spacetime
• String Theory  noncommutative geometry (NCG)
Pictures from:
http://commons.wikimedia.org/wiki/File:Black_Hole_in_the_universe.jpg
http://en.wikipedia.org/wiki/File:Universe_expansion2.png
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Strings
Strings
Quantise
Particles and Fields
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D-Branes
Boundary Conditions
• Neumann
• Dirichlet
Fields: NO
Background: Flat
String: Neutral
Commutation Relations
𝑥 𝑖 , 𝑝𝑗 = 𝑖ℏδ𝑖 𝑗 ,
𝑥 𝑖 , 𝑥 𝑗 = 𝑝𝑖 , 𝑝𝑗 = 0
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Noncommutative D-Brane
Boundary Conditions
• Neumann
• Dirichlet
Fields: constant NS-NS B-field
Background: Flat
String: Charged
Commutation Relations
𝑥 𝑖 , 𝑝𝑗 = 𝑖ℏ𝛿 𝑖 𝑗 ,
𝑥 𝑖 , 𝑥 𝑗 = 𝑖𝜃 𝑖𝑗 ,
𝑝𝑖 , 𝑝𝑗 = 0
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Topics in Quantum Field
Theory in Noncommutative
Spacetime
• UV/IR mixing
• Morita Equivalence
etc.
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D-Brane in pp-wave Background
Boundary Conditions
• Neumann
• Dirichlet
Fields: constant NS-NS B-field
Background: pp-wave
String: Charged
Commutation Relations
𝑥 𝑖 , 𝑝𝑗 = 𝑖ℏ𝛿 𝑖 𝑗 ,
𝑥 𝑖 , 𝑥 𝑗 = 𝑖𝜃 𝑖𝑗 ,
𝑝𝑖 , 𝑝𝑗 = 𝑖ϕ𝑖𝑗
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To Study Physics in
Noncommutative Phase Space
• Goal: Quantum Field Theory
• Quantum Field Theory  Lots of
Simple Harmonic Oscillators
|𝑥〉 |𝑝〉
• Problem: Coordinate and Momentum
Space Representation no longer works
• Need to view phase space as a whole
• Study Phase Space Quantisation
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Two Dimensional Simple
Harmonic Oscillator
• Hamiltonian
1 2
𝐻 = 𝑥 + 𝑦 2 + 𝑝𝑥2 + 𝑝𝑦2
2
• Commutation Relations
𝑥 𝑖 , 𝑝𝑗 = 𝑖δ𝑖 𝑗 ,
𝑥 𝑖 , 𝑥 𝑗 = 𝑝𝑖 , 𝑝𝑗 = 0,
𝑖, 𝑗 = 1,2
• Spectrum
𝑛 = 0,1,2, … ,
• Degeneracies
𝐸𝑛,𝑚 = 1
𝐸𝑛,𝑚 = 2
𝐸𝑛,𝑚 = 3
𝐸𝑛,𝑚 = 𝑛 + 1,
𝑚 = −𝑛, −𝑛 + 2, … , 𝑛 − 2, 𝑛
1 state
2 states
3 states
⋮
𝑚=0
𝑚 = −1,1
𝑚 = −2,0,2
2
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Two Dimensional Simple Harmonic Oscillator
in Noncommutative Phase Space
• Hamiltonian
1 2
𝐻 = 𝑥 + 𝑦 2 + 𝑝𝑥2 + 𝑝𝑦2
2
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• Commutation Relations
𝑥 𝑖 , 𝑝𝑗 = 𝑖δ𝑖 𝑗 , 𝑥, 𝑦 = 𝑖𝜃, 𝑝𝑥 , 𝑝𝑦 = 𝑖𝜙,
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𝑖, 𝑗 = 1,2
• Spectrum
𝑛+1
𝜃+𝜙 𝑚
2
𝐸′𝑛,𝑚 =
4+ 𝜃−𝜙 −
,
2
2
𝑛 = 0,1,2, … ,
𝑚 = −𝑛, −𝑛 + 2, … , 𝑛 − 2, 𝑛
• This analysis valid for 𝜃𝜙 < 1
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Two Dimensional Simple Harmonic Oscillator
in Noncommutative Phase Space
𝜃𝜙 < 1
• Small 𝜃, 𝜙
′
𝐸𝑛,𝑚
− 𝐸𝑛,𝑚
𝑚
≈ − (𝜃 + 𝜙)
2
2
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Two Dimensional Simple Harmonic Oscillator
in Noncommutative Phase Space
• 𝜙 = 0 (noncommutative spacetime)
𝜃𝜙 < 1
2
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Two Dimensional Simple Harmonic Oscillator
in Noncommutative Phase Space
• General 𝜃, 𝜙
𝜃𝜙 < 1
𝜙=1
2
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Two Dimensional Simple Harmonic Oscillator
in Noncommutative Phase Space
• Assume
𝜃𝜙 ≥ 1
𝑛+1
𝜃+𝜙 𝑚
2
𝐸′𝑛,𝑚 =
4+ 𝜃−𝜙 −
2
2
continues to work for 𝜃𝜙 ≥ 1
• 𝜃𝜙 = 1: Degenerate vacuum with
𝜃+𝜙
′
𝐸𝑣𝑎𝑐𝑢𝑢𝑚 =
2
• 𝜃𝜙 > 1: No vacuum
𝐸′𝑛,𝑚 → −∞ as 𝑛 → ∞
2
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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
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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 Dbrane, 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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