Magnetic monopoles in noncommutative spacetime                                      Tapio Salminen                        ...
Quantizing spacetime                      MotivationBlack hole formation in the process of measurement at smalldistances (...
Quantizing spacetime                      MotivationBlack hole formation in the process of measurement at smalldistances (...
Quantizing spacetime                      MotivationBlack hole formation in the process of measurement at smalldistances (...
Quantizing spacetime        Implementation  Impose [ˆµ , x ν ] = iθµν and          x ˆ    choose the frame where         ...
Quantizing spacetime                  Implementation           Impose [ˆµ , x ν ] = iθµν and                   x ˆ        ...
Wu-Yang monopole                     Commutative spacetimeFind potentials AN and AS such that:                 µ      µ   ...
Wu-Yang monopole                               Commutative spacetime Solution:  N/S                    N/SAt   = AN/S = Aθ...
Wu-Yang monopole                      Commutative spacetimeSolution:Single-valuedness of            2ige                 φ...
Wu-Yang monopole                            NC spacetimeFind potentials AN and AS such that:                 µ      µ    N...
Wu-Yang monopole                 Maxwell’s equations              1. NC Maxwell’s equations                   µνγδ        ...
Wu-Yang monopole                                       Maxwell’s equations                      Task: Expand to second ord...
Wu-Yang monopole      Maxwell’s equationsTask: Expand to second order in θ
Wu-Yang monopole                     Gauge transformations                   2. NC gauge transformations          N/S     ...
Wu-Yang monopole                                     Gauge transformations                      Task: Expand to second ord...
Wu-Yang monopole     Gauge transformationsTask: Expand to second order in θ
Wu-Yang monopole                 ContradictionComparing the two sets of equations for AN2 − AS2                           ...
Wu-Yang monopole                                         Contradiction         Comparing the two sets of equations for AN2...
Wu-Yang monopole                                         Contradiction         Comparing the two sets of equations for AN2...
Wu-Yang monopole                      Conclusion There does not exist potentials AN and AS that would                     ...
Wu-Yang monopole                      Conclusion There does not exist potentials AN and AS that would                     ...
Wu-Yang monopole                           DiscussionPossible causes for the failure of the DQC:   Rotational invariance, ...
Wu-Yang monopole                         DiscussionPossible causes for the failure of the DQC:   Rotational invariance, 3D...
Wu-Yang monopole                           DiscussionPossible causes for the failure of the DQC:   Rotational invariance, ...
Wu-Yang monopole                           DiscussionPossible causes for the failure of the DQC:   Rotational invariance, ...
BonusCovariant source
Wu-Yang monopole                   Covariant source                NC Maxwell’s equations                    Dµ F µν = J ν...
Wu-Yang monopole                          Covariant source                   NC Maxwell’s equations                       ...
Wu-Yang monopole                                     Covariant source       Using this requirement we get two covariant so...
Thank you
Upcoming SlideShare
Loading in …5
×

Monopole zurich

441 views

Published on

Seminar talk given in Quantum Theory and Gravitation, Zurich, June 2011.

Published in: Education
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
441
On SlideShare
0
From Embeds
0
Number of Embeds
8
Actions
Shares
0
Downloads
4
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Monopole zurich

  1. 1. Magnetic monopoles in noncommutative spacetime Tapio Salminen University of Helsinki In collaboration with Miklos L˚ngvik and Anca Tureanu a [arXiv:1104.1078], [arXiv:1101.4540]
  2. 2. Quantizing spacetime MotivationBlack hole formation in the process of measurement at smalldistances (∼ λP ) ⇒ additional uncertainty relations forcoordinates Doplicher, Fredenhagen and Roberts (1994)
  3. 3. Quantizing spacetime MotivationBlack hole formation in the process of measurement at smalldistances (∼ λP ) ⇒ additional uncertainty relations forcoordinates Doplicher, Fredenhagen and Roberts (1994)Open string + D-brane theory with an antisymmetric Bij fieldbackground ⇒ noncommutative coordinates in low-energylimit Seiberg and Witten (1999)
  4. 4. Quantizing spacetime MotivationBlack hole formation in the process of measurement at smalldistances (∼ λP ) ⇒ additional uncertainty relations forcoordinates Doplicher, Fredenhagen and Roberts (1994)Open string + D-brane theory with an antisymmetric Bij fieldbackground ⇒ noncommutative coordinates in low-energylimit Seiberg and Witten (1999) A possible approach to Planck scale physics is QFT in NC space-time
  5. 5. Quantizing spacetime Implementation Impose [ˆµ , x ν ] = iθµν and x ˆ choose the frame where   0 0 0 0  0 0 θ 0  θµν =   0 −θ 0 0   0 0 0 0
  6. 6. Quantizing spacetime Implementation Impose [ˆµ , x ν ] = iθµν and x ˆ choose the frame where   0 0 0 0  0 0 θ 0  θµν =   0 −θ 0 0   0 0 0 0 This leads to the -product of functions i ← − µν − → (f g ) (x) ≡ f (x)e 2 ∂ µ θ ∂ν g (y ) |y =xInfinite amount of derivatives induces nonlocality
  7. 7. Wu-Yang monopole Commutative spacetimeFind potentials AN and AS such that: µ µ N/S1. Bµ = × Aµ N/S2. Aµ are gauge transformable to each other in the overlap δ N/S3. Aµ are nonsingular outside the origin
  8. 8. Wu-Yang monopole Commutative spacetime Solution: N/S N/SAt = AN/S = Aθ = 0 r N gAφ = (1 − cos θ) r sin θ gAS = − φ (1 + cos θ) r sin θ that gauge transformAN/S → UAN/S U −1 = Aµ µ µ S/N 2ige φ U=e c
  9. 9. Wu-Yang monopole Commutative spacetimeSolution:Single-valuedness of 2ige φ U=e c implies 2ge = N = integer c Dirac Quantization Condition (DQC)
  10. 10. Wu-Yang monopole NC spacetimeFind potentials AN and AS such that: µ µ N/S1. Aµ satisfy NC Maxwell’s equations N/S2. Aµ are gauge transformable to each other in the overlap δ N/S3. Aµ are nonsingular outside the origin
  11. 11. Wu-Yang monopole Maxwell’s equations 1. NC Maxwell’s equations µνγδ Dν Fγδ = 0 Dµ F µν = J ν 1 µνγδwhere Fµν = 2 Fγδ is the dual field strength tensor and Fµν = ∂µ Aν − ∂ν Aµ − ie[Aµ , Aν ] Dν = ∂ν − ie[Aν , ·] Task: Expand to second order in θ
  12. 12. Wu-Yang monopole Maxwell’s equations Task: Expand to second order in θ2 N S 4θ 2 xz h 2 2 3 2 2 2 2 4 2 2 6 i (B 2 − B 2 )1 = − 375(x + y ) + 131z (x + y ) − 2z (x + y ) − 4z (x 2 + y 2 )3 r 10 N S − ∂1 ρ 2 + ∂1 ρ 22 N S 4θ 2 yz h 2 2 3 2 2 2 2 4 2 2 6 i (B 2 − B 2 )2 = − 375(x + y ) + 131z (x + y ) − 2z (x + y ) − 4z (x 2 + y 2 )3 r 10 N S − ∂2 ρ 2 + ∂2 ρ 22 N S 4θ 2 h 2 2 5 2 2 4 2 2 2 3 4 (B 2 − B 2 )3 = 120(x + y ) − 900(x + y ) z − 1285(x + y ) z (x 2 + y 2 )4 r 10 i 2 2 2 6 2 2 8 10 N S − 1289(x + y ) z − 652(x + y )z − 132z − ∂3 ρ 2 + ∂3 ρ 2
  13. 13. Wu-Yang monopole Maxwell’s equationsTask: Expand to second order in θ
  14. 14. Wu-Yang monopole Gauge transformations 2. NC gauge transformations N/S S/N Aµ should transform to Aµ (x) under U (1)AN/S (x) → U(x) AN/S (x) U −1 (x)−iU(x) ∂µ U −1 (x) = AS/N (x) µ µ µ with groups elements U(x) = e iλ Task: Expand to second order in θ
  15. 15. Wu-Yang monopole Gauge transformations Task: Expand to second order in θ2 N S GT 4θ 2 xz “ 2 2 3 2 2 2 2 2 2 4 6 ” (B 2 − B 2 )1 = − 321(x + y ) + 205(x + y ) z + 26(x + y )z + 4z (x 2 + y 2 )3 r 102 N S GT 4θ 2 yz “ 2 2 3 2 2 2 2 2 2 4 6 ” (B 2 − B 2 )2 = − 321(x + y ) + 205(x + y ) z + 26(x + y )z + 4z (x 2 + y 2 )3 r 102 N S GT 4θ 2 “ 2 2 5 2 2 4 2 2 2 3 4 (B 2 − B 2 )3 = 144(x + y ) − 564(x + y ) z − 455(x + y ) z (x 2 + y 2 )4 r 10 ” 2 2 2 6 2 2 8 10 − 403(x + y ) z − 188(x + y )z − 36z
  16. 16. Wu-Yang monopole Gauge transformationsTask: Expand to second order in θ
  17. 17. Wu-Yang monopole ContradictionComparing the two sets of equations for AN2 − AS2 i i After some algebra we get...
  18. 18. Wu-Yang monopole Contradiction Comparing the two sets of equations for AN2 − AS2 i i N S 24θ 2 x “ 2 2 4 2 2 3 2 2 2 2 40 = (∂x ∂z − ∂z ∂x )(ρ 2 − ρ 2 ) = 41(x + y ) + 426(x + y ) z + 704(x + y ) z (x 2 + y 2 )5 r 8 ” 2 2 6 8 + 496(x + y )z + 128z N S 24θ 2 y “ 2 2 4 2 2 3 2 2 2 2 40 = (∂y ∂z − ∂z ∂y )(ρ 2 − ρ 2 ) = 41(x + y ) + 426(x + y ) z + 704(x + y ) z (x 2 + y 2 )5 r 8 ” 2 2 6 8 + 496(x + y )z + 128z
  19. 19. Wu-Yang monopole Contradiction Comparing the two sets of equations for AN2 − AS2 i i N S 24θ 2 x “ 2 2 4 2 2 3 2 2 2 2 40 = (∂x ∂z − ∂z ∂x )(ρ 2 − ρ 2 ) = 41(x + y ) + 426(x + y ) z + 704(x + y ) z (x 2 + y 2 )5 r 8 ” 2 2 6 8 + 496(x + y )z + 128z N S 24θ 2 y “ 2 2 4 2 2 3 2 2 2 2 40 = (∂y ∂z − ∂z ∂y )(ρ 2 − ρ 2 ) = 41(x + y ) + 426(x + y ) z + 704(x + y ) z (x 2 + y 2 )5 r 8 ” 2 2 6 8 + 496(x + y )z + 128z These equations have no solution!
  20. 20. Wu-Yang monopole Conclusion There does not exist potentials AN and AS that would µ µsimultaneously satisfy Maxwell’s equations and be gauge transformable to each other.
  21. 21. Wu-Yang monopole Conclusion There does not exist potentials AN and AS that would µ µsimultaneously satisfy Maxwell’s equations and be gauge transformable to each other. ⇒ The DQC cannot be satisfied
  22. 22. Wu-Yang monopole DiscussionPossible causes for the failure of the DQC: Rotational invariance, 3D vs 2D Aharonov-Bohm effect works Vortex line quantization has problems CP violation and the Witten effect Perturbative method used
  23. 23. Wu-Yang monopole DiscussionPossible causes for the failure of the DQC: Rotational invariance, 3D vs 2D Aharonov-Bohm effect works Vortex line quantization has problems CP violation and the Witten effect Perturbative method used
  24. 24. Wu-Yang monopole DiscussionPossible causes for the failure of the DQC: Rotational invariance, 3D vs 2D Aharonov-Bohm effect works Vortex line quantization has problems CP violation and the Witten effect Perturbative method used
  25. 25. Wu-Yang monopole DiscussionPossible causes for the failure of the DQC: Rotational invariance, 3D vs 2D Aharonov-Bohm effect works Vortex line quantization has problems CP violation and the Witten effect Perturbative method used
  26. 26. BonusCovariant source
  27. 27. Wu-Yang monopole Covariant source NC Maxwell’s equations Dµ F µν = J νThe lhs transforms covariantly under gauge transformations⇒ also the rhs must transform nontrivially
  28. 28. Wu-Yang monopole Covariant source NC Maxwell’s equations Dµ F µν = J νThe lhs transforms covariantly under gauge transformations⇒ also the rhs must transform nontriviallyFrom this one gets the gauge covariance requirement up tothe 2nd order correction (J 0 = ρ = ρ0 + ρ1 + ρ2 + O(θ3 ))ρ1 → ρ1 + θij ∂i λ∂j ρ0 θij θklρ2 → ρ2 + θij ∂i λ∂j ρ1 + ∂k λ∂i λ∂j ∂l ρ0 − ∂j λ∂l ρ0 ∂i ∂k λ 2
  29. 29. Wu-Yang monopole Covariant source Using this requirement we get two covariant sources „ “ ” ρ = 4πg δ 3 (r ) − θkl ∂k Al δ 3 (r ) + θij A1 ∂i δ 3 (r ) j » “ ” 1 – « +θij θkl A0 ∂k ∂i A0 δ 3 (r ) + A0 ∂i δ 3 (r ) + A0 A0 ∂j ∂l δ 3 (r ) + O(θ3 ) j l l i k 2 „ « 3 ij 1ρ = 4πg δ (r ) − θ A0 ∂i δ 3 (r ) j −θ ij A1 ∂i δ 3 (r ) j + θij θkl A0 A0 ∂j ∂l δ 3 (r ) + O(θ3 ) i k 2 All of the coefficients are uniquely fixed!
  30. 30. Thank you

×