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Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
Monopole zurich
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Monopole zurich

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

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

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  • 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. Quantizing spacetime MotivationBlack hole formation in the process of measurement at smalldistances (∼ λP ) ⇒ additional uncertainty relations forcoordinates Doplicher, Fredenhagen and Roberts (1994)
  • 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. 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. 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. 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. 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. 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. Wu-Yang monopole Commutative spacetimeSolution:Single-valuedness of 2ige φ U=e c implies 2ge = N = integer c Dirac Quantization Condition (DQC)
  • 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. 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. 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. Wu-Yang monopole Maxwell’s equationsTask: Expand to second order in θ
  • 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. 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. Wu-Yang monopole Gauge transformationsTask: Expand to second order in θ
  • 17. Wu-Yang monopole ContradictionComparing the two sets of equations for AN2 − AS2 i i After some algebra we get...
  • 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. 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. 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. 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. 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. 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. 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. 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. BonusCovariant source
  • 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. 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. 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. Thank you

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