The document summarizes research on magnetic monopoles in noncommutative spacetime. It begins by motivating noncommutative spacetime as a way to incorporate quantum gravitational effects. It then shows that attempting to quantize spacetime by imposing noncommutativity of coordinates leads to inconsistencies when trying to define a Wu-Yang magnetic monopole in this framework. Specifically, the potentials describing the monopole fail to simultaneously satisfy Maxwell's equations and transform correctly under gauge transformations when expanded to second order in the noncommutativity parameter. This suggests the Dirac quantization condition cannot be satisfied in noncommutative spacetime. Possible reasons for this failure and directions for future work are discussed.

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
Motivation
Black hole formation in the process of measurement at small
distances (∼ λP ) ⇒ additional uncertainty relations for
coordinates
Doplicher, Fredenhagen and Roberts (1994)

3. Quantizing spacetime
Motivation
Black hole formation in the process of measurement at small
distances (∼ λP ) ⇒ additional uncertainty relations for
coordinates
Doplicher, Fredenhagen and Roberts (1994)
Open string + D-brane theory with an antisymmetric Bij field
background ⇒ noncommutative coordinates in low-energy
limit
Seiberg and Witten (1999)

4. Quantizing spacetime
Motivation
Black hole formation in the process of measurement at small
distances (∼ λP ) ⇒ additional uncertainty relations for
coordinates
Doplicher, Fredenhagen and Roberts (1994)
Open string + D-brane theory with an antisymmetric Bij field
background ⇒ noncommutative coordinates in low-energy
limit
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 =x
Infinite amount of derivatives induces nonlocality

7. Wu-Yang monopole
Commutative spacetime
Find potentials AN and AS such that:
µ µ
N/S
1. Bµ = × Aµ
N/S
2. Aµ are gauge
transformable to each
other in the overlap δ
N/S
3. Aµ are nonsingular
outside the origin

8. Wu-Yang monopole
Commutative spacetime
Solution:
N/S N/S
At = AN/S = Aθ = 0
r
N g
Aφ = (1 − cos θ)
r sin θ
g
AS = −
φ (1 + cos θ)
r sin θ
that gauge transform
AN/S → UAN/S U −1 = Aµ
µ µ
S/N
2ige
φ
U=e c

9. Wu-Yang monopole
Commutative spacetime
Solution:
Single-valuedness of
2ige
φ
U=e c
implies
2ge
= N = integer
c
Dirac Quantization
Condition (DQC)

10. Wu-Yang monopole
NC spacetime
Find potentials AN and AS such that:
µ µ
N/S
1. Aµ satisfy NC
Maxwell’s equations
N/S
2. Aµ are gauge
transformable to each
other in the overlap δ
N/S
3. 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 µνγδ
where 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 θ
2 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 2
2 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 2
2 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 equations
Task: 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 θ
2 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 10
2 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 10
2 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 transformations
Task: Expand to second order in θ

17. Wu-Yang monopole
Contradiction
Comparing 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 4
0 = (∂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 4
0 = (∂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 4
0 = (∂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 4
0 = (∂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 µ
simultaneously 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 µ
simultaneously satisfy Maxwell’s equations and be gauge
transformable to each other.
⇒ The DQC cannot be satisfied

22. Wu-Yang monopole
Discussion
Possible 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
Discussion
Possible 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
Discussion
Possible 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
Discussion
Possible 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. Bonus
Covariant source

27. Wu-Yang monopole
Covariant source
NC Maxwell’s equations
Dµ F µν = J ν
The 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 ν
The lhs transforms covariantly under gauge transformations
⇒ also the rhs must transform nontrivially
From this one gets the gauge covariance requirement up to
the 2nd order correction (J 0 = ρ = ρ0 + ρ1 + ρ2 + O(θ3 ))
ρ1 → ρ1 + θij ∂i λ∂j ρ0
θij θkl
ρ2 → 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 1
ρ = 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