Monopole zurich
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Transcript of Monopole zurich
Magnetic monopoles innoncommutative spacetime
Tapio SalminenUniversity of Helsinki
In collaboration with Miklos L̊angvik and Anca Tureanu
[arXiv:1104.1078], [arXiv:1101.4540]
Quantizing spacetimeMotivation
Black 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)
VA possible approach to Planck scale physics isQFT in NC space-time
Quantizing spacetimeMotivation
Black 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)
VA possible approach to Planck scale physics isQFT in NC space-time
Quantizing spacetimeMotivation
Black 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)
VA possible approach to Planck scale physics isQFT in NC space-time
Quantizing spacetimeImplementation
Impose [x̂µ, x̂ν ] = iθµν andchoose the frame where
θµν =
0 0 0 00 0 θ 00 −θ 0 00 0 0 0
This leads to the ?-product of functions
(f ? g) (x) ≡ f (x)ei2
←−∂ µθµν
−→∂ νg(y) |y=x
Infinite amount of derivatives induces nonlocality
Quantizing spacetimeImplementation
Impose [x̂µ, x̂ν ] = iθµν andchoose the frame where
θµν =
0 0 0 00 0 θ 00 −θ 0 00 0 0 0
This leads to the ?-product of functions
(f ? g) (x) ≡ f (x)ei2
←−∂ µθµν
−→∂ νg(y) |y=x
Infinite amount of derivatives induces nonlocality
Wu-Yang monopoleCommutative spacetime
Find potentials ANµ and AS
µ such that:
1. Bµ = ∇× AN/Sµ
2. AN/Sµ are gauge
transformable to eachother in the overlap δ
3. AN/Sµ are nonsingular
outside the origin
Wu-Yang monopoleCommutative spacetime
Solution:
AN/St = AN/S
r = AN/Sθ = 0
ANφ =
g
r sin θ(1− cos θ)
ASφ = − g
r sin θ(1 + cos θ)
that gauge transform
AN/Sµ → UAN/S
µ U−1 = AS/Nµ
U = e2ige~c φ
Wu-Yang monopoleCommutative spacetime
Solution:
Single-valuedness of
U = e2ige~cφ
implies
2ge
~c= N = integer
Dirac QuantizationCondition (DQC)
Wu-Yang monopoleNC spacetime
Find potentials ANµ and AS
µ such that:
1. AN/Sµ satisfy NC
Maxwell’s equations
2. AN/Sµ are gauge
transformable to eachother in the overlap δ
3. AN/Sµ are nonsingular
outside the origin
Wu-Yang monopoleMaxwell’s equations
1. NC Maxwell’s equations
εµνγδDν ? Fγδ = 0
Dµ ? Fµν = Jν
where Fµν = 12εµνγδFγδ is the dual field strength tensor and
Fµν = ∂µAν − ∂νAµ − ie[Aµ,Aν ]?
Dν = ∂ν − ie[Aν , ·]?
Task: Expand to second order in θ
Wu-Yang monopoleMaxwell’s equations
Task: Expand to second order in θ
∇2(BN2 − BS2 )1 =4θ2xz
(x2 + y2)3r10
h− 375(x2 + y2)3 + 131z2(x2 + y2)2 − 2z4(x2 + y2)− 4z6
i− ∂1ρ
N2 + ∂1ρS2
∇2(BN2 − BS2 )2 =4θ2yz
(x2 + y2)3r10
h− 375(x2 + y2)3 + 131z2(x2 + y2)2 − 2z4(x2 + y2)− 4z6
i− ∂2ρ
N2 + ∂2ρS2
∇2(BN2 − BS2 )3 =4θ2
(x2 + y2)4r10
h120(x2 + y2)5 − 900(x2 + y2)4z2 − 1285(x2 + y2)3z4
− 1289(x2 + y2)2z6 − 652(x2 + y2)z8 − 132z10i− ∂3ρ
N2 + ∂3ρS2
Wu-Yang monopoleMaxwell’s equations
Task: Expand to second order in θ
Wu-Yang monopoleGauge transformations
2. NC gauge transformations
AN/Sµ should transform to A
S/Nµ (x) under U?(1)
AN/Sµ (x)→ U(x)?AN/S
µ (x)?U−1(x)−iU(x)?∂µU−1(x) = AS/N
µ (x)
with groups elements U(x) = e iλ?
Task: Expand to second order in θ
Wu-Yang monopoleGauge transformations
Task: Expand to second order in θ
∇2(BN2 − BS2 )GT1 =
4θ2xz
(x2 + y2)3r10
“− 321(x2 + y2)3 + 205(x2 + y2)2z2 + 26(x2 + y2)z4 + 4z6
”
∇2(BN2 − BS2 )GT2 =
4θ2yz
(x2 + y2)3r10
“− 321(x2 + y2)3 + 205(x2 + y2)2z2 + 26(x2 + y2)z4 + 4z6
”
∇2(BN2 − BS2 )GT3 =
4θ2
(x2 + y2)4r10
“144(x2 + y2)5 − 564(x2 + y2)4z2 − 455(x2 + y2)3z4
− 403(x2 + y2)2z6 − 188(x2 + y2)z8 − 36z10”
Wu-Yang monopoleGauge transformations
Task: Expand to second order in θ
Wu-Yang monopoleContradiction
Comparing the two sets of equations for AN2i − AS2
i
After some algebra we get...
Wu-Yang monopoleContradiction
Comparing the two sets of equations for AN2i − AS2
i
0 = (∂x∂z − ∂z∂x )(ρN2 − ρS2 ) =24θ2x
(x2 + y2)5r8
“41(x2 + y2)4 + 426(x2 + y2)3z2 + 704(x2 + y2)2z4
+ 496(x2 + y2)z6 + 128z8”
0 = (∂y∂z − ∂z∂y )(ρN2 − ρS2 ) =24θ2y
(x2 + y2)5r8
“41(x2 + y2)4 + 426(x2 + y2)3z2 + 704(x2 + y2)2z4
+ 496(x2 + y2)z6 + 128z8”
These equations have no solution!
Wu-Yang monopoleContradiction
Comparing the two sets of equations for AN2i − AS2
i
0 = (∂x∂z − ∂z∂x )(ρN2 − ρS2 ) =24θ2x
(x2 + y2)5r8
“41(x2 + y2)4 + 426(x2 + y2)3z2 + 704(x2 + y2)2z4
+ 496(x2 + y2)z6 + 128z8”
0 = (∂y∂z − ∂z∂y )(ρN2 − ρS2 ) =24θ2y
(x2 + y2)5r8
“41(x2 + y2)4 + 426(x2 + y2)3z2 + 704(x2 + y2)2z4
+ 496(x2 + y2)z6 + 128z8”
These equations have no solution!
Wu-Yang monopoleConclusion
There does not exist potentials ANµ and AS
µ that wouldsimultaneously satisfy Maxwell’s equations and be gauge
transformable to each other.
⇒ The DQC cannot be satisfied
Wu-Yang monopoleConclusion
There does not exist potentials ANµ and AS
µ that wouldsimultaneously satisfy Maxwell’s equations and be gauge
transformable to each other.
⇒ The DQC cannot be satisfied
Wu-Yang monopoleDiscussion
Possible causes for the failure of the DQC:
Rotational invariance, 3D vs 2DAharonov-Bohm effect worksVortex line quantization has problems
CP violation and the Witten effect
Perturbative method used
Wu-Yang monopoleDiscussion
Possible causes for the failure of the DQC:
Rotational invariance, 3D vs 2DAharonov-Bohm effect worksVortex line quantization has problems
CP violation and the Witten effect
Perturbative method used
Wu-Yang monopoleDiscussion
Possible causes for the failure of the DQC:
Rotational invariance, 3D vs 2DAharonov-Bohm effect worksVortex line quantization has problems
CP violation and the Witten effect
Perturbative method used
Wu-Yang monopoleDiscussion
Possible causes for the failure of the DQC:
Rotational invariance, 3D vs 2DAharonov-Bohm effect worksVortex line quantization has problems
CP violation and the Witten effect
Perturbative method used
BonusCovariant source
Wu-Yang monopoleCovariant source
NC Maxwell’s equations
Dµ ? Fµν = Jν
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 (J0 = ρ = ρ0 + ρ1 + ρ2 +O(θ3))
ρ1 → ρ1 + θij∂iλ∂jρ0
ρ2 → ρ2 + θij∂iλ∂jρ1 +θijθkl
2
(∂kλ∂iλ∂j∂lρ0 − ∂jλ∂lρ0∂i∂kλ
)
Wu-Yang monopoleCovariant source
NC Maxwell’s equations
Dµ ? Fµν = Jν
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 (J0 = ρ = ρ0 + ρ1 + ρ2 +O(θ3))
ρ1 → ρ1 + θij∂iλ∂jρ0
ρ2 → ρ2 + θij∂iλ∂jρ1 +θijθkl
2
(∂kλ∂iλ∂j∂lρ0 − ∂jλ∂lρ0∂i∂kλ
)
Wu-Yang monopoleCovariant source
Using this requirement we get two covariant sources
ρ = 4πg
„δ3(r)− θkl∂k
“Alδ
3(r)”
+ θijA1j ∂iδ
3(r)
+θijθkl
»A0
j ∂k
“∂iA
0l δ
3(r) + A0l ∂iδ
3(r)”
+1
2A0
i A0k∂j∂lδ
3(r)
–+O(θ3)
«
ρ′ = 4πg
„δ3(r)− θijA0
j ∂iδ3(r)− θijA1
j ∂iδ3(r) +
1
2θijθklA0
i A0k∂j∂lδ
3(r) +O(θ3)
«
All of the coefficients are uniquely fixed!
Thank you