Upload
tapio-salminen
View
425
Download
0
Embed Size (px)
DESCRIPTION
Seminar talk given in Quantum Theory and Gravitation, Zurich, June 2011.
Citation preview
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