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 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


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


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



∇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 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?




∇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 monopoleContradiction

Comparing the two sets of equations for AN2i − AS2

i

After some algebra we get...



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!



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
















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

)


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

)


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

Monopole zurich

Education