Magne&c Monopoles. David Tong

Transcription

1 Magne&c Monopoles David Tong Trinity Mathema&cal Society, February 2014

2 North and South Poles

3 It s the law: B =0 Magne&c monopoles are not allowed: B = g 4πr 2 ˆr

4 And the Law is Non- Nego&able B = A B =0

5 Quantum Mechanics Doesn t Seem to Help The vector poten&al is necessary in quantum mechanics. H = 1 2m p e A c 2 This has physical consequences.

6 Gauge Symmetry In general the electric and magne&c fields can be wripen as E = φ A t B = A These are leq invariant under the gauge symmetry φ φ + χ t A A + χ The physical quan&&es are anything that you can build from the gauge fields that are invariant under this gauge symmetry. This includes E and B. But there is more as well

7 Aharonov- Bohm Effect Take a quantum par&cle. Move it along some closed trajectory C in space. C The wavefunc&on picks up a phase. ψ(x) e ieα ψ(x) with α = C A dx

8 Aharonov- Bohm Effect The phase shiq is measurable even if the par&cle never goes near a magne&c field B ψ(x) e ieα ψ(x) α = C A dx = B d S

9 Looking for a loophole None of this bodes well for magne&c monopoles Quantum mechanics needs the gauge poten&al A. The existence of the gauge poten&al means no monopoles. What goes wrong if we just try to write down a gauge poten&al for a monopole? A ϕ = g 4πr 1 cos θ sin θ B = g 4πr 2 ˆr singular at θ = π But should we be working with a singular A? [Dirac, 1931]

10 Gauge Symmetry as the Loophole Gauge poten&als related by A A + χ are the same. But no one says that a gauge poten&al has to be defined everywhere. valid everywhere but θ = π valid everywhere but θ =0 A N ϕ = A S ϕ = g 1 cos θ 4πr sin θ g 1 + cos θ 4πr sin θ B = g 4πr 2 ˆr Where they overlap, these two poten&als differ by a gauge transforma&on. [argument due to Wu and Yang]

11 Dirac Quan&sa&on Suppose that we have a magne&c monopole B = g 4πr 2 ˆr B d S = g

12 Move an electron around in its vicinity Dirac Quan&sa&on ψ(x) e ieα ψ(x) α = C A dx

13 Move an electron around in its vicinity Dirac Quan&sa&on ψ(x) e ieα ψ(x) α = C A dx = S B d S

14 Dirac Quan&sa&on But there s an ambiguity in how we choose the surface S ψ(x) e ieα ψ(x) α = C A dx = S B d S

15 These must give the same answer Dirac Quan&sa&on ψ(x) e ieα ψ(x) α = B ds S ψ(x) e ieα ψ(x) α = S B d S e ieα = e ieα = e ie(g α)

16 Dirac Quan&sa&on e ieg =1 eg =2πn n Z A rela&onship between electric and magne&c charge [Dirac, 1931]

17 The Next Step Why monopoles have to appear [ t HooQ; Polyakov 1974]

18 Par&cles Arise From Fields Usually par&cles appear as quantum fluctua&ons of fields

19 Par&cles Arise From Fields But par&cles could appear in another way..as solitons

20 t HooQ- Polakov Monopole Magne&c monopoles appear as solitons in certain field theories. These are solu&ons to classical par&al differen&al equa&ons. (Non- linear versions of Maxwell s equa&ons)

21 Monopoles in Our Universe? Our best theory of the Universe is the Standard Model SU(3) SU(2) U(1) The Standard Model does not contain monopoles. But anything that goes beyond the Standard Model does! e.g. SU(3) SU(2) U(1) SU(5)

22 Where are They? We expect them to be very heavy: way beyond what the LHC can produce. But MoEDAL detector part of LHCb

23 Monopoles and PDEs Monopoles are solu&ons to par&al differen&al equa&ons. Here are the simplest B i = D i φ B 1 = 2 A 3 3 A 2 i[a 2,A 3 ] B 2 = 3 A 1 1 A 3 i[a 3,A 1 ] B 3 = 1 A 2 2 A 1 i[a 1,A 2 ] A i and φ are Hermi&an 2x2 matrices D i φ = i φ i[a i, φ]

24 Monopoles and PDEs These equa&ons have a rather special property. They admit many many solu&ons. Fix the magne&c charge n. Then the most general solu&on has 4n parameters. This should be thought of as n monopoles. Each can sit anywhere in R 3. Each has an extra degree of freedom showing how it s embedded in the 2x2 matrix

25 Monopole ScaPering Looks like we have to solve nasty, &me dependent par&al differen&al equa&ons

26 Monopoles and Geometry Idea: If the scapering is very slow, a photograph at any &me will look like a sta&c monopole configura&on [Manton]

27 The Moduli Space This is the space of solu&ons to the monopole equa&ons of fixed charge, n It is a manifold (space) of dimension 4n

28 ScaPering on the Moduli Space Low- energy scapering of monopoles traces out a path on the moduli space

29 ScaPering on the Moduli Space Manton s Idea: Make the moduli space curved so that this is the natural path

30 It s Like General Rela&vity but in higher dimensions

31 Manton Metric Make the moduli space curved so that this is the natural path There is a metric on the moduli space that does this.

32 Proper&es of the Metric These curved manifolds turn out to have lots of interes&ng proper&es Related to quaternionic manifolds (hyperkahler) [A&yah and Hitchin]

33 Monopoles and Geometry This is the beginning of a long story connec&ng monopoles and geometry The story is likely not yet finished [ Sen, Seiberg and WiPen, Donaldson, Taubes ]

34 Thank you for your apen&on