Solitons and instantons in gauge theories

Transcription

1 Solitons and instantons in gauge theories Petr Jizba FNSPE, Czech Technical University, Prague, Czech Republic ITP, Freie Universität Berlin, Germany March 15, 2007 From: M. Blasone, PJ and G. Vitiello, Quantum Field Theory and its Macroscopic Manifestations (WS, 2007) Praha,

2 Outlook: 2

3 Outlook: 2

4 Outlook: Introduction & motivations 2

5 Outlook: Introduction & motivations Korteweg de Vries solitons 2

6 Outlook: Introduction & motivations Korteweg de Vries solitons Topological solitons in relativistic field theories 2

7 Outlook: Introduction & motivations Korteweg de Vries solitons Topological solitons in relativistic field theories 2

8 Outlook: Introduction & motivations Korteweg de Vries solitons Topological solitons in relativistic field theories sine-gordon solitons 2

9 Outlook: Introduction & motivations Korteweg de Vries solitons Topological solitons in relativistic field theories sine-gordon solitons 2

10 Outlook: Introduction & motivations Korteweg de Vries solitons Topological solitons in relativistic field theories sine-gordon solitons φ 4 solitary waves 2

11 Outlook: Introduction & motivations Korteweg de Vries solitons Topological solitons in relativistic field theories sine-gordon solitons φ 4 solitary waves Topological solitons in gauge theories with SSB 2

12 Outlook: Introduction & motivations Korteweg de Vries solitons Topological solitons in relativistic field theories sine-gordon solitons φ 4 solitary waves Topological solitons in gauge theories with SSB 2

13 Outlook: Introduction & motivations Korteweg de Vries solitons Topological solitons in relativistic field theories sine-gordon solitons φ 4 solitary waves Topological solitons in gauge theories with SSB Nielsen Olesen vortex solution 2

14 Outlook: Introduction & motivations Korteweg de Vries solitons Topological solitons in relativistic field theories sine-gordon solitons φ 4 solitary waves Topological solitons in gauge theories with SSB Nielsen Olesen vortex solution 2

15 Outlook: Introduction & motivations Korteweg de Vries solitons Topological solitons in relativistic field theories sine-gordon solitons φ 4 solitary waves Topological solitons in gauge theories with SSB Nielsen Olesen vortex solution t Hooft Polyakov magnetic monopole solution 2

16 Outlook: Introduction & motivations Korteweg de Vries solitons Topological solitons in relativistic field theories sine-gordon solitons φ 4 solitary waves Topological solitons in gauge theories with SSB Nielsen Olesen vortex solution t Hooft Polyakov magnetic monopole solution 2

17 Outlook: Introduction & motivations Korteweg de Vries solitons Topological solitons in relativistic field theories sine-gordon solitons φ 4 solitary waves Topological solitons in gauge theories with SSB Nielsen Olesen vortex solution t Hooft Polyakov magnetic monopole solution Homotopy theory and soliton classification 2

18 Introduction & motivation In QM bound states describe extended structures (nucleons, atoms, molecules) that are inherently quantum (they are stable against radiation only in the QM setting). 3

19 Introduction & motivation In QM bound states describe extended structures (nucleons, atoms, molecules) that are inherently quantum (they are stable against radiation only in the QM setting). Q.: 3

20 Introduction & motivation In QM bound states describe extended structures (nucleons, atoms, molecules) that are inherently quantum (they are stable against radiation only in the QM setting). Q.: Can we have stable bound states in (relativistic) QFT which are not inherently quantum? 3

21 Introduction & motivation In QM bound states describe extended structures (nucleons, atoms, molecules) that are inherently quantum (they are stable against radiation only in the QM setting). Q.: Can we have stable bound states in (relativistic) QFT which are not inherently quantum? A: 3

22 Introduction & motivation In QM bound states describe extended structures (nucleons, atoms, molecules) that are inherently quantum (they are stable against radiation only in the QM setting). Q.: Can we have stable bound states in (relativistic) QFT which are not inherently quantum? A: Yes. In a non-linear field theory, with an appropriate amount of non-linearity, stable bound states can exist on a classical, as well as quantum level. Such bound states are called solitons. 3

23 Introduction & motivation In QM bound states describe extended structures (nucleons, atoms, molecules) that are inherently quantum (they are stable against radiation only in the QM setting). Q.: Can we have stable bound states in (relativistic) QFT which are not inherently quantum? A: Yes. In a non-linear field theory, with an appropriate amount of non-linearity, stable bound states can exist on a classical, as well as quantum level. Such bound states are called solitons. solitons can be viewed as elementary objects in much the same way as elementary particles are (sol. are stabilized by topological charge). 3

24 Other properties 4

25 Other properties S. have localized energy density (they do not dissipate) 4

26 Other properties S. have localized energy density (they do not dissipate) S. have sharply defined mass and momentum 4

27 Other properties S. have localized energy density (they do not dissipate) S. have sharply defined mass and momentum S. dynamics obeys (Poincaré) Galileo symmetry 4

28 Other properties S. have localized energy density (they do not dissipate) S. have sharply defined mass and momentum S. dynamics obeys (Poincaré) Galileo symmetry After solitons scatter they retain their shape they are stable against collisions 4

29 Other properties S. have localized energy density (they do not dissipate) S. have sharply defined mass and momentum S. dynamics obeys (Poincaré) Galileo symmetry After solitons scatter they retain their shape they are stable against collisions Technical note: Solitons vs. solitary waves 4

30 Other properties S. have localized energy density (they do not dissipate) S. have sharply defined mass and momentum S. dynamics obeys (Poincaré) Galileo symmetry After solitons scatter they retain their shape they are stable against collisions Technical note: Solitons vs. solitary waves 4

31 Other properties S. have localized energy density (they do not dissipate) S. have sharply defined mass and momentum S. dynamics obeys (Poincaré) Galileo symmetry After solitons scatter they retain their shape they are stable against collisions Technical note: Solitons vs. solitary waves Solitons: system has multi-soliton solutions that are stable against collisions 4

32 Other properties S. have localized energy density (they do not dissipate) S. have sharply defined mass and momentum S. dynamics obeys (Poincaré) Galileo symmetry After solitons scatter they retain their shape they are stable against collisions Technical note: Solitons vs. solitary waves Solitons: system has multi-soliton solutions that are stable against collisions Solitary waves: system does not have multi-soliton solutions or they do not obey cluster decomposition 4

33 Korteweg de Vries solitons 5

34 Korteweg de Vries solitons Solitary wave was first observed in 1834 by a Scottish engineer John Scott Russell on a canal near Edinburgh the Report of the British Association for the Advancement of Science 5

35 Theoretical explanation by D.J. Korteweg and G. de Vries in 1895 Propagation of waves in the x-direction on the surface of shallow water has the form η t η + u 0 x }{{} non-dispersive evolution + α η η x }{{} finite amplitude effects + β 3 η x 3 }{{} dispersive term = 0 u 0, α and β are constants α, β > 0 Passing to the frame S u0 moving with the speed u 0 Galileo transformations is; x x u 0 t and t t canonical KdV equation for the surface elevation η t + α η η x + β 3 η x 3 = 0 6

36 By wanting the localized wave solution we must require that η 0, η 0 and η 0 as x The solution can be find explicitly: ( ) η(t, x) = a sech 2 aα 12β (x x 0 ut), a = 3u 0 α where the soliton velocity is ( u = u aα ) 3u 0 ( ) u 1 u 0 propagation velocity increases with the amplitude of the hump. As the soliton width is (aα/12β) 1/2 the smaller the width the larger the height taller KdV solitons travel faster. 7

37 Note: The localized behavior results from a balance between nonlinear steepening and dispersive spreading and so it cannot be attained in linear equations because an appropriate amount of nonlinearity is necessary.!!! KdV equation allows for a superposition of solitons and hence it can describe an interaction between solitons. Numerical investigations of the interaction of two solitons reveal that if a high, narrow soliton is formed behind a low, broad one, it will catch up with the low one, they undergo a non-linear interaction the high soliton passes through the low one and both emerge with their shape unchanged. Note: KdV equation can be generalized. Ensuing equations are called higher-order KdV equations and they have form η t + u 0 η x p 2 + (p + 1)ηp η x + 3 η η 3 = 0, 8

38 They admit single-soliton solutions η(x, t) = a sech 2/p (b(x x 0 ut)) The same analysis as for ordinary KdV solitons gives a = [ p u 0 ( u u 0 1)] 1/p and b = p a p 2(p + 2) slimmer higher-order KdV solitons travel faster than smaller and thicker ones Numerical simulations indicate that for p > 2 the higher-order KdV soliton equations do not allow for a nonlinear superposition of solitons 9

39 η(x) u x Profile of the KdV soliton at t = 0. Soliton parameters are chosen to be: a = 1, x 0 = 0 and the soliton width is 1. b) 8 4 a) η t x Interaction of two KdV solitons. 10

40 Topological solitons in relativistic field theories Important class of systems which often exhibit solitonic solutions are dimensional relativistic field theories. For the single scalar field φ the Lagrangians reads L = 1 2 ( φ) (φ ) 2 V (φ) and the corresponding E L equation is φ φ dv (φ) = dφ The energy is obtained from the 00 component of the E-M tensor E[φ] = dx T 00 (φ) = dx ( 1 2 ( φ) ) 2 (φ ) 2 + V (φ) 11

41 To find the solitons we solve E-L equation in the soliton s rest frame and then boosting the solution with a Lorentz transformation. The rest-frame soliton obeys the equation φ = dv (φ) dφ 1 2 ((φ ) 2 ) = dv (φ) dφ φ 1 2 (φ ) 2 = V (φ) + A Conditions: Localization of the solitonic solution φ ( x ) = 0 Energy localization E[φ( x )] = 0 V (φ( x )) = 0. Note: V (φ) = 0 and φ = 0 at x A = 0 Take now the square root and integrate ± φ(x) φ(x 0 ) dφ 2V (φ) = x x 0 which gives a solitonic solution φ = φ(φ(x 0 ), x x 0 ). 12

42 a) sine-gordon solitons The sine-gordon system that is described by the Lagrangian L = 1 2 ( µφ)( µ φ) + m4 λ [ cos ( ) λ m φ 1 ] (µ = 0, 1) The ensuing E L equations read ( ) φ(t, x) + m3 λ sin φ(t, x) λ m = 0 Taking x µ mx µ and φ λφ/m the equation boils down to φ(t, x) + sin φ(t, x) = 0 T.H.R. Skyrme, Proc.Roy.Soc. (1958),(1961); D. Finkelstein & J. Rubinstein 13

43 By inserting the sine-gordon potential to we obtain ± φ(x) φ(x 0 ) dφ 2 sin(φ/2) = x x 0 By resolving this equation w.r.t. φ we have φ(x) = ±4 arctan [exp(x x 0 )] After boosting to the frame that moves with the velocity u φ(t, x) = ±4 arctan exp x x 0 ut 1 u 2 Soliton solutions with + are called kinks, with antikinks. 14

44 Resulting static kink/antikink energy density is e 2(x+x 0) ε(x) = 1 2 (φ ) 2 + V (φ) = 2V (φ) = 16( 2 e 2x + e 2x 0) or in the original (unscaled) variables ε(x) = 16m4 λ e 2m(x+x 0) ( e 2mx + e 2mx 0) 2 Total energy of the sine-gordon kink/antikink is E[φ] = dx T 00 (t, x) = 1 8m3 1 u 2 λ M 1 u 2 M is the so called sine-gordon kink/antikink mass 15

45 a) b) ε(x) φ(x) x x Energy density a) and kink solution b) for the sine-gordon system. We chose x 0 = 0. Stability of s-g solitons can be conveniently characterized by a conserved quantity known as topological charge. Note: The kink profile has the asymptotes; φ(t, x ) = π/2 and φ(t, x ) = 0 at any fixed time t. Note: When φ is solution of E-L equation, then also φ ± 2πN; N Z!!! It takes an infinite energy to change kink to one of its constant vacuum configurations φ = 2πN. 16

46 As S. are finite-energy solutions they must at x ± tend towards one of vacua, labeled by N define the conserved charge Q as Q = dx J0 (t, x) 1 2π dx φ (t, x) = 1 2π [φ(t, x ) φ(t, x )] = N 2 N 1 N 1 and N 2 are integers corresponding to asymptotic values of the field. J µ that satisfies the continuity equation can be defined as J µ (t, x) = 1 2π εµν ν φ(t, x) (ε µν is the 2-dimensional Levi Civita tensor) Q = const. S. with one Q cannot evolve into S. with a different Q. Note: Topological charge Q cannot be derived from Noether s theorem since it is not related to any continuous symmetry of the Lagrangian. 17

47 b) φ 4 solitary waves For small field elevations, we can expand in s-g sin(...) up to the third order. After rescaling λ/3! to λ the resulting E L equation is φ(t, x) + m 2 φ(t, x) λφ 3 (t, x) = 0 Ensuing Lagrange density can be written as L = 1 2 ( µφ)( µ φ) λ 4 ( φ 2 m2 λ ) 2 Inserting the V (φ) into static kink equation ± φ(x) φ(x 0 ) dφ ( λ2 φ 2 ) = x x 0 m2 λ 18

48 The boosted solution then reads φ(x) = ± m λ tanh [ m 2 (x x 0) φ(t, x) = ± m tanh m (x x 0 ut) λ 2 1 u 2 One calls solutions with + as kinks and sign solutions as antikinks. Resulting kink/antikink energy density T 00 is ] T 00 (t, x) = 1 2 ( φ) (φ ) 2 + V (φ) = m 4 2λ(1 u 2 ) sech4 m (x x 0 ut) 2 1 u 2 19

49 Energy density of the static kink/antikink reads ε(x) = 1 2 (φ (x)) 2 + V (φ(x)) = 2V (φ(x)) [ ] = m4 m 2λ sech4 2 (x x 0) ε(x) 0.5 a) b) φ(x) x x -1 Energy density a) and kink solution b) for the φ 4 system. We chose m = λ = 1 and x 0 = 0. 20

50 The kink/antikink energy is E[φ] = dx T 00 (t, x) = 1 m u 2 3λ M 1 u 2 M is the φ 4 kink/antikink mass Note: φ(t, x ) = m/ λ and φ(t, x ) = m/ λ at any fixed t define the topological charge Q as λ Q = dx J0 (t, x) 2m dx φ (t, x) λ λ = [φ(t, x ) φ(t, x )] = 2m 2m [ φ+ φ ] φ + and φ correspond to the asymptotics 21

51 The associated topological current J µ can be defined as J µ (t, x) = λ 2m εµν ν φ(t, x) possible values of Q are { 1, 0, 1} Note: Values N > 1 are not a allowed because the corresponding field configurations do not have boundary conditions compatible with the finite energy no multi-solitonic solutions with N > 1 can exist Note: Field configurations containing a finite mixture of kinks and antikinks alternating along the x-direction preserving N = { 1, 0, 1} can be (numerically) constructed but there are no static solutions of this type 22

52 Domain walls/kinks are associated with models in which there is more than one separated mimimum. 23

53 Topological solitons in gauge theories with SSB Among prominent systems exhibiting solitonic solutions are the spontaneously broken gauge theories a) Abelian Higgs model Nielsen Olesen soliton (vortex) Vortex is the simplest soliton in Yang Mills gauge theory with scalar fields Abelian gauge models Action for the dimensional Abelian Higgs model is S = d 4 x [ 1 4 F µνf µν + D µ φ 2 g 2 ( φ 2 v 2) 2 ] Originally found by A.A. Abrikosov (1957) in type II superconductors (flux lines, flux tubes or fluxons). Independently by H.B. Nielsen and P. Olesen (1973) in the context of the dim. U(1) Higgs model 24

54 A µ is a U(1) gauge field and φ a complex Higgs field of charge e The covariant derivative and the field strength are D µ φ = ( µ + iea µ ) φ, F µν = µ A ν ν A µ. Note: S is invariant under the U(1) gauge transformations φ(x) e iα(x) φ(x), A µ (x) A µ (x) + 1 e µα(x) We look for a static string-like soliton configurations (vortices).!!! Static configuration = configuration in the Weyl (i.e., temporal) gauge A 0 = 0 with A i and φ not dependent on t, i.e., A i = A i (x), φ = φ(x) 25

55 Energy functional has then the form E[F µν, φ] = [ d 3 1 x 4 F ijf ij + D i φ 2 + g ] 2 ( φ 2 v 2 ) 2 Assume that the prospective solitons are straight, static vortices along the z-direction energy functional per unit length is E[F µν, φ] = [ d 2 1 x 2 B2 + D x φ 2 + D y φ 2 + g ] 2 ( φ 2 v 2 ) 2 B = x A y y A x is the z-component of the magnetic field Note: Necessary conditions for finite-energy configurations are φ 2 x = v 2, D x,y φ x = O(1/r), B x = O(1/r) 26

56 Note: (classical) vacuum manifold for the φ field is a circle Svac 1 parametrized by the phase angle ϑ (φ = φ e iϑ ) φ x = φ(n) = v e iϑ(n) with n = (x/r, y/r) with each finite-energy static conf. φ is associated a map S 1 S1 vac Because the field is single-valued ϑ(2π) = ϑ(0) + 2πn for some n Z. Note: n is known as the winding number, and it is the topological charge/number of the field configuration Note: This is equivalent to saying that the map S 1 S1 vac is topologically characterized by the fundamental homotopy group π 1 (S 1 ) = Z To construct a vortex solution we set an Ansatz for φ and A x,y 27

57 Q.: How can I find an appropriate Ansatz? A.: Note two points: a) The most symmetric form of φ(n) with the winding number n is φ(n) = v e inϑ b) The covariant derivative gives D x,y φ x = 0 A x,y x = i e x,y log(φ(n)) A x,y x has the azimuthal component A ϑ x = i e 1d log(φ(n)) r dϑ = n er 28

58 Ansatz can be assumed in the form φ(r, ϑ, z) = vf(r)e inϑ, A ϑ (r, ϑ, z) = n er h(r) B(r, ϑ, z) = n er h (r) f and g should be smooth with f(0) = g(0) = 0, f( ) = g( ) = 1 E L equations for f and h read f + 1 r f n2 r 2 (1 h)2 f + gv 2 (1 f 2 )f = 0, h 1 r h + 2e 2 v 2 f 2 (1 h) = 0 Note: Suitability of the Ansatz is justified when E-L eq. yield a nontrivial solution. Although, no analytic solution is known so far, the consistency of the Ansatz can be checked numerically 29

59 (Cosmic) strings/vortices are associated with models in which the set of minima are not simply-connected, that is, the vacuum manifold has holes in it. The minimum energy states on the left form a circle and the string corresponds to a non-trivial winding around this. 30

60 !!!A non-trivial insight into the structure of the solutions follows from the asymptotic behavior φ x = v The second equation can be solved exactly 1 h(r) x 2 π 2ev r1/2 e 2evr The first equation in can be linearized at large r. behavior for f satisfying the condition f x = 1 is The asymptotic 1 f(r) x r 1 e 2 2evr r 1/2 e 2gvr if 2e < g if 2e > g 31

61 Note: There are two length scales governing the large r behavior of f and h, namely the inverse masses of the scalar and vector excitations (Higgs and gauge particles), i.e., m 2 s = 2gv 2, m 2 v = 2e 2 v 2 Note: Asymptotic behavior depends on the ration (Ginzburg Landau parameter) κ = m2 s m 2 v = g e 2 for κ < 4 the behavior of f is controlled by m s for κ > 4, m v controls behaviors of f and h 32

62 Note: In superconductors, the two length scales are known as the correlation length ξ = 1/m s and the London penetration depth λ = 1/m v Values of κ distinguish type II and type I superconductors In type II superconductors ξ < λ superconductors can be penetrated by magnetic flux lines Note: Vortices with n > 1 are unstable repulsive force between parallel n = 1 lattice of vortices (Abrikosov vortex lattice) In type I superconductors ξ > λ no penetration by magnetic flux lines (Meissner Ochsenfeld effect) 33

63 The Kibble mechanism for the formation of (cosmic) strings/vortices. Note: There is no immediate meaning of κ in in cosmic strings 34

64 By utilizing Stoke s theorem we obtain for a magnetic flux Φ = 2π d 2 xb = lim dl r ia i = lim r r 0 ra ϑ dϑ = 2πn e Magnetic flux is quantized Note: Similar flux quantization exists in type II superconductors. True flux quantization in superconductors following from quantum theory reads Φ = 2πn 2q 2q corresponds to the charge of a Cooper pair!!! It is only at the quantum level that the coupling constant e and the charge of the Cooper pair 2q are related through 2q = e 35

65 Stnapshot of a cosmic string network (B. Allen & E. P. Shellard) 35

66 b) Georgi Glashow SO(3) model t Hooft Polyakov soliton (magnetic monopole) In 1974 t Hooft and independently Polyakov discovered a topologically nontrivial finite energy solution in the SO(3) Georgi Glashow model t Hooft Polyakov monopole or non-abelian magnetic monopole Georgi Glashow SO(3) model: SO(3) gauge group triplet of real Higgs scalar fields (isovector) φ Lagrangian L = 1 2 Tr(F µνf µν ) (D µφ) (D µ φ) g 4 ( φ φ v 2 ) 2 = 1 4 F a µν F µνa (D µφ) a (D µ φ) a g 4 ( φ a φ a v 2) 2 36

67 Note: Convention µ, ν = 0, 1, 2, 3 (Killing) normalization for T a s.t. Tr(T a T b ) = 1 2 δab 3 3 matrices T a are the SO(3) generators F µν = F a µνt a, A µ = A a µt a Note: Some elements of non-abelian gauge theories Recall that the algebraic structure of d dim. the commutators Lie algebra is given by [T a, T b ] = ic c ab T c C ab c are the structure constants and a = 1,..., d 37

68 Adjoint representation is defined so that (T a ) c b = ic c ba or equivalently (T a ) bc = ic abc T a is d d matrix Example: adjoint representation for SU(2) = SO(3) has d = 3 representation space is 3 dim. vector space with matrix elements (T a ) bc : (T a ) bc = iε abc Note: Gauge fields correspond to elements of a Lie algebra in its adjoint representation Fundamental representation corresponds to the defining matrix representation 38

69 Example: FR of SO(3) corresponds to 3 3 orthogonal matrices of determinant 1 representation space is 3 dim. FR of SU(2) has 2 dim. representation space The covariant derivative is D µ φ = µ φ iea a µ T a φ e is a real constant (gauge coupling constant) Under the gauge transformation φ(x) g(x)φ(x), A µ (x) g(x)a µ (x)g 1 (x) + ie 1 ( µ g(x))g 1 (x) 39

70 the covariant derivative transforms covariantly D µ φ(x) g(x)d µ φ(x) The field strength F µν transforms under the gauge transformation covariantly F µν = µ A ν ν A µ + ie[a µ, A ν ] = ie 1 [D µ, D ν ] gf µν g 1 L is invariant under the SO(3) gauge group To show that topological soliton exist in this model we consider static field configurations with finite energy 40

71 Static configuration are configurations in the Weyl gauge A a 0 = 0 and A a i and φ a are t indep., i.e. A a i = A a i (x), φa = φ a (x) energy functional is E[F µν, φ] = d 3 x [ 1 4 F a ij F a ij (D iφ) a (D i φ) a + g 4 ( φ a φ a v 2) 2 ] Necessary condition for the finiteness of the energy is that (φ a φ a ) x = v 2 The vacuum φ a may depend on the direction in the physical space, i.e. φ a x = φ a (n) n = x/r 41

72 each configuration of fields with finite energy is associated with the mapping S 2 S2 vac Note: From algebraic topology the second homotopy group π 2 (S 2 ) = Z mapping S 2 S2 vac n = 0, ±1, ±2,... is classified by integer topological numbers Q.: How can I choose the soliton Ansatz A.: Note that the most symmetric form of φ a (n) corresponding to a mapping S 2 S2 vac is φ a x = n a v As for the asymptotic behavior A a i (x), note the covariant derivative 42

73 D µ φ = µ φ iea a µt a φ must vanish at r. At the same time i φ a x = 1 r (δai n a n i )v Requirement of zero covariant derivative asymptotic gauge field: A a i (x) x = 1 er εaij n j Here we have used that (T a ) bc = iε bac and ε ijk ε ilm = δ jl δ km δ jm δ kl 43

74 To obtain a smooth soliton solution in R 3 we take the Ansatz φ a = n a vf(r), A a i = 1 er εaij n j (1 h(r)) f(r) and h(r) are smooth with f( ) = h(0) = 1, f(0) = h( ) = 0 Static energy functional reads E[f, h] = 4π [ 1 dr 0 e 2(h ) 2 + r2 v 2 2 (f ) e 2 r 2(1 h2 ) 2 + v 2 f 2 h 2 + gr2 v 4 static equations of motion are 4 (f 2 1) 2 f + 2 r f = 2f r 2 h2 + gv 2 f(f 2 1), h = h r 2(h2 1) + e 2 v 2 f 2 h ] 44

75 Note: Note: The exact analytic solutions are difficult to find Existence of non-trivial solutions is confirmed numerically Solution for φ a is also known as a hedgehog to stress that the Higgs isovector φ at a given point of space is directed along the radius vector Only the three-dimensional hedgehog configuration on the left corresponds to a monopole. Note: Presence of gauge fields is essential for hedgehog to be a soliton 45

76 Without gauge fields the Higgs field would be linearly divergent corresponding to the infinite rather than finite energy solution Q.: Why is t Hooft Polyakov soliton a magnetic monopole? A.: Notice that we have theory with SSB (SO(3) SO(2)). One may then identify the unbroken SO(2) = U(1) symmetry with the electromagnetic field To be able to identify the (physically observable) electromagnetic field that is embedded in the original SO(3) theory t Hooft proposed the gauge invariant definition of the electromagnetic field in in the form F µν = φa v F a µν 1 e v 3ε abc φ a (D µ φ b )(D ν φ c ) 46

77 In the broken phase one can chose φ a = δ 3a v then F µν usual Maxwell equations fulfills the F µν = µ A 3 ν νa 3 µ The massless gauge boson corresponding to the unbroken U(1) group can be identified with the photon i.e., A 3 µ = A µ provided we fix φ a = δ 3a v F µν is a 4 4 combination of magnetic and electric fields with the components F ij = ε ijk B k, F 0i = E i = 0 static soliton configuration carries only magnetic field 47

78 Using that F a µν has at large distances the asymptotic behavior: we have Fij a = nk n a v x er 2 ε ijk = nk er 2 ε ijk φ a x, F0i a = 0 B k x = 1 2 εkij F ij x = nk er 2 Applying Gauss s law, the total magnetic-monopole charge is Q = lim r B ds = 4π Sr 2 e t Hooft Polyakov m. satisfies the Dirac-like quantization condition Qe = 2πn, n N + for n = 2 48

79 Note: As in the Nielsen Olesen case, the origin of the quantization condition is purely topological and is not related to the dynamics nor quantum theory True Dirac s quantization condition is Qq = 2π n (in the CGS units Qq = n/2) Dirac s quantization condition emerges only at the quantum level which enforces the Lagrange coupling constant e to be related with the electric charge q via q = e 49