Lectures on Quantum sine-gordon Models Juan Mateos Guilarte, Departamento de Física Fundamental (Universidad de Salamanca IUFFyM (Universidad de Salamanca Universidade Federal de Matto Grosso Cuiabá, Brazil, 00
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The sine-gordon action Scalar field and R, Minkowski space-time conventions φ : R, R, x µ R,, µ = 0,, x 0 = t, x = x x µ x µ = g µνx µ x ν, g = = g, g = g = 0 µ = x µ, µ µ = = t x, dx = dx 0 dx S = { ( ( } dx µφ µ φ m4 λ cos λ m φ Re-scaling the fields and coordinates: S[φ] = = m λ = m λ dt {T[φ(t] V[φ(t]} { dt dx λ m φ φ, mxµ x µ [ φ φ t t φ φ x x } dx { µφ µ φ ( cos φ ]} ( cos φ (
The field equation Field energy φ(t, x = The sine-gordon equation ( t x φ(t, x + sin φ(t, x = 0 ( φ u + v (u, v = sin φ(u, v, x = u v E[φ] = T[φ(t] + V[φ(t] = m3 λ Configuration space Asymptotic conditions φ lim x ± x (t 0, x = 0, Topological current dx [ φ φ + t t, t = u v C = {φ(t 0, x Maps(R, R/V[φ(t 0 ] < + } lim φ(t 0, x = πn ±, n +, n Z, x ± ] φ φ + ( cos φ x x C = C n+ n j µ T = π εµν νφ(t, x, ε 0 = ε 0 =, ε 00 = ε = 0, µj µ T = 0
Topological charge Q T = Q T = t The stress tensor The sine-gordon invariants dx j 0 t (t, x = φ(t, + φ(t, π π φ φ (t, + + (t, t t = 0 L = µφ µ φ + cos φ, T µν = µφ νφ g µνl T 00 = ( φ φ + φ φ + cos φ, T 0 = φ φ t t x x t x T = ( φ φ + φ φ + cos φ, T 0 = φ φ t t x x x t Conserved quantities: Energy and momentum T µν = g µρ g νσ T ρσ, T 00 = T 00, T = T, T 0 = T 0 t T00 x T0 = 0 t P0 = t dx T 00 = 0 x µ Tµν = 0 dx T 0 = 0 t T0 x T = 0 t P = t
Painlevè property: φ(u, v = f (z, z = uv zf (z+f (z = sin f (z g (z (g Static homogeneous solutions The sine-gordon soliton g (z+ g (z g (z + z if (z = 0, g(z = e φ n = πn, T 00 [φ n] = cos φ n = 0, Q T [φ n] = P 0 [φ n] = P [φ n] = 0 φ n+ = (n + π, T 00 [φ n+ ] = cos φ n+ =, n Z Kink traveling waves T 00 [φ(x] = dφ dφ dφ + cos φ(x = 0 dx dx dx = ± ( cos φ(x dφ sin φ = ±(x x c φ Kn (x = 4 arctan[exp(±(x x c] + πn Lorentz transformations φ Kn ( x xc γt γ ±(x xc γt = 4 arctan[exp[ ]] + πn, γ R γ > γ > 0 = a a +, a = ± + γ γ, C = exp[ x c ], γ xc R
The sine-gordon soliton Kink topological charge, energy, and momentum Q T [φ Kn (t, x] =, P 0 8 [φ Kn (t, x] =, 8 γ P [φ Kn (t, x] = γ γ
Two-soliton solutions: two-kinks Two step Bäcklund : C = C = [ a + a φ 4 (t, x = 4arctan a a [ { }] [ exp (a + x + (a a t exp a [ + exp { (a + }] x + (a a t a }] { (a + a + a + a x + (a + a a a t Two sine-gordon kinks centered at the origin + γ + γ a =, a =, Q T [φ KK (t, x] = γ γ +γ +γ γ φ KK (t, x = 4 arctan γ +γ +γ + exp exp [ γ + x+γ t γ ] γ exp [ x γ t γ [ γ γ ( γ ( γ ( x + (γ γ t ] ]
Energy and momentum P 0 8 [φ KK ] = + γ 8 γ Two-soliton solutions: Kink-Antikink, P 8 [φ KK ] = γ γ 8 γ One sine-gordon kink and one anti-kink centered at the origin a = + γ + γ, a = γ γ, Q T [φ KA (t, x] = 0 φ KA (t, x = +γ + +γ γ 4 arctan γ +γ +γ γ γ [ ] [ ] x γ exp t exp x γ t γ γ [ ] γ + exp γ (x (γ γ t ( γ ( γ γ
Energy and momentum P 0 8 [φ KA ] = + γ Two-soliton solutions: Breathers 8 γ Soliton-antisoliton bound states [ φ B (t, x = 4 arctan tan θ sin(cos θ t ] L cosh(sin θ x L Center of mass γ = 0: a = sin θ i cos θ, Topological charge, energy, momentum, and period, P 8 [φ KA ] = γ γ 8 γ γ, t L = t γx γ, x L = x γt γ a = sin θ + i cos θ Q T [φ B (t, x] = 0, P 0 [φ B (t, x] = 6 γ sin θ P [φ B (t, x] = 6 γ sin θγ, T = π sin θ
The sine-gordon Hamiltonian Canonical momenta and Hamiltonian density π(t, x = δl φ = (t, x δ φ t H[π, φ] = π φ L = t π(t, xπ(t, x + φ φ + cos φ(t, x x x Poisson brackets {F[π, φ], G[π, φ]} = {π(t, x, φ(t, y} = ( λ δf m δπ The sine-gordon Hamiltonian H[π, φ] = dx H[π, φ] = dk ρ(k k + + l δg δφ δf δφ δg δπ λ δ(x y, Ḟ[φ, φ] = {H, F} m p l + 8 + n p n + 6 sin θ n
Lax pair The sine-gordon Lax pair X = i x φ t T3 + k cos φ T + ω sin φ T, T a = σa, a =,, 3 Y = i t φ x T3 + ω cos φ T + k sin φ T, (ω, k R,, ω k = XY YX = [ ] φ i t φ x + sin φ T 3, [T, T b ] = iε abc T c Zero curvature and flat connections D t = t + iat(t, x = iy, Dx = + iax(t, x = ix x F tx = [D t, D x] = i [ ] φ t φ x sin φ T 3 = 0, g(t, x = exp[iθ a(t, xt a ] A t(t, x = g (t, x tg(t, x, A x(t, x = g (t, x xg(t, x Linear spectral problem Xψ = 0, Yψ = 0, X[φ(0, x, π(0, x, ω]ψ k (x = 0 ( i x k ( π(0, x cos φ (0, x ψ k (x k cos φ (0, x i x + π(0, x ψ k (x = ( ( 0 i ω = sin φ (0, x ψ k (x i ω φ (0, x ψ sin k (x
Scattering data Jost matrices XF = 0 df ( π dx = i T3 k cos φ T + ω sin φ T F(x, k { x } F(x, k = Pexp i dya x[π(0, y, φ(0, y, k], F ± (x, k x ± exp [i( n ± kxt ] Scattering amplitudes ( a(k b(k F(x, k = b (k a (k T(k = a(k, R(k = a(k b(k Reading the spectrum. Discrete spectrum: zeroes of a(k Bound states: k = iκ l, 0 < κ l R, ω = κ l ( ( ψ l x e κlx, ψ l x + F (x, k, a(k + b(k =, a (k = a(k, C l e κ lx, C l R Resonances: k = ξ m, ξ m C, ±Reξ m > 0, Imξ m > 0, ω = + ξm ( ( ψ m x e iξmx, ψ m x + d me iξmx, d m C
Discrete plus continuous spectrum Action-angle variables S (b(k, κ l, C l, ξ m, d m ( a(k b(k Evolution of scattering data: (lim x + Y b (k a (k F (x, k = 0 b(k, t = exp(iωtb(k, 0, a(k, t = a(k, 0 C l (t = exp(iω l tc l (0, κ l (t = κ l (0 d m(t = exp(iω mtd m(0, ξ m(t = ξ m(0 One-kink scattering: half-bound state ( d i dx (k tanh x + i ω cosh x ( ψ ω (k tanh x i cosh x d ψ i dx ω = 0 k = i, ψ(x exp[ dx tanh x] = ( ψ x exp[ ( x], ψ x + = 0, ψ = ψ = ψ cosh x exp[ x]
Kink phase shifts One-kink scattering: half-bound state ( d i dx (k tanh x + i ω cosh x ( ψ ω (k tanh x i cosh x d ψ i dx ω = k = 0, ψ(x exp[ ( ψ 0 x ( + i i + e x i ( i + i + e x i ( ψ 0 x + ( + i i + e x i ( i + i + e x i One-kink scattering: continuous spectrum ( d i dx k ( ψ k d ψ i dx =x 0 = 0, ψ = ψ = ψ dx cosh x ] = exp[ arctan[tanh[ x ]]] ψ (x =x e ikx + R(ke ikx, R(k = b(k a(k ψ (x =x (e ikx R(ke ikx ψ = dψ i dx d ψ dx + k ψ = 0
One-kink scattering: continuous spectrum ( d i dx k ( ψ k d ψ i dx Transparent scattering Kink phase shifts =x 0 ψ (x =x T(ke ikx, T(k = a(k ψ (x =x T(ke ikx ψ = dψ i dx d ψ dx + k ψ = 0 a(k = ik + ik b(k = 0, δ(k = arctan k ( ( F(k, x e iδ 0 e iδ 0 =x 0 e iδ F (k, x = 0 e iδ exp[ikxt ] Evolution of the kink scattering data ( Y(t, x = i t sechx (ω tanh x + i k cosh x k (ω tanh x i cosh x i t + sechx ( Y ± = Y(t, x ± = i t ω ω i t
Y ( Y + ( T(ke ikx f (t = 0 df dt e ikx f (t + R(ke ikx g(t (e ikx f (t R(ke ikx g(t Small deformations = iωf (t f (t = eiωt = 0 { df = iωf (t dt dg = iωg(t dt ψ k (t, x =x e ikx+iωt + R(k, te ikx+iωt, R(k, t = e iωt R(k, 0 ψ k (t, x =x T(k, te ikx+iωt, T(k, t = T(k, 0 f (t = e iωt g(t = e iωt Close solutions [ ( X φ(t, x, φ t = ( φ i ( (t, x, Y φ(t, x, φ (t, x t t (t, x φ (t, x + sin φ(t, x x ] ψ = T 3 ψ φ(t, x φ S (t, x + δφ(t, x + O[(δφ φ (t, x], t φ x + sin φ = 0 ( [φ s(t, x]δφ(t, x = t x + cos φ S(t, x δφ(t, x = 0
Two-soliton well at t = 0 Two-soliton ground states Zero-modes at t = 0 The ground states for any time x φ KK = t φ KK = 3(5 cosh[x] + 4 cosh[ (3t + 5x] 4 8 + 9 cosh[/4(3t + x] + cosh[3/4(t + 3x] 9 cosh[x] 8 + 9 cosh[/4(3t + x] + cosh[3/4(t + 3x]
Two-soliton well at t = 0 Two-soliton ground state Zero modes at t = 0
Bibliography L. D. Faddeev and V. E. Korepin, Quantum theory of solitons", Physics Reports C4 (978-87 R. Rajaraman, Solitons and instantons", North Holland, Amsterdam, 98 P. Drazin and R. Jhonson, Solitons: an introduction", Cambridge University Press, Cambridge U. K., 996