Fluctuations for su(2) from first principles

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1 Fluctuations for su(2) from first principles Benoît Vicedo DAMTP, Cambridge University, UK AdS/CFT and Integrability Friday March 14-th, 2008

2 Outline Semiclassical quantisation The zero-mode problem Finite dimensions Wentzel-Kramers-Brillouin-Maslov method Finite-gap strings

3 Outline Semiclassical quantisation The zero-mode problem Finite dimensions Wentzel-Kramers-Brillouin-Maslov method Finite-gap strings

4 Semiclassical quantisation Classical theory Action functional S[φ]

5 Semiclassical quantisation Classical theory Action functional S[φ] Quantum theory Partition function Z = Dφe i S[φ]

6 Semiclassical quantisation Classical theory Action functional S[φ] Semiclassical SPA Z 0 e i S[φ cl] 1 det S φ cl :S [φ cl ]=0 [φ cl ] Quantum theory Partition function Z = Dφe i S[φ]

7 Semiclassical quantisation Classical theory Action functional S[φ] Semiclassical SPA Z 0 e i S[φ cl] 1 det S φ cl :S [φ cl ]=0 [φ cl ] Quantum theory Partition function Z = Dφe i S[φ] Remark Particle spectrum:

8 Semiclassical quantisation Classical theory Action functional S[φ] Semiclassical SPA Z 0 e i S[φ cl] 1 det S φ cl :S [φ cl ]=0 [φ cl ] Quantum theory Partition function Z = Dφe i S[φ] Remark Particle spectrum: quanta of φ

9 Semiclassical quantisation Classical theory Action functional S[φ] Semiclassical SPA Z 0 e i S[φ cl] 1 det S φ cl :S [φ cl ]=0 [φ cl ] Quantum theory Partition function Z = Dφe i S[φ] Remark Particle spectrum: quanta of φ + all classical solutions φ cl.

10 Semiclassical quantisation Classical theory Action functional S[φ] Semiclassical SPA Z 0 e i S[φ cl] 1 det S φ cl :S [φ cl ]=0 [φ cl ] Quantum theory Partition function Z = Dφe i S[φ] Remark Particle spectrum: quanta of φ + all classical solutions φ cl. Aim: Give φ cl a quantum interpretation.

11 Zero-mode problem Quantise around background φ cl :

12 Zero-mode problem Quantise around background φ cl : φ = φ cl + δφ. Z = D(δφ)e i S[φ cl]+ 1 2 δφs [φ cl ]δφ+...

13 Zero-mode problem Quantise around background φ cl : φ = φ cl + δφ. Z = D(δφ)e i S[φ cl]+ 1 2 δφs [φ cl ]δφ+... Propagator G = (S [φ cl ]) 1 if S [φ cl ] invertible.

14 Zero-mode problem Quantise around background φ cl : φ = φ cl + δφ. Z = D(δφ)e i S[φ cl]+ 1 2 δφs [φ cl ]δφ+... Propagator G = (S [φ cl ]) 1 if S [φ cl ] invertible. Definition δφ is a zero-mode of φ cl if S [φ cl ]δφ = 0.

15 Zero-mode problem Quantise around background φ cl : φ = φ cl + δφ. Z = D(δφ)e i S[φ cl]+ 1 2 δφs [φ cl ]δφ+... Propagator G = (S [φ cl ]) 1 if S [φ cl ] invertible. Definition δφ is a zero-mode of φ cl if S [φ cl ]δφ = 0. OK φ = φ 0

16 Zero-mode problem Quantise around background φ cl : φ = φ cl + δφ. Z = D(δφ)e i S[φ cl]+ 1 2 δφs [φ cl ]δφ+... Propagator G = (S [φ cl ]) 1 if S [φ cl ] invertible. Definition δφ is a zero-mode of φ cl if S [φ cl ]δφ = 0. OK BAD φ = φ 0 φ = φ cl

17 Interpretation If φ cl has zero-mode δφ then φ cl belongs to a family φ cl (a).

18 Interpretation If φ cl has zero-mode δφ then φ cl belongs to a family φ cl (a). Ψ[φ] will spread over family φ cl (a): φ cl (a)

19 Interpretation If φ cl has zero-mode δφ then φ cl belongs to a family φ cl (a). Ψ[φ] will spread over family φ cl (a): φ cl (a) Hence need to quantise family φ cl (a) and NOT φ cl = φ cl (0).

20 Interpretation If φ cl has zero-mode δφ then φ cl belongs to a family φ cl (a). Ψ[φ] will spread over family φ cl (a): φ cl (a) Hence need to quantise family φ cl (a) and NOT φ cl = φ cl (0). Definition Parameter a is called a collective coordinate.

21 zero-modes & broken symmetries Lemma Let v be an infinitesimal transformation.

22 zero-modes & broken symmetries Lemma Let v be an infinitesimal transformation. If (1) v is a symmetry, i.e. v(s [φ]) = S [φ](vφ),

23 zero-modes & broken symmetries Lemma Let v be an infinitesimal transformation. If (1) v is a symmetry, i.e. v(s [φ]) = S [φ](vφ), (2) φ cl breaks v, i.e. vφ cl 0,

24 zero-modes & broken symmetries Lemma Let v be an infinitesimal transformation. If (1) v is a symmetry, i.e. v(s [φ]) = S [φ](vφ), (2) φ cl breaks v, i.e. vφ cl 0, then (vφ cl ) is a zero-mode of φ cl.

25 zero-modes & broken symmetries Lemma Let v be an infinitesimal transformation. If (1) v is a symmetry, i.e. v(s [φ]) = S [φ](vφ), (2) φ cl breaks v, i.e. vφ cl 0, then (vφ cl ) is a zero-mode of φ cl. So for every symmetry broken by φ cl :

26 zero-modes & broken symmetries Lemma Let v be an infinitesimal transformation. If (1) v is a symmetry, i.e. v(s [φ]) = S [φ](vφ), (2) φ cl breaks v, i.e. vφ cl 0, then (vφ cl ) is a zero-mode of φ cl. So for every symmetry broken by φ cl : Lemma φ cl belongs to family φ cl (x 0 ),

27 zero-modes & broken symmetries Lemma Let v be an infinitesimal transformation. If (1) v is a symmetry, i.e. v(s [φ]) = S [φ](vφ), (2) φ cl breaks v, i.e. vφ cl 0, then (vφ cl ) is a zero-mode of φ cl. So for every symmetry broken by φ cl : Lemma φ cl belongs to family φ cl (x 0 ), Noether φ cl (x 0 ) belongs to family φ cl (x 0, p 0 ).

28 zero-modes & broken symmetries Lemma Let v be an infinitesimal transformation. If (1) v is a symmetry, i.e. v(s [φ]) = S [φ](vφ), (2) φ cl breaks v, i.e. vφ cl 0, then (vφ cl ) is a zero-mode of φ cl. So for every symmetry broken by φ cl : Lemma φ cl belongs to family φ cl (x 0 ), Noether φ cl (x 0 ) belongs to family φ cl (x 0, p 0 ). (x 0, p 0,...) φ cl (φ, π) = (φcl (x, 0), φ cl (x, 0))

29 Sine-Gordon kink S-G: µ µ φ + m 3 /λ sin[ λφ/m] = 0,

30 Sine-Gordon kink S-G: µ µ φ + m 3 /λ sin[ λφ/m] = 0, 1-parameter family of sol ( )] φ u (x, t) = 4 tan [exp 1 x ut 1 u 2

31 Sine-Gordon kink S-G: µ µ φ + m 3 /λ sin[ λφ/m] = 0, 1-parameter family of sol ( )] φ u (x, t) = 4 tan [exp 1 x ut 1 u 2 Periodic box: x [ L 2, L 2 ) φ u(x, t) periodic.

32 Sine-Gordon kink S-G: µ µ φ + m 3 /λ sin[ λφ/m] = 0, 1-parameter family of sol ( )] φ u (x, t) = 4 tan [exp 1 x ut 1 u 2 Periodic box: x [ L 2, L 2 ) φ u(x, t) periodic. Lemma φ u has one zero-mode, φ u / x (note φ u / t φ u / x).

33 Sine-Gordon kink S-G: µ µ φ + m 3 /λ sin[ λφ/m] = 0, 1-parameter family of sol ( )] φ u (x, t) = 4 tan [exp 1 x ut 1 u 2 Periodic box: x [ L 2, L 2 ) φ u(x, t) periodic. Lemma φ u has one zero-mode, φ u / x (note φ u / t φ u / x). Corollary φ u (x, t) φ u (x + x 0, t)

34 Sine-Gordon kink S-G: µ µ φ + m 3 /λ sin[ λφ/m] = 0, 1-parameter family of sol ( )] φ u (x, t) = 4 tan [exp 1 x ut 1 u 2 Periodic box: x [ L 2, L 2 ) φ u(x, t) periodic. Lemma φ u has one zero-mode, φ u / x (note φ u / t φ u / x). Corollary φ u (x, t) φ u (x + x 0, t) x 0 u (φ u (x + x 0, 0), φ u (x + x 0, 0)) P S G.

35 The Breather solution 2-parameter family of solutions φ τ,v (x, t) = 4m tan 1 λ (τm 2π [ ] ) 2 1 sin 2π t vx τ 1 v 2 ] [ (τm ) 2 cosh 2π 1 2π x vt τ 1 v 2.

36 The Breather solution 2-parameter family of solutions φ τ,v (x, t) = 4m tan 1 λ (τm 2π [ ] ) 2 1 sin 2π t vx τ 1 v 2 ] [ (τm ) 2 cosh 2π 1 2π x vt τ 1 v 2.

37 The Breather solution 2-parameter family of solutions φ τ,v (x, t) = 4m tan 1 λ (τm 2π [ ] ) 2 1 sin 2π t vx τ 1 v 2 ] [ (τm ) 2 cosh 2π 1 2π x vt τ 1 v 2. Periodic box: x [ L 2, L 2 ) φ τ,v quasi-periodic.

38 Breather zero-modes Lemma φ τ,v has two zero-modes, φ τ,v / x and φ τ,v / t.

39 Breather zero-modes Lemma φ τ,v has two zero-modes, φ τ,v / x and φ τ,v / t. Corollary φ τ,v (x, t) φ τ,v (x + x 0, t + t 0 )

40 Breather zero-modes Lemma φ τ,v has two zero-modes, φ τ,v / x and φ τ,v / t. Corollary φ τ,v (x, t) φ τ,v (x + x 0, t + t 0 ) t 0 x 0 (φ τ,v (x + x 0, t 0 ), φ τ,v (x + x 0, t 0 )) τ P S G. v

41 Outline Semiclassical quantisation The zero-mode problem Finite dimensions Wentzel-Kramers-Brillouin-Maslov method Finite-gap strings

42 WKB method Consider a 1-dimensional system. Classical: Hamiltonian H(x, p)

43 WKB method Consider a 1-dimensional system. Classical: Hamiltonian H(x, p) conserved trajectories H = E. p H = E x

44 WKB method Consider a 1-dimensional system. Classical: Hamiltonian H(x, p) conserved trajectories H = E. p H = E x Quantum: Ĥ = H(x, i / x). Spectrum Ĥψ = Eψ.

45 WKB method Consider a 1-dimensional system. Classical: Hamiltonian H(x, p) conserved trajectories H = E. p H = E x Quantum: Ĥ = H(x, i / x). Spectrum Ĥψ = Eψ. WKB ansatz: ψ WKB = e is(x)/ ρ(x) + O( ) ( To order O(1) : H x, S(x) ) = E. x

46 WKB method (continued) Lemma Let ι : H 1 (E) T X and α = pdx, then dι α = 0.

47 WKB method (continued) Lemma Let ι : H 1 (E) T X and α = pdx, then dι α = 0. Proof. Locally, ι α = S x dx = ds is exact.

48 WKB method (continued) Lemma Let ι : H 1 (E) T X and α = pdx, then dι α = 0. Proof. Locally, ι α = S x dx = ds is exact. H 1 (E) is Lagrangian submanifold.

49 WKB method (continued) Lemma Let ι : H 1 (E) T X and α = pdx, then dι α = 0. Proof. Locally, ι α = S x dx = ds is exact. H 1 (E) is Lagrangian submanifold. Next order O( ): Theorem (Maslov) Can define ψ WKB globally on H 1 (E), patchwise.

50 WKB method (continued) Lemma Let ι : H 1 (E) T X and α = pdx, then dι α = 0. Proof. Locally, ι α = S x dx = ds is exact. H 1 (E) is Lagrangian submanifold. Next order O( ): Theorem (Maslov) Can define ψ WKB globally on H 1 (E), patchwise. Theorem (Bohr-Sommerfeld) Single-valuedness of ψ WKB on H 1 (E) implies 1 ι α = N + 1 2π 2 + O( ). H 1 (E)

51 Bohr-Sommerfeld quantisation Consider an n-dimensional system. Classical: Integrals F (F 1,...,F n )

52 Bohr-Sommerfeld quantisation Consider an n-dimensional system. Classical: Integrals F (F 1,...,F n ) motion on Λ f F 1 (f). Λ f T X X

53 Bohr-Sommerfeld quantisation Consider an n-dimensional system. Classical: Integrals F (F 1,...,F n ) motion on Λ f F 1 (f). Λ f T X X Quantum: ψ WKB localised near Λ f.

54 Bohr-Sommerfeld quantisation Consider an n-dimensional system. Classical: Integrals F (F 1,...,F n ) motion on Λ f F 1 (f). Λ f T X X Quantum: ψ WKB localised near Λ f. Theorem (Einstein-Brillouin-Keller) Single-valuedness of ψ WKB on Λ f implies 1 ι α = N(γ) + µ γ 2π 2 + O( ), γ H 1(Λ f ). γ

55 Degenerate torii Recall, Definition (T X, ω, H) is integrable if F 1,...,F n C(T X) s.t.

56 Degenerate torii Recall, Definition (T X, ω, H) is integrable if F 1,...,F n C(T X) s.t. (1) {F i, F j } = 0, i, j = 1,...,n,

57 Degenerate torii Recall, Definition (T X, ω, H) is integrable if F 1,...,F n C(T X) s.t. (1) {F i, F j } = 0, i, j = 1,...,n, (2) H = H(F 1,...,F n ),

58 Degenerate torii Recall, Definition (T X, ω, H) is integrable if F 1,...,F n C(T X) s.t. (1) {F i, F j } = 0, i, j = 1,...,n, (2) H = H(F 1,...,F n ), (3) df 1... df n 0, almost everywhere.

59 Degenerate torii Recall, Definition (T X, ω, H) is integrable if F 1,...,F n C(T X) s.t. (1) {F i, F j } = 0, i, j = 1,...,n, (2) H = H(F 1,...,F n ), (3) df 1... df n 0, almost everywhere. So F is regular almost everywhere.

60 Degenerate torii Recall, Definition (T X, ω, H) is integrable if F 1,...,F n C(T X) s.t. (1) {F i, F j } = 0, i, j = 1,...,n, (2) H = H(F 1,...,F n ), (3) df 1... df n 0, almost everywhere. So F is regular almost everywhere. If c regular value, then F 1 (c) = T n (as in Arnold-Liouville).

61 Degenerate torii Recall, Definition (T X, ω, H) is integrable if F 1,...,F n C(T X) s.t. (1) {F i, F j } = 0, i, j = 1,...,n, (2) H = H(F 1,...,F n ), (3) df 1... df n 0, almost everywhere. So F is regular almost everywhere. If c regular value, then F 1 (c) = T n (as in Arnold-Liouville). We will be interested in critical values of F, e.g. F i = 0 some i.

62 Quantising isolated orbits Consider an isolated periodic orbit γ in 2-dimensional system.

63 Quantising isolated orbits Consider an isolated periodic orbit γ in 2-dimensional system. p 0 γ S 1 R 2

64 Quantising isolated orbits Consider an isolated periodic orbit γ in 2-dimensional system. p 0 γ S 1 R 2 Quantise neighbouring 2-torus using EBK, Theorem (Voros) 1 ι α = N + µ ( γ 2π γ 2 + n + 1 ) ν 2 2π + O( ).

65 2-d harmonic oscillator H = p ω2 1 x p ω2 2 x 2 2 = H 1 + H 2, ω 1 ω 2.

66 2-d harmonic oscillator H = p ω2 1 x p ω2 2 x 2 2 = H 1 + H 2, ω 1 ω 2. Consider orbit γ with H 2 = 0 on H 1 (E): p 1 p 2 x 1 ωx 2 T 1 = 2π ω 1 T 2 = 2π ω2

67 2-d harmonic oscillator H = p ω2 1 x p ω2 2 x 2 2 = H 1 + H 2, ω 1 ω 2. Consider orbit γ with H 2 = 0 on H 1 (E): p 1 p 2 x 1 ωx 2 T 1 = 2π ω 1 T 2 = 2π ω2 Perturb γ within H 1 (E): p 1 p 2 x 1 ωx 2 ν T 1 = 2π ω 1 T 2 = 2π ω2

68 2-d harmonic oscillator (continued) Voros 1 I 1 = ( n ) + ( n ) ν 2π + O( )

69 2-d harmonic oscillator (continued) Voros 1 I 1 = ( n ) + ( n ) ν 2π + O( ) E = ω 1 I 1 = ( n ) ω1 + ( n ) ω2 + O( 2 ).

70 2-d harmonic oscillator (continued) Voros 1 I 1 = ( n ) + ( n ) ν 2π + O( ) E = ω 1 I 1 = ( n ) ω1 + ( n 2 + 2) 1 ω2 + O( 2 ).

71 2-d harmonic oscillator (continued) Voros 1 I 1 = ( n ) + ( n ) ν 2π + O( ) E = ω 1 I 1 = ( n ) ω1 + ( n 2 + 2) 1 ω2 + O( 2 ). What have we done? E 2 E 2 E 1 E 1

72 2-d harmonic oscillator (continued) Voros 1 I 1 = ( n ) + ( n ) ν 2π + O( ) E = ω 1 I 1 = ( n ) ω1 + ( n 2 + 2) 1 ω2 + O( 2 ). What have we done? E 2 E 2 E 1 E 1 Remark Could have applied EBK to 2-torii with E 1 0, E 2 0.

73 Quantising degenerate torii Consider a family Λ f of p-torii in n-dimensional system.

74 Quantising degenerate torii Consider a family Λ f of p-torii in n-dimensional system. Theorem (Voros) Let k = 1,...,p and γ k H 1 (Λ f ) then 1 ( ι α = N k + µ ) γ k 2π γ k 4 + n α=p+1 ( n α + 1 ) ν k α 2 2π + O( ). γ 2 γ 1

75 Outline Semiclassical quantisation The zero-mode problem Finite dimensions Wentzel-Kramers-Brillouin-Maslov method Finite-gap strings

76 Application to finite-gap strings Recall, A 2 (ˆγ 0 ) J(Σ, ± ) M (2g+2) C A 1 (ˆγ 0 ) ι P S 2 S 1

77 Application to finite-gap strings Recall, A 2 (ˆγ 0 ) J(Σ, ± ) M (2g+2) C A 1 (ˆγ 0 ) ι P S 2 S 1 Ingredients of Voros formula are:

78 Application to finite-gap strings Recall, A 2 (ˆγ 0 ) J(Σ, ± ) M (2g+2) C A 1 (ˆγ 0 ) ι P S 2 S 1 Ingredients of Voros formula are: (1) ι α, where ω = dα is symplectic structure on P.

79 Application to finite-gap strings Recall, A 2 (ˆγ 0 ) J(Σ, ± ) M (2g+2) C A 1 (ˆγ 0 ) ι P S 2 S 1 Ingredients of Voros formula are: (1) ι α, where ω = dα is symplectic structure on P. (2) Non-zero stability angles.

80 Symplectic structure Proposition The pullback of the Poisson bracket on P by the finite-gap solution ι is (with z = x + 1 x ) ( g+1 g+1 ) λ λ ˆω 2g+2 = dp(ˆγ i ) dz(ˆγ i ) = d 4πi 4πi p(ˆγ i)dz(ˆγ i ). i=1 i=1

81 Symplectic structure Proposition The pullback of the Poisson bracket on P by the finite-gap solution ι is (with z = x + 1 x ) ( g+1 g+1 ) λ λ ˆω 2g+2 = dp(ˆγ i ) dz(ˆγ i ) = d 4πi 4πi p(ˆγ i)dz(ˆγ i ). i=1 Definition Let α = λ 4πi pdz. i=1

82 Symplectic structure Proposition The pullback of the Poisson bracket on P by the finite-gap solution ι is (with z = x + 1 x ) ( g+1 g+1 ) λ λ ˆω 2g+2 = dp(ˆγ i ) dz(ˆγ i ) = d 4πi 4πi p(ˆγ i)dz(ˆγ i ). i=1 Definition Let α = λ 4πi pdz. Then action variables are S I = 1 2π a I α i=1 a 1 a 2 a 3 + p (x) p (x)

83 Perturbations What are the perturbations of a finite-gap solution?

84 Perturbations What are the perturbations of a finite-gap solution? a 1 a 2 a + 3 a 0 p (x) p (x) B 0

85 Perturbations What are the perturbations of a finite-gap solution? a 1 a 2 a + 3 a 0 p (x) p (x) B 0 Lemma Location of small cut determined by B 0 dp = 2πn 0, n 0 Z.

86 Perturbations What are the perturbations of a finite-gap solution? a 1 a 2 a + 3 a 0 p (x) p (x) B 0 Lemma Location of small cut determined by B 0 dp = 2πn 0, n 0 Z. Lemma Associated stability-angles defined by ν (I) 2π B 0 dq (I), I = 1,...,g + 1,

87 Perturbations What are the perturbations of a finite-gap solution? a 1 a 2 a + 3 a 0 p (x) p (x) B 0 Lemma Location of small cut determined by B 0 dp = 2πn 0, n 0 Z. Lemma Associated stability-angles defined by ν (I) 2π B 0 dq (I), I = 1,...,g + 1, where a J dq (I) = 0, B J dq (I) = δ IJ.

88 Action spectrum Apply Voros formula (n N): S I = N I α=g+2 ( n α ) B 0 dq (I) + O( ).

89 Action spectrum Apply Voros formula (n N): Remark S I = N I At O(1): S I Z. α=g+2 ( n α + 1 ) dq (I) + O( ). 2 B 0

90 Action spectrum Apply Voros formula (n N): Remark S I = N I At O(1): S I Z. α=g+2 ( n α + 1 ) dq (I) + O( ). 2 B 0 At O( ): S I receive 1-loop corrections.

91 Energy spectrum Now [ŜI, ŜJ] = O( 3 ) and classically E = E cl [S 1,...,S g+1 ], hence semiclassical energy spectrum is just E = E cl N I ( n α + 1 ) dq (I) + O( 2 ). 2 B 0 α=g+2

92 Energy spectrum Now [ŜI, ŜJ] = O( 3 ) and classically E = E cl [S 1,...,S g+1 ], hence semiclassical energy spectrum is just E = E cl N I ( n α + 1 ) dq (I) + O( 2 ). 2 B 0 α=g+2 Proposition The energy spectrum can be rewritten, to order O( ) as [( E = E cl N ) (,..., N g ) (, n ) ],..., where E cl is the classical energy of an infinite-gap solution.

93 Recall, Conclusion 1 +1 p (x) p (x)

94 Recall, Conclusion 1 +1 p (x) p (x) Semiclassical spectrum energy of classical solution with ( ) 1 α 2 + N γ for contour γ around all cuts and all singular points.

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