Quantization of the open string on exact plane waves and non-commutative wave fronts

Transcription

1 Quantization of the open string on exact plane waves and non-commutative wave fronts F. Ruiz Ruiz (UCM Madrid) Miami 2007, December arxiv: [hep-th], with G. Horcajada

2 Motivation On-going issue in the NC community: Does non-commutativity play a rôle in connection with gravity?... The hope (=? expectation) is the answer to this question to be yes!... It seems reasonable to look for an answer within string theory. One step along this direction is to invoke the Penrose limit and look for non-commutative spaces in pp strings. Genuine interest in pp strings within string theory. 2

3 Problem In particular, I will consider the pp background ds 2 = dx + dx + m 2 [ (x 1 ) 2 (x 2 ) 2 ] (dx + ) 2 + (dx i ) 2 + (dx a ) 2 i = 1, 2 a = 3,... D 1 B 12 = B, any other B µν = 0 Φ = const. and quantize the open string on it. This is a Non-singular exact solution to all orders in α. The results to be presented trivially extend to e.g. 10 dimensions 4 [ ] 8 ds 2 = dx + dx + (x k ) 2 (x k+4 ) 2 (dx + ) 2 + (dx k ) 2 k=1 B 15, B 26, B 37, B m 2 k Compare with maximally supersymmetric pp IIB superstring. [ 4 8 ] 8 ds 2 = dx + dx + m 2 (x k ) 2 + (x k ) 2 (dx + ) 2 + (dx i ) 2 k=1 F 5 = m dx + ( dx 1 dx 2 dx 3 dx 4 + dx 5 dx 6 dx 7 dx 8). k=5 k=1 k=1 3

4 Strategy: use canonical quantization in lightcone and conformal gauge: 1. Solution to the equations of motion for the classical string. 2. Canonical quantization: by inverting the symplectic form. 3. The Hamiltonian spectrum and the Fock-Hilbert space Let us anticipate some results: 4. Commutation relations (in light-cone gauge x + = κτ wave fronts) [ X i (τ, σ), P j (τ, σ ) ] = iα δ i j δ(σ σ ) [ X 1 (τ, σ), X 2 (τ, σ ) ] = i Θ(σ, σ ) Θ(σ, σ ) 0 for all σ, σ (i.e. all along the string, not only at endpoints). Non-commutativity without/outside branes. As m 0, Minkowski spacetime results are recovered. Non-commutativity is confined back to the string endpoints. 4

5 1. Classical string. S = 1 4πα ( γ dτ dσ γ rs G µν r X µ s X ν +ε rs B µν r X µ s X ν +α ) γ R Φ. In lightcone and conformal gauge: X + = κτ and (from the action) Equations of motion Boundary conditions X + 4m 2 κ ( X 1 τ X 1 X 2 τ X 2) = 0 σ X σ=0,π = 0 X 1 + m 2 κ 2 X 1 = 0 σ X 1 B τ X 2 σ=0,π = 0 X 2 m 2 κ 2 X 2 = 0 σ X 2 + B τ X 1 σ=0,π = 0 X a = 0 σ X a σ=0,π = 0 Virasoro constraints κ τ X = m 2 κ 2 [ (X 1 ) 2 (X 2 ) 2 ] + ( τ X i ) 2 + ( τ X a ) 2 + [ ( σ X i ) 2 + ( σ X a ) 2] κ σ X = 2 ( τ X i)( σ X i) + 2 ( τ X a)( σ X a) a-directions: same as in Minkowski spacetime. Solution is well-known. Equations for } X1, X 2 Equation for X Virasoro constraints +. 5

6 The solution for the boundary problem for X 1, X 2 is X 1 (τ, σ) = i α ( sin βπ ) c λ cos βσ + λb cos βπ 1 sin βσ e iλτ λ Λ ± X 2 (τ, σ) = ( cos απ ± 1 ) cos ασ + sin ασ e iλτ sin απ λ Λ ± c λ Λ + = {solutions of F + (λ) = 0, sin απ sin βπ 0} Λ = {solutions of F (λ) = 0, sin απ sin βπ 0} α = λ 2 m 2 κ 2 β = λ 2 + m 2 κ 2 F ± (λ) = c λ = arbitrary constants of integration αβ (cos απ ± 1) (cos βπ 1) λ 2 B2 sin απ sin βπ Λ + and Λ are disjoint: F + (λ) and F (λ) do not have common roots. 6

7 The eigenvalue equations F ± (λ) = 0 have infinitely many solutions: Come in pairs (λ, λ) since F ± are functions of λ 2. λ 2 > 0 Real λ e iλτ = oscillations in τ Finite number ( 1, depends on the value of mκ) of them with λ < mκ. Infinitely many with λ > mκ. In fact, for λ mκ, there are 2 solutions in the neighborhood of every large enough integer n λ (1,2) n = n [1 ± m2 κ 2 2 n 2 1 B B 2 + O ( m 4 κ 4 ) ]. λ 2 < 0 Imaginary λ e iλτ = non-oscillatory τ-dependence Finite number ( 1, depends on the value of mκ) of them, all with λ < mκ. Conclusion: infinitely many oscillations (real λ) and a finite number of nonoscillatory position-momentum type degrees of freedom (imaginary λ). Comment. This solution exists for all values of mκ. (i) If m 2 κ 2 = even integer, there are a few additional oscillation modes. (ii) If mκ = integer, there is in addition one more position-momentum type mode. To keep formulas simple, I will not write them here. n 4 7

8 Momenta. P i (τ, σ) = δs δ ( τ X i (τ, σ) ) = 1 ( τ X i B ɛ 2πα ij σ X j) = λ Λ ± From the resulting mode expansions the total momentum p i = p 1 (τ ) = 1 B c λ cos απ 1 πα β 2 sin απ λ Λ p 2 (τ ) = 1 πα λ Λ + c λ λα e iλτ p i (τ) are not conserved. Expected! e iλτ π 0 dσp i is p i (τ) are collective. Receive contributions from all modes. Upon quantization, they will not play a significant rôle. 8

9 3. Quantization. S = dτ dσ [ ( P i τ X i + P a τ X a) + F ] 1 [ (2πα P 2πα i + B ε ij σ X j) 2 ( ) + 2πα 2 ( P a + σ X i) 2 ( + σ X a) ] 2 π Symplectic form: Ω = dσ ( dp i dx i + dp a dx a) =: Ω i + Ω a 0 Ω i = 1 π dσ d( 2πα τ X i ) dx i + B dx 1 dx 2 2πα 0 Using the mode expansions and integrating over dσ, Ω i = 1 Ω λ, λ dc λ dc λ 2 λ Λ ± Ω λ, λ = i [ ] λα (cos απ ± 1) 2 (mκ) 4 ( ) πα sin απ λ 2 α 2 β ± π 2 α sin απ π β sin βπ }{{} =: f(λ) 0 for all λ Λ ± σ=π σ=0 9

10 Ω is non-singular. Its inverse exists Quantization: Amplitudes c λ become operators, with c λ = c λ c λ = c λ [ cλ, c λ λ real λ imaginary ] ( ) = i Ω 1 = πα λλ f(λ) δ λ+λ,0 Comment. Symplectic form has canonical form (only λ + λ = 0 contributes). 10

11 4. Hamiltonian and Fock-Hilbert space We want to solve H ψ = E ψ, where H = H i + H a H i = 1 4πα H a = 1 4πα π 0 π 0 { ( τ dσ X i) 2 ( + σ X i) 2 m 2 κ 2[ (X 1 ) 2 (X 2 ) 2 ] } [ ( τ dσ X a) 2 ( + σ X a) ] 2 H a ψ a = E a ψ a Solution is well-known plane waves of momentum p a harmonic oscillators of integer frequency. E a has a normal ordering constant D

12 H i ψ i = E i ψ i H i = 1 2πα = 1 πα λ f(λ) c λ c λ λ Λ ± λ f(λ) : c λ c λ: λ Λ ± Re λ>0 }{{} Harmonic oscillators of frequency λ 1 λ + sign [ λf(λ) ] ( ) ˆp 2 λ ˆq λ 2 2 λ Λ ± λ Λ ± Re λ>0 Im λ>0 }{{}}{{} ) πα normal c ±λ = 2 λf(λ) (ˆqλ ± ˆp λ ordering Im λ > 0 For every λ (Imλ > 0), Hamiltonian for an inverted harmonic oscillator ψ I = ψ λ ( Imλ>0 d 2 ) E I = sign [ λf(λ) ] E λ dqλ 2 + qλ 2 + E λ ψ λ (q λ ) = 0 Imλ>0 Does not have bound sates. Its solutions are scattering states 12

13 ψ λ (q λ ) = C 1 e iq2 λ /2 q λ Φ ( ie 4, 3 2 ; ) iq2 λ + C2 e iq2 λ /2 Φ ( ie 4, 1 2 ; ) iq2 λ regular at q λ = 0 [ 1 at q λ superpositions of exp ( ± i qλ 4 Eλ ln qλ 2 + ) ] 2q2 λ ie ψ λ (q λ ) e λτ scattering states in 1,2-directions 13

14 4. Canonical commutation relations and non-commutativity [ X 1 (τ, σ), X 2 (τ, σ ) ] = 1 2B λ Λ ± ( cos απ ± 1 sin απ α λ f(λ) ( cos βσ + sin βπ ) cos βπ 1 sin βσ cos ασ + sin ασ ) =: i Θ(σ, σ ) 0 Case mκ 1. The mode eigenvalues λ can be found as power series in mκ: λ I = λ R = Λ = {±iλ I, λ (n) mκ 1 + B 2 mκ 1 + B 2 } [1 + (mκ)2 12 [1 (mκ)2 12 Λ + = {±λ R, λ (n) + } n = ±1, ±2,... π 2 B B + (mκ) π 2 B B + (mκ) π 4 B 2 (5 B 2 24) (1 + B 2 ) 2 + O ( m 6 κ 6) π 4 B 2 (5 B 2 24) (1 + B 2 ) 2 + O ( m 6 κ 6) λ (n) ± = n [1 ± ( ) n m 2 κ 2 1 B 2 2 n B m4 κ 4 B 4 6B n 4 + O ( m 6 κ 6) (1 + B 2 ) 2 ]. ] ] 14

15 Θ(σ, σ ) = k=0 Θ 2k (σ, σ ) (mκ) 2k Functions of σ and σ Involve powers of σ and σ and Θ 0 (0, 0) = Θ 0 (π, π) = α πb 1 + B 2 Θ 0 (σ, σ ) = 0 for σ + σ 0, 2π n=1 ( ) cos nσ sin nσ ( ) cos nσ sin nσ positive integer n As in Minkowski spacetime [Chu-Ho, Seiberg-Witten] It comes out of the calculation! Θ 2 (σ, σ α B { [ ) = (1 + B 2 ) 2 B 2 σ ( σ 2 3σ 2 ) + π ( σ 2 σ 2 2σσ ) π ( )] σ 3σ σ ( 7σ 2 + 9σ 2 ) + π ( 7σ 2 + 3σ 2 + 6σσ ) π2 ( ) 3σ + σ + π π [ 8 σ σ 2B ( 2 σ 2 + σ π ) ] } + 5σ σ 2π. 0 for arbitrary σ, σ Non-commutativity outside without branes! As m 0, NC gets confined to endpoints. 15

16 Similarly [ X i (τ, σ), P j (τ, σ ) ] = C i j,0 = iα δ i j δ(σ σ ) C i j,2 = 0... k=0 C i j,2k(σ, σ )(mκ) 2k (Total) normal ordering constant is also a power series in mκ: K = D 4 24 = D λ Λ ± Re λ>0 λ mκ B 2 [1 (mκ)2 12 π 2 B B + O( m 3 κ 3) ] 2 16

17 It looks as if there is room for non-commutativity in connection with gravity within string theory. The low energy-limit (Seiberg-Witten maps). The Penrose limit. Implications for spacetime singularities, etc? 17