Quantization of the open string on plane-wave limits of dS_n x S^n and non-commutativity outside branes
aa r X i v : . [ h e p - t h ] F e b Quantization of the open string on plane-wave limits of dS n × S n and non-commutativity outside branes G. Horcajada and F. Ruiz Ruiz
Departamento de F´ısica Te´orica I, Universidad Complutense de Madrid,28040 Madrid, Spain
Abstract
The open string on the plane-wave limit of dS n × S n with constant B and dilatonbackground fields is canonically quantized. This entails solving the classical equa-tions of motion for the string, computing the symplectic form, and defining fromits inverse the canonical commutation relations. Canonical quantization is provedto be perfectly suited for this task, since the symplectic form is unambiguously de-fined and non-singular. The string position and the string momentum operators areshown to satisfy equal-time canonical commutation relations. Noticeably the stringposition operators define non-commutative spaces for all values of the string world-sheet parameter σ , thus extending non-commutativity outside the branes on whichthe string endpoints may be assumed to move. The Minkowski spacetime limit issmooth and reproduces the results in the literature, in particular non-commutativitygets confined to the endpoints. Solutions to the Einstein equations in general relativity have been known for a long timeto have plane waves as limits [1]. These limits, known as Penrose limits, give a plane wavespacetime approximation for the full spacetime along a null geodesic. This observationled in the sixties and seventies to a detailed study of the geometric properties of plane-wave metrics and of matter fields defined on them [2]. Already within string theory,it soon became clear that higher-dimensional plane waves give exact solutions to stringtheory, provided the Kalb-Ramond and dilaton fields satisfy certain conditions [3] [4].The generalization of the Penrose limiting procedure relating higher dimensional planewaves with more complicated solutions to string theory [5] further triggered the interestin such space-times.By now, there is a very extensive literature on plane waves in string theory. Motivatedby the fact that
AdS × S and AdS × S are solutions to M-theory and AdS × S is solution of IIB supergravity, and by the AdS/CFT correspondence, special attentionhas has been given the Penrose limit [6, 7, 8] AdS k × S n pp -limit: ds = − dx + dx − − m x k + n − ( dx + ) + d x k + n − (1.1)of AdS k × S n spaces. Two milestones in this regard are (i) the quantization [9] of the R-Rsector of the closed superstring on this background for k = n = 5, and (ii) the derivationof its spectrum from that of U ( N ) N = 4 super Yang-Mills theory [10]. The interesthas extended also to type IIB superstring models in 6 dimensions [11] describing gener-alizations of the Nappi-Witten model. As a matter of fact, the Nappi-Witten model [12]is itself the Penrose limit of AdS × S . There has been as well interest on strings on4-dimensional homogenous plane-wave backgrounds [13]. These have the form (1.1) with m replaced by a function C | x + | − and, for different values of the constant C , occuras the Penrose limit of the spatially flat FRW metric, near horizon regions of Dp -branebackgrounds and fundamental strings backgrounds [8, 14]In this paper we consider quantization of the open string on the Penrose limit of dS n × S n with non-zero constant 2-form B . To date, no background p -forms have beenfound that support dS n × S n as a solution to IIB supergravity. Yet there are indicationsthat de Sitter space may occur in type IIA theories [15]. In any case, the Penrose limit of dS n × S n is an exact solution of string theory in the critical dimension [4]. There are othermotivations for taking de Sitter space-time: its “apparent” simplicity when it comes toquantum gravity [16], the dS/CFT correspondence [17] and the fact that the non-existenceof a positive conserved energy indicates that there cannot be unbroken supersymmetry,so it seems a good starting point to go down in the number of supersymmetries. Themotivation for taking B = 0 comes from an interest in understanding non-commutativityin relation with gravity. As is well-known, string theory gives explicit realizations of non-commutative spaces. The simplest example is provided by an open string in Minkowskispacetime with endpoints moving on a D -brane on which a magnetic field is defined: uponquantization, the string position operators generate a non-commutative space along thebrane [18, 19, 20]. Since non-commutativity is postulated as a candidate to reconcilequantum mechanics with general relativity [21], and the low energy limit of string theoryincludes general relativity, it seems natural to explore the non-commutativity/gravityconnection within string theory. One way to push forward this approach is to examinenon-commutativity for plane wave backgrounds. As a matter of fact, this program hasalready started for the open string on plane-wave limits of AdS n × S n . In 10 dimensionswith a constant non-zero B in ref. [22], and in 4 dimensions with a Nappi-Witten 2-formin [23]. In both instances, the string endpoints define non-commutative spaces. Here weinvestigate non-commutativity for the Penrose limit of dS n × S n .More precisely, we will quantize the open string interacting through a plane-wave2etric ds = − dx + dx − + m (cid:2) ( x ) − ( x ) (cid:3) ( dx + ) + X i =1 ( dx i ) + D − X a =3 ( dx a ) (1.2)and constant antisymmetric and dilaton fields B ij = ǫ ij B B ab = 0 Φ = Φ . (1.3)It will come out that the string position operators X ( τ, σ ) and X ( τ, σ ′ ) do not commutefor arbitrary σ and σ ′ . This is in contrast with the results available so far for open stringson AdS n × S n plane-wave limits supported by a non-zero B [19, 22, 23], for which non-commutativity is restricted to the brane manifold on which the string endpoints move.Our results are consistent with those in the literature for Minkowski spacetime [19], sincethe latter are recovered in the limit m →
0, and in particular non-commutativity getsconfined to the string endpoints.We will work in light-cone and conformal gauges. The paper is organized as follows.In Section 2 we derive the equations of motion for the classical string and solve them.The solution turns out to be an infinite sum over modes, with a highly non-trivial de-pendence on the parameter m . As compared to the open string in Minkowski spacetime,two important differences are encountered. The first one is that the string has a finitenumber of non-oscillating degrees of freedom associated to modes exponentially growingand decaying in τ . The second one is that the string total momentum is not an indepen-dent degree of freedom but receives contributions from all the modes. In Section 3 thestring is canonically quantized. This is done by calculating the symplectic form and thenusing it to find the commutation relations for the operators associated to all the stringmodes. The symplectic form is unambiguous and non-singular, not being necessary toprovide additional constraints or to modify its definition so as to fix the commutators.As a check it is shown that the string momentum and the string position operators sat-isfy equal-time canonical commutation relations. Section 4 shows that the string positionoperators X and X do not commute for arbitrary values of σ and σ ′ , thus definingnon-commutative waves fronts. In Section 5 we find the eigenstates and spectrum of thehamiltonian. Section 6 contains our conclusions. We have included two Appendices withsome of the details of the calculations of Sections 2 and 4. Due to its length, this section is divided into five parts. In the first one, we study thebackground metric (1.2). The second subsection contains the derivation of the equa-tions of motion and of the boundary conditions for the classical open string in the back-ground (1.2)-(1.3). The equations of motion are solved in the thir part, where expressions3or the string coordinates as sums over modes ready to be quantized are found. Thefourth subsection presents a brief discussion of the string center of mass coordinates andthe string total momentum. Finally, in the fifth part we discuss the case m κ ≪ dS n × S n The metric (1.2) is the Penrose limit of dS × S × E D − , with E D − euclidean space in D − k -dimensional de Sitter space-time times an n -sphere, dS k × S n , both or radius ℓ . Itsmetric can be written as ds = ℓ h − (1 − ρ ) dt + dρ − ρ + ρ d Ω k − + (1 − r ) dχ + dr − r + r d Ω ′ n − i , (2.1)where d Ω k − and d Ω ′ n − are the round metrics on the unit ( k −
2) and ( n − χ in the vicinity of ρ = r = 0. Making the changes u ± = t ± χ , rescaling u + = x + u − = x − ℓ ρ = ¯ ρℓ r = ¯ rℓ with ℓ → ∞ , (2.2)and introducing a mass scale x + → mx + , x − → x − / m , one arrives at dS k × S n pp -limit: ds pp = − dx + dx − + m (cid:0) x k − − y n − (cid:1) ( dx + ) + d x k − + d y n − . (2.3)Here cartesian coordinates x k − = ( ¯ ρ, Ω k − ) and y n − = (¯ r, Ω n − ) have been introduced.Backgrounds ds = ds pp + ds (E D − n − k ) H = dB = A ij ( x + ) dx i ∧ dy j are solutions to all orders in α ′ for the bosonic/fermionic string in D = 26 /
10 provided A ij satisfies the condition [4] 4 m ( n − k ) = A ij A ij .H vanishes for k = n , in which case one may take B = B ij dx i ∧ dy j , with B ij constant.The metric (1.2) is recovered for k = n = 2 and is non-singular, meaning it is geodesicallycomplete. The results in this paper are trivially extended to the case k = n = 5.It is important to note the positive sign in front of x k − in the metric coefficient g ++ in eq. (2.3). This has its origin in the fact that we have started with de Sitter space-time,rather than anti-de Sitter, and implies that the metric (2.3) does not admit a conservedpositive energy. To understand this we recall that in de Sitter space there is no positiveconserved energy since there is no generator of its isometry group, SO (1 , d ), which istimelike everywhere. In the coordinates (2.1), the generator ∂/∂t is timelike for ρ < ρ = 1. Hence, ∂/∂t and its associated hamiltoniancan only be used to define time evolution in the region 0 ≤ ρ ≤ dS n × S n and taking the Penrose limit, this implies that for themetric (2.3) the sign of the energy depends on the sign of x k − − y n − . This is a propertyof the background considered. Our starting point is the bosonic part of the classical action S = 14 πα ′ Z dτ dσ (cid:16) √− γ γ rs G µν ∂ r X µ ∂ s X ν + ǫ rs B µν ∂ r X µ ∂ s X ν + α ′ √− γ R Φ (cid:17) for the open string on the D -dimensional background G µν ( X ) , B µν ( X ) , Φ( X ) in eqs. (1.2)-(1.3). Greek letters µ, ν, . . . denote spacetime indices, while lower case letters r, s, . . . fromthe end of the Roman alphabet denote world-sheet indices. Here γ rs is the metric onthe string world-sheet, R its scalar curvature and ǫ rs is defined by ǫ = 1. As usualthe world-sheet coordinates τ and σ take values on the intervals −∞ < τ < ∞ and0 ≤ σ ≤ π . We are using units in which string coordinates have dimensions of length and τ, σ are dimensionless. From now on we will use capital case letters X ′ s for the stringcoordinates.If wished, the string endpoints may be assumed to lie on a Dp -brane on which amagnetic field F ij lives . This amounts to adding to the action a term δS = 12 πα ′ Z dτ A i ∂ τ X i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ = πσ =0 , with A i ( X ) the U (1) gauge field on the brane. If this term is included in the action,the analysis in this paper goes through with the only difference that the field B ij mustbe replaced by the Born-Infeld field strength B ij = B ij − F ij , where F ij is the U (1) fieldstrength on the brane.The string action has three world-sheet symmetries. We will fix one of them byworking in light-cone gauge [24] X + = κτ , with κ a parameter with dimensions of length. The other two will be fixed by choosingconformal gauge h rs = √− γ γ rs = diag ( − , +1) . In this gauge, the classical action becomes S = Z dτ L , p is 1 for k = n = 2 in (2.3) and 4 for k = n = 5. L is given by L = p − ∂ τ x − − πα ′ Z π dσ n m κ (cid:2) ( X ) − ( X ) (cid:3) + (cid:0) ∂ τ X i (cid:1) + (cid:0) ∂ τ X a (cid:1) − (cid:0) ∂ σ X i (cid:1) − (cid:0) ∂ σ X a (cid:1) − B (cid:2) ∂ τ X ∂ σ X − ∂ σ X ∂ τ X (cid:3)o , with p − = − κ α ′ the momentum conjugate to x − ( τ ), defined [25] as the average over σ at a given τ of X − ( τ, σ ) x − ( τ ) = 1 π Z π dσ X − ( τ, σ ) . Here we have reserved the subscript i for the 1 and 2 directions, while a runs from 3 to D −
2, a convention that we will follow from now on.The field equations and boundary conditions are obtained by varying the action withrespect to X i and X a . They take the form ✷ X + m κ X = 0 (2.4) ✷ X − m κ X = 0 (2.5) ✷ X a = 0 , (2.6)with ✷ = − ∂ τ + ∂ σ the 2-dimensional d’Alambertian, and ∂ σ X − B ∂ τ X (cid:12)(cid:12)(cid:12) σ =0 ,π = 0 (2.7) ∂ σ X + B ∂ τ X (cid:12)(cid:12)(cid:12) σ =0 ,π = 0 (2.8) ∂ σ X a (cid:12)(cid:12)(cid:12) σ =0 ,π = 0 . (2.9)To quantize the theory we will need the momenta. In our case, these are given by p − = − κ α ′ (2.10) P i = 12 πα ′ (cid:0) ∂ τ X i − B ǫ ij ∂ σ X j (cid:1) (2.11) P a = 12 πα ′ ∂ τ X a . (2.12)6n terms of them, the lagrangian L can be written as L = − p − ∂ τ x − + Z π dσ (cid:2) − (cid:0) P i ∂ τ X i + P a ∂ τ X a (cid:1) + H (cid:3) , where the hamiltonian density H has the form4 πα ′ H = (cid:0) πα ′ P i + Bǫ ij ∂ σ X j (cid:1) + (cid:0) πα ′ P a (cid:1) + ( ∂ σ X i ) + ( ∂ σ X a ) − m κ (cid:2) ( X ) − ( X ) (cid:3) . (2.13)We note that H is not positive definite because of the negative sign in front of ( X ) . Asexplained in Subsection 2.1, this originates in the fact that in de Sitter space-time thereis no positive conserved energy and implies that H can only be used to account for timeevolution in the region where it is non-negative. The solution for X a is the well-known mode sum X a ( τ, σ ) = c a + d a τ + ∞ X n =0 i c an n cos nσ e − i nτ , (2.14)where c an are complex constants of integration (mode amplitudes). Reality of X a impliesthat c a and d a are real and that ( c an ) ⋆ = c a − n .The solution for X and X is more involved. To find it we use separation of variables X i ( τ, σ ) = T i ( τ ) S i ( σ ). This gives¨ T T = S ′′ S − m κ = − λ ¨ T T = S ′′ S + m κ = − λ , where the dot and prime indicate differentiation with respect to τ and σ respectively. Theboundary conditions (2.7) and (2.8) imply that non-trivial solutions are only possible for λ = λ . We therefore set λ := λ = λ , introduce α = √ λ − m κ β = √ λ + m κ (2.15)and distinguish several cases.Case 1.: λ = 0. It is straightforward to see that non-trivial solutions only exist if mκ is an integer. In particular, for mκ an odd integer the solution reads X ( τ, σ ) = h a o + b o τ sinh (cid:16) mκ π (cid:17) i cos( mκ σ ) (2.16) X ( τ, σ ) = Bmκ b o cosh h mκ (cid:16) π − σ (cid:17)i , (2.17)7hereas for mκ an even integer the solution takes the form X ( τ, σ ) = (cid:20) a e + b e τ cosh (cid:16) mκ π (cid:17) (cid:21) cos( mκ σ ) (2.18) X ( τ, σ ) = Bmκ b e sinh h mκ (cid:16) π − σ (cid:17)i , (2.19)with a o , b o and a e , b e arbitrary constants of integration in every instance.Case 2.: λ = ± m κ . This corresponds to either α or β zero and it is very easy toshow that the only solution for X and X is the trivial one.Case 3.: λ = 0 , ± m κ . Solving then for T i and S i and imposing the boundaryconditions, it follows that the eigenvalues λ must satisfy the equation (cid:0) λ B + α β (cid:1) sin απ sin βπ − λ B αβ (cid:0) cos απ cos βπ − (cid:1) = 0 . (2.20)Solutions to this equation may occur either because both its terms vanish or because noneof them vanishes but their sum does. We therefore consider two subcases: Subcase 3.1.
Both terms in eq. (2.20) vanish. Since α and β are non-zero, wemust have sin απ sin βπ = cos απ cos βπ − . (2.21)It is very easy to see then that the modes for X and X have the form X k,l ) ( τ, σ ) = iλ (cid:16) a λ ( k,l ) αB cos βσ + b λ ( k,l ) sin βσ (cid:17) e − iλτ (2.22) X k,l ) ( τ, σ ) = − (cid:16) b λ ( k,l ) βλ B cos ασ + a λ ( k,l ) sin ασ (cid:17) e − iλτ , (2.23)where a λ ( k,l ) and b λ ( k,l ) are arbitrary constants of integration. It follows from eqs. (2.21)that α and β must be integers and that their difference must be an even integer. Hencewe write α = k β = k + 2 l , (2.24)with k and l arbitrary positive integers since β ≥ α and α and β are defined as positive.With this, equations (2.15) imply m κ = 2 l ( k + l ) > λ = ± p l + ( l + k ) . (2.25)The first one of these equations states that m κ is an even integer. We thus concludethat for m κ an even integer, there are as many modes of type (2.22)-(2.23) as pairs( k, l ) of positive integers solving the equation m κ = 2 l ( k + l ), which is clearly a finitenumber. Subcase 3.2
We now look at solutions λ to equation (2.20) such thatsin απ sin βπ = 0 . (2.26)8n this case the case the modes for X and X read X λ ( τ, σ ) = i c λ λB (cid:16) α cos βσ + K λ β sin βσ (cid:17) e − iλτ (2.27) X λ ( τ, σ ) = − (cid:16) K λ λ B cos ασ + sin ασ (cid:17) e − iλτ . (2.28)where c λ is a arbitrary constant of integration and K λ is given by K λ = λ B sin απ + αβ sin βπ cos βπ − cos απ . (2.29)Let us study the solutions of equation (2.20) under condition (2.26). Equation (2.20) isan equation in λ , so its solutions come in pairs ( λ, − λ ). Solutions with λ > λ and oscillating degrees of freedom. By contrast, solutions with λ < λ , for which the τ -exponentials are real.For λ > x = m κ /λ ≪
1, with result(1 + B ) sin λπ − x (cid:20) λ π − B ) (2.30)+ (1 + B ) (cid:16) λπ λπ + sin λπ (cid:17)(cid:21) + O ( x ) = 0 . The left-hand side is, up to order x , negative for integer λ and positive for non-integer λ .It follows that the left-hand side of equation (2.20), to which (2.30) is an approximationfor large λ , must change its sign twice in the vicinity of every integer n ≫ | mκ | , thusproving the existence of two solutions around n . These solutions can be found as powerseries in mκ/n by making for λ in the neighborhood of n the ansatz λ n = n ∞ X k =0 a k (cid:16) mκn (cid:17) k a = 1 , where the coefficient a has been taken equal to 1 since λ = n solves equation (2.20) tolowest order. Substituting this ansatz in eq. (2.20) and solving order by order in mκ/n ,one obtains two different sets of solutions for the coefficients { a k } , leading to λ (1 , n = n (cid:20) ± m κ n − B B + O (cid:18) m κ n (cid:19) (cid:21) This confirms the existence of two real eigenvalues for every large enough integer n , thusshowing that there are infinitely many real solutions with | λ | > | mκ | .By contrast, there is only a finite number of real solutions with | λ | < | mκ | and thisnumber depends on the value of mκ . This can be seen as follows. Assume, without loss9f generality, that mκ is in between two consecutive integers, so that N ≤ | mκ | < N + 1,with N a positive integer. Denote by N ′ the integer such that N ′ < √ | mκ | ≤ N ′ + 1.Study the sign of the right-hand side of equation (2.20) as a function of β by dividing theinterval for β in subintervals [0 , , [1 , . . . , [ N ′ , N ′ + 1]. It is not then very difficult toprove that(i) for N ′ even there are 2( N ′ − N + 1) real solutions, and(ii) for N ′ odd the number of solutions is also 2 ( N ′ − N + 1) if2 √ | mκ | πB sin (cid:0) √ | mκ | π (cid:1) + cos (cid:0) √ | mκ | π (cid:1) + 1 > N ′ − N ) otherwise.We come now to imaginary solutions. For λ <
0, with | λ | > mκ , the left-handside of equation (2.20) is positive definite and never vanishes. Hence imaginary solutionsmust have | λ | < mκ . Using similar arguments to those employed for real λ , it can be seenthat in this case the number of solution for a given mκ is 2( N + 1), with N the integersuch that N < | mκ | ≤ N + 1. We note that imaginary λ ′ s occur due to the differentsigns with which ( X ) and X (2) enter the background metric (1.2) and account forexponential growth of X and X at τ → ±∞ . This is reminiscent of de Sitter space, forwhich space expands so fast that light rays cannot follow.This analysis shows that there are infinitely many modes of type (2.27)-(2.28), ofwhich a finite number of them have imaginary λ with | λ | < | mκ | , a finite number havereal λ with | λ | < | mκ | , and infinitely many of them have real λ with | λ | > | mκ | . It isimportant to emphasize that this is so for arbitrary values of mκ , since equation (2.20)and condition (2.26) do not place any limitation on mκ . These modes can also be writtenin the following way, which will be very useful in some parts of this paper. The eigenvalueequation (2.20) can be recast as F + ( λ ) F − ( λ ) = 0 , with F ± ( λ ) functions given by F ± ( λ ) = αβλ B − (cos απ ±
1) (cos βπ ∓ απ sin βπ . (2.31)Condition (2.26) and the observation that F + ( λ ) and F − ( λ ) do not have common zerosimply that set of solutions to the eigenvalue equation (2.20) is the union of the disjointsets Λ + = { λ + } and Λ − = { λ − } of solutions of the equations F ± ( λ ± ) = 0 . (2.32)10t is then a matter of algebra to write X and X as X iλ ( τ, σ ) = X i + ( τ, σ ) if λ ∈ Λ + X i − ( τ, σ ) if λ ∈ Λ − i = 1 , , (2.33)with X i ± given by X ± ( τ, σ ) = i c λ αλB (cid:16) cos βσ + sin βπ cos βπ ∓ βσ (cid:17) e − iλτ (2.34) X ± ( τ, σ ) = − c λ (cid:16) cos απ ± απ cos ασ + sin ασ (cid:17) e − iλτ . (2.35)Putting all cases together, we conclude that the solution for the boundary problemfor X , X is:(1) If mκ is not an integer and its square is not an even integer, the only modes thatoccur are those in eqs. (2.34)-(2.35), corresponding to λ ∈ Λ ± .(2) If mκ is not an integer but its square is an even integer, one has in addition themodes ( k, l ) in (2.22)-(2.23).(3) If mκ is an even integer, there is one additional mode, X , X in (2.18)-(2.19).(4) Finally, if mκ is an odd integer, the only modes that occur are those in (1) and X , X in (2.16)-(2.17).We summarize all these situations by writing X i ( τ, σ ) = X λ ∈ Λ ± X iλ + δ m κ , even X ( k,l ) X i ( k,l ) + δ mκ, even X i e + δ mκ, odd X i o . (2.36)The mode expansions for the momenta P i , P a follow from their expressions (2.11)-(2.12) in terms of string coordinates and the mode expansions for the string coordinates.For the flat a -directions it is trivial to arrive at2 πα ′ P a = d a + ∞ X n =0 c an cos nσ e − i nτ . For the i -directions we have P i ( τ, σ ) = X λ ∈ Λ ± P i,λ + δ m κ , even X ( k,l ) P i, ( k,l ) + δ mκ, even P i, e + δ mκ, odd P i, o , (2.37)where the explicit expressions for the various contributions to the right-hand side can befound in Appendix A. 11 .4 The string center of mass coordinates and the string total momentum The string center of mass coordinates x i,a cm ( τ ) = 1 π Z π dσ X i,a ( τ, σ )and the string total momentum p i,a ( τ ) = Z π dσ P i,a ( τ, σ )are straightforward to calculate from the mode expansions in the previous subsection. Letus consider for instance the total momentum. For the flat a -directions integration over dσ gives the standard result p a = d a / α ′ . The a -component of the total string momentumis thus given by one of the string modes in that direction. The situation for the 1 and2-component is very different. Indeed, integration over dσ of the equations in AppendixA yields p ( τ ) = δ mκ, even πα ′ B mκ b e sinh (cid:16) mkπ (cid:17) − m κ πα ′ (cid:20) δ m κ , even X ( k,l ) k odd b λ ( k,l ) βλ e − iλτ + X λ ∈ Λ − B c λ β cos απ − απ e − iλτ (cid:21) (2.38)and p ( τ ) = δ mκ, odd πα ′ (cid:20) B b o τ sinh (cid:16) mkπ (cid:17) − a o B (cid:21) + im κ πα ′ (cid:20) δ m κ , even X ( k,l ) k odd λ a λ ( k,l ) α e − iλτ + X λ ∈ Λ + c λ λα e − iλτ (cid:21) . (2.39)The components p and p receive contributions from all the string modes in those direc-tions. More importantly, p and p are not conserved since their derivatives with respectto τ do not vanish. This is not a surprise, for the plane-wave metric (1.2) is not invariantunder translations in the 1 and 2-directions. Upon quantization, we therefore do not ex-pect the eigenvalues of the corresponding operators to play a significant rˆole. It is trivialto convince oneself that this collective nature of p i is also true for the string center ofmass coordinates, whose explicit expression can be trivially obtained through integrationover dσ . 12 .5 Case | mκ | ≪ We finish by considering the regime | mκ | ≪
1. Since mκ is not an integer, nor m κ is an even integer, the only modes that exist in this case are those in (2.27)-(2.28),or equivalently (2.34)-(2.35). Furthermore, the mode eigenvalues λ can be explicitlyfound as formal power series in mκ by making the ansatz λ = P ∞ b k ( mκ ) k and solvingequation (2.20) for the coefficients b k order by order. Proceeding in this way we obtain: (i) Imaginary eigenvalues. As already mentioned, they have | λ | < | mκ | . The algebrashows that there are only two of them, Λ I = {± iλ I } , given by λ I = mκ √ B (cid:20) mκ ) π B B + ( mκ ) π B (5 B − B ) + O (cid:0) m κ (cid:1) (cid:21) . (2.40)In terms of equations (2.32), they happen to solve F − ( λ ) = 0, thus belong to Λ − . (ii) Real eigenvalues with | λ | < | mκ | . There are also two of them, Λ R = {± λ R } , where λ R = mκ √ B (cid:20) − ( mκ ) π B B + ( mκ ) π B (5 B − B ) + O (cid:0) m κ (cid:1) (cid:21) . (2.41)They are now solutions of F + ( λ ) = 0, thus are in Λ + . (iii) Real eigenvalues with | λ | > | mκ | . They read (cid:26) λ n ˜ λ n (cid:27) = n (cid:20) ± m κ n − B B − m κ n B − B + 1(1 + B ) + O (cid:0) m κ (cid:1) (cid:21) , (2.42)where n is a non-zero integer and the + / − signs on the right-hand side correspond to λ n / ˜ λ n on the left side. We will use the notation Λ := { λ n } and ˜Λ := { ˜ λ n } . Theseeigenvalues can be reorganized in terms of solutions of equations (2.32) as λ ( n )+ = ( λ n if n even˜ λ n if n odd λ ( n ) − = ( ˜ λ n if n even λ n if n odd . (2.43)It is instructive to compare the mode eigenvalues with those for the open string inflat space-time and zero antisymmetric field, i.e. with m = B = 0. In that case, X and X have the same expansion as in (2.14) and the mode eigenvalues are the integers.The flat zero mode λ flat = 0 has multiplicity four in the 1,2-directions, for there are fourarbitrary constants of integration, which in our notation would be denoted c , c , d , d .Every pair ( n, − n ) of non-zero flat modes is also 4-degenerate in these directions, for ineach direction there are two complex coefficients c i − n and c in and one complex constraint( c i ) ⋆n = c i − n . If m and B are switched on, the flat zero mode unfolds into two non-zeroimaginary modes ( iλ I , − iλ I ) and two non-zero real modes ( λ R , − λ R ), and every pair offlat modes ( n, − n ) unfolds into four modes ( λ n , ˜ λ n , λ − n , ˜ λ − n ). Whereas in Minkowskispace-time, the string center of mass and string total momentum are independent degreesof freedom associated to the 4-degenerate zero mode, in our plane-wave background theyare collective quantities. 13 Quantization
There is a discussion in the literature for m = 0 as for how to quantize the open string withnon-trivial boundary conditions like those in (2.7) and (2.8). It seems to be a widespreadbelieve that these boundary conditions impeach the use of canonical quantization. In fact,for m = 0, Dirac quantization, with the boundary conditions regarded as constraints, hasbeen used as an alternative. The problem that arises then is whether the boundaryconditions should be regarded as first or second class, and this is not a trivial choice forthey lead to different results [26] [27].We will use plain canonical quantization and show that there is nothing wrong withit. Our approach consists of two steps. In the first one we compute the symplectic formin terms of the modes. This is straightforward, since the action is first order in timederivatives and it is well-known how to proceed in these cases [28] [29]. The resultingsymplectic form will be non-singular, so it has an inverse. Its inverse defines, uponstandard canonical quantization [28], the commutation relations for the quantum theory.We emphasize that the calculation of the symplectic form may be involved but ,as pointedout in refs. [28] [29], as far as it is non-singular there is nothing wrong with canonicalquantization and there is no need to introduce constraints of any type. It is also worthnoting in this respect that the boundary conditions have already been taken into accountin solving the classical equations of motions, so one would na¨ıvely expect the symplecticform to already account for them. We will see that this quantization method is consistentwith the equal-time commutation relations (cid:2) X i ( τ, σ ) , P j ( τ, σ ′ ) (cid:3) = i δ ij δ ( σ − σ ′ ) . (3.1)In Section 5 we will explicitly construct the Fock-Hilbert space for the theory and findthe hamiltonian spectrum. The symplectic form Ω = Z π dσ (cid:0) d P i ∧ d X i + d P a ∧ d X a (cid:1) . is the sum of two contributions, which we will call Ω pp and Ω flat . They respectively arisefrom the modes in the i -directions and the flat a -directions. Since they do not mix, thesymplectic form can be studied by separately looking at each one of these two sectors.Let us first look at Ω pp . Recalling the mode expansions for X a and P a , one easilyarrives atΩ flat = Z π dσ d P a ∧ d X a = 12 α ′ D − X a =3 (cid:16) d d a ∧ d c a − X n =0 i n d c an ∧ d c a − n (cid:17) . (3.2)14his can be written as Ω flat = 12 Ω MM ′ d A M ∧ d A M ′ (3.3)where { A M } = { d a , c a , c an } and a summation over indices M = ( a, n ) and M ′ = ( a ′ , n ′ ) isunderstood. The form Ω flat is non-singular and can be inverted. Upon quantization, theamplitudes { A M } become operators with commutation relations given by the inverse ofΩ as [ A M , A M ′ ] = i (cid:0) Ω − (cid:1) MM ′ . (3.4)This yields the standard commutation relations[ c a , d b ] = 2 iα ′ δ ab [ c an , c bm ] = 2 α ′ n δ ab δ n + m, . Reality of the field operators X a imply that c a and d a are hermitean and that c a − n = ( c an ) † .So far, this is the same analysis as for Minkowski spacetime.To compute Ω pp it is most convenient to use equation (2.11) and write P i in terms ofderivatives of X i with respect to τ and σ . This givesΩ pp = Z π dσ d P i ∧ d X i = ˜Ω pp + ¯Ω pp , where ˜Ω pp and ¯Ω pp read ˜Ω pp = 12 πα ′ Z π dσ d ( ∂ τ X i ) ∧ d X i (3.5)and ¯Ω pp = B πα ′ d X ∧ d X (cid:12)(cid:12)(cid:12)(cid:12) σ = πσ =0 . (3.6)As compared to the flat a -directions, for which 2 πα ′ P a = ∂ τ X a , the nontrivial boundaryconditions not only modify the modes but also add a boundary term ¯Ω pp to the symplecticform. Computation of the boundary piece ¯Ω pp is straightforward. To calculate ˜Ω pp , weuse the mode expansion (2.36), integrate over dσ , rearrange the mode sums and employthat the eigenvalues λ ∈ Λ ± are solutions of equations (2.32). After some very long, butalso very straightforward algebra, we obtain thatΩ pp = Ω Λ ± + δ m κ , even Ω { ( k,l ) } + δ mκ, even Ω e + δ mκ, odd Ω o . (3.7)15he various contributions in this equation are given byΩ Λ ± = i πα ′ X λ ∈ Λ ± f ( λ ) d c λ ∧ d c − λ (3.8)Ω { ( k,l ) } = − i α ′ B X ( k,l ) (cid:2) f a ( λ ) d a λ ( k,l ) ∧ d a − λ ( k,l ) + f b ( λ ) d b λ ( k,l ) ∧ d b − λ ( k,l ) (cid:3) (3.9)Ω e = − α ′ cosh (cid:16) mκπ (cid:17) d a e ∧ d b e (3.10)Ω o = − α ′ sinh (cid:16) mκπ (cid:17) d a o ∧ d b o , (3.11)where f ( λ ) , f a ( λ ) and f b ( λ ) read f ( λ ) = − λα (cos απ ± απ (cid:20) mκ ) λ α β ± πα sin απ ∓ πβ sin βπ (cid:21) (3.12) f a ( λ ) = λB (cid:16) B − m κ λ (cid:17) (3.13) f b ( λ ) = 1 λB (cid:16) B + m κ λ (cid:17) . (3.14)In accordance with the notation that we are using, the double signs ± on the right of theequation for f ( λ ) apply, respectively, to the eigenvalues λ ± solving the equations (2.32).We make at this point two comments concerning the computation of Ω pp . The first oneis that the only non-zero components Ω MM ′ of the symplectic form have M + M ′ = 0, where M labels all the existing mode { A M } = { a o , b o , a e , b e , a λ ( k,l ) , b λ ( k,l ) , c λ } . Some authorscall this orthogonality of modes. Note in particular that there is not any mixing ofthe modes for m κ = even, mκ = even and mκ = odd among themselves, nor withmodes λ ∈ Λ ± . The second comment is to emphasize that the result above for Ω pp follows straightforwardly from eqs. (3.5)-(3.6) after plain integration over dσ , without anyassumption whatsoever.The form Ω pp is non-singular and has an inverse Ω − pp . Canonical quantization is thenstraightforward. The amplitudes { A M } become operators. Hermiticity of X i impliesthat a o , b o , a e , b e and c λ ( λ ∈ Λ ± imaginary) are hermitean and that a † λ ( k,l ) = a − λ ( k,l ) b † λ ( k,l ) = b − λ ( k,l ) c † λ = c − λ ( λ ∈ Λ ± real) . The commutation rules are obtained from the inverse of Ω pp as in (3.3)-(3.4), the only16on-trivial commutation relations being (cid:2) c λ , c † λ (cid:3) = − πα ′ f ( λ ) (3.15) (cid:2) a λ ( k,l ) , a † λ ( k,l ) (cid:3) = 2 α ′ f a ( λ ) (cid:2) b λ ( k,l ) , b † λ ( k,l ) (cid:3) = 2 α ′ f b ( λ ) (3.16) (cid:2) a e , b e (cid:3) = − iα ′ cosech (cid:16) mκπ (cid:17) (3.17) (cid:2) a o , b o (cid:3) = − iα ′ sech (cid:16) mκπ (cid:17) . (3.18)We note that f ( λ ) is real for λ real and imaginary for λ imaginary. The space of stateson which these operators act and their action is given in the Section 5. Let us move onto study the consistency of this quantization with the canonical commutation relations(3.1). The commutator [ X i ( τ, σ ) , P j ( τ, σ ′ )] can be computed by replacing X i and P j with theirmode expansions and using the relations (3.15)-(3.18) for the mode operators in them. Indoing so, the τ -dependence of the commutator is removed and a mode sum is left. Thissum involves in particular an infinite sum over mode eigenvalues λ ∈ Λ ± whose terms areproducts of sines and cosines at ασ, βσ, ασ ′ , βσ ′ with complicated coefficients involvingthe function f ( λ ). We do not see a way to perform this sum in closed form and obtaina compact expression for the commutator. We will instead expand the commutator inpowers of mκ and perform the mode sums order by order in mκ . We do this in the sequel.If | mκ | ≪
1, the only modes that exist are those in eqs. (2.34)-(2.35). We recallfrom Subsection 2.4 that in this case the mode eigenvalues are given by Λ I = {± iλ I } ,Λ R = {± λ R } , Λ = { λ n } and ˜Λ = { ˜ λ n } in eqs. (2.40)-(2.42), with n = ± , ± , . . . We denoteby { c I ± } , { c R ± } , { c n } and { ˜ c n } the corresponding annihilation and creation operators, forwhich hermiticity of the string position operators implies( c I ± ) † = c I ± ( c R+ ) † = c R − ( c n ) † = c − n (˜ c n ) † = ˜ c − n . Expanding the right-hand-side of eq. (3.15) in powers of mκ , we obtain the following17ommutations relations for them:[ c I+ , c I − ] = − iα ′ B B ) (1 + B ) / mκ (cid:20) π B ( mκ ) B ) + O (cid:0) m κ (cid:1) (cid:21) (3.19)[ c R+ , c R − ] = − α ′ π B B ) / mκ (cid:20) π (1 − B ) ( mκ ) B ) + O (cid:0) m κ (cid:1) (cid:21) (3.20)[ c n , c k ] = α ′ π B n (1 + B ) ( mκ ) (cid:20) − (3 − B ) ( mκ ) n (1 + B ) + O (cid:0) m κ (cid:1) (cid:21) δ n + k, (3.21)[˜ c n , ˜ c k ] = α ′ B n (1 + B ) (cid:20) mκ ) n (1 + B ) + O (cid:0) m κ (cid:1) (cid:21) δ n + k, , (3.22)all other commutators being zero. The commutator [ X i ( τ, σ ) , P j ( τ, σ ′ )] can then bewritten as a sum [ X i ( τ, σ ) , P j ( τ, σ ′ )] = X ω = I,R,n, ˜ n C ij ( ω ; σ, σ ′ )of four contributions C ij ( ω ; σ, σ ′ ) arising from the four sets in which the modes have beenorganized. Each one of these contributions is a power series in mκ , depends on σ and σ ′ and can be computed with relative ease order by order. To illustrate this, let us take asan example i = j = 2. After some algebra we obtain C (I; σ, σ ′ ) = − iα ′ B ( mκ ) B ) (cid:16) π − πσ − πσ ′ + 2 σσ ′ (cid:17) + O (cid:0) m κ (cid:1) (3.23) C (R; σ, σ ′ ) = iα ′ − iα ′ B ( mκ ) B ) (cid:16) π πσ − σ − πσ ′ + σ ′ (cid:17) + O (cid:0) m κ (cid:1) (3.24) C (Λ; σ, σ ′ ) = 2 iα ′ ∞ X n =1 cos nσ ′ (cid:20) cos nσ + B ( mκ ) B (cid:16) cos nσn + σ sin nσn − π nσn (cid:17)(cid:21) + O (cid:0) m κ (cid:1) (3.25) C ( ˜Λ; σ, σ ′ ) = − iα ′ B ( mκ ) B ∞ X n =1 (2 σ ′ − π ) sin nσ cos nσ ′ n + O (cid:0) m κ (cid:1) . (3.26)It follows from inspection of these formuli that only C (R) and C (Λ) carry contri-butions of order zero in mκ . These are easily summed by recalling that, for functionsdefined on [0 , π ] with vanishing derivatives at the boundary, Dirac’s delta function hasthe representation π δ ( σ − σ ′ ) = 1 + 2 ∞ X n =1 cos nσ cos nσ ′ . X ( τ, σ ) , P ( τ, σ ′ )] = iα ′ δ ( σ − σ ′ ) , where the subscript 0 refers to the order in mκ . To sum the order-two in mκ contributions,it is convenient to introduce variables σ ± = σ ± σ ′ , which take values σ − ∈ [ − π, π ] and σ + ∈ [0 , π ]. In terms of these, we have h C (Λ) + C ( ˜Λ) i = iα ′ B ( mκ ) B ) (cid:2) F ( σ − ) + F ( σ + ) + σ − F ( σ − ) + σ − F ( σ + ) (cid:3) , where F and F stand for the Fourier series F ( σ − ) := 2 ∞ X n =1 sin nσ − n = π | σ − | σ − − σ − if 0 < | σ − | < π σ − = 0 , ± π (3.27) F ( σ + ) := 2 ∞ X n =1 sin nσ + n = ( π − σ + if 0 < σ + < π σ + = 0 , π . (3.28) F ( σ − ) := 2 ∞ X n =1 cos nσ − n = σ − − π | σ − | + π F ( σ + ) := 2 ∞ X n =1 cos nσ + n = σ − π σ + + π . (3.30)Putting together all contributions of order two in eqs. (3.23)-(3.26), we obtain (cid:2) X ( τ, σ ) , P ( τ, σ ′ ) (cid:3) = 0 , in agreement with (3.1). Proceeding in the same way, it is straightforward to see that thecommutation relations in (3.1) also hold for other values of i and j , so we can write[ X i ( τ, σ ) , P k ( τ, σ ′ )] = iα ′ δ ij δ ( σ − σ ′ ) + O (cid:0) m κ ) . This proves the consistency of the quantization procedure used here with equal-timecanonical commutation relations, at least up to order m κ .We find quite surprising the asymmetric rˆole that each type of mode plays in thisanalysis, yet all combine to produce the desired result. It is also worth noting that C ij (Λ) will involve to any order in mκ polynomials in σ ± multiplied with convergentFourier series of σ ± , thus becoming a question of algebra force to go to higher orders in mκ . It is by now clear that canonical quantization works and that it does because thesymplectic form is non-singular. 19 Non-commutative wave fronts
The plane-wave metric (1.2) foliates spacetime by null surfaces X + = const. We shownext that these spaces are non-commutative. The commutator (cid:2) X ( τ, σ ) , X ( τ, σ ′ ) (cid:3) canbe computed by replacing X and X with their mode expansions and using the com-mutation relations (3.15)-(3.18) for the mode operators. This results in (cid:2) X ( τ, σ ) ,X ( τ, σ ′ ) (cid:3) = i h Θ Λ ± ( σ, σ ′ )+ δ m κ , even Θ { ( k,l ) } ( σ, σ ′ ) + δ mκ, even Θ e ( σ, σ ′ ) + δ mκ, odd Θ o ( σ, σ ′ ) i (4.1)where the contribution Θ Λ ± ( σ, σ ′ ) is given byΘ Λ ± ( σ, σ ′ ) = 12 B X λ ∈ Λ ± αλ f ( λ ) (cid:16) cos βσ + sin βπ cos βπ ∓ βσ (cid:17) × (cid:16) cos απ ± απ cos ασ ′ + sin ασ ′ (cid:17) (4.2)and Θ { ( k,l ) } ( σ, σ ′ ) , Θ e ( σ, σ ′ ) and Θ o ( σ, σ ′ ) readΘ { ( k,l ) } ( σ, σ ′ ) = − α ′ B X ( k,l ) (cid:20) α cos βσ sin ασ ′ λ (1 + B ) − m κ + β sin βσ cos ασ ′ λ (1 + B ) + m κ (cid:21) (4.3)Θ e ( σ, σ ′ ) = 4 α ′ Bmκ cosech (cid:16) mκπ (cid:17) cos( mκσ ) sinh h mκ (cid:16) π − σ ′ (cid:17) i (4.4)Θ o ( σ, σ ′ ) = 4 α ′ Bmκ sech (cid:16) mκπ (cid:17) cos( mκσ ) cosh h mκ (cid:16) π − σ ′ (cid:17) i . (4.5)We recall that the sum in Θ { ( k,l ) } ( σ, σ ′ ) is over the finite number of solutions ( k, l ) ofequation (2.25) and that α and β in this sum are as in (2.24), so the contributions (4.3)(4.5)do not pose any problems.The most complicated piece to understand is the contribution Θ Λ ± ( σ, σ ′ ). We mayproceed as in Section 3 and consider | mκ | ≪
1. In this case only Θ Λ ± ( σ, σ ′ ) contributes tothe commutator [ X , X ]. Expanding the right-hand side of equation (4.2) in powers of mκ , the sum over modes can then be performed order by order in mκ , so that Θ( σ, σ ′ )becomes a power series Θ( σ, σ ′ ) = ∞ X k =0 Θ k ( σ, σ ′ ) ( mκ ) k whose coefficients are explicit functions of σ and σ ′ . The first two terms of this series arecalculated in Appendix B. We exhibit here the result. At the string endpoints we obtainΘ(0 ,
0) = − Θ( π, π ) = α ′ πB B (cid:20) π ( mκ ) B ) + O (cid:0) m κ (cid:1)(cid:21) , (4.6)20hereas at σ + σ ′ = 0 , π we haveΘ( σ, σ ′ ) = α ′ B ( mκ ) (1 + B ) (cid:26) B h − σ (cid:0) σ − σ ′ (cid:1) + π (cid:0) σ − σ ′ − σσ ′ (cid:1) − π (cid:0) σ − σ ′ (cid:1)i − σ (cid:0) σ + 9 σ ′ (cid:1) + π (cid:0) σ + 3 σ ′ + 6 σσ ′ (cid:1) − π (cid:0) σ + σ ′ (cid:1) + π π | σ − σ ′ | h B (cid:0) σ + σ ′ − π (cid:1) + 5 σ − σ ′ − π i (cid:27) + O (cid:0) m κ (cid:1) . (4.7)The limit m → m → m = 0, two novelties are found: non-commutativity atthe string endpoints receives m -dependent corrections, and non-commutativity occurs forarbitrary values of σ and σ ′ , so that it extends all along the string. Even for σ = σ ′ = 0 , π non-commutativity pervades, since in that caseΘ( σ, σ ) = α ′ B ( mκ ) B ) (2 σ − π ) h B σ ( σ − π ) − (cid:0) σ − π (cid:1) i + O (cid:0) m κ (cid:1) = 0 . At the string midpoint σ = σ ′ = π/ m κ ≪ mκ since the right-hand side of (4.2) vanishes. Note also that commutativity isrecovered as B → B = B and that the string endpoints move freely,except for the boundary conditions imposed by the presence of the B field. The endpointsare then not distinguished by non-commutativity. The second one is to assume that B ij vanishes but that the string endpoints are constrained to move on a D x a on which a constant magnetic field F = B is defined. The boundary conditionsfor X and X then remain unchanged while those for X a become X a (cid:12)(cid:12) σ =0 ,π = x a . Theonly difference with the situation discussed here is that the mode expansion for X a is nolonger (2.14) but rather X a ( τ, σ ) = x a + X n =0 i c aa n sin nσ e − inσ . This only introduces some trivial modifications in the analysis of the flat a -directions [19].From this point of view, the plane-wave metric extends non-commutativity outside the D The Fock-Hilbert space and the spectrum
We want to solve the eigenvalue problem H | ψ i = E | ψ i , where the hamiltonian is the integral over σ of the hamiltonian density H in equa-tion (2.13). As discussed in Subsections 2.1 and 2.3, the classical hamiltonian is notpositive. This translates, upon quantization, into an unbounded hamiltonian operatorfrom below. It will become explicit below that it is precisely the modes with imaginary λ that make the hamiltonian unbounded, as otherwise was to be expected. Hence notall the states to be constructed in this Section are within reach for an observer but onlythose with positive eigenenergies.It is convenient to split H as the sum H = H flat + H pp of a contribution H flat = 14 πα ′ Z π dσ h (cid:0) ∂ τ X a (cid:1) + (cid:0) ∂ σ X a (cid:1) i from the flat a -directions and a contribution H pp = 14 πα ′ Z π dσ n(cid:0) ∂ τ X i (cid:1) + (cid:0) ∂ σ X i (cid:1) − m κ (cid:2) ( X ) − ( X ) (cid:3)o . from the 1,2-directions. The eigenstates of H are then of the form | ψ i = | ψ flat i ⊗ | ψ pp i and the eigenenergies read E = E flat + E pp , with {| ψ flat i , E flat } and {| ψ pp i , E pp } thesolutions to the eigenvalue problems H flat | ψ flat i = E flat | ψ flat i H pp | ψ pp i = E pp | ψ pp i . H flat Apart from the number of dimensions, it is the same problem as for the open string inMinkowski spacetime. Using the mode expansions for X a , one obtains for H flat a sum H flat = 12 α ′ ∞ X n =1 : c an † c an : + α ′ p a + D − n > a . As usual, : AB : denotes normal ordering of AB and the sum P n> n entering thenormal ordering constant has been regulated using ζ -regularization, so that it takes the22alue ζ ( −
1) = − /
12. The solution for H flat is well known. The Fock space is formed bystates | ψ flat i = | ψ { k an } i = D − O a =3 (cid:12)(cid:12) { k an } ∞ n =1 , p a (cid:11) k an = 0 , , . . . , (5.1)with k an the occupancy number of the harmonic oscillator of frequency n in the a -direction.The energies of these states are E flat = E { k an } = D − X a =3 ∞ X n =1 n k an + α ′ p a + D − . We note that the sum over n is actually finite, since for every eigenstates there is a finitenumber of non-zero occupancy numbers k an . The action of c bn † and c bn on |{ k ar } , p a i is √ α ′ n times the usual one of creation and annihilation harmonic oscillator operators. H pp Employing the mode expansions for X i , we obtain after some work that H pp = H Λ ± + δ m κ , even H { ( k,l ) } + δ mκ, even H e + δ mκ, odd H o , (5.2)where H Λ ± is given by H Λ ± = 12 πα ′ X λ ∈ Λ ± λ f ( λ ) c λ c − λ (5.3)and H { ( k,l ) } H e and H o take the form H { ( k,l ) } = 14 α ′ X λ ( k,l ) λ h f a ( λ ) a λ a − λ + f b ( λ ) b λ b − λ i (5.4) H e , o = 14 πα ′ h π mκπ ) + B mκ sinh( mκπ ) − i b , o . (5.5)We first study the problem for H Λ ± and postpone the solution for the pathological modes m κ = even , mκ = even , mκ = odd.We recall that an infinite number of the modes λ ∈ Λ ± have real λ and that a finitenumber of them have imaginary λ . We separate their contributions H R and H I to H Λ ± and write H Λ ± = H R + H I . The eigenstates and eigenvalues of H Λ ± are | ψ Λ ± i = | ψ R i ⊗ | ψ I i and E Λ ± = E R + E I , with {| ψ R i , E R } and {| ψ I i , E I } solutions to the problems H R | ψ R i = E R | ψ R i H I | ψ I i = E I | ψ I i . .2.1 Solution for H R The commutation relations (3.15) for the operators c λ , yield for H R H R = 1 πα ′ X λ ∈ Λ ± Re λ> λ f ( λ ) : c † λ c λ : + K R , where K R is the normal ordering constant K R = − X λ ∈ Λ ± Re λ> λ . (5.6)The hamiltonian H R is a sum of harmonic oscillators, one for every real λ >
0. Theeigenstates of H R are then harmonic oscillator states | ψ R i = (cid:12)(cid:12) { k λ } Re λ> (cid:11) k λ = 0 , , . . . , (5.7)with k λ the occupancy number for the harmonic oscillator of frequency λ , while theeigenenergies read E R = E { k λ , Re λ> } = X Re λ> λ k λ + K R . The action of c † λ and c λ on the states |{ k λ ′ }i is p πα ′ /f ( λ ) times the usual action ofannihilation and creation harmonic oscillator operators.Since there are infinitely many positive real λ with no accumulation point, the normalordering constant K R needs regularization. For every mκ we can always take a sufficientlylarge integer N such that m κ ≪ N and split the sum for K R into two sums: one over0 < λ < N and one over N < λ . Since ( mκ/N ) ≪
1, the λ ′ s in the second sum are givenby equation (2.42), so that K R can be written as K R = − X Re λ
1) = − mκ √ B (cid:20) − ( mκ ) π B B + O (cid:0) m κ (cid:1)(cid:21) . .2.2 Solution for H I It is convenient to introduce for every imaginary λ operators ˆ q λ and ˆ p λ defined by c ± λ = s πα ′ | λf ( λ ) | (cid:0) ˆ q λ ± ˆ p λ (cid:1) Im λ > . (5.8)They are hermitean and satisfy commutation relations [ˆ q λ , ˆ p λ ] = i sign (cid:2) λf ( λ ) (cid:3) . In termsof them, H I takes the form H I = X λ ∈ Λ ± Im λ> sign (cid:2) λf ( λ ) (cid:3) (cid:0) ˆ p λ − ˆ q λ (cid:1) . It is clear that the H I is not bounded from below. Let us forget for a moment about thisand formally solve the eigenvalue problem for H I . The solution is given by | ψ I i = Q | ϕ λ i and E I = P E λ , with the product and the sum extended over all imaginary λ withIm λ >
0, and {| ψ λ i , E λ } being solutions of (cid:0) ˆ p λ − ˆ q λ (cid:1) | ψ λ i = E λ | ψ λ i Im λ > . (5.9)To solve (5.9) we work in a position representation, in which the wave function for | ψ λ i is ψ λ ( q λ ) and the operators ˆ q λ and ˆ p λ act on it through multiplication and derivation, i.e.ˆ q λ → q λ and ˆ p λ → i ddq λ . Equation (5.9) then becomes (cid:16) d dq λ + q λ + E λ (cid:17) ψ λ ( q λ ) = 0 Im λ > . This is the time-independent Schr¨odinger equation for a particle in an inverted harmonicpotential. Such equation does not have bound states and for every real E λ admits ψ λ, ( q λ ) = e − iq λ / q λ Φ (cid:0) + iE , ; iq λ (cid:1) and ψ λ, ( q λ ) = e − iq λ / Φ (cid:0) + iE , ; iq λ (cid:1) as two linearly independent solutions, Φ( µ, ν ; z ) being the degenerate hypergeometricfunction. Both ψ λ, ( q λ ) and ψ λ, ( q λ ) are regular at q λ = 0, while at | q λ | → ∞ aresuperpositions of the oscillating exponentials1 p | q λ | exp h ± i (cid:0) E λ ln q λ + 2 q λ (cid:1)i . The most general solution for ψ λ ( q λ ) is then an arbitrary linear combination ψ λ ( q λ ) = C ψ λ, ( q λ ) + C ψ λ, ( q λ ) . ψ λ ( q λ ) e iE λ τ is a scattering state which in this position representationis asymptotically formed by one incoming and one outgoing traveling wave. It is worthnoting that these waves are not plane and that the effect of the inverted harmonic potentialis felt at | q λ | → ∞ . The eigenstates of H I are then | ψ I i = (cid:12)(cid:12) { E λ } Im λ> (cid:11) → Y Im λ> ψ λ ( q λ ) E λ real and arbitrary, (5.10)and the energies read E I = X Im λ> sign (cid:2) λf ( λ ) (cid:3) E λ . The action of c ± λ on ψ λ ( q λ ) is through (5.8) and multiplication and derivation. The states | ψ I i play in the 1 and 2-directions the equivalent rˆole to that of the plane wave states | p a i in the flat a -directions. One way to ensure that the eigenenergies are non-negative is torestrict to scattering states with E λ = sign (cid:2) λf ( λ ) (cid:3) | E λ | for every imaginary λ .Putting everything together, the eigenstates and eigenvalues of H Λ ± are | ψ Λ ± i = | ψ { k an } i ⊗ |{ k λ } Re λ> }i ⊗ |{ E λ } Im λ> i E Λ ± = D − X a =3 ∞ X n =1 n k an + X Re λ> λ k λ + α ′ p a + X Im λ> sign (cid:2) λf ( λ ) (cid:3) E λ + D −
224 + ∆ K ( m ) . H { ( k,l ) } , H e , H o If mκ is such that it squares to an even integer, or is itself an even or odd integer, thehamiltonian also receives the contributions H { ( k,l ) } , H e and H o in (5.4)-(5.5). In case m κ = even, it is trivial to see that H { ( k,l ) } = 12 α ′ X λ ( k,l ) > λ (cid:2) f a ( λ ) a † λ a λ + f a ( λ ) b † λ b λ (cid:3) + X λ ( k,l ) > λ . This only adds to the total hamiltonian two harmonic oscillators for every λ ( k, l ), one forthe a λ -mode and one for the b λ -mode, and contributes to the normal ordering constantwith a finite quantity. The eigenstates and eigenenergies are trivial to write. Assumefor example m κ = 6. There is then only one solution for ( k, l ), namely k = 2 , l = 1and λ = √
10. This adds to oscillators to the total hamiltonian and √
10 to the normalordering constant.For mκ = even and mκ = odd, the hamiltonian adds an m -dependent momentum-likecontribution to the energy. 26 Conclusion and outlook
In this paper we have canonically quantized the open string on the Penrose limit of dS n × S n supported by constant antisymmetric B and a constant dilaton. Canonicalquantization has proved perfectly suited for the task, thus making unnecessary to resortto Dirac quantization and avoiding the problem of whether the boundary conditions forthe string endpoints should be regarded as first or second class constraints. The positionoperators for the quantized string define non-commutative spaces, the wave fronts, for allvalues of the string parameter σ . Noticeably non-commutativity is not restricted to thestring endpoints but extends outside the brane on which the endpoints may be assumedto move. The Minkowski limit is smooth and reproduces the results in the literature [19].We think that further investigation of strings on plane-wave backgrounds is worthto understand non-commutativity in relation with gravity. The low-energy field-theorylimit looks particularly interesting since it may shed light on an effective theory for non-commutative gravity. It must be mentioned in this regard that there is a vast literature [30]on the formulation of Seiberg-Witten maps for gravity and effective non-commutativecorrections to general relativity solutions, plane waves among them [31].From a purely string theory point of view, the strings considered here may be thoughtof as “in” or “out” states to study string scattering on more complicated spaces, whichin turn will have a Penrose limit, and strings near spacetime singularities [32]. Acknowledgments
The authors are grateful to MEC and CAM, Spain for partial support through grantsNos. FIS2005-02309 and UCM-910770. The work of GHR was supported by an MEC-FPU fellowship.
Appendix A. Explicit expression for the string momentum
We collect here the contributions to the string momentum components P i ( τ, σ ) in equa-tion (2.37) of the various existing modes. They are obtained by using (2.11) for X i odd , X i even , X i ( k,l ) and X iλ . For the modes X i o and X i e , in (2.16)-(2.17) and (2.18)-(2.19), we have2 πα ′ P , o = − mκB b o (cid:26) sinh (cid:16) mκπ (cid:17) cos( mκσ ) − B sinh h mκ (cid:16) π − σ (cid:17)i(cid:27) πα ′ P , o = − mκB (cid:20) a o − mκB b o τ sinh (cid:16) mκπ (cid:17)(cid:21) sin( mκσ )27nd 2 πα ′ P , e = mκB b e (cid:26) cosh (cid:16) mκπ (cid:17) cos( mκσ ) + B cosh h mκ (cid:16) π − σ (cid:17)i(cid:27) πα ′ P , e = − mκB (cid:20) a e + mκB b e τ cosh (cid:16) mκ π (cid:17) (cid:21) sin( mκ σ ) . The contribution of the modes X i ( k,l ) in (2.22)-(2.23) in turn reads2 πα ′ P , ( k,l ) = (cid:20) αB a λ ( k,l ) (cid:16) cos βσ + B cos ασ (cid:17) + b λ ( k,l ) (cid:0) sin βσ − αβλ sin ασ (cid:17)(cid:21) e − iλτ πα ′ P , ( k,l ) = iλ (cid:20) a λ ( k,l ) (cid:16) sin ασ − αβλ sin βσ (cid:17) + βBλ b λ ( k,l ) (cid:16) cos ασ + B cos βσ (cid:17) (cid:21) e − iλτ . Finally, the modes X iλ in (2.33)-(2.35) yield the contributions2 πα ′ P , ± ( τ, σ ) = c λ αB (cid:20) cos βσ + sin βπ cos βπ ∓ βσ − B (cid:16) cos απ ± απ sin ασ − cos ασ (cid:17)(cid:21) e − iλτ and 2 πα ′ P , ± ( τ, σ ) = i c λ λ (cid:20) cos απ ± απ cos ασ + sin ασ (A.1) − αβλ (cid:16) sin βσ − sin βπ cos βπ ∓ βσ (cid:17)(cid:21) e − iλτ . (A.2) Appendix B. Derivation of eqs. (4.6)-(4.7)
Organizing the modes in the four sets Λ I , Λ R , Λ , ˜Λ introduced in Section 3 and expand-ing (4.2) in powers of m κ ≪
1, the function Θ( σ, σ ′ ) becomes a sum i Θ( σ, σ ′ ) = X k = I,R, Λ , ˜Λ i Θ( k ; σ, σ ′ )of four contributions Θ( k ; σ, σ ′ ), each one of which is a power series in mκ . Up to orderfour in mκ , these contributions read i Θ(I; σ, σ ′ ) = iα ′ B B ) ( π − σ ) (B.1) − iα ′ B ( mκ )
12 (1 + B ) (cid:2) σ ( σ − π ) (2 + B ) − B σ ′ ( σ ′ − π ) − π (cid:3) , + O (cid:0) m κ )28 Θ(R; σ, σ ′ ) = iα ′ B B ) ( π − σ ′ ) (B.2)+ iα ′ B ( mκ )
12 (1 + B ) (cid:2) σ ′ ( σ ′ − π ) (2 + B ) − B σ ( σ − π ) − π (cid:3) + O (cid:0) m κ ) ,i Θ(Λ; σ, σ ′ ) = − iα ′ B B ∞ X n =1 cos nσ sin nσ ′ n (B.3)+ iα ′ B ( mκ ) (1 + B ) ∞ X n =1 (cid:20) B (2 σ − π ) sin nσ sin nσ ′ n + (2 σ ′ − π ) cos nσ cos nσ ′ n − − B ) cos nσ sin nσ ′ n (cid:21) + O (cid:0) m κ )and i Θ( ˜Λ; σ, σ ′ ) = − iα ′ B B ∞ X n =1 sin nσ cos nσ ′ n (B.4) − iα ′ B ( mκ ) (1 + B ) ∞ X n =1 sin nσ cos nσ ′ n + O (cid:0) m κ ) . Summing all the contributions of order zero in mκ in these equations, we have i Θ ( σ, σ ′ ) = iα ′ B B (cid:2) π − σ + − F ( σ + ) (cid:3) where F ( σ + ) is the Fourier series (3.28). This trivially leads to the order zero contribu-tions in eqs. (4.6) and (4.7). To sum the order two contributions, we first note that (cid:2) i Θ(Λ) + i Θ( ˜Λ) (cid:3) = − iα ′ B B ) n(cid:2) B ( σ + + σ − − π ) + ( σ + − σ − − π ) (cid:3) F ( σ − ) − (cid:2) B ( σ + + σ − − π ) − ( σ + − σ − − π ) (cid:3) F ( σ + )+ (2 B + 1) F ( σ + ) + (5 − B ) F ( σ − ) o , (B.5)where the Fourier series F ( σ ± ) are as in (3.29)-(3.30) and F ( σ ± ) read F ( σ − ) := 2 ∞ X n =1 sin nσ − n = σ − − π σ − | σ − | + π σ − F ( σ + ) := 2 ∞ X n =1 sin nσ + n = σ − π σ + π σ + . Eqs. (B.1), (B.2) and (B.5) then lead to the second order contributions in Section 5.29 eferences [1] R. Penrose, in “Differential geometry and relativity”, edited by M. Cahen and M.Flato, Reidel (Dordrecht 1976), pp. 271.[2] R. Penrose, Rev. Mod. Phys. (1965) 215.For a summary of the geometric properties, see H. Stephani, D. Kramer, M. Mac-callum, C. Hoenselaers and E. Herlt, “Exact solutions to Einstein’s field equations”,2nd edition (Cambridge University Press, Cambridge, 2003).G. W. Gibbons, Commun. Math. Phys. (1975) B219 (1989) 443.[4] G. T. Horowitz and A. R. Steif, Phys. Rev. Lett. (1990) 260.[5] R. G¨uven, Phys. Lett. B482 (2000) 255 [arXiv:hep-th/0005061].[6] J. Kowalski-Glikman, Phys. Lett.
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