String with deformed dispersion relation

We study string theory with momentum living on de Sitter space. We argue that consistency of the theory implies that the global momentum can be defined on de Sitter space. Then we perform careful canonical analysis and we we also show that this presumption leads to the string with deformed dispersion relation.


Introduction and summary
There was an intensive study of the particle in the curved momentum space going back to 60's when these theories were studied as possible examples of divergence free theories [1]. Another very interesting situation was analysed in [2] and in [3] where the particle with the four dimensional de Sitter space with curvature κ was investigated. Explicitly, in term of five dimensional Minkowski space-time coordinates P A it has the form − P 2 0 + P 2 1 + P 2 2 + P 2 3 + P 2 4 = κ 2 .
Further the Casimir for the first four coordinates forming SO (3,1) subgroup of SO(4, 1) that leaves the quadratic form given above invariant is With these two main ingrediences the authors in [2] derived an action for massive particle with de Sitter momentum space. It was shown that this action has deformed dispersion relation and that it possesses non-trivial symplectic structure. We would like to see whether similar idea can be applied in case of string theory where the fundamental object is two dimensional string instead of one dimensional point particle. Explicitly, we consider D-dimensional string theory and impose the condition that momenta lives on D-dimensional de Sitter space. With this presumption we expect that we a e-mail: klu@physics.muni.cz (corresponding author) get string theory with deformed disperssion relation whose existence was firstly proposed in [4]. We studied this proposal from Hamiltonian point of view in [5] and we showed that the formulation of string theory presented in [4] cannot lead to string theory with deformed dispersion relation. We also suggested in [5] that the way how to define string with deformed dispersion relation is to follow analysis performed in [2,3]. As we argued above we have to impose the condition that momenta lives on D-dimensional de Sitter space. However the question is whether this constraint should restrict momentum density that depends on world-sheet spatial coordinate. We argue that such constraint cannot be imposed due to the fact that its Poisson bracket with spatial diffeomorphism constraint does not vanish on the constraint surface that would imply that they are not the first class constraints breaking consistency of the string theory. For that reason we should rather impose the condition that the global momentum is restricted to live on D-dimensional de Sitter space. In other words we consider string with additional first class global constraint. Then, following [2], we fix this global constraint and introduce appropriate variables that parametrize reduced phase space. As a consequence of this procedure we find string theory with modified dispersion relation between global momenta. Note that since the inner part of the string is not affected by this modification clearly consistency and the spectrum of this string is the same as in case of undeformed one. In other words we can very easily implement the modification of dispersion relation into string theory. Then we can study properties of such a modified string theory in the same way as in point particle with all consequences. This paper is organized as follows. In the next Sect. 2 we introduce string theory with global momentum to be restricted on de Sitter manifold. Then in Sect. 3 we study its canonical form and determine corresponding Dirac brackets.

String theory with deformed dispersion relation
Our starting point is the action for bosonic string in Ddimensions. In order to find its form with deformed dispersion relation it is convenient to write it in the canonical form together with an additional constraint whose explicit form will be specified below. Explicitly, we consider string theory action in the form where H τ ≈ 0 , H σ ≈ 0 are standard Hamiltonian and diffeomorphism constraints of bosonic string in the form where μ, ν = 0, 1, . . . , D and where A, B = 0, 1, . . . , D + 1. We further presume closed string with the length L. An important ingredient is the constraint that enforces momentum to be de Sitter This is natural generalization of the constraint used in [2]. However an important point is that this local constraint S does not have weakly vanishing Poisson bracket with H σ ≈ 0 on the constraint surface. Such a situation would imply that H τ , H σ were not the first class constraints with the breaking of the consistency of string theory. For that reason we would rather presume that the total momentum is forced to live on de Sitter space and that p D+1 depends on τ only. Explicitly we consider following constraint In order to demonstrate consistency of the theory let us introduce smeared form of spatial diffeomorphism and Hamiltonian constraints Then using canonical Poisson brackets x μ (σ ), p ν (σ ) = δ μ ν δ(σ − σ ) we obtain following Poisson brackets between all constraints The formulas given in (8) show that all Poisson brackets between constraints vanish on the constraint surface and hence they are the first class constraints. Hence the action has the form Since S is the first class constraint it is appropriate to fix it. In fact, since S ≈ 0 is global constraint that does not depend on σ we can presume that its gauge fixing eliminates global variables only. For that reason we split p μ and x μ in the following way where where L is the length of the string and we have chosen the pre factor L −1 in the expression for X μ so that it has correct dimension [length]. Note that (10) implies Further, using (10) and the canonical Poisson brackets Finally, using (10) we can rewrite the Hamiltonian constraint and spatial diffeomorphism constraint in the form As we argued above S Poisson commutes with all constraints so it is appropriate to gauge fix it. It is important to stress that the inner variablesx μ ,p μ are not affected by this procedure. Following [2] we introduce gauge fixing function G that does not depend on σ and that obeys so that S and G are two second class constraints and hence we reduce global degrees of freedom from 2(D + 1) into 2D. Following [2] we parametrize reduced phase space with variables y μ , k μ that have vanishing Poisson brackets with constraints Now we find that Dirac brackets of k μ and y μ with arbitrary functions are equal to Poisson brackets since As the next step we introduce gauge fixing function G that, following [2], has the form where T is real number. Then we have In what follows we will consider AN (D)-group submanifold defined by two conditions S = 0 , P 0 + P D+1 > 0 . Let us choose k μ coordinates in the following way where i = 1, . . . , D. Then using the first and the third equation in (20) we can express k 0 as Inserting this relation into the second equation in (20) we can express k i as In other words we have k μ = k μ (P A ). Then using these relations we immediately get In the same way we can show that {k i , G} = 0 since k i depend on the linear combination P 0 + P D+1 only. We can further argue that due to the fact that k μ are functions of P μ and P μ Poisson commute withx μ by definition. To proceed further we introduce generator of rotations that for the full string has the form Then it is easy to see that Following [2] we define y μ using components of J AB as and calculate (recall that X 0 = −X 0 , X D+1 = X D+1 ) and we see that this Poisson bracket is zero for T = 0. Let us further calculate Poisson bracket between y i and G and we get where we used the fact that we have Then with the help of definition (27) and the Poisson bracket (30) we get If we finally take into account the gauge fixing function G = 0 we can find inverse relation between X A and y μ in the form Then inserting these relations into string theory action we obtain where The action (33) is final form of the string theory action with modified dispersion relation. We see that the inner part has the same form as in non-deformed case. In the next section we perform canonical analysis of the action (33).

Hamiltonian analysis
In this section we perform canonical analysis of the action (33), following [2]. To do this we tread y μ and k μ as independent variables. In other words we can interpret the action (33) as Lagrangian form of the action and determine conjugate momenta from it. We define l μ as momentum conjugate to y μ an μ as momentum conjugate to k μ where we have canonical Poisson brackets Further, from (35) we get following primary constraints Then with the help of (36) we obtain Poisson brackets between constraints in the form It is convenient to write these Poisson brackets into matrix form introducing common notation for all constraints A ≡ (φ 0 , φ i , 0 , i ) and hence we can rewrite (38) into more symmetric form Then it is easy to determine inverse matrix M AB from (39) Now we are ready to determine Dirac brackets between canonical variables (Note that due to the fact that A are second class constraints we can solve them for μ and l μ so that independent variables will be y μ and k μ ). First of all we have so that we have These Dirac brackets are the same as in the point particle case derived in [2] and that are counterparts of κ-deformed phase space [6]. Once again we should stress that k μ , y μ have zero Dirac brackets withx μ ,p μ . In fact, there are still Hamiltonian and diffeomorphism constraints H τ = P μ (k)η μν P ν (k) +H τ , H σ =H σ + P μ (k)∂ σx μ (43) that are still first class constraints. Finally, since inner part of the string represented by variablesx μ ,p μ decouple from k μ , y μ it is clear that the quantum consistency of the string is the same as in undeformed case.
Let us outline main result that was derived in this paper. The goal was to find string theory with deformed dispersion relation in an alternative way to the proposal presented in [4]. We showed that it can be done when we generalized procedure suggested in [2,3] from one dimensional object to two dimensional object-string. More explicitly, we argued that the only possibility is to have global momentum of the string to live on de Sitter space-time while the inner part of momenta obeys standard dispersion relation. As a result the mass spectra of the string is the same as in non-deformed case. In other words such a string theory can be consistently defined and offers new possibility of the study of alternative string theories.