Semi-classical strings in $(2+1)-$dimensional backgrounds

This study analyzes the geometrical relationship between a classical string and its semi-classical quantum model. From an arbitrary $(2+1)-$dimensional geometry, a specific ansatz for a classical string is used to generate a semi-classical quantum model. In this framework, examples of quantum oscillations and quantum free particles are presented that uniquely determine a classical string and the space-time geometry where its motion takes place.


INTRODUCTION
Quantization schemes in string theory are characterized by their background dependency.
From this standpoint, space-time is something more fundamental than the strings, and thus cannot be framed in terms of them. This conceptual impediment seems to prevent string theory from being a quantum model of gravity. However, even if we disregard philosophical questions, the technical difficulties in string theory are also great, and a general quantization procedure for string theory is as yet unknown.
On the other hand, quantization of string theory is possible in specific cases, such as the semi-classical method developed for the pulsating string in AdS 5 × S 5 [1,2], a method that has been applied to various backgrounds [3][4][5][6][7][8]. The procedure is linked to a particular geometry where classical string motion takes place, and in this article a generalization of the method that enables a description of a wider class of classical strings in various (2 + 1)dimensional spaces is presented. Strings can be understood to move in an effective space, which is a subspace of the ten dimensional space-time where string theory is consistently defined.
The basis for this generalization is the observation that metric tensor elements determine the potential of the classical Hamiltonian which is used to build a quantum model. From this simple idea, it is possible to vary the potential and to establish correspondence between the classical model and the quantum model based on the geometry of the effective (2 + 1)−dimensional space-time where the string moves. The correspondence between the quantum model and the classical model does not mean equivalence or duality, as in the AdS/CFT correspondence, but an association between a classical motion and a quantum model with a common space-time geometry. It is not clear if there are any other classical strings that can generate an identical quantum model, but this question is not posed here, because the aim is to demonstrate the existence of correspondence only. Potential is the central element relating the quantum model to the classical model, and a specific potential requires a particular space-time. The results show that quantum oscillation and quantum free particles occur at different space-time topologies. A space-time that could be used to construct a quantum oscillation and a quantum free particle for the same string has not been found.
The article is organized as follows: section (2) describes dynamics of a string in a (2 + 1)−dimensional and the embedding of the string world-sheet in a 3−dimensional plane space. In section (3), string motion and space-time geometry are determined from specific potentials of the quantum model, and the quantum spectrum is also studied. Section (4) contains the the author's conclusions.

THE STRING AND SPACE
(2 + 1)−dimensional space-time where the string moves is described by the line element where x is a general coordinate, which can be a distance or an angle, ϕ is an angular coordinate and f is an arbitrary function of the coordinates. In this space, a classical string performs an arbitrary motion. Depending on the string, the equations of motion generate the conditions that must be fulfilled by f , and the first task is to choose a string.

classical and quantum dynamics
The string of interest is described by the ansatz where κ is a constant and the string is wrapped m times along ϕ and executes some motion along x. If f = f (x), the Nambu-Goto action for this string, namely determines its equation of motion where the prime denotes a derivative with respect to x and the dot denotes a derivative with respect to τ . The above equation is non-linear and difficult to use, thus it will be substituted by the Virasoro constraintẋ From (3) and (5), the canonical momentum and energy are whose canonical Hamiltonian is This classical formalism can be used to semi-classically quantize the string using the square of the Hamiltonian (7) to express the Schrödinger equation as H 2 Ψ = E 2 κ 2 Ψ, where E 2 is the squared quantum energy. From this analysis, it follows that the coefficient f of the line element (1) determines the geometry of the space as well as the classical potential and the quantum wave-function.

space geometry
The spatial motion of the string is constrained by the metric (1), which defines the worldsheet of the string. Consequently, the world-sheet of a string moving through the whole of space is identical to space, and thus the embedding of the 2−dimensional surface into a 3−dimensional plane space enables both the geometry of the space and the motion of the string to be visualized. Expressing (1) as ds 2 = −dt 2 + ds 2 , the two dimensional sub-space generated by x and ϕ must be embedded into a three-dimensional plane space with the metric It is assumed that the coordinates of the plane space have a cylindrical symmetry, so that where ρ = ρ(x), z = z(x) and φ = φ(ϕ). The embedding is obtained through the identity Using (9) in the metric tensor h ab , we obtained the system of differential equations where indices x and ϕ represent derivatives with respect to these coordinates. It can immediately be seen that φ = C ϕ, and as both of the coordinates have the same range, C = 1 and ρ = f . As f is known from the beginning, the embedding must be obtained from the

MOVING STRINGS
In the preceding section the general formalism which associates a moving string in a (2 + 1)−dimensional space to a quantum model has been presented. In order to relate the geometry and the topology of the space to the classical motion of the string, f and x are specialized, and then we expect some quantum-geometry relation to arise. Maintaining a rotational symmetry for ϕ, x may be either a radial or an angular coordinate. For a radial x = r > 0, the motion of the string is constrained to an open surface, and for an angular coordinate, the surface can be closed. However, this division does not exhaust the possibilities, because these two categories can be subdivided. Below, the radial coordinate case has been divided into the pulsating string case and the falling string case.

radial coordinate pulsating string
In this case, it will be used the ansatz f = ℓ r n 2 , where n a positive integer and ℓ is a dimensional constant responsible for f having length dimension. By choosing y = m √ λ f κ 2 and κτ = r g(y), we obtain the equation of motion from (5) g + n y g y = 1 Using the hypergeometric function relation F (a, b; b; z) = (1 − z) −a and contiguous hypergeometric function relations, it is possible to ascertain that the general solution for (13) is For n = 1, κτ ∝ √ 1 − y, which implies that the time is limited; hence, this can be discarded as a physical solution. The inverse of (14) gives r = r(τ ) for each n. Although this inverse function is unknown, some considerations can be stated by observing the figure   1 below.
For n = 2, t = arcsin(y), and thus periodic oscillatory behavior for r is warranted. For other values of n, the graph shows that the maximum value of y is one and that the maximum of y is reached at a value of t, which approaches one the greater n. Another aspect of the problem to be considered is the geometry of the two dimensional surface where the string moves. Defining the variable w = f 2 r = n 2 ℓ 2 4 r n−2 and the coordinate z = r h(w), from (12) we obtain and consequently (16) is valid for n > 2. If n = 1, w ∝ 1/r, so that r > 1 and as the string vibrates for r < 1, there is no physical solution for (15). For n = 2, z is constant, and the whole plane is allowed for vibration of the string. This is an expected result coherent with the known solutions [1,3]. The graph of (16) in figure 4 gives an idea of the surface where the string is allowed to move Of course, for each particular n, the above graph has an identical reflected line in the negative z direction. Besides this, the surface is cylindrically symmetric, so that the whole surface is similar to a cone. As n increases, the graph approaches a straight line and the whole surface approaches a rectangular cone. It is also interesting to note that the space is finite. Only the n = 2 case generates an infinite plane surface.
The description of the space and the motion of the string concludes the analysis of the classical behavior of the string. The next goal is semi-classically study the quantum features of this system. For each n, a specific quantum model can be obtained, and it is described by the Scrödinger equation Except for the non-physical n = 1 case, the exactly solvable solution of (17) only occurs when n = 2. For this particular situation, the wave-function and the energy spectrum are where N is the normalization constant, N is a positive integer and L N are the Laguerre polynomials. In this case, ℓ is dimensionless and can be set to one.
The n > 2 solutions must be studied perturbatively, and the non-perturbed case is calculated by excluding the potential term of (17). The solutions are where The index Z is due to the fact that the zeros of the Bessel functions J and Y are not common and then there are two independent wave-functions and two energy spectra. Both of the wave-functions are normalizable, except only the Y wave function for n = 6, 10, 14, etc.
The perturbative calculations for energy need a wave-function given by an orthogonal set, and thus only the quantized energy wave-functions can be used. The orthogonal set can be obtained from Bessel function J, which obeys the condition which is zero if α and β are different Bessel function zeros. From (21) we also get the normalization condition Thus, the energy in the first order of perturbation for the potential f 2 = ℓ 2 r n is where 2 F 3 is a generalized hypergeometric function. As the series defined by this object converges for every finite argument, (23) is expected to be a well-behaved value that does not diverge for any zero of J.
The result rounds off the analysis, which comprises the geometrical correspondence between the classical dynamics and quantum dynamics of a string. Of course, there is no correspondence in the terms of gauge/gravity duality, as the classical string does not have a quantized spectrum and so the models are not identical in this sense. However, the example does show that a classical pulsating string and a quantum oscillation are connected through a specific geometry, which determines the string motion and the quantum energy spectrum.
Another example of this correspondence is provided in the next section.

free falling string
This model is constructed using f 2 = ℓ 2 r −n , with n ∈ N, and the analysis follows the manner developed for the pulsating string, comprising of the classical string motion, the geometry of the space and semi-classical quantization.
Choosing y = m √ λℓf κ 2 and κτ = r g(y), we ascertain from (5) that the classical motion obeys g − n y g y = 1 From the hypergeometric function relation F (a, b; b; z) = (1 − z) −a and the contiguous hypergeometric function relations, it is possible to ascertain that the general solutions for (24) are The greater the n value, the more the solutions approach the straight line κτ = r. This string goes continuously to infinity, asymptotically approaching a constant velocity of a free particle, hence it can be described as a string in free fall.
The other aspect of the classical picture, the geometry of the space, is obtained from (12), in a same manner that it was obtained for the pulsating string case, and it is described by whose positive part can be seen in figure 6.
Of course, each surface has a cylindrical symmetry and it consists of two infinite sheets with a hole in the center. The existence of the hole is naturally predictable, as the metric is not defined at r = 0, and then the puncture in space is expected. The existence of the two sheets where the free fall of the string can occur seems somewhat surprising. However, this kind of situation has already been observed in a sphere [1,3], where the string independently pulsates in each hemisphere.
After the classical description, the quantum fluctuations are studied through the Schrödinger equation The n = 1 has already been observed not to have a classical physical solution, and then the analysis comprises n ≥ 2. There is an exact solution for the n = 2 case, namely where a = √ 1 + m 2 λℓ 2 A and B are integration constants. For n > 2, the exact solutions are unknown and the free particle solutions are very similar to the aforementioned exact solution, given by Although the intensity of the wave-function increases with r and diverges at infinity, the solutions are indeed free particles. One manner of visualizing this is to see that it comes from the fact that the non normalizable free particle solutions define a Dirac delta function and then obey a localization condition [9], The Dirac delta function in terms of Bessel functions in a (d + 1)−dimensional space, and it fits perfectly with (30). Thus, even the exact solution for n = 2 is a free particle, and for other values of n, the same interpretation holds.
As for the previous case, there are continuous and quantized energies. The quantized energies obey the condition that the wave-function is zero at the edge of space, and then (20) is valid in this case also. The energy of the n > 2 cases are calculated using perturbation theory, so that The exact expression of the above integral, given in terms of generalized 2 F 3 functions is complicated and not really pertinent to this study. However, as in the former case, the series that represents the function is convergent for any value of the argument, and this is enough to assure a finite correction to the energy. Although the results extend the range of quantum models that can be obtained from a string motion, from the point of view of the author of this article, it is somewhat frustrating that the potential that goes with the inverse of the distance is not permitted in the models presented. The string motion that could model the relevant physical phenomena described by this potential, namely gravity and electromagnetism remains unknown. On the other hand, the results are evidence that the link between quantum theory and general relativity through geometry seems not to be merely a myth.