Semiclassical Strings in (2+1)-Dimensional Backgrounds

Abstract

This study analyzes the geometrical relationship between a classical string and its semiclassical 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.

1. 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 semiclassical method developed for the pulsating string in 𝐴𝑑𝑆₅×𝑆⁵ [1, 2], a method that has been applied to various backgrounds [3–8]. The procedure is linked to a particular geometry where classical string motion takes place, and in this paper 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. The 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. There is an important question about the consistency of the proposed geometries as supergravity backgrounds. This point has not been developed, and the results show that it is worthwhile to study it in the future, particularly because there is no reason for preventing the proposed geometries from being string backgrounds.

The paper 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 author’s conclusions.

2. The String and Space

(2+1)-dimensional space-time where the string moves is described by the line element 𝑑𝑠²=−𝑑𝑡²+𝑑𝑥²+𝑓²𝑑𝜑², (1) 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.

2.1. Classical and Quantum Dynamics

The string of interest is described by the ansatz 𝑡=𝜅𝜏,𝑥=𝑥(𝜏),𝜑=𝑚𝜎, (2) 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, 𝒜=−𝑚√𝜆𝑑𝜏𝑓√𝜅²−𝑥̇², (3) determines its equation of motion 𝑓^′𝑓𝑥̈=−𝜅²−𝑥̇², (4) where the prime denotes a derivative with respect to 𝑥 and the dot denotes a derivative with respect to 𝜏. The above equation is nonlinear and difficult to use; thus, it will be substituted by the Virasoro constraint 𝑥̇²=𝜅²−𝑚²𝑓². (5) From (3) and (5), the canonical momentum and energy are 𝑚√Π=𝜆√𝜅²−𝑥̇²𝑓𝑥̇,𝓔=√𝜆𝜅 (6) whose canonical Hamiltonian is ℋ=𝜅Π²+𝑚²𝜆𝑓². (7) This classical formalism can be used to semiclassically quantize the string using the square of the Hamiltonian (7) to express the Schrödinger equation as ℋ²Ψ=(ℰ²/𝜅²)Ψ, where ℰ² 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.

2.2. Space Geometry

The spatial motion of the string is constrained by the metric (1), which defines the world-sheet 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 𝑑𝑠²=−𝑑𝑡²+𝑑𝓼², the two-dimensional subspace generated by 𝑥 and 𝜑 must be embedded into a three-dimensional plane space with the metric 𝑑𝑆²=𝑔_𝑖𝑗𝑑𝑥^𝑖𝑑𝑥^𝑗=𝑑𝑥¹²+𝑑𝑥²²+𝑑𝑥³². (8) It is assumed that the coordinates of the plane space have a cylindrical symmetry, so that 𝑥¹=𝜌sin𝜙,𝑥²=𝜌cos𝜙,𝑥³=𝑧, (9) where 𝜌=𝜌(𝑥), 𝑧=𝑧(𝑥), and 𝜙=𝜙(𝜑). The embedding is obtained through the identity 𝑑𝓼²=ℎ_𝑎𝑏𝑑𝜉^𝑎𝑑𝜉^𝑏=𝑔_𝑖𝑗𝜕𝑥^𝑖𝜕𝜉^𝑎𝜕𝑥^𝑗𝜕𝜉^𝑏𝑑𝜉^𝑎𝑑𝜉^𝑏,where(𝑥,𝜗)=𝜉¹,𝜉². (10) Using (9) in the metric tensor ℎ_𝑎𝑏, we obtained the system of differential equations 𝜌²_𝑥+𝑧²_𝑥=1,𝜌²𝜙²_𝜑=𝑓², (11) where indices 𝑥 and 𝜑 represent derivatives with respect to these coordinates. It can immediately be seen that 𝜙=𝐶𝜑, 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 integration 𝑧=±𝑑𝑥1−𝑓^′2. (12)

3. 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.

3.1. Radial Coordinate Pulsating String

In this case, the ansatz 𝑓=ℓ𝑟^𝑛/2 will be used, where 𝑛 is a positive integer and ℓ is a dimensional constant responsible for 𝑓 having length dimension. By choosing 𝑦=(𝑚√𝜆𝑓/𝜅)² and 𝜅𝜏=𝑟𝑔(𝑦), we obtain the equation of motion from (5) 𝑔+𝑛𝑦𝑔_𝑦=1.1−𝑦 (13) 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 1𝜅𝜏=𝑟𝐹2,1𝑛1;1+𝑛;𝑚²𝜆ℓ𝜅²𝑟^𝑛. (14)

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 Figure 1.

Figure 1 

Time for the pulsating string. 𝑛=2 solid line; 𝑛=5 dotted line; 𝑛=50 dashed line.

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. For an even number 𝑛, 𝑡 is an odd function where negative values are allowed for the argument, indicating a range of 𝑦 in the interval [−1,1] for the function. This fact may be observed in Figure 2 by formally inverting the first terms of the infinite series generated by (14). The same fact cannot be stated for odd 𝑛, where 𝑡 does not have a definite parity for the negative values of the argument. On the other hand, the inversion of the first terms confirms the existence of the maximum value of 𝑟 to be one in Figure 3. However, as the negative values are not actually allowed in both of the cases in this particular problem, the behavior of 𝑟(𝜏) given by the positive 𝑦(𝑡) can be described as oscillatory. Of course, at 𝑦=0, the string changes its direction and the derivative of 𝑦(𝑡) is not defined there. Only when negative values of 𝑦 are allowed for even values of 𝑛, the derivative of 𝑦(𝑡) is well defined at these points. The conclusion is that the classical motion of 𝑟(𝜏) is possibly oscillatory for any 𝑛≥2.

Figure 2 

Formal inversion of 𝑡(𝑦). 𝑛=2 solid line; 𝑛=6 dashed line; 𝑛=8 dotted line.

Figure 3 

Formal inversion of 𝑡(𝑦); 𝑛=3 dotted line; 𝑛=5 dashed line; 𝑛=7 solid line.

Another aspect of the problem to be considered is the geometry of the two-dimensional surface where the string moves. Defining the variable 𝑤=𝑓²_𝑟=(𝑛²ℓ²/4)𝑟^𝑛−2and the coordinate 𝑧=𝑟ℎ(𝑤), from (12), we obtain ℎ+(𝑛−2)𝑤ℎ_𝑤=√1−𝑤, (15) and consequently −1𝑧=𝑟𝐹2,11𝑛−2;1+;𝑛𝑛−2²ℓ²4𝑟^𝑛−2. (16) Equation (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.

Figure 4 

Embedded space for the pulsating string. 𝑛=3 dotted line; 𝑛=10 solid line; 𝑛=25 dashed line.

Of course, for each particular 𝑛, the pervious 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 to semiclassically 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 −Ψ^′′−𝑛Ψ2𝑟^′+𝑚²𝜆ℓ²𝑟^𝑛ℰΨ=²𝜅²Ψ. (17) Except for the nonphysical 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 Ψ_𝑁=𝒩𝑒^{√−(𝑚𝜆ℓ/2)𝑟2}𝐿_𝑁𝑚√𝜆ℓ𝑟²,ℰ²_𝑁𝜅²𝑚√𝜆ℓ=4𝑁+2, (18) 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 nonperturbed case is calculated by excluding the potential term of (17). The solutions are 1Ψ=𝑟^(𝑛−2)/4𝐴𝐽_(𝑛−2)/4ℰ𝜅𝑟+𝐵𝑌_(𝑛−2)/4ℰ𝜅𝑟, (19) where 𝐴 and 𝐵 are integration constants. The space is finite and the range of the radial coordinate can be assumed to be [0,𝑅]. Solution (19) describes quantum free particles; however, the permitted energies can be either continuous or quantized. Wave functions where Ψ(𝑟=𝑅)≠0 have continuous energies and wave functions where Ψ(𝑟=𝑅)=0 have a quantized energy spectrum given according to the zeros of the Bessel functions, ℰ_𝑁𝜅=𝑅_𝒵^(𝑁)𝑅,sothat𝑁∈ℕ,𝒵={𝐽,𝑌}. (20) 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, and so forth.

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 the Bessel function 𝐽, which obeys the condition ¹₀𝑑𝑥𝑥𝐽_𝜇(𝛼𝑥)𝐽_𝜇=𝛽𝑥𝛽𝐽_𝜇(𝛼)𝐽_𝜇−1𝛽−𝛼𝐽_𝜇−1(𝛼)𝐽_𝜇𝛽𝛼²−𝛽², (21) which is zero if 𝛼 and 𝛽 are different Bessel function zeros. From (21), we also get the normalization condition ¹₀𝑑𝑥𝑥𝐽²_𝜇𝑅^(𝑁)𝑥=𝐽²_𝜇+1𝑅_𝐽^(𝑁)2𝑅_𝐽^(𝑁)2. (22) Thus, the energy in the first order of perturbation for the potential 𝑓²=ℓ²𝑟^𝑛 is 𝛿ℰ²_𝑁𝜅²𝑚²𝜆ℓ²=𝑅^{(𝑁)𝑛+11}₀𝑑𝑥𝑥^𝑛+1𝐽²_(𝑛−2)/2𝑅^(𝑁)𝑥=𝜆𝑅^{(𝑁)2𝑛+2}𝑛Γ(𝑛/2)2^𝑛−2𝐽²_()𝑛−2/2𝑅^(𝑁)2𝐹₃×𝑛,𝑛−12,𝑛𝑛−1,2,𝑛+1,−𝑅^(𝑁)2, (23) where ₂𝐹₃ 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.

3.2. Free Falling String

This model is constructed using 𝑓²=ℓ²𝑟^−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 semiclassical quantization.

Choosing 𝑦=(𝑚√𝜆ℓ𝑓/𝜅)² and 𝜅𝜏=𝑟𝑔(𝑦), we ascertain from (5) that the classical motion obeys 𝑔−𝑛𝑦𝑔_𝑦=1.1−𝑦 (24) 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 1𝜅𝜏=𝑟𝐹21,−𝑛1;1−𝑛;𝑛²𝜆ℓ²𝜅²𝑟^−𝑛. (25) This solution holds for 𝑛>1, because 𝑛=1 allows negative values for 𝜏, thus comprising an unphysical solution. These solutions are different from the former pulsating case, because the radial coordinate and the time coordinate continuously increase, as can be seen in Figure 5.

Figure 5 

Time for the free falling string. 𝑛=2 dashed line; 𝑛=5 dotted line; 𝑛=50 solid line.

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 the same manner that it was obtained for the pulsating string case, and it is described by −1𝑧=±𝑟𝐹21,−1𝑛+2;1−;𝑛𝑛+2²ℓ²4𝑟^−𝑛−2, (26) whose positive part can be seen in Figure 6.

Figure 6 

Embedded space for the free falling string. 𝑛=2 dotted line; 𝑛=4 solid line; 𝑛=50 dashed line.

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 −Ψ^′′+𝑛Ψ2𝑟^′+𝑚²𝜆ℓ²𝑟^𝑛ℰΨ=²𝜅²Ψ. (27) 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, Ψ=𝑟𝐴𝐽_𝑎ℰ𝜅𝑟+𝐵𝑌_𝑎ℰ𝜅𝑟, (28) where 𝑎=1+𝑚²𝜆ℓ² and 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 Ψ=𝑟^(𝑛+2)/4𝐴𝐽_(𝑛+2)/4ℰ𝜅𝑟+𝐵𝑌_(𝑛+2)/4ℰ𝜅𝑟. (29) 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 nonnormalizable free particle solutions define a Dirac delta function and then obey a localization condition [9], ^∞_∞ΨΨ^⋆√−𝑔𝑑𝑥=𝛿(𝑥). (30) The Dirac delta function in terms of the Bessel functions in a (𝑑+1)-dimensional space is 𝛿^𝑑+1𝜖²−𝜂²=^∞₀𝑑𝑟𝑟𝐽_𝜇(𝜖𝑟)𝐽_𝜇𝜂𝑟, (31) 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 energies of the 𝑛>2 cases are calculated using perturbation theory, so that 𝛿ℰ²_𝑁𝜅²𝑚²𝜆ℓ²=𝑅^{(𝑁)𝑛+11}₀𝑑𝑥𝑥^𝑛+1𝐽²_()𝑛−2/2𝑅^(𝑁)𝑥. (32) The exact expression of the above integral, given in terms of generalized ₂𝐹₃ 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.

4. Conclusion

In this paper, examples of classical strings were presented that can be semiclassically quantized through a well-known prescription. The examples demonstrate that the classical string and its quantum fluctuations are connected through the space where the motion takes place. The geometry and the topology of the space determine both the classical string and the quantum Hamiltonian.

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 paper, 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.

Acknowledgments

S. Giardino is thankful for the financial support of Capes and for the facilities provided by the IFUSP.