This study analyzes the geometrical relationship between a classical
string and its semiclassical quantum model. From an arbitrary

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

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

The paper is organized as follows. Section

The string of interest is described by the ansatz

The spatial motion of the string is constrained by the metric (

In the preceding section the general formalism which associates a moving string in a

In this case, the ansatz

For

Time for the pulsating string.

For

Formal inversion of

Formal inversion of

Another aspect of the problem to be considered is the geometry of the two-dimensional surface where the string moves. Defining the variable

Embedded space for the pulsating string.

Of course, for each particular

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

The

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

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.

This model is constructed using

Choosing

Time for the free falling string.

The greater the

The other aspect of the classical picture, the geometry of the space, is obtained from (

Embedded space for the free falling string.

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

After the classical description, the quantum fluctuations are studied through the Schrödinger equation

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 (

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.

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