String theory in the garden

How to quantize gravity is a puzzle for theoretical physicists. String theory is currently the only relatively mature quantum gravity model, and a new work has recently demonstrated the potential of this model. The researchers used the "bootstrap" method commonly used in theoretical physics to give a lower limit on the α parameter of the modified gravity model, which is surprisingly similar to the value given by string theory.

Written by Zhou Siyi (Postdoctoral Fellow, Kobe University)

In 1915, Einstein proposed the theory of general relativity, which is a very successful theory. So far, this theory has achieved amazing success in describing all phenomena on large scales. However, on small scales, Einstein's gravity faces problems. For example, when we consider quantum gravity, the loop diagram of gravitons is divergent, which means that gravity cannot be renormalized. Einstein's theory of gravity on small scales needs to be revised.

However, theoretical physicists do not know how to make the corrections, or even whether gravity needs to be quantized. However, after all, gravitational waves and electromagnetic waves have amazing similarities in many properties, so it is natural for theoretical physicists to believe in the existence of quantum gravity. Exploring the microscopic nature of gravity has become an important research goal for many scholars. String theory is currently the only relatively mature quantum gravity model, and of course there are some other quantum gravity models.

Parameterizing the theory

On a large scale, the theory must return to general relativity, so theoretical physicists parameterize the theory on a small scale and set some correction parameters for Einstein gravity, such as α and β. Among them, α is the lowest order and most important correction of quantum gravity to Einstein gravity at low energy; the coefficients of other smaller correction terms are β, and so on. Different quantum gravity models will give different values ​​of α and β. The larger these parameters are, the greater the correction of this quantum gravity model to Einstein at low energy.

Starting from string theory, we can calculate the exact value of α. So is string theory the only correct model of quantum gravity? We cannot answer this question now, but a recent work has added stronger evidence for it. Andrea Guerrieri of Tel Aviv University in Israel, João Penedones of the Swiss Federal Institute of Technology in Lausanne, and Pedro Vieira of the Perimeter Institute for Theoretical Physics in Canada published a paper in Physical Review Letters (PRL). They gave a lower limit for the parameter α, indicating that string theory is in a "garden".

Limitations of Positive Definiteness

In general, a correct quantum gravity model requires that the scattering amplitude has good analytical properties. On the other hand, the scattering amplitude must be finite at high energies. Andrea Guerrieri et al. constructed such a function

Guerrieri et al. constructed such a function

The optical theorem can be used to derive that α≥0. This is what is generally known as the positive definiteness limit. Obviously, the correct quantum gravity model has α≥0. The region where α＜0 is likened by the authors to a "desert".

Bootstrap

In their work on the quantization of gravity, the three physicists used the Bootstrap method, which is a common method in theoretical physics. The word Bootstrap itself refers to the thing on the back of a shoe used to lift it, and also means to develop by one's own ability. In physics, it is generally translated as "boot" or "self-bootstrap". It means starting from the self-consistency conditions of the theory itself, automatically making some restrictions on the parameters of the theory, which is a non-perturbative processing method. Bootstrap is widely used in theoretical physics, and most of its applications are concentrated in the field of conformal field theory.

The main idea of ​​the Bootstrap method is not to start from a specific high-energy model, but to focus on the properties that the assumed high-energy model must have. A few years ago, Andrea Guerrieri and others first successfully applied this method to the interaction of pions, this time their target was gravity. They used the scattering amplitude Bootstrap: first write out the form of the scattering amplitude that satisfies certain conditions in the high-energy theory, and then expand this scattering amplitude at low energy.

Here is a general form of the scattering amplitude:

The garden is where all quantum gravity models must be, and string theory fills the entire garden.

summary

The purpose of this article is to find the position of string theory in all high-energy theoretical models that meet the requirements of quantum gravity. The parameters in the scattering amplitude of the low-energy effective theory of gravity give the characteristics of the specific high-energy theory. Different high-energy theoretical models can give different values ​​of α. Starting from the simplest, the analyticality and unitary properties that the scattering amplitude must satisfy, we can quickly exclude the desert area where α<0, which is the so-called positive definiteness condition. The work of Andrea Guerrieri and others hopes to impose stricter constraints on the high-energy scattering amplitude, thereby giving stricter conditions that α must satisfy. We can see that string theory occupies almost all the areas allowed by the Bootstrap method. This provides stronger evidence for string theory as a self-consistent quantum gravity model.

References

[1] Andrea Guerrieri, João Penedones, and Pedro Vieira, Phys. Rev. Lett. 127, 081601 e-Print: 2102.02847 [hep-th]

[2] https://www.quantamagazine.org/a-correction-to-einstein-hints-at-evidence-for-string-theory-20220121/

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