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This article comes from the official account of Wechat: back to Park (ID:fanpu2019), author: Zhou Siyi (postdoctoral fellow, Kobe University)

How to quantize gravity has always been a difficult problem for theoretical physicists. At present, string theory is the only mature model of quantum gravity, and recently a new work has proved the potential of this model. The researchers used the "bootstrap" method, which is widely used in theoretical physics, to give the lower bound of the α parameter of the modified gravity model, which is strikingly similar to the value given by string theory.

In 1915, Einstein put forward the general theory of relativity, which is a very successful theory. At present, this theory has achieved amazing success in describing all phenomena on a large scale. However, on a small scale, Einstein's gravity is faced with problems, such as when we consider quantum gravity, the circle diagram of the graviton is divergent, which means that gravity is irreversible. Einstein's theory of gravity needs to be revised on a small scale.

However, theoretical physicists do not know exactly how to correct it, or even whether gravity needs to be quantized. However, after all, gravitational waves and electromagnetic waves are strikingly similar in many properties, so it is natural for theoretical physicists to believe in quantum gravity. Exploring the micro nature of gravity has become an important goal for many scholars. String theory is the only mature quantum gravity model at present, of course, there are also some other quantum gravity models.

To parameterize the theory on a large scale, the theory must return to the general theory of relativity, so the practice of theoretical physicists is to parameterize the theory on a small scale and set some modified parameters α, β for Einstein's gravity. Among them, α is the lowest order and the most important correction of quantum gravity to low-energy Einstein gravity; 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, the greater the correction of the 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 can't answer this question yet, but a recent work adds stronger evidence to it. Andrea Guerrieri of Tel Aviv University in Israel, Jo ã o Penedones of Lobsang Federal Institute of Technology in Switzerland and Pedro Vieira of the Canadian circumferential Institute of theoretical Physics published a paper in physical Review KuaiBao (PRL). They gave the lower bound of the parameter α, indicating that string theory is in a "garden".

The article recently has a work called "Where is String Theory in the Space of Scattering Amplitudes". In this work, the IIA type string theory which can be calculated accurately is given.

It is finite when z is infinity. The picture shows the contour of the function g (z) on the complex plane. The relationship between α and scattering amplitude can be deduced from the contour integral.

α ≥ 0 can be derived from the optical theorem. This is what is commonly known as positive definiteness. Obviously, the correct quantum gravity model α ≥ 0. In the area where α < 0, the author likens it to "desert".

Bootstrap in the gravitational quantization work of three physicists, they used Bootstrap's method, which is commonly used in theoretical physics. The word Bootstrap itself refers to something behind a shoe that is used to lift a shoe, and it also means to develop on one's own ability, which is generally translated as "boot tripping" or "bootstrapping" in physics. It means that starting from the self-consistent condition of the theory, it automatically makes some restrictions on the parameters of the theory, and it is a non-perturbative processing method. Bootstrap is widely used in theoretical physics, and the most applications are concentrated in the field of conformal field theory.

The main idea of the Bootstrap method is not to start from the specific high-energy model, but to focus on the properties that the 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 with the goal of gravity. They use the scattering amplitude Bootstrap: first write out the form of the scattering amplitude that the high-energy theory satisfies certain conditions, and then expand the scattering amplitude at low energy.

The following is a general form of scattering amplitude:

Where GN is Newtonian constant. S, t, u are Mandelstein invariants and ρ is a function consisting of Mandelstein invariants. Starting from the properties that must be satisfied by the high energy theory (Lorentz invariance, cross symmetry, analytic and unitary), they derive some properties that must be satisfied by the coefficient α (abc). Then they expand the function which satisfies the good properties of the high-energy theory under the low energy, that is, the very small limit of s. In this case, the scattering amplitude is in the form of:

Because the form of the high energy expansion of the scattering amplitude is very long, they must make a finite truncation of N, get the α value of the low energy theory, and then interpolate, and get the conditions that α must meet when N tends to infinity. After numerical calculation, α ≥ 0.13 ±0.02 is obtained.

Because there are five kinds of string theories, Type IIA,Type IIB,Type Ipenol so (32) hybrid strings and E8 × E8 hybrid strings, these string theories are unified under the framework of M theory. In this work, they also worked out the alpha parameters of IIA-type string theory and IIB-type string theory, which can be accurately calculated by string theory, respectively.

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

The purpose of this article is to find out the position of string theory in all the 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 specific characteristics of the high-energy theory. Different models of high energy theory can give different values of α. Starting from the simplest analytic and unitary property that the scattering amplitude must satisfy, we can quickly exclude the desert region where α < 0, which is the so-called positive definite condition. Andrea Guerrieri et al.'s work hopes to impose stricter restrictions on the amplitude of high-energy scattering, so as to give the conditions that must be satisfied for a stricter α. As we can see, string theory occupies almost all the areas allowed by the Bootstrap method. This provides stronger evidence for string theory as a self-consistent model of quantum gravity.

reference

Andrea Guerrieri, Jo ã o 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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