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This article comes from Weixin Official Accounts: Fanpu (ID: fanpu2019), by Dong Weiyuan

Two-dimensional conformal field theory has always been one of the important frontier research tools in theoretical physics, especially Liouville conformal field theory, which is closely related to quantum gravity. By means of conformal bootstrap method, Liouville conformal field can be solved exactly without perturbation. However, its key equations turned out to be guesswork, and it was only in recent years that mathematicians gave a rigorous proof. Mathematicians and physicists have gained a little more insight into the implications of quantum field theory.

Quantum gravity theory is the accepted holy grail of physics, and it has always attracted the best minds on our planet. Intelligent scientists have long been able to skillfully apply quantum theory and general relativity, and even find their shadows in daily life. However, the mysteries of the universe hidden behind these two theories are still so elusive.

In 2003, American physicist Lee Smolin, one of the founders of loop quantum gravity, Lee Smolin wrote in his popular science book The Origin of the Universe. Three Roads to Quantum Gravity concludes with an optimistic outlook: "By 2010, at the very most by 2015, we should have the basic framework of a quantum theory of gravity... within ten years of having this theory, new experiments capable of testing it will have been invented... by the end of the twenty-first century, high school students around the world will be learning the quantum theory of gravity." "Looking back now, Smolin's prediction was clearly too optimistic.

Perhaps the clearest illustration of the gap between quantum theory and gravity is the concept of cosmic dark energy. According to general relativity, the accelerating expansion of the universe implies that vacuum has energy, that is, the "cosmological constant" in Einstein's equations is not zero. According to quantum field theory, vacuum also has nonzero energy, which has been confirmed by experiments on the Casimir effect. Thus, both theories seem to give the same vacuum energy, but in fact the values given by the two are 120 orders of magnitude different! Note that it is not 120 times, but 120 orders of magnitude, that is, 10120 times. Attempts to explain the cosmological constant by vacuum zero-point energy became the most outrageous guess in physics.

However, our universe cannot have two kinds of vacuum, so the concept of "cosmic dark energy" was proposed to bridge the huge differences between the two theories on the description of vacuum energy. Dark energy is called "dark" because it is neither within the framework of quantum theory nor explained by gravity. This mysterious gap, which accounts for 70% of the total energy of the universe, may have to wait for future quantum gravity theories to stitch it up.

"Dimensional reduction" is full of obstacles that existing mathematical tools cannot overcome on the way to explore quantum gravity, so researchers are trying to build new tools while trying to simplify the problem. Two-dimensional models are one of the most commonly used detours.

The most obvious benefit of reducing high dimensions to two dimensions is the dramatic simplification of processing. For example, in a two-dimensional plane, the order of several rotation operations can be arbitrarily exchanged, and the final operation result will not be affected by the change of order. In three-dimensional or higher dimensional spaces, the order of rotation operations cannot be exchanged at will. It can be seen that two-dimensional space is less restricted than high-dimensional space, and there is more room for manoeuvre when dealing with complex calculations.

Of course, the rotation operation is just one example of a bad idea. What researchers really like is an operation called "Conformal transformations." This operation is also known as conformal transformation, which as its name implies can keep the angle between any two lines unchanged when twisting deformation. For example, the transformation shown below is a typical conformal transformation. After the transformation, each blue line remains perpendicular to each red line.

If this is the first time you've heard the term conformal transformation, don't be intimidated by the bluff. Look at the curved lines in the picture above. Do they remind you of electric and magnetic field lines in high school textbooks? Think back to that little experiment when I was a kid where iron filings on paper showed magnetic field lines. In fact, when you hold two magnets and move them freely, the change of the iron filings pattern on the paper is a conformal transformation.

Physicists are in great need of conformal transformations when studying fields. The invariant in each conformal transformation is essentially a symmetry, just like the symmetry of mirror inversion or spatial translation. Symmetry is a favorite of physicists, and for every symmetry added, physicists can write one more equation that constrains the system. The number of unknowns does not increase, but the number of equations increases, and the hope of finding an answer certainly increases.

Conformal transformations and conformal symmetries are so important that CFT (conformal field theory) has become a widely used fundamental subject. It is an indispensable tool not only in quantum field theory and gravity theory, but also in condensed matter physics and thermodynamics. Especially after the discovery of the Ads / CFT duality at the end of the 20th century, the importance of CFT was further enhanced.

Although conformal fields are not limited to two dimensions, two-dimensional conformal fields are undoubtedly the friendliest object for researchers eager to solve equations. Because there are infinitely many conformal transformations only on two-dimensional surfaces, and only a finite number of conformal transformations in higher-dimensional spaces, two-dimensional conformal fields are especially powerful. In some cases, the symmetries alone are sufficient to solve the problem accurately, regardless of other factors.

As early as the 1970s, Russian physicist Alexander Polyakov was attracted by the powerful power of two-dimensional conformal fields and proposed a new method for solving quantum fields-conformal bootstrap. The basic idea of this method is to disassemble the solution process into stair climbing step by step. First select a three-point structure as the basis, then add a fourth point, and then add a fifth point... This seemingly cumbersome process of solving, but in fact solved a problem that has long plagued professionals.

The basic idea of solving quantum fields traditionally, either directly or indirectly, inherits from the ancient analytical mechanics and classical field theory, that is, starting from Lagrangian or Hamiltonian. The techniques used, such as canonical quantization and Feynman path integration, are also based on Lagrangian and Harderian quantities. This method is very practical and durable. Many key links have been repeatedly polished and prepared by the ancient sages. For the latecomers, we are almost left with brainless calculations.

However, this formula has a flaw in quantum field theory, which is that the interaction between fields cannot be too strong, and it is best to have free fields with no interaction at all. This is like a set of methods for solving the state of motion of an object, which can only solve uniform linear motion. When dealing with uniform circular motion, the acceleration perpendicular to the direction of motion is added as a higher order correction term. If you encounter variable speed circular motion, you have to add more correction terms.

This patch-on-patch approach is known in technical terms as "perturbation." This means that all field interactions and other constraints are treated as "small perturbations" of the free field, and the resulting effects are reflected only in those corrections. Obviously, when we encounter very strong interactions, the perturbation method fails to provide conclusions that are consistent with reality.

The conformal bootstrap method mentioned above is a non-perturbative routine that can solve many strongly coupled quantum fields. In the early 1980s, Polyakov and his collaborators Belavin and Zamolodchikov published an important paper, which gave a framework for solving a series of two-dimensional conformal fields and showed researchers the power of this method. Since then, the BPZ equation, named after the three authors, has become a milestone in the development of CFT.

The BPZ equation from N − 1 points to N points looks like this:

It doesn't matter if you don't understand it. This article doesn't really intend to explain the meaning of this equation. The equation is listed purely to satisfy some readers 'curiosity. And show off the author's ability to use search engines.

Along the ladder built by BPZ equations, many treasures that cannot be mined by traditional perturbation methods can now be mined by conformal bootstrapping. Among these treasures, there is a special two-dimensional conformal field that is very closely related to quantum gravity theory: it is the Liouville field.

As a two-dimensional conformal field, the Liouville field is of course an absolute quantum field. At the same time, the classical limit of the Liouville field naturally gives a two-dimensional version of Einstein's equations. So the Liouville field itself is a beautiful two-dimensional theory of quantum gravity. Moreover, Liouville fields can describe the excitation of Bose strings in two-dimensional planes, and thus can be regarded as part of the quantum gravity model constructed by string theory. Furthermore, through the Ads / CFT duality, the Liouville field is also a description of gravity in a three-dimensional curved spacetime.

The above passage may confuse non-theoretical physics readers, but despite all technical terminology, Liouville fields are present in many studies related to quantum gravity theory. So we know by feeling that this Liouville field must be very closely related to quantum gravity. To learn more about quantum gravity, Liouville fields must be a valuable entry point.

Since conformal bootstrapping is a powerful tool, the solution of Liouville field seems to be within reach, but there is still a difficult problem that hinders the progress of research, that is, the three-point structure that starts with BPZ ladder must be accurately described and a series of constraints must be satisfied. If only path integral and perturbation methods are used to calculate, the theme of "non-perturbation" is lost from the source. However, the search for this structural constant took some effort. It wasn't until the 1990s that two groups of researchers came up with identical formulas for determining this structure constant. This formula was named DOZZ formula, representing two groups of researchers Dorn, Otto and Zamolodchikov.

There is no clerical error here, and the last two are indeed surnamed Zamolodchikov, one of whom is the "Z" in BPZ, the full name is Alexander Zamolodchikov, and the other is his twin brother Alexei Zamolodchikov. By the way, although the last names of the three BPZ are different, their names are all Alexander, which is also an interesting coincidence.

Returning to the DOZZ formula, with this formula as a starting point, the researcher can finally solve the correlation function of the Liouville field. However, the origin of this DOZZ formula was still unsatisfactory. This was because this complicated formula had not been deduced, but had been guessed. It could be said that it inherited the fine tradition of top physicists. In a paper published in 1996, the authors, the Zamolodchikov brothers, admitted bluntly:

"It is important to emphasize that the arguments in this section are irrelevant to derivation. These are more likely to be dynamics, and we'll take the proposed expressions as guesses, which we'll try to confirm in later chapters. This conjecture seems natural, and may even seem obvious to those concerned. "

（ "It should be stressed that the arguments of this section have nothing to do with a derivation. These are rather some motivations and we consider the expression proposed as a guess which we try to support in the subsequent sections. This guess appears quite natural and might even be thought obvious to those concerned with the problem. " ）

If we cannot deduce this formula logically, we do not understand its meaning. Even if it can help us a lot in the concrete calculation of Liouville fields, it will not provide a role in revealing the nature of the physical world.

So some researchers began to work hard again, trying to figure out from which angle the DOZZ formula could be derived. The task was harder than many had expected, and there had been no significant progress in the more than a decade since the DOZZ formula had been proposed. It wasn't until a few years after 2014 that several papers appeared one after another.

These DOZZ formula proof processes obtained in recent years are quite cross-border, using probability theory language and tools, mainly including GMC (Gaussian Multiplicative Chaos) and GFF (Gaussian Free Field).

From these proofs, we have also gained a lot of new knowledge. It was thought that the gravitational field caused by random fluctuations was strongly coupled and could not be reconciled with the path integral. However, with the help of some tools from probability theory, those fluctuations could be polished smooth enough and compatible with the path integral smoothly.

The goal of physicists, of course, was not merely to tame the brute fluctuations of gravity in the two-dimensional plane, but to bring their experience and tools to bear on gravity in real spacetime.

Attached:

DOZZ formula looks like:

wherein

A more intuitive way of saying that

Such a complicated formula was actually guessed by feeling! What a powerful intuition!

references

[1] Kupiainen, Antti; Rhodes, Rémi; Vargas, Vincent (2017). "Integrability of Liouville theory: Proof of the DOZZ Formula". arXiv:1707.08785 [math.PR].

[2] Vargas, Vincent (2017). "Lecture notes on Liouville theory and the DOZZ formula". arXiv:1712.00829 [math.PR]

[3] A.B.Zamolodchikov; Al.B.Zamolodchikov (1996). "Structure Constants and Conformal Bootstrap in Liouville Field Theory". DOI: 10.1016/0550-3213(96)00351-3. arXiv:hep-th/9506136

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