The core mystery of physics-only mathematics can solve it


“It’s totally boring, because you describe a lonely field without any interaction, so it’s kind of academic practice,” Rejzner said.

But you can make it more interesting. Physicists dial the interaction, trying to maintain mathematical control over the picture, because they make the interaction stronger.

This method is called perturbative QFT because you allow small changes or perturbations in the free field. You can apply the perturbation perspective to quantum field theory similar to the theory of freedom. It is also very useful for verification experiments. “You get amazing accuracy, amazing experimental consistency,” Rejzner said.

However, if you continue to increase the interaction, the perturbation method will eventually overheat. It did not produce more and more accurate calculations close to the real physical universe, but became more and more inaccurate. This shows that although the perturbation method is a useful guide for experiments, in the end it is not the correct way to try and describe the universe: it is actually useful but theoretically unstable.

Gaiotto said: “We don’t know how to add up everything to get something reasonable.”

Another approximation is to try to sneak up on mature quantum field theory in other ways. In theory, quantum fields contain infinitely fine-grained information. To construct these fields, physicists start with a grid or grid and limit the measurement to where the grid lines cross each other. Therefore, you cannot measure the quantum field anywhere, but only at selected locations at a fixed distance.

From there, physicists increased the resolution of the lattice, pulling the wires closer to form finer and finer weaves. As it becomes tighter, the number of points you can measure will increase, approaching an idealized concept, and you can measure anywhere.

“The distance between the points becomes very small, and something like this becomes a continuous field,” Seiberg said. In mathematical terms, they say that the continuous quantum field is the limit of the shrinking lattice.

Mathematicians are used to dealing with limits and know how to determine that certain limits do exist. For example, they have proved that the limit of the infinite sequence 1/2 + 1/4 + 1/8 +1/16… is 1. The physicist wants to prove that the quantum field is the limit of this lattice process. They just don’t know how.

“It is unclear how to reach this limit and what it means in mathematics,” Moore said.

Physicists do not doubt that the compact lattice is moving towards the idealized concept of quantum fields. The close fit between QFT’s predictions and experimental results strongly suggests that this is indeed the case.

“There is no doubt that all these limitations do exist, because the success of quantum field theory is truly amazing,” Seiberg said. But there is strong evidence that something is correct, and ultimately proves that it is two different things.

This is a degree of imprecision, inconsistent with other great physical theories that QFT wants to replace. Isaac Newton’s laws of motion, quantum mechanics, Albert Einstein’s special and general relativity-they are all just part of the larger story that QFT wants to tell, but unlike QFT, they can all be used with precision Write down the mathematical terms.

“Quantum field theory appears as an almost universal language of physical phenomena, but its mathematical form is terrible,” Dijkgraaf said. For some physicists, this is the reason for the suspension.

“If the whole house relies on this core concept that itself cannot be understood mathematically, why are you so confident that this is describing the world? This exacerbates the whole problem,” Dijkgraaf said.

External stirrer

Even in this incomplete state, QFT has contributed to many important mathematical discoveries. The general pattern of interaction is that physicists using QFT stumble upon surprising calculations, and then mathematicians try to explain these calculations.

“This is a machine that generates ideas,” Tong said.


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