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“It’s totally boring because you described a lonely area that has nothing to interact with, so it’s an academic exercise,” Rejzner said.
But you can make it more interesting. Physicists mark interactions by trying to maintain mathematical control of the image as interactions intensify.
This approach is called a disruptive QFT when you allow small changes or disturbances in a free area. You can apply a disruptive approach to quantum field theories that are similar to a free theory. It is also very useful for verifying experiments. “You get amazing accuracy, amazing experimental agreement,” Rejzner said.
But if you continue to reinforce the interactions, the disruptive approach eventually warms up. Instead of making calculations that approach the real physical universe, they are becoming more and more accurate. This suggests that although the perturbation method is a useful guide for experiments, it is ultimately not the right way to describe the universe: it is almost useful, but theoretically it is vibrating.
“We don’t know how to add everything and get something that makes sense,” Gaiotto said.
Another approximate scheme, in other ways, attempts to arrive at the whole theory of the quantum field. In theory, a quantum field contains fine-grained information. To cook these areas, physicists start with a grid or grid, and the measurements are limited to places where the lines of the grid intersect. So instead of being able to measure the quantum field everywhere, you can only measure it in some places that are initially at a fixed distance.
From there, physicists improve the resolution of the network by bringing the threads closer together and creating a finer texture. As you tighten, the number of points you can make increases, bringing you closer to the idealized notion of an area where you can take measurements everywhere.
“The distance between the points becomes very small and something like that becomes a constant area,” Seiberg said. In terms of mathematics, they say that the continuous quantum field is the limit of the tightening network.
Mathematicians are accustomed to working with boundaries and know that some actually exist. For example, they have proved that 1/2 + 1/4 + 1/8 +1/16 … infinite sequence limit 1. Physicists would like to prove that quantum fields are the limit of this network procedure. They don’t know how.
“It’s not so clear how to take that limit and what it means mathematically,” Moore said.
Physicists do not doubt that constriction networks go towards an idealized notion of the quantum field. The close link between QFT predictions and experimental results suggests that this is the case.
“There’s no doubt that all of these limitations actually exist because the successful theory of quantum fields has been truly amazing,” Seiberg said. But having strong evidence that something is right and proving it decisively are two different things.
It is a level of accuracy that is not comparable to other excellent physical theories that QFT seeks to replace. Isaac Newton’s laws of motion, quantum mechanics, Albert Einstein’s theories of special and general relativity – these are just parts of the larger story that QFT wants to tell, but unlike QFT, they can all be written in specific mathematics.
“Quantum field theory was almost universal in the language of physical phenomena, but mathematics looks bad,” Dijkgraaf said. For some physicists, this is a reason to pause.
“If the whole house is based on this basic concept that is not mathematically understood, why are you so sure that it is describing the world? That sharpens the whole problem,” Dijkgraaf said.
External Stirrer
Even in this incomplete situation, QFT has led to some important mathematical discoveries. The general model of interaction has been for physicists using QFT to encounter surprising calculations that mathematicians try to explain.
“It’s a machine for generating ideas,” Tong said.
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