The Mystery at the Heart of Physics – What Only Mathematics Can Solve

“It’s totally boring, because you describe a lonely field with nothing to interact with, so it’s a bit of an academic exercise,” Rejzner said.

But you can make it more interesting. Physicists call the interaction, trying to maintain mathematical control of the picture while making the interaction stronger.

This approach is called perturbative QFT, in the sense that it allows for small changes, or perturbations, in a free field. You can apply perturbative perspective to quantum field theories that are similar to a free theory. It is also extremely useful for verifying experiments. “You have an amazing accuracy, an amazing experimental arrangement,” Rejzner said.

But if you continue to make the interaction stronger, the disruptive approach ends up overheating. Instead of producing ever more accurate calculations that approximate the true physical universe, it becomes less and less accurate. This suggests that while the method of perturbation is a useful guide for experiments, it is ultimately not the right way to test and describe the universe: It is practically useful, but theoretically tremendous.

“We don’t know how to add it all up and make any sense,” Gaiotto said.

Another approximation scheme attempts to build on a complete quantum field theory by other means. In theory, a quantum field contains information at infinite grain. To cook these fields, physicists start with a grid, or grid, and limit the measurements to places where the grid lines intersect. So instead of being able to measure the quantum field at any location, in the beginning it can only be measured at selected locations at a fixed distance.

From there, physicists increase the resolution of the network, bringing the threads closer together to create an ever-thinner tissue. As you narrow down, the number of points where you can take measurements increases, approaching the ideal notion of a field where you can take measurements anywhere.

“The distance between the points becomes very small, and such a thing becomes a continuous field,” Seiberg said. In mathematical terms, they say that the continuous quantum field is the limit of the string network.

Mathematicians are accustomed to working with limits and fully establish that some really exist. For example, they have proved that the limit of the infinite sequence 1/2 + 1/4 + 1/8 +1/16 … is 1. Physicists would like to show that quantum fields are the limit of this lattice procedure. They don’t know how.

“It’s not so clear how to take that limit and what it means mathematically,” Moore said.

Physicists have no doubt that the reinforcement network moves toward the idealized notion of a quantum field. The tight fit between QFT predictions and experimental results strongly suggests that this is the case.

“There is no doubt that all of these limits really exist, because the success of quantum field theory has been truly amazing,” Seiberg said. But having strong evidence that something is correct and conclusively demonstrating that it is are two different things.

It is a degree of inaccuracy that is out of step with the other major physical theories that QFT aspires to replace. Isaac Newton’s laws of motion, quantum mechanics, Albert Einstein’s theories of special and general relativity – they are all just pieces of the major story that QFT wants to tell, but unlike QFT, they can all be written. in exact mathematical terms.

“Quantum field theory emerges as an almost universal language of physical phenomena, but it is in poor mathematical form,” Dijkgraaf said. And for some physicists, this is a reason to pause.

“If the whole house rests on this fundamental concept that he himself doesn’t understand mathematically, why are you so sure he’s describing the world? That adds up to the whole problem,” Dijkgraaf said.

Shaker out

Even in this incomplete state, QFT provoked a number of important mathematical discoveries. The general pattern of the interaction was that physicists using QFT stumbled upon surprising calculations that mathematicians try after explaining.

“It’s a machine that generates ideas,” Tong said.

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