“It’s totally boring, because you’ve described an isolated area with nothing to interact with, so it’s a bit of an academic exercise,” Rejzner said.
But you can make it more interesting. Physicists compose the interactions, trying to maintain mathematical control of the image as they reinforce the interactions.
This approach is called perturbative QFT, in the sense that you allow small changes, or disturbances, in a free field. You can apply the perturbative perspective to quantum field theories which are similar to a free theory. It is also extremely useful for checking experiences. “You get incredible precision, incredible experimental tuning,” Rejzner said.
But if you keep strengthening interactions, the disruptive approach ends up overheating. Instead of producing more and more precise calculations that come closer to the real physical universe, it becomes less and less precise. This suggests that while the perturbation method is a useful guide for experiments, it’s ultimately not the right way to try to describe the universe: it’s practically useful, but theoretically fragile.
“We don’t know how to add it all up and get something sane,” Gaiotto said.
Another approximation scheme tries to squeeze into a full-fledged quantum field theory by other means. In theory, a quantum field contains infinitely fine information. To prepare for these fields, physicists start with a grid, or lattice, and limit measurements to where the lines of the lattice intersect. So instead of being able to measure the quantum field anywhere, at first you can only measure it at selected places at a fixed distance.
From there, physicists improve the resolution of the network, bringing the wires together to create an increasingly fine weave. As it tightens, the number of points at which you can take action increases, approaching the idealized notion of a field where you can take action 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 constriction network.
Mathematicians are used to working with limits and know how to establish that some really exist. For example, they proved that the limit of the infinite sequence 1/2 + 1/4 + 1/8 +1/16… is 1. Physicists would like to prove that quantum fields are the limit of this network procedure. They just 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 tightening of the network is moving towards the idealized notion of a quantum field. The close fit between QFT’s predictions and experimental results strongly suggests that this is the case.
“There is no doubt that all of these limitations 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 proving that it is are two different things.
It is a degree of imprecision out of step with other major physical theories that QFT aspires to supplant. Isaac Newton’s laws of motion, quantum mechanics, Albert Einstein’s theories of special and general relativity – these are just parts of the bigger story QFT wants to tell, but unlike QFT, they can all be written in exact mathematical terms.
“Quantum field theory has emerged as an almost universal language of physical phenomena, but it is in a mathematical bad state,” said Dijkgraaf. And for some physicists, that’s a reason to take a break.
“If the packed house is built on this central concept which itself is not understood mathematically, why are you so confident that it describes the world? This sharpens the whole problem, ”said Dijkgraaf.
Even in this incomplete state, QFT sparked a number of important mathematical discoveries. The general pattern of interaction is that physicists using QFT come across some surprising calculations that mathematicians then attempt to explain.
“It’s an idea-generating machine,” Tong said.