Different maths might help quantum theory and relativity talk to each other

Time can be a bit of a problem besides the problem of not having enough of it.

Relativity sees time as another dimension to add to the X, Y and Z space dimensions, forming what's sometimes called a block universe. The evolution of such a universe is like clockwork and everything follows from the initial conditions. If a suitably powerful demon knew all the properties of every particle in the universe, it could precisely predict the future. In such a universe the distinction between past, present and future is merely an illusion and all the information a universe needs is present at the very beginning.

Diagram of the block universe.
Block universe. I can't draw in four dimensions, so only three are shown.

Quantum mechanics sees things a bit differently. Quantum states are described by a wavefunction and, whilst the evolution of the wavefunction in time can be predicted, the outcomes of individual measurements cannot. Particles exist in superpositions — combinations of states — and there's no telling what you'll find until you make the actual measurement. Such a universe is not clockwork and the future cannot be completely predicted.

A physicist called Nicolas Gisin thinks the problem is mathematical and can be resolved by using a different type of mathematics.

The problem, says Gisin, is how we treat real numbers in normal mathematics. Real numbers are just the numbers we use all the time: 42, -6.3, 9.9999… etc. But the problem with them is that most have a number of decimals that, at any given time, we can only know to a certain precision. We have to zoom in further and make a more precise measurement to get the next decimal digit.

This process of zooming in on a number with more precision is called a choice sequence. Standard maths says none of this a problem. Even though we may not know the absolute precision of a number, we can treat the number as if it nevertheless exists and use it on that basis. Importantly, all numbers follow the law of the excluded middle, which says that either something is true or its negation is true. Either x equals 1 or x does not equal 1 and that seems logical at first blush.

There is however another type of maths called intuitional mathematics and the law of the excluded middle does not apply there. Maybe we can't say x equals 1 and can't say x is not equal to one either. If we have a number like 0.999999 and we were sure the 9s would continue, we could say that x = 1 (because x differs from 1 by less than any finite distance), but we're not sure how that sequence will continue. The next digit might be a 3, for example, and then x would certainly not equal 1.

Don't get bogged down with the maths, the important thing is that a lack of any law of the excluded middle gives us an imprecision that can only possibly be worked out in the future. In the present we only have 0.999999 and, right now, the future of that number is indeterministic, or at least that's the case with intuitional maths. In standard maths we'd say that number already exists in full and we can work with it as we please. It is already either 1 or not 1 even if we don't know which. It's a philosophical difference in some ways.

The point of all this is that by framing both quantum theory and relativity via a different type of mathematics, it may be possible to join them together, which is something that has been elusive for nigh on a century. In doing so we'd resolve the different ideas of time the theories have.