General Relativity and Quantum Mechanics are famously incompatible theories—no one yet knows how to combine them in a self-consistent way.
And yet, there is something that GR and QM have in common. They both give us elegant, simple equations that defy solution, even with the largest computers available.
The fundamental GR equation looks like this:
Hidden in those Greek subscripts are 64 interdependent partial differential equations in 64 unknowns.
The fundamental equation of QM is the Schrodinger Equation, even simpler:
But looks are deceiving! This is not an equation about how particles move but about how entire systems of particles evolve, as a single connected system. The equation has the property that it gives a beautifully precise solution for one electron in the isolated Hydrogen atom. But adding a second electron makes the equation a million times harder to solve, and each subsequent electron you add requires a million times as much computing power as the previous one.
The most complicated classical equation that has ever been solved exactly is a computer simulation of 10 billion galaxies.
The most complicated quantum equation that has ever been solved exactly is a computer simulation of 2 electrons (in this case, the Helium atom).
It’s as if God had said to physicists, “You want an equation that tells you how the world works—OK, try this one!”
What does it mean to have a theory that you believe describes the world very accurately, but you can never compare it to experiment because you can’t solve the damn equations? It means that quantum physicists are always coming up with new and ever more clever approximations, then comparing with experiment to adjust the solutions and make a better fit.
It also means that quantum physicists can’t (or don’t) make bold predictions, and are subject to experimental surprises. High-temperature superconductors. Cold fusion. Quantum biology? Extra-sensory perception?