The nature of theoretical models

Question

Mathematics is exact. It is a beautiful language that allows us to express quantities that aren't possible to be represented physically. We build theoretical models of physical systems that work out great on paper, yet there are cases where we cannot solve these models exactly. We need these tools, such as perturbation theory, to get almost exact answers.

For example, in Quantum Mechanics we use Perturbation Theory to solve the Schrodinger equation for the hydrogen atom.

Shouldn't our theoretical models be refined if they cannot be solved exactly? Let us imagine we needed the exact answer to whatever our problem is; an approximation, no matter how good, just won't do. Shouldn't this be a fundamental rule of accepted theories; it needs to work out exactly?

Answer 1

No.

There is nothing wrong with perturbation theory, or with theories with known, restricted accuracy. The point of theory is to explain the results of observation from as simple an initial theoretical standpoint as possible. Therefore:

1. Since experiment always has a finite uncertainty, one can only ask that theory match the experimental value within its uncertainty range. Exact answers are not very useful in physics, because we know that other, smaller effects will always get in the way before an experiment matches the infinite set of significant figures we can draw out of an 'exact' answer. Ignoring these effects, or trying to minimize them, is very often the best way to hide your head in the sand while some amazing physics walks by.

2. The point of theory is to make the initial standpoint as simple and understandable as possible, and to try and get some intuition for how that standpoint implies experimental results. It is a fact that most simple models will in general not be solvable. More importantly, it is not a given that the universe is describable by solvable models. We're not out to impose structure on nature, we're out to find the structure that's there. If it turns out it's not an exactly solvable model, then that's the way it is.

Answer 2

There will always be solutions that can't be analytical. For example, any model of more than two bodies without any special constraints, cannot be solved analytically. From the gravitational interactions between three planets to three particles interacting (electromagnetically or otherwise) in quantum theory. To have mathematically analytical solutions, certain simple constraints need to be applied, regardless of the system you are modeling. This is why physical theories or analysis (calculus) in general are taught using simple models. Spherical, cubic, or cylindrical models, generally with two bodies, or otherwise special coordinate constraints are placed on larger numbers of bodies.

Some problems, whether higher degree polynominals, certain integrals, or many differential equations simply lack a closed mathematical expression to solve them. This is when numerics steps in (pun intended). Numerics is a huge branch of mathematics that finds solutions to problems through approximating them. How exact you want to approximate it is up to you (check out the Newtonian approximation to solve x-intercepts as an example). You can run your algorithm as long as you like, and you can also calculate your error and even prove that it converges to zero. If anything, this is more like the real world. If you repeat an experiment over and over, even trying to repeat it exactly, you will usually get slightly different values. You need to calculate your errors and see if these measurements fall within your values given by your theoretical model.

Now interestingly enough there exists a hybrid of the two systems. Finding an approximation of a problem that is an analytical function. This covers perturbation theory. One good example of this is the nuclear shell model. To reproduce the "magic numbers" a couple of analytical functions were combined that gave good results. Take a look at the deformed harmonic oscillator model (Woods-Saxon is interesting also).

Are these models flawed or wrong because they are approximated? No, they give the results needed for calculations, within a tolerance that can be calculated. They are no more wrong than Newtonian mechanics is for a slow body moving in a flat space-time.

To conclude, there will always be problems that cannot be solved analytically, a proof can even be offered to argue that for many of them, and there will be some surprises occasionally (a new analytical solution to a given problem when only some very special constraints are applied), but in the end we have to accept that the universe isn't a perfect sphere, in a vacuum, with only two bodies in it to calculate, and that a constraint free many-body problem will always have to be solved numerically.