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When people say that the three-body problem is "unsolvable" they really mean that there's not a way to write out an analytic solution to the problem. In the sam

When people say that the three-body problem is "unsolvable" they really mean that there's not a way to write out an analytic solution to the problem. In the same way, you can call quintic polynomials "unsolvable", because there's no way to take arbitrary quintic polynomial equations and express the solutions as ordinary algebraic formulas. However, it's very misleading to call quintic equations unsolvable, because we know where the solutions are, and we can use various numeric methods to calculate the solution with arbitrary precision. Any time we can calculate the answer with as much precision as we want, I'd like to say that the problem is "solved" in a very real and meaningful way. The problem is worse with quantum mechanics. With quantum mechanics, not only do we lack analytic solutions to many of the equations used in QM, but we also lack good numeric solutions (using real hardware, at least).

ethbr0 5y ago
Following the analogy, as a layperson, what's the nature of the difference with quantum mechanics that doesn't allow it to be solved to arbitrary precision (even if through brute force iteration)? Is it that we haven't discovered the solution generating algorithms? That the state/probabilities of quantum mechanics are fundamentally untenable to similar calculation? Or something else entirely?
- klodolph 5y ago
  It's nothing fundamental, as far as I am aware. Depends on what you mean by "fundamental". There is no known way to simulate a quantum computer, using a classical computer, in polynomial time. A quantum computer is just a kind of quantum system, so we know that some quantum systems cannot be efficiently modeled (barring revolutionary advances in simulation algorithms). When your simulations take superpolynomial time, it tends to be easy to find problems which you simply do not have the computational resources to solve, and you may not be able to solve interesting versions of the problem. There are lots of examples of problems like this. However, I don't consider this to be a fundamental difference. For example, satellite navigation systems are just fine calculating directions for driving all the way across the continental US, even though that's a very "large" instance of the problem that they are solving. But if you try to find the fastest route for a delivery driver to make a hundred deliveries within one city, good luck. This is just an analogy, and I'd like to emphasize that "no KNOWN algorithm" efficiently solves these problems, and that we haven't proven whether such an algorithm exists.
  - ethbr0 5y ago
    Thanks! By "fundamental," I was curious about the nature of the problem's difficulty, moreso than the progress on solutions. And computational intractability due to system properties makes sense!
- zardo 5y ago
  Also a layperson, but as I understand it. We do have the algorithms to simulate quantum systems with n states to arbitrary precision through brute force. But they require that you can put ~n bits into a superposition state. If your computer doesn't have that capability, you can simulate it to arbitrary precision with ~2^n bits.
- evanb 5y ago
  Many problems in physics are perturbative. Then, approximate methods suffice to get reliable answers. In Asimov’s “relativity of wrong” it’s the perturbative nature of gravity—-gravity is very weak—-that makes “the Earth is a sphere” less wrong than “the Earth is flat”. So, if you need an answer to only such-and-such precision, the approximation scheme lets me know how hard I must work to achieve reliability at that scale. Once I get the precision I need I can stop. Electrodynamics is like that. There is a number, the fine structure constant, about 1/137, that gives the natural scale for how big the next step in the approximation is, compared to the size of the current step. So, if I need to know the answer to 1 part in 10^9, I’m going to need to do 4 or 5 steps of fixing up the approximation (each fix, of course, being a great deal more arduous). QCD, and other “strongly coupled” or “non-perturbative” problems are not like that. If you make the dumbest approximation (flat-earth) and then fix it up a bit (sphere earth), answers don’t change just a little. They change completely. In QCD the number that characterizes the “obvious” approximation (the Feynman diagram approach)—-the number that’s 1/137 for electrodynamics—-is about 1.5. That’s a disaster! The approximation scheme is obviously no good—-you learn that you can never stop improving your approximation, because if you only worked “a little harder” your answer could change completely. Other approaches are required.

ethbr0 5y ago

Following the analogy, as a layperson, what's the nature of the difference with quantum mechanics that doesn't allow it to be solved to arbitrary precision (even if through brute force iteration)? Is it that we haven't discovered the solution generating algorithms? That the state/probabilities of quantum mechanics are fundamentally untenable to similar calculation? Or something else entirely?

klodolph 5y ago
It's nothing fundamental, as far as I am aware. Depends on what you mean by "fundamental". There is no known way to simulate a quantum computer, using a classical computer, in polynomial time. A quantum computer is just a kind of quantum system, so we know that some quantum systems cannot be efficiently modeled (barring revolutionary advances in simulation algorithms). When your simulations take superpolynomial time, it tends to be easy to find problems which you simply do not have the computational resources to solve, and you may not be able to solve interesting versions of the problem. There are lots of examples of problems like this. However, I don't consider this to be a fundamental difference. For example, satellite navigation systems are just fine calculating directions for driving all the way across the continental US, even though that's a very "large" instance of the problem that they are solving. But if you try to find the fastest route for a delivery driver to make a hundred deliveries within one city, good luck. This is just an analogy, and I'd like to emphasize that "no KNOWN algorithm" efficiently solves these problems, and that we haven't proven whether such an algorithm exists.
- ethbr0 5y ago
  Thanks! By "fundamental," I was curious about the nature of the problem's difficulty, moreso than the progress on solutions. And computational intractability due to system properties makes sense!
zardo 5y ago
Also a layperson, but as I understand it. We do have the algorithms to simulate quantum systems with n states to arbitrary precision through brute force. But they require that you can put ~n bits into a superposition state. If your computer doesn't have that capability, you can simulate it to arbitrary precision with ~2^n bits.
evanb 5y ago
Many problems in physics are perturbative. Then, approximate methods suffice to get reliable answers. In Asimov’s “relativity of wrong” it’s the perturbative nature of gravity—-gravity is very weak—-that makes “the Earth is a sphere” less wrong than “the Earth is flat”. So, if you need an answer to only such-and-such precision, the approximation scheme lets me know how hard I must work to achieve reliability at that scale. Once I get the precision I need I can stop. Electrodynamics is like that. There is a number, the fine structure constant, about 1/137, that gives the natural scale for how big the next step in the approximation is, compared to the size of the current step. So, if I need to know the answer to 1 part in 10^9, I’m going to need to do 4 or 5 steps of fixing up the approximation (each fix, of course, being a great deal more arduous). QCD, and other “strongly coupled” or “non-perturbative” problems are not like that. If you make the dumbest approximation (flat-earth) and then fix it up a bit (sphere earth), answers don’t change just a little. They change completely. In QCD the number that characterizes the “obvious” approximation (the Feynman diagram approach)—-the number that’s 1/137 for electrodynamics—-is about 1.5. That’s a disaster! The approximation scheme is obviously no good—-you learn that you can never stop improving your approximation, because if you only worked “a little harder” your answer could change completely. Other approaches are required.