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Chiral Anomaly clarified the following as: [do] we have a mathematical model that would probably account for everything we've observed about matter if we had the ability to do all of the calculations?

About: I am not a physicist. Here is an explanation of what I am asking. There are many models of matter behavior that simplify the simulation calculation by selecting only certain aspects of the scene and/or creating particles that represent many actual physical particles. I suppose quantum mechanics goes to a fine level of description. However it also seems QM is not concerned with every aspect of matter, being supplemented by QCD and other things. So my question is, with every avalaible theories, is it possible to do a complete simulation of matter? Take an atom, for example. How complete would be the model of an iron atom? Gravity not being merged with QM, I suspect this makes all simulations incomplete. But other than that? What are our modeling limits, computer power apart?

In a nutshell: can we model without making any approximation.

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  • $\begingroup$ Don't you think it would be better to ask about a hydrogen atom than an iron atom? $\endgroup$
    – JEB
    Commented Jan 6, 2021 at 3:22
  • $\begingroup$ We can’t even do hydrogen without approximation, as far as I know. For example, the Dirac equation doesn’t explain the Lamb shift. And it doesn’t take the enormously complicated structure of the proton into account. But so what? We can calculate to the accuracy we can measure. $\endgroup$
    – G. Smith
    Commented Jan 6, 2021 at 3:28
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    $\begingroup$ @Exocytosis To clarify: Are you asking if we have a mathematical model that would probably account for everything we've observed about matter if we had the ability to do all of the calculations? Or are you asking if we have the ability to actually do all of those calculations? I'm asking for this clarification because theorists often use the word "model" for a mathematical model that we may or may not have the ability to solve or to accurately simulate. Is the question really focused on our ability to do simulations? $\endgroup$
    – Chiral Anomaly
    Commented Jan 6, 2021 at 3:34
  • $\begingroup$ @Chiral Anomaly: the former question. I am appending it to the question text, thanks. $\endgroup$
    – Winston
    Commented Jan 6, 2021 at 9:51
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    $\begingroup$ @Exocytosis By "subtler things," I really just meant: (1) gravity, (2) astrophysical clues (dark matter and dark energy), (3) the fact that the standard model has lots of adjustable knobs, which suggests that it's not the end of the story, and (4) the fact that the standard model requires unnaturally-fine tuning, which again suggests that it's not the end of the story. Points (1) and (2) are from real-world observation, and points (3) and (4) are aesthetic considerations. $\endgroup$
    – Chiral Anomaly
    Commented Jan 7, 2021 at 14:46

1 Answer 1

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No

The mathematics of quantum mechanics is not easy, and there are very, very few problems that can be solved in closed form without approximations (the infinite square well is one of them, but that's not a physical system).

What happens in practice is that you start with some system, say the Hydrogen atom, which every physics student encounters in their quantum mechanics course. This is the simplest atom out there so it's an important test case for quantum mechanics. It's not difficult to write down the Hamiltonian (this is a technical term for the operator that corresponds to the total energy of the system). If you're familiar with classical mechanics this is quite simple, it's just the momentum operator $p^2/2m$ plus the Coulomb potential $e/4\pi\epsilon_0 r$. From there we can solve this for the wavefunctions of the electron, which also gives the energy orbitals, etc that correspond to observations very well. But the correspondence is not perfect. As observations get more and more exquisite, we need to incorporate more and more effects:

As you might expect, the math gets more and more involved as more and more interactions are taken into account. I'm pretty sure the standard undergraduate physics curriculum stops before reaching the anomalous magnetic dipole moment, for example, or the Lamb Shift. And all this is only for the hydrogen atom, the simplest of all atoms. The more complicated atoms probably can't even be attacked analytically.

If you're looking for an approximation-free, includes-every-effect description of matter, we don't have that. But do we need such a description? It only matters if you need exquisite precision. If you're trying to model things such as throwing a baseball at a batter, you don't need general relativity or quantum mechanics, because classical mechanics is more than capable of producing the correct result to the level of accuracy that you desire. That's the case with quantum physics, as well.

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  • $\begingroup$ Thanks, valuable answer. My question is not if we need it or not. It is whether we have an exhaustive description of all physical laws governing matter or if we are still approximating, not because of insufficient computing power but because we are unable to give a complete description of all rules involved. $\endgroup$
    – Winston
    Commented Jan 6, 2021 at 9:58
  • $\begingroup$ @Exocytosis that sounds rather vague to me - we do not have an exhaustive description of all physical laws because we don't have a theory of quantum gravity. Is that the answer you are looking for? $\endgroup$
    – Allure
    Commented Jan 6, 2021 at 10:39
  • $\begingroup$ no, I already mentioned gravity. I am asking about the rest. Do we know everything about all observed phenomena involving matter, except how gravity fits in with the rest? $\endgroup$
    – Winston
    Commented Jan 6, 2021 at 11:42
  • $\begingroup$ @Exocytosis I still find the question quite vague I'm afraid. For example, in studying the hydrogen atom, there will be a correction to the Hamiltonian due to the gravitational field of the nucleus, it's just really small and beyond the experimental limit. We neglect gravity not because we don't have a theory of quantum gravity, but because it's unnecessary. Is the answer yes or no then? $\endgroup$
    – Allure
    Commented Jan 6, 2021 at 12:02
  • $\begingroup$ No, I would have hoped someone tells me there are phenomena that are not entirely described (apart from gravity), or the contrary. Like do we have a complete description of internal organisation and evolution inside a neutron? Is it only a matter of computation or are there fundamental pieces missing? $\endgroup$
    – Winston
    Commented Jan 6, 2021 at 12:47

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