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Typically, the ground truth is taken to be the continuous model. Numerical simulations are taken to be the approximation. These simulations deviate from the continuous model due to both a constant rounding error and truncation error that accumulates with time. You can see an example here:

https://math.stackexchange.com/q/3235194/

In the answer there you can see how these types of errors combine to lead to a "sweet spot" of highest accuracy. Ie, the approximation becomes less accurate at very small or large scales. This reminded me of General Relativity, which yields very good predictions at solar system scales but has issues at galaxy (~10^12 larger) and atomic (~10^24 smaller) scales.

I'm wondering if anyone has explored the reverse happening. Ie, we take a discrete model to be the ground truth.

I'd guess that with current technology the continuous model would still be a better approximation than discrete, since people aren't running simulations with 10^30 steps per second. But perhaps deviations from the continuous model observed in nature can be taken to support discrete over continuous spacetime.

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  • $\begingroup$ I would point out that we already know from modelling solids that periodic lattices of discrete elements interacting, how some of the properties would change from a continuum version of the same. We simply haven't been able to observe any such deviation. $\endgroup$
    – naturallyInconsistent
    Commented Jul 24, 2023 at 1:48
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    $\begingroup$ Does this answer your question? Is spacetime discrete or continuous? $\endgroup$
    – hft
    Commented Jul 24, 2023 at 2:18
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    $\begingroup$ @hft Nope, I am wondering specifically about inverting the concepts of rounding and truncation error. In particular, if that would imply modelling a discrete spacetime as continuous would lead to deviations at very large and small scales. GR, for example, was developed to explain observations of the solar system. It was before we knew even about galaxies. My question is whether the errors due to assuming a continuum (if reality is discrete) could be negligible at solar system scale but accumulate to cause deviations from observation at the galaxy-level. Likewise at very small scales. $\endgroup$
    – Livid
    Commented Jul 24, 2023 at 2:30
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    $\begingroup$ it is categorically impossible for these approximations to fail to capture the large scale behaviour properly. It is not only possible, it is the main difficulty that numerical modellers face. Look up leapfrog integration or see: en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods $\endgroup$
    – Livid
    Commented Jul 24, 2023 at 13:01
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    $\begingroup$ A numerical computation vs a continuous function, I wonder if that is the same as discrete vs continuous physical behavior. After all, water or air is fundamentally a discrete substance of molecules. But the continuous fluid model makes basically perfect predictions for macroscopic behavior, except in fringe cases like near-vacuum. For spacetime (if Planck scale quantization is true) there are far more than 10^23 discrete units per macroscopic unit. So the behavior may only manifest in fringe cases we can't observe yet. $\endgroup$
    – RC_23
    Commented Jul 24, 2023 at 20:44

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Typically, the ground truth is taken to be the continuous model.

I would dispute that point. In e.g. the Ising model (and many other lattice models), the fundamental degrees of freedom of the theory are discrete, and it is the continuum formulation which is the approximation - typically arrived at via some coarse graining procedure. In solid state physics, all physical samples have a finite volume and therefore the electronic states have a discrete energy spectrum, but in the interest of simplicity one takes the thermodynamic limit in which the volume goes to infinity and e.g. the density of states becomes continuous.

The kinds of "errors" that can arise from this procedure depend on how coarse the approximation actually is. One interesting example is when one considers magnetic skyrmions, which are topological solitions in the magnetization field of a magnetic material. The magnetization $\mathbf m(\mathbf x)$ is treated as a continuous function, from which one can calculate the energy of a magnetic configuration. Since there is a term in the energy functional which depends on the spatial gradient of $\mathbf m$, a discontinuity would require infinite energy - which suggests that topological solitons like magnetic skyrmions should be completely stable.

Of course, this analysis discounts the fact that the magnetization is not continuous when you zoom in close enough, and indeed one of the annihilation modes for magnetic skyrmions is that they may shrink to the point where the continuum approximation fails and then simply pop out of existence with the flipping of a handful of spins.

On a more philosophically interesting note, if you study the Ising model on a finite domain, then you will find that there is no such thing as ferromagnetism or permanent magnetization. It is only in the limit as the number of sites goes to infinity - in which the average magnetization becomes a continuously-varying quantity rather than a discrete one - that a magnetic phase transition can occur. This suggests that the existence of permanent magnetism is itself an "error" which we obtain by taking a limit in which a certain quantity becomes continuous.

Interestingly enough, we want to make this kind of error. In reality, there are no such thing as permanent magnets - but if the chunk of iron in my hand will retain its stable magnetization for a hundred trillion lifetimes of the universe, then I would far prefer to make the "error" that the magnetism is permanent rather than to make the technically correct statement that in the limit of infinite time, all magnetization is transient.

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  • $\begingroup$ This seems related but doesn't quite answer my question. If the phenomenon under study is discrete, and you can scale your discrete model to match, then it will perform better than continuous. We cannot currently do this for spacetime because if it really is discrete, it is at a level many orders of magnitude smaller than we are capable of simulating. One question that may help though, what happens when you apply the continuous Ising model to something much larger? Like a neutron star? $\endgroup$
    – Livid
    Commented Jul 24, 2023 at 2:45
  • $\begingroup$ @Livid To be clear, what I mean to say is that we very often start with discrete models (or more generally, models in which certain features take discrete values) and then, in the interest of simplification, approximate those features as continuous ones. In other words, sometimes it is discrete -> continuous which is the approximation, and the errors in this approximation typically take the form of infinities (which arise when when considering gradients in our only-approximately continuous function) or non-analytic behavior which is provably absent with a finite number of degrees of freedom. $\endgroup$
    – J. Murray
    Commented Jul 24, 2023 at 2:55
  • $\begingroup$ I'm not quite sure what you mean when you talk about applying the Ising model to a neutron star. $\endgroup$
    – J. Murray
    Commented Jul 24, 2023 at 2:58
  • $\begingroup$ I used a neutron star as an object much larger than originally considered with a strong magnetic field. If that is inappropriate imagine an iron bar the size of a planet, or whatever makes sense. Wikipedia gives an equation after saying: The energy of a configuration σ is given by the Hamiltonian function. Does the continuous approximation become less accurate at predicting the energy this scale? $\endgroup$
    – Livid
    Commented Jul 24, 2023 at 3:08
  • $\begingroup$ @Livid If you have a continuous function which, for computational purposes, you approximate as being discrete, then the discretization errors occur because you are sampling a continuous function at a finite set of points and can accumulate over large scales if e.g. you are integrating. I get the sense that you are searching for an analogous phenomenon with the discrete -> continuous approximation, but in many cases it's not really like that. When you coarse grain a discrete theory to obtain a continuous one, you're fundamentally changing the theory itself, not merely performing a calculation. $\endgroup$
    – J. Murray
    Commented Jul 24, 2023 at 3:27

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