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At the end of this article: https://hal.archives-ouvertes.fr/hal-00542373/document

it is said that: "For example, several works simulate quantum field theoretical equations in the continuous limit of a QCA dynamics."

But a user on reddit told me that:

"the continuous limit is not the same as being continuous. The "continuous limit" is a trick of physics that goes back to Boltzmann's work on thermodynamics in the 19th century, and the people who work with QCA in the continuous limit use the same trick he came up with. Discrete dynamic models often present simpler math for some systems, than do the more accurate continuum dynamics models. So you find a way to get an answer to a question or prediction for an event using discrete dynamics, but have the quantized units' quanta appear in the solution. At this point you might think, "ok, just make the quanta 0 to represent a continuum, and problem solved." However, typically, if those values are zero, you get some kind of undefined answer like 0/0, 1infinity, or something else like that. So instead you do something familiar to anyone who's taken calculus - you take the limit of the solution as the quantized units approach zero. Boom, done. This is not actually a continuum simulation. It let's you predict what a continuum simulation would yield, if you could build one. It gives you the mathematical power to calculate the state of a system at any moment in continuous time. But you are not actually calculating the state of the system at every moment in time. Depending on the system, you might get an algebraic solution for the state of the system with respect to continuous time, in which case you do actually have the state of the system at every moment in time, but this is not guaranteed to occur - it may be a computational solution instead. And it's still not the same as having literally simulated the continuum."

I sent an email to the author and I have contacted with other physicists/mathematicians, and they told me that what this user said were sloppy words and nothing tangible/meaningful.

Then, was this user wrong? Is continuous limit the same as continuum? Can the cellular automata mentioned in the article really simulate (quantum) continuous processes/behaviour (with perfect accuracy, not only just to an arbitrary accuracy)?

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  • $\begingroup$ Normally one makes time and space discrete for purposes of numerically solving PDEs/ODEs. Something to do with computers being discrete counting machines. $\endgroup$ – Kyle Kanos Jun 28 '18 at 11:03
  • $\begingroup$ @KyleKanos Thanks! and could you tell me whether in that particular article time is discrete or not? $\endgroup$ – user199226 Jun 28 '18 at 12:18
  • $\begingroup$ No idea, haven't read it. I suppose the point I was trying to make is that in computers, we have to use discrete time/space due to intrinsic properties of computers, even when the problem we're solving is continuous. There may very well be a map between the two, but I don't see that it's a problem. $\endgroup$ – Kyle Kanos Jun 28 '18 at 12:22
  • $\begingroup$ I haven't read it either, but it is discrete time. Just look at the beginning of Section 2.1 or at the definition of $\dot\psi$ after eqs (3)-(4)... $\endgroup$ – Yvan Velenik Jun 28 '18 at 12:23
  • $\begingroup$ And, of course, there are no problems in deriving continuous macro-evolution from discrete micro-evolution. (But I don't know what is done in this particular paper.) $\endgroup$ – Yvan Velenik Jun 28 '18 at 12:41

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