# Why is it hopeless to view differential geometry as the limit of a discrete geometry?

This is a follow-up question to Introductions to discrete space-time:

Why is this line of thought hopeless?

Classical mechanics can be understood as the limit of relativistic mechanics $RM_c$ for $c \rightarrow \infty$.

Classical mechanics can be understood as the limit of quantum mechanics $QM_h$ for $h \rightarrow 0$.

As a limit of which discrete geometry $\Gamma_\lambda$ can classical mechanics be understood for $\lambda \rightarrow 0$?

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Why special relativity and quantum mechanics are discrete geometries? – kennytm Nov 23 '10 at 20:39
They are not, of course. They are just examples of giving rise to limits. – Hans Stricker Nov 23 '10 at 20:52
This question is clearly about maths and not physics (even though it has physical implications). I think it has a much better chance to be answered on math.stackexchange.com – Sklivvz Nov 23 '10 at 21:45
@Mark: Thank you for having made the title more compelling. – Hans Stricker Nov 23 '10 at 22:01

To begin with, what is $\lambda$? And why would this particular limit be the relevant one (and, say, not $\lambda\rightarrow\infty$)?

Secondly, this notion that $\hbar\rightarrow 0$ recovers Classical Mechanics (from Quantum Mechanics) is not quite correct.

So, i think this question is a bit too vague...

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@dmckee: i just saw the other question... it would have been nice to know this beforehand though, ie, it would have been nice to have had this information in the question, so we're not lost. Thanks for the 'heads up'. :-) – Daniel Nov 23 '10 at 20:47
@Hans: you can look for decoherence in quantum mechanics and also Bose-Einstein condensates, etc. – Daniel Nov 23 '10 at 20:49
@Hans: There is nothing wrong with pop-sci descriptions of physical laws. But (1) they are often developed after the mathematical version and (2) it is rarely possible to reason correct from them because they are cartoons. There are occasional exceptions to the second point such as Feynman's description of QED, which is why I'd like to be that smart. – dmckee Nov 23 '10 at 20:53
@Hans: when you say that $\hbar\rightarrow 0$ recovers classical mechanics, i objected, correct? So, i provided a link to explain the tip of the iceberg behind my objection. – Daniel Nov 23 '10 at 21:05
@dmckee @Hans: Discrete Differential Geometry is the name of the game — there's much that can be done. – Daniel Nov 23 '10 at 21:05

Not exactly sure what you are asking. For a cubic lattice, the limit as the lattice spacing goes to 0 recovers all of classical physics, and, if you want to discretize time, then as $\Delta{}t$ approaches 0, then finite difference equations become differentials.

For example, Low energy sound waves in a crystal don't see the discreteness, and it can be treated as a continuous medium.

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I think you might be interested in the following important distinction: relativistic mechanics can be built up upon it's own first principles and then derives classical mechanics as a limiting case; and although QM reduces to classical mechanics in a limiting case, QM actual requires classical mechanics for it's very formulation, and thus QM occupies a peculiar place among physical theories. See Landau, volume 3, chapter 1.

Now I will attempt to answer your question, even though in my opinion this question should be asked at math.stack exchange. In the broadest sense, a continuous manifold IS the limit of a discrete set of points, just like the set of real numbers is the limit set of the set of fractions. But after viewing your previous post that you linked to, I would like you to consider the following:

The cardinality of a discrete set of points is the cardinality of the integers, and the cardinality of the set of points on a continuous manifold is at least that of the continuum. There are certainly many consequences of this fact which distinguish the two cases.

Here is another thing you should bear in mind, the FT defines functions by a countable set of coefficients, but FT's only converge up to a set of measure zero.

In QM the FT plays a very important role, and I think the discrete nature of QM is not related to a discrete space time necessarily, but rather to the mathematics of continuous symmetries, i.e. Lie groups. This explains for example the quantum numbers associated with angular momentum.

A Lie group is a group which is also a continuous manifold, and thus we see that already, a discrete set of points can never be a Lie group, because a discrete set of points is by definition not continuous.

Certainly computer methods which make use of discrete approximation have produced good results, but the thing about computer models is you have to be so careful all the time to make sure you are not introducing an error by choosing a bad parametrization for your discrete "mesh".

Let's take a look at the case of fluid mechanics simulations. It is well known, and should be intuitively clear, that for the case of turbulent flows, the density of the mesh at regions of intense turbulence is going to have a strong influence on the results of your computer simulation. There are so many subtle things that are involved with mathematical convergence for these types of situations. It turns out, that there are lots of situations where you just can't simulate the turbulence because the chaotic nature of the turbulence is so sensitive to the initial conditions and the mesh density, that it's very hard to get your simulation to compile (converge) in any meaningful way. I think this example of simulating fluid mechanics really shows the limitations of using discrete methods to approximate solutions of certain problems.

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