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Supposed space and time were to be discrete, then how would i go about modelling this inside a computer simulation?

In a simple 2D world, taking a square for example with side length $A$, then if would want to reach the top right corner of this square, coming from the bottom left corner, I would have to walk twice the distance $A$, if I first walked to the right, then up. If I wanted to walk this distance diagonally, in the real world it would be $\sqrt{2A^2}$.

However, if I walked the same distance inside my computer model which is quantized/discrete, hence the square is subdivided into several smaller squares, the distance would be $2A$ still. More precisely, I would have to pass through as many squares when walking right, then up, as if I walked diagonally, within this simple discrete model. Diagonally, as in first moving one square to the right, then up and repeat until I reach the top right square.

Do physicists have a discrete model of the world which works, when pondering time and space being quantized, where the distance between two quantized space/time units is close to $\sqrt{a^2+b^2}$ on the macro-scale and if yes, what is its concept?

edit: I will give an example of what i am trying to achieve to hopefully clarify a bit more what i am asking for.

Let's say i program some digital world simulation inside my computer. The whole world is subdivided into discrete units of space.

I then draw a square inside this simulated digital world.

I will then place a simulated human on the bottom left corner of the square.

The human will walk to the top right corner of the square in two different ways.

scenario 1) human walks right, towards the bottom right corner, then up to the top right corner.

scenario 2) human walks "diagonally" to the top right corner.

In both cases, the human is allowed to move only by jumping from one discrete unit of space to the next unit of space of this subdivided square.

When counting the units of space the human traversed, he should count 2*A units of space in scenario a) while in scenario b) it should be close/approach $\sqrt{2A^2}$. The more finely grained our subdivision of this model into discrete units would be, the closer we would get to this value.

So i am asking for which mathematical model of discrete space(time) would allow for this, should it be possible at all.

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    $\begingroup$ 1. Time and space may be discretized, but not quantized. (Quantization is a technical term for turning classical things into quantum things) 2. Every simulation of the physical world essentially works as you describe it. 3. No currently uncontroversially accepted theory posits that time and space are fundamentally discretized. 4. I'm not quite sure what you are actually asking for - are you asking if there exist theories that think time and space are fundamentally discrete? $\endgroup$ – ACuriousMind Aug 2 '15 at 21:28
  • $\begingroup$ @ACuriousMind I want a model of reality which is composed of discrete space and time units which i could program into my computer. In this model, when i "walk" from the bottom left corner to the top right corner of a square which has been subdivided into many discrete units of space, the amount of units i passed when walking diagonally should be approximately sqrt(2*A^2) on the macro-scale, hence, if i walked a sufficient amount of units. $\endgroup$ – pZombie Aug 2 '15 at 21:34
  • $\begingroup$ pZombie, as ACuriousMind said, many simulations already do this. That is, if you want to model some system in physical space and time (say electromagnetic or fluid dynamics), you set up a grid and time vector. Whatever quantity you're interested in calculating is done on a discrete grid in discrete time, as you're proposing. It's interesting to think that space and time might actually be discrete, and to look for observable effects of this. I'm sure someone is already doing this (I'm not personally familiar with this in a research context). $\endgroup$ – anon01 Aug 2 '15 at 21:45
  • $\begingroup$ @anon0909 I don't think that in those simulations they calculate the distance between two objects by counting the discrete units of this grid one would have to traverse in order to get to from object A to object B. If they did that, i would think the results would be far from the real thing they are trying to simulate. The distance between two objects in those popular simulations is calculated via non-discrete math OR if you believe otherwise, then please show me how those simulations calculate the distance. $\endgroup$ – pZombie Aug 2 '15 at 22:03
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    $\begingroup$ Conservation of angular momentum poses a very serious problem for attempts to discretize space. Viewed as a Noetherian conserved current, there is a requirement that the underlying space be invariant on arbitrary rotations. Indeed, working around these issues has been a major challenge for lattice computations. $\endgroup$ – dmckee Aug 2 '15 at 23:16
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This is just a short answer, from Wikipedia, which maybe familiar to you already.

It may be useful as an analogy of what you are trying to do, if I have understood your question correctly.

Lattice Field Theory

In physics, lattice field theory is the study of lattice models of quantum field theory, that is, of field theory on a spacetime that has been discretized onto a lattice. Although most lattice field theories are not exactly solvable, they are of tremendous appeal because they can be studied by simulation on a computer. One hopes that, by performing simulations on larger and larger lattices, while making the lattice spacing smaller and smaller, one will be able to recover the behaviour of the continuum theory.

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Firstly, you could be discrete without being a lattice, for instance you could make a bunch of different shaped tetrahedra such that each region of space was inside a tetrahedra. If the tetrahedra were randomly sheared but all had a kind of regularity where they were all big enough to contain a sphere of radius A and be contained in a sphere of radius B, then I think the best distance along edges might approach a fixed multiple of the true distance as the continuous crow flies. And I mention this because this is a not uncommon way to discretize space.

However, you basically asked not just for a discrete space, but for a lattice space. And if you truly want that have that at every time you are located on the corner of some cube that is nicely packed inside a larger cube with identical cubes and that larger cube is itself nicely packed inside a still larger cube with identical cubes and so on. Then it is fairly easy to get some physics similar to the real world by allowing things to jump from neighbor to neighbor at different speeds.

For instance we don't know where particles are located exactly. We can measure them accurately enough to say they are in some box at one time, and then in a possibly different box at a later time. And we can say that they must have had a certain average velocity to get from here the there in that time interval.

And based on the velocity (where you ended up relative to where you started all divided by how long it took you to get from here to there) you can infer the distance travelled. But you can't actually see them perfectly so you don't know how far they actually traveled. You can use fancy equipment and place them in very small boxes but you still don't know where in the box they are or how they are moving, just that at one time they are in this box and then at another time they are in that box.

That is exactly what reporting position to finite accuracy means. So you really effectively have boxes based on how well you are measuring things. So what if you imagined smaller boxes inside those and yet smaller boxes inside those. Are the particles moving faster along the x and y axis going from corner to corner of those super small boxes inside the small boxes inside the actual boxes that you can measure? Who knows (or cares) what is happening when you aren't looking.

No one thinks they are doing that because we would be surprised if someday we started measuring things to better accuracy and all of a sudden things behaved differently and have preferred directions, in fact not having preferred direction is part of how we predict what to expect.

But even if we don't expect it, it could happen because we don't know what is going on faster than we look and in regions smaller than we can look inside.

So if you get scientific about distance travelled you see that really it is about being in one place at a time and being at another place at a different time and us tracking what you did as much as we felt like and then saying that more accurately that we just don't know.

When you don't know what the particle did you can base your answer on the geometry of the boxes instead. You literally imagine every line that started in one box and ended on the other box and take the average. Or you imagine every possible path from one vox to the other and take minimum (this will underestimate distance travelled but the error goes away as the boxes get smaller, it's like rounding down to the nearest \$1000 then rounding down to the nearest \$100 then nearest \$10 then nearest \$1 you are always underpaying but the error gets smaller, and unlike money we could make the boxes as small as we want).

So you can do the same. To go at a certain velocity just take certain ratios of steps left versus forward and take them faster based on the overall velocity. But then rethink distance not as the unknowable how do things actually move but instead on what is the geometrical relationship between where you started and stopped to someone that can't see exactly how you move.

Is this related to actual models physicists make? Not really. lattice theories in Physics are usually for fields that have values at every single lattice point and you are computing how they change, so it is already everywhere and you just want to know how it is changing.

Like if you have temperature, you could imagine a value of temperature at all the lattice points, and then if you have a hot point surrounded by cooler points you could have it get colder and have an equal amount of heat flow to the nearby points.

That's something that happens on some physicists models, but really just because the other models were too hard.

If you just have some particles it is easier yo just say where they are rather than have a bunch of points most of which are empty.

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