Comment by kbr- | Hacker News Reader

Comment by kbr- | Hacker News Reader

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> The point is that it can’t be simulated on any lattice of any density. It doesn’t matter how fine the lattice is compared to the sensitivity of the measurement.

It feels like you want to do sth like: first discretize (choose a lattice), then simulate (do calculations on this lattice).

I want to do the opposite: first simulate (do calculations on a continuous domain), then discretize (restrict my results to a lattice).

> For your case you showed, it would be like if x^2 simply didn’t exist on a lattice, and couldn’t be calculated by a computer no matter how much memory you threw at the problem.

So the functions we're talking about do exist on continuous domains - but they don't have a corresponding definition on a lattice?

Couldn't we embed a lattice in the continuous domain, then restrict the function along the embedding, thus getting a definition on a lattice?

Unless it's not possible to embed the lattice in a continuous domain - then my reasoning breaks.

(note: I know nothing about physics, I'm a programmer with math education, talk to me like I'm an idiot)

I apologize I am actually running out the door at this point, so I can’t reply in a ton of detail (I’ll try to remember to do so later)

But the point to remember is that these aren’t normal algebraic equations, they’re based on the quantum operators, right? And so we can always do symbolic math on them, but to get numeric results, they have to be instantiated at some point. The operators exist at every point in space, and some of the operators can be approximated on a lattice discretization, but some of them cease to be well defined as soon as there is any distance between the operators (so they require true real numbers, not floating point numbers - ergo, infinite memory).

One point that i think is missing is that there’s a bit of a difference between numerical solutions to QFT equations (like the calculation of g-2 referenced in the paper) and lattice calculations in that in general those numerical calculations are giving averaged quantities. We couldn’t, for instance, take that average quantity and use it instead of the dynamically fluctuating quantity in a lattice simulation. We could run a lattice simulation and estimate the value of g-2 from the lattice to see how well our discretization -> continuum extrapolation worked. But we couldn’t go backwards from the numerical solution to the lattice, so to speak.

If you have an oracle that can tell you the answer, you can subsample that at lattice points.

But how do you compute the continuous answer in the first place?

I guess the (crazy, I know) assumption that I made is that I have some analytical, symbolic expression for a function that describes the state of the universe at every point. This "state" describes some fundamental quantity (not necessarily a quantity we have a name for yet).

Then we express the value of any particle field at every point as a (potentially very complex) symbolic expression that only uses the state function from my previous paragraph.

All of these expressions need only finite memory to store. They describe functions with domain R^n to some co-domain of operators or whatever.

Then I can calculate the value of this complex function at any point with any precision I like, with finite memory, although unbounded - I need to allocate more memory when I want more precision.

Point is, I delay the process of "latticization" (calculating the numerical values at each point of a chosen lattice) to the very end - only then I have to choose how fine-grained my lattice is.

> I guess the (crazy, I know) assumption that I made is that I have some analytical, symbolic expression for a function that describes the state of the universe at every point.

This is the error. All you have is some partial differential equation. It has no known symbolic solution.

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