5 ms·

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.

by atty 4y ago

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. 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. Certain parts of QFT work fine in lattice based calculations (lattice-QCD, for instance, where it’s simulated on a certain lattice scale, and then extrapolated to the continuous limit), and some seemingly don’t work at all.

kbr- 4y ago
> 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)
- zmgsabst 4y ago
  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?
  - kbr- 4y ago
    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.
    - evanb 4y ago
      > 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.
- atty 4y ago
  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.
criddell 4y ago
Do analog computers have the same fundamental limitations?
- atty 4y ago
  Honestly I’ve never considered an analog computer, and I’m a physicist but not a lattice expert. While an analog computer would let you use real numbers, you’d still need a way to store state at every point in space, which would lead to the infinite memory problem (and you’d need infinite compute to operate on the infinite memory). Perhaps there’s a clever way to get around that, but my suspicion is that if it were possible someone would have done it already.
  - MacsHeadroom 4y ago
    The complex plane does have infinite memory. A Turing machine can perform all discrete computations with a single bit. An idealized analog computer can compute anything with a "single" complex plane. Our (mis)conception of there being an "infinite memory problem" when the complex plane is infinite is analogous to flatlanders getting hung up about a "discrete bit memory problem."
- klyrs 4y ago
  Pulling on this thread leads directly to quantum computation.
- evanb 4y ago
  I do lattice QCD. It's not that we have a problem because of float/double inexact and limited arithmetic. It's that we have a finite amount of RAM. So the thing we'd want is not continuous values but continuous registers. Maybe this is possible with some very clever engineering but I'd wager that your computation will develop other problems, such as thermal noise causing problems (whereas digital computers have error correction).
  - effie 4y ago
    So, lattice QCD is often mentioned as an example of computationally well-defined quantum field theory with some reliable results. Wikipedia talks about "lattice QCD is a way to solve the theory". What does this really mean? What questions did lattice QCD answer? Is it possible to state and solve initial value problem for, say, proton-proton scattering?

by atty 4y ago

kbr- 4y ago

> 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)

zmgsabst 4y ago
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?
- kbr- 4y ago
  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.
  - evanb 4y ago
    > 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.
atty 4y ago
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.

criddell 4y ago

Do analog computers have the same fundamental limitations?

atty 4y ago
Honestly I’ve never considered an analog computer, and I’m a physicist but not a lattice expert. While an analog computer would let you use real numbers, you’d still need a way to store state at every point in space, which would lead to the infinite memory problem (and you’d need infinite compute to operate on the infinite memory). Perhaps there’s a clever way to get around that, but my suspicion is that if it were possible someone would have done it already.
- MacsHeadroom 4y ago
  The complex plane does have infinite memory. A Turing machine can perform all discrete computations with a single bit. An idealized analog computer can compute anything with a "single" complex plane. Our (mis)conception of there being an "infinite memory problem" when the complex plane is infinite is analogous to flatlanders getting hung up about a "discrete bit memory problem."
klyrs 4y ago
Pulling on this thread leads directly to quantum computation.
evanb 4y ago
I do lattice QCD. It's not that we have a problem because of float/double inexact and limited arithmetic. It's that we have a finite amount of RAM. So the thing we'd want is not continuous values but continuous registers. Maybe this is possible with some very clever engineering but I'd wager that your computation will develop other problems, such as thermal noise causing problems (whereas digital computers have error correction).
- effie 4y ago
  So, lattice QCD is often mentioned as an example of computationally well-defined quantum field theory with some reliable results. Wikipedia talks about "lattice QCD is a way to solve the theory". What does this really mean? What questions did lattice QCD answer? Is it possible to state and solve initial value problem for, say, proton-proton scattering?