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).
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?
AND! I forgot to mention the LQCD calculations of the muon's anomalous magnetic moment, g-2, which is a precision observable currently being studied by an experiment at Fermilab.
This quantity (g-2) is dominated by electrodynamic effects. The QCD contribution is on the scale of 1e-10 compared to the full result. But it is the leading theoretical uncertainty.
There are two ways to get a prediction for the QCD contribution to this process. One is to try to back it out of a variety of data from unrelated experiments. In other words: try to fit it by requiring consistency. The other is a LQCD calculation.
The experimental data-driven fit to the QCD contribution and the LQCD calculation differ significantly. This difference is the difference between the current experiment claiming a sizeable new physics effect (the data-driven approach) and perfect agreement with the Standard Model (LQCD calculation).
"Solve the theory" is bit of a wish-washy expression. It depends on what you are interested in. What it really means is that it provides an approximation-free prescription for extracting observables from the QCD action.
Given some experimentally-measured observables (the pion mass, the kaon mass, etc. as long as you include 1 observable per parameter in QCD, meaning 1 gauge coupling + 1 per quark flavor you include in your simulation) you can calculate all other quantities---in a world that is pure QCD. We are now getting to an era where we can do lattice QCD+QED, but we cannot do lattice-standard-model because of the issue with chiral gauge theories mentioned in the article. However, since the weak sector is perturbative, we can combine pen-and-paper/Feynman-diagram-style calculations with LQCD calculations to get what we want: first-principles Standard Model predictions.
To discover new physics you need both an experimental signal and a precision prediction from the Standard Model. If they differ that difference is something not in the Standard Model.
What kinds of quantities can we calculate? The first major success was a calculation of the hadronic spectrum [spectrum]. We've also seen precision determinations of the QCD phase diagram (vacuum properties as a function of temperature) [phase diagram] and other properties that are important for the early universe, although there is a computational resources / scaling problem for nonzero density. With QED included there's a calculation of the proton-neutron mass splitting [mass splitting]. In terms of hadronic matrix elements we can compute the axial coupling of the neutron [gA], neutral kaon mixing, matrix elements needed for nuclear calculations of neutrinoless double beta decay [0vbb], all sorts of stuff. The FLAG review [flag] collects and evaluates individual results and tries to construct a community consensus for our "best values" for certain observables.
> Is it possible to state and solve initial value problem for, say, proton-proton scattering?
You picked one of the most difficult problems in lattice QCD (and one I spent a lot of time on as a postdoc): extracting baryon-baryon scattering. At high energy actually there is a lot of progress because QCD factorizes and you get a parton picture [PDFs]. But at low energy the situation is worse. There are a few issues:
1. We calculate in a finite box, typically with periodic boundary conditions (think PacMan). So there is no way to make protons "asymptotically far apart", and once they're heading apart, they're heading towards their next encounter. This conceptual difficulty was solved in the late 1980s: what you can do is take the energy of the finite-volume standing waves and turn that into phase shifts (quantum-mechanical scattering data).
2. Our calculations are done in Euclidean time, for reasons I have explained / touched on on hn before
So we don't have real Minkowski time; you can think: instead of things evolving like exp(i H t) things evolve like exp(- H t), where H is the Hamiltonian.
3. Baryons suffer from a signal-to-noise issue, where the longer we look in time the worse the variance gets. The source of this issue is understood and we have variational methods to get reliable signals earlier in time. The signal-to-noise problem is easier at heavier quark (or pion) masses (which we, as computational physicists can change even though experimentalists cannot).
For a long time we sharpened our tools at ~800 MeV pion masses. Current state of the art is down around 300 MeV.
This sounds dismal but I want to say: we have the whole pipeline working end-to-end. There are phenomenal results in the meson sector [meson scattering]. There's very-close-to-physical-parameters meson-baryon scattering [n-pi].
SO: is it possible to solve the initial value problem for proton-proton scattering using LQCD?
Not yet but work is in progress.
What these calculations give are phase shifts (and inelasticities) as a function of scattering momentum.
Once these are under control we can try to get a handle on the three-body sector; the three-body force is needed for accurate calculations in nuclear physics.
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.
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."
