# Why can precomputed sets of lattice QFT field configurations be used to measure arbitrary observables?

My knowledge of quantum mechanics is rusty and my understanding of (lattice) quantum field theory on a very novice level at best, so it is likely my whole question is based on completely wrong assumptions and a lack of understanding.

Most introductory texts about QFT give some sort of a translation table between quantities in QM and QFT, see e.g. top of page 16 here. A given particle path (over all of which you integrate in the path integral formulation of QM to get the amplitude of a given process) is translated into a given field configuration in QFT (over all of which, again, you integrate in the path integral formulation of QFT).

As far as I understand, lattice quantum field theory calculations on big high-power computing clusters are effectively generating lots of field configurations (for a given lattice size, spacing, boundary conditions etc, but independent of any "starting conditions"). These field configurations are generated using metropolis MC methods (or similar more advanced importance sampling schemes), based on the action calculated for a given field configuration. In the "list" of output field configurations, the occurence of field configurations is then already weighted by their effective action, so that summing up over them yields the most relevant results without near infinite amounts of practically irrelevant field configurations.

To extract an observable from such a set of field configurations, one simply sums the value of the observable for all field configurations.

I wonder why it is possible and reasonable to extract any observable from a pre-computed number of field configurations that did not include any information about what observable one would like to obtain. In other words: how can the action of a field configuration be independent of the process I want to extract afterwards?

To maybe clarify a bit further, consider a simple double slit experiment: I want to calculate the QM amplitude of an electron at position A (on one side of the double slit) at time t0 to appear at position B (on the other side of the double slit) at time t1. For this I randomly generate a bunch of paths the satisfy the conditions of my observable (position A at t0, position B at t1) and evaluate their actions. If I want to be smart about it, I do some importance sampling of paths (Metropolis or whatever). However in this scenario, I only generated paths that were connected to the observable I knew I was looking for from the beginning (propagation A->B). I could not change the observable to a different transition amplitude afterwards and use the same paths.

So how do I unify these two pictures in my head? The only thing I can think of would be to not restrict the generation of QM paths to starting point A and ending point B, instead generating QM paths for all possible starting and end points. Afterwards, to calculate the desired transition amplitude, I could only sum up over paths going from A to B, which would have made the very most of my generated paths absolutely unnecessary. If that should be the case, why to LQFT calculations not restrict the generation of field configurations to such that give a meaningful contribution to a predefined observable?

• I'm not sure what exactly you're asking, but: The boundary of the path corresponds to the initial and final states w.r.t. to which you determine the expectation value of the operator. The initial and final states in (lattice) quantum field theory are usually the vacuum, and the path integral without boundary conditions yields the expectation value in the vacuum (see every good introduction to the path integral formalism). Jul 1, 2016 at 13:31
• @ACuriousMind : Just copied from my other comment further below: I have no idea how an actual LQCD observable would look like, but say we want to measure the pion mass. Why is this fully done in measuring vacuum fluctuations, why is the lattice not "initialised" with a pion on it? Jul 1, 2016 at 16:41

The path integral form of the Gell-Mann-Low theorem says that $<0|T\{ \pi(x,t) \pi(x,t+T)\}|0>$ is equal to the statistical average of $\pi(x,t) \pi(x,t+T)$ in the sum over all gauge field paths. Here $\pi(x,t)$ is some lattice opertaor that has a non-zero matrix element (proprtional to $\sqrt Z$) between the vacuum and the pion state (just as in the continuum). Then for large $T$ we have $$<0|T\{ \pi(x,t) \pi(x,t+T)\}|0> \sim Z e^{-m_\pi T}$$ where $m_\pi$ is the lighest mass (giving the slowest decay in imaganinary time $T$) eigenstate in the pion channel. The paths (gauge field configuartions) in the path integral don't care about what is being averaged, so a precomputed table of configuartions can be used.