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I remember hearing at some point that pertubation QFT using Feynman diagrams can be thought of as certain limits of lattice QFT. Is there a precise statement of this fact? Or is it just a heuristic with no mathematical formulation/justification?

Alternatively, is there a way to do calculation on lattices, that gives the same result as calculating Feynman diagrams, taking into account things like Pauli Villars/dimensional regularizations and renormalizations?

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Perturbative QFT using feynman diagrams is a method to calculate observables quantities by expanding them in a serie of terms that correspond to graphical representations of the interactions between particles and fields. These diagrams are useful for organizing and simplifying the calculations, but they also have a physical interpretation as the possible ways that a process can occur. Lattice QFT is an alternative approach to QFT that discretizes space and time into a finite grid of points and defines the fields and their dynamics on this lattice. This allows for numerical simulations of QFT using computers, which can handle non-perturbative effects that are difficult or impossible to capture with Feynman diagrams.

There is a relation between perturbation QFT using Feynman diagrams and lattice QFT, but it is not straightforward or exact. One way to understand this relation is to consider the continuum limit of lattice QFT, where the lattice spacing goes to zero and the lattice QFT approaches the original continuous QFT.

In this limit, one can recover the Feynman diagrams from the lattice QFT by applying certain rules and approximations. However, this procedure may introduce errors or ambiguities, and it may not work for all types of QFTs or Feynman diagrams. Another way to relate perturbation QFT using Feynman diagrams and lattice QFT is to use quantum simulation techniques, where one uses a quantum system (such as a quantum computer) to simulate another quantum system (such as a QFT). In this case, one can construct quantum circuits that are equivalent to either the real-time path integral from the discrete-time Lagrangian formulation of lattice QFT or the Feynman diagrams from the perturbative expansion of continuous QFT.

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The Lattice is just a particular regularization for Quantum Field Theories. You can do perturbation theory on the lattice, for example see this review. Perturbation theory relies on the parameter being expanded in to be small, we know however that for interesting QFTs (such as QCD) the strong coupling constant is not small on the scale of hadrons. We also know that there are in general renormalon problems that impede direct resummation of perturbative series. So the whole point of using the lattice for lattice QCD is to study the nonperturbative behaviour.

Oftentimes, if you are calculating some matrix element of interest on the lattice, phenomenologists will care about the value in MS/MSbar schemes. So there is some (perturbative) matching required to match your lattice results to MS/MSbar. Often this involves using some intermediate scheme, such as RI-MOM.

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