Quantum field theories are notoriously hard to define. A uniquely well established way of defining QFTs is to "put them on the lattice," i.e. to view them as effective, low-energy descriptions of some quantum theory on a finite but huge lattice. This is a subtle endeavor: simple phenomena in the continuum may be tricky to see on the lattice, and vice versa. For instance, chiral theories are much more natural in the continuum, whereas confinement is more natural on the lattice.
What's more, the mere connection between lattice and continuum operators is severely ill-understood. I will explain this issue in detail, and I will propose a robust method for consistently isolating the continuum (low-energy) operators when starting from a theory of lattice fermions. This will lead to a new perspective on many fundamental issues in QFT, and in particular the notion of an operator product expansion will naturally emerge.