As an organizational scheme for approximate numerical calculations, the path-integral formulation is fine. But path integrals can't serve as a rigorous foundati

As an organizational scheme for approximate numerical calculations, the path-integral formulation is fine. But path integrals can't serve as a rigorous foundation for QFTs, because for the most part they don't actually exist. The high-frequency behavior Woit describes is one obstacle; the fact that no infinite-dimensional Lebesgue measure is possible is another. There are others. Presumably there's some mathematical object out there that captures all the data a path integral ought to have without its pathologies, but no one knows what it is yet.