Geometry and Quantum Theory Seminar: From Random Walks to the Free Quantum Field

The free quantum field is, heuristically, a Gaussian measure with density exp(−S_E) on a space of fields. We make this precise by building the Gaussian free field from discrete objects: random walks, Brownian motion, and the GFF on a lattice. Along the way we see why its covariance is a Green’s function, and why in dimension two and higher it exists only as a distribution. We then pass to the continuum free field of Chatterjee’s Chapter 3 and its Markov property in time. We close with a tour of how this gives the quantum theory, the Feynman propagator, and the Wightman functions.