Quantum field theory (QFT) was originally developed as a set of rules describing elementary particles moving at relativistic speeds. It worked — often to unprecedented accuracy. QFT has since outgrown its initial remit and found uses in many areas of physics, mathematics, chemistry, and computer science. At its core, it is now a framework for studying systems with many degrees of freedom and finding simple, effective descriptions of their correlations, phase transitions, and other collective phenomena.

Perhaps surprisingly, theoretical physicists today use two fundamentally different approaches to QFT, much like two people may use different languages to describe the world. The continuum approach, based on analysis and differential geometry, is usually used in particle physics, string theory, and cosmology. The lattice approach, based on algebra and discrete mathematics, is more often seen in condensed matter, statistical mechanics, quantum information science, and in various numerical simulations. We know relatively little about the dictionary between the lattice and continuum “languages,” even though they can often describe the same physical phenomenon, such as the collision of two particles or the onset of superconductivity. In this talk I will describe a new and very general way to write such a dictionary. This formalizes decades of intuition, leads to long-sought mathematically rigorous definitions, and provides new insights into phase diagrams, dualities, and inherent limitations of well known examples of QFTs. Perhaps most importantly, this dictionary opens the door to realizing many QFTs as finite quantum systems that can be numerically or experimentally simulated.