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I recently learned of lattice field theory wherein quantum fields are defined on a discrete spacetime. Naively this sounds to me a lot like certain numerical methods for solving differential equations (e.g. the finite-difference method) where you start with continuum field equations and solve them approximately on a discrete grid.

What is the relationship between these two methods, if any? Suppose I had some Lagrangian of a theory composed of a scalar field and a gauge field. I could vary the Lagrangian with respect to the fields to get the continuum equations of motion, then approximately solve those equations using finite differences. However, I could also define so-called lattice links to obtain a discretized version of the field equations that can be solved on a computer (as is done in this paper). Apparently this second approach has certain advantages such as being manifestly gauge invariant and not requiring the enforcement of constraint equations.

Are these just different approaches to the same problem, or is lattice field theory fundamentally different from the application of grid-based numerical methods?

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The difference is that lattice field theory is used to solve quantum field problems. It is insufficient to consider just one configuration of the field. We need to do a path integral over all configurations of the field. Since the number of possible field configurations is enormous even for a small lattice, we evaluate the integral using the Monte Carlo method. We essentially generate a random set of field configurations that obey the probability distribution associated with the integral we want to evaluate and then perform a sum. It can get a lot harder than just solving a differential equation on a lattice. Solving the differential equations is what you would have to do in classical field theory.

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