Can we do physics with a path integral and lagrangian density?

Let us start a new blog entry about the path integral and lagrangian density method which is used in quantum field theory textbooks.

In Chapter 8 of Mark Srednicki's online book (https://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf), we move from the physics of single particle, to the physics of a whole field by replacing the position coordinate q(t) of the particle with a whole field φ(x, t). That is, we jump from a path in 2 dimensions (q, t) to a path in uncountably many dimensions, where we have a distinct value of φ(x, t) for uncountably many x.

If we are dealing with a single particle in a path integral, we restrict ourselves to continuous paths in R^4 and calculate the integral of the lagrangian formula over the length of the path. Note that the value of the lagrangian is determined by the full information of the particle position at each time t: we can use all spatial coordinate values simultaneously as arguments of a function to determine the value of the lagrangian at that point.

That is very different from the lagrangian density method, where we do not have the full information of φ(x, t) to determine the value of the lagrangian at a time t. We can only use "coordinates" φ(x, t), x, and t at each point x to calculate the lagrangian density. We cannot use a function of all values φ(x', t) for every point x' in the space.

Thus, in the case of lagrangian density, we are dealing with a very restricted class of functions for calculating the whole path integral. The analogous situation in the case of a single particle in 4 dimensions would be that the lagrangian would have a form f(x, t) + g(y, t) + h(z, t). There would be no interdependent term k(x, y, z, t).

This raises the question when is the lagrangian density method useful? What kind of physical systems can we model with the lagrangian density method?