# From classical mechanics to classical field theory

Suppose I have a system of $N$ classical particles described by the Lagrangian $\mathcal{L}(\mathbf{q}_i,\dot{\mathbf{q}}_i,t)$. Similarly, we can introduce the Hamiltonian of such system via the Legendre transform. So, our phase space is given by $2d \cdot N\cdot$ degrees of freedom: $\mathbf{q}_i$, $\mathbf{p}_i$. The dynamics is described by the Hamilton's equations. Now we consider the limit $N \rightarrow \infty$. In this case, the claim is that one can derive the corresponding classical field theory. What people usually write is that, roughly speaking, the correspondence is given by $$\mathbf{q}_i(t) \rightarrow \phi(\mathbf{x},t),$$ $$\mathbf{p}_i(t) \rightarrow \pi(\mathbf{x},t).$$

Often, people just give an example of 1D chain in a continuum limit. But this one is rather obvious and, in some sense, a bit different: here we go from a lattice field theory to a continuum field theory rather than from classical mechanics to a classical field theory. So, my questions are:

1. Is there general prescription for promoting $N$-particle classical mechanics to a classical field theory? If so, what is this prescription?

2. If one then quantizes the obtained classical field theory, is it safe to say that the particles associated with creation/annihilation operators $\hat{a}_k$, $\hat{a}_k^{\dagger}$ are those of classical mechanics, i.e. in some limit their dynamics is depicted by the Hamilton's equations?

3. With the above (naive) prescription, I can only see how one can derive a classical scalar field theory. What about vector/spinor fields? For instance, how to derive Maxwell's EM from classical mechanics?

• Whether $N$ is finite or not is not so important as whether the system is continuous or not. I would classify the lattice as a discrete particle classical mechanical system, no matter if it is infinite or not. – Javier May 14 '18 at 19:06