nLab
action functional

In classical physics, every classical mechanical system (possibly even with infinitely many degrees of freedom?, as for a classical field theory) has an action functional on the space of paths? in the configuration space?. This determines the laws of motion by the action principle: the motion of a particle corresponds to a path in configuration space which minimizes (or at least extremises) the action functional.

The action functional is normally obtained by integrating the Lagrangean function along the path; in the case of a continuum system, the Lagrangean function is obtained by integrating the Lagrangean density on space. For a more manifestly relativistic? formulation, we may obtain the action functional by integrating the Lagrangean density on spacetime?.

1. $S(q)={\int }_a^{b}L(q,\dot{q})\mathrm{d}t$, where $q$ is a path through configuration space, on the time interval $[a,b]$, with derivative $\dot{q}=\mathrm{d}q/\mathrm{d}t$. When minimising the action, we fix the values of $q(a)$ and $q(b)$.
2. $L(q,\dot{q})={\int }_Sℒ(q,\dot{q})\mathrm{d}x\mathrm{d}y\mathrm{d}z$, where now $q$ is a configuration of fields on $S$, which is a region of space. We fix boundary conditions on the boundary of $S$ (typically that $q$ and $\dot{q}$ go to zero if $S$ is all of space).
3. $S(q)={\int }_Rℒ(q,\dot{q})\mathrm{d}x\mathrm{d}y\mathrm{d}z\mathrm{d}t$, where now $q$ is a configuration of fields on $R$, which a region of spacetime, with time derivative $\dot{q}=\partial q/\partial t$. We fix boundary conditions on the boundary of $R$.

The formulation of (3) above is still not manifestly coordinate indepdendent. However, $\mathrm{d}x\mathrm{d}y\mathrm{d}z\mathrm{d}t$ is simply the volume form? on spacetime and $\dot{q}$ is merely one choice of coordinate on state space? and could just as easily be replaced by a derivative with respect to any timelike coordinate on spacetime (or drop coordinates altogether).

The path integral version of quantization emphasises the Lagrangean formalism and path integrals. Most other approaches to quantization favor the Hamiltonian approach and phase space?.) In the path integral approach, the amplitudes in quantum field theory are expressed in terms of the path integral of the action functional over the space of paths in the space of field configurations.