I've been reading through Matthew Schwartz's book "Quantum Field Theory and the Standard Model" and in chapter 24 there is a section on locality (section 24.4). In it he defines locality in terms of the Lagrangian stating
" We take locality to mean that the Lagrangian is an integral over a Lagrangian density that is a functional of fields and their derivatives evaluated at the same point."
Which is all well and good, but then he further elaborates to say that
"To be clear, this definition is mathematical, not physical: it is a property of our calculational framework, not of observables."
This part confuses me as I thought that the whole motivation was physical, i.e. that objects at different spacetime points should not be able to directly interact with one another? It makes sense to me that, as the Lagrangian density characterises the dynamics of a physical system at a given spacetime point, and locality demands that the dynamics of the system at that given point should depend only on the state of the system at that point (i.e. the field configuration and how it's changing at that point), then clearly the Lagrangian density should depend on no more than the state of the system at that point, i.e. the field configuration and its rate of change (in spacetime)?! Maybe I'm missing something?