Lagrangian density: What assumption are we making and how could it change?

Question

I'm having a hard time picturing the physical system under consideration in QFT. Scattering and propagators are understandable since we a priori assume we have certain particles given by the hamiltonian and are calculating probabilities.

But when we write down Lagrangian densities for certain theories like scalar field theory, are we a priori assuming the presence of a particle described by it? If so, how could there be change in Lagrangian from spontaneous symmetry breaking occur which leads to excitation of other particles?

In essence I would like to know what we are assuming about a physical system when we write down the Lagrangian density for a theory.

Answer 1

One practical way of thinking about it would be considering the general Lagrangian densities you could build for a given set of local fields belonging to particular representations of the Lorentz group (scalars, spinors, vectors, etc). The sensible Lagrangian densities would need to obey Lorentz invariance, and you could strongly constrain the set of allowed interactions by requiring renormalizability of the theory. After constructing some Lagrangian density one could look (at least if the interactions are relatively weak) at its ground state and one-particle excitations. This is how you would learn what kinds of particles does this theory describe - and compare it to the real-world particles.

Historically it has sometimes happened like this. However, sometimes people have started from the known equations of motion for a field describing a particle and then constructed the Lagrangian density which produces such equations of motion. The ultimate beauty of nature is that they arrived to simple Lagrangian densities, which would be the first natural candidates to look at if you were to construct all possible Lagrangian densities.

However, constructing the Lagrangian density to reproduce the equations of motion of observed particles won't always bring you to the fundamental description. Strong interactions are the most prominent example: in experiments we see different baryons and mesons, but the underlying microscopic description is in terms of different particles: quarks and gluons. Because of confinement we could not observe isolated quarks and gluons, and their Lagrangian density was constructed based on symmetry considerations.