Why do we consider Lagrangian densities in field theory (as opposed to Lagrangians as in point mechanics)?

Question

My question is: Why do we consider Lagrangian densities in field theory (as opposed to Lagrangians as in point mechanics)?

Is it simply because of the following?

We wish the theories to be Lorentz invariant and as in Special Relativity space and time are no longer independent of one another we must consider the physics at each spacetime point $x^{\mu}$, implying that the theory should be locally described by a density.

Additionally, to obey Poincare invariance the Lagrangian density should not be explicitly dependent on $x^{\mu}$, and for it to be Lorentz invariant it must be dependent on spatial derivatives $\nabla\phi$ of the fields as well as temporal derivatives $\partial_{t}\phi$ (as time derivatives in one reference frame will correspond to a combination of time and spatial derivatives in another frame).

(Physically does the fact that $\mathscr{L}$ depends on $\nabla\phi$ as well as $\partial_{t}\phi$ because the field $\phi$ is defined at each spacetime point and so a fluctuation in $\phi$ at a point $x^{\mu}$ will produce temporal and spatial derivatives at that point which will affect fields in its immediate neighbourhood (i.e. infinitesimally close to it)?)

Answer 1

The difference is that in classical mechanics positions are exactly the fields you are looking at, whereas in general field theories they are the variables the actual fields depend on.

In classical mechanics the solution of the dynamics is given by the knowledge of the position and the velocity $(q(t),\dot{q}(t))$ at any time $t$. Time plays the role of the path parameter that describes the solution on the fibre bundle of the configuration space and therefore it is the only free parameter you are allowed to integrate on; hence the action must be of the form $$ S[q,\dot{q}]=\int_{\gamma}dt\,L(q(t),\dot{q}(t);t). $$ In field theories you essentially play the same game: fields are maps $x^{\mu}\to\phi(x^{\mu})$ depending on both positions and time (whereas previously the only field was the position itself, depending on time, notice the difference). You may want to mirror the above action as $$ S[\phi,\dot{\phi}] = \int_{\mathcal{D_t}}dt\,L(\alpha(t),\dot{\alpha}(t);t) $$ having in mind that the new dummy variables $\alpha(t),\dot{\alpha}(t)$ result in turn from the integration of the fields on the position variables as $$ S[\phi,\partial\phi] = \int_{\mathcal{D_t}}dt\, \left(\int_{\mathcal{D'}}d^3x\,\mathscr{L}(\phi(x,t),\partial\phi(x,t);x,t) \right) $$ collecting the above you may simply define the action as $$ S[\phi,\partial\phi] = \int_{\mathcal{D}}d^4x\,\mathscr{L}(\phi(x,t),\partial\phi(x,t);x,t). $$ We call Lagrangian (density) whatever function appears in the integral. Density just means that you have something you want to integrate over the variables that you have, no more than that. The standard Lagrangian in classical mechanics is a density as well with respect to the time variable, only we do not call it such (although it is). Then, as you pointed out, for the action to be Lorentz invariant you can drop the explicit dependencies on $(x,t)$, so that the Lagrangian (or the density, as you want to call it) ends up being invariant under translations in the Poincaré group. Further requirements can be added to include any other invariance we want.

Answer 2

The use of the Lagrangian density is a convenience, and it is not directly related to causality or relativity, and neither strictly to quantum theories. What I mean is that it is possible to formulate non-relativistic (quantum or classical) field theories using exactly the same language.

The difference between "mechanics" and "field theory" is that, instead of using particles as the fundamental objects of the theory, we use fields i.e. functions $\phi:X\to \mathbb{C}$. So, while in mechanics the phase space is a finite dimensional structure (e.g. a manifold) in field theory it is an infinite dimensional one (e.g. the space of measurable functions).

The physical motivation is that there are systems that are impossible to describe with a finite number of degrees of freedom (e.g. the elctromagnetic field). Special relativity adds an important motivation, on the quantum side: particles can be created and destroyed, therefore your relativistic quantum space must contain all the possible configurations with an arbitrary number $n$ of particles; this is conveniently described mathematically considering the one-particle QM space as the classical phase space, and build the so called Fock-Cook (second) quantization upon it. Again, an infinite dimensional phase-space ($L^2(\Omega)$, for some suitable $\Omega$) has to be considered.

Once you are in an infinite-dimensional phase space setting, the introduction of the Lagrangian density is quite natural, and it is a matter of convenience. Let $\mathbb{C}^X$ be the set of functions from $X$ to $\mathbb{C}$, and your infinite dimensional phase space (or position-velocity space for lagrangian formulation). Mimicking the finite dimensional situation, you want to build up a function(al) of the variables of the system $\phi\in \mathbb{C}^X$ that full encodes the dynamical informations. We call it Lagrangian function $L:\mathbb{C}^X\to \mathbb{R}$ (usually taken to be real-valued). For the moment it is not important to worry about the distinction between variables (fields $\phi$) and their derivatives ($\dot{\phi}$); think of them as "independent variables" inside $\mathbb{C}^X$ (as position and velocity are in finite dimensions).

Now, we have an object $L(\cdot)$ that encodes the dynamical informations, and has to be evaluated on functions $\phi\in \mathbb{C}^X$. What happens in most situations in practice, is that the Lagrangian consists of two formal operations, one that provides a pointwise information for each $x\in X$ (depending on $\phi$), and another that puts all those informations together providing the complete evaluation $L(\phi)$. The first is the Lagrangian density $\mathscr{L}:\mathbb{C}^X\to \mathbb{R}^X$ (because usually it is real-valued), the second is the "integration" or we may call it in general the global evaluation $\mathbb{E}_{gl}: \mathbb{R}^X\to \mathbb{R}$. This splitting results in $L=\mathbb{E}_{gl}\circ \mathscr{L}$, and it is convenient if you want to separate the "functional pointwise manipulations", done in the lagrangian density, from the global evaluation of these manipulations w.r.t. the totality of points.

However, apart from convenience, this is just the standard setting of infinite dimensional phase spaces; no relativistic or quantum considerations involved a priori.