coordinate systems - Why is position considered a label in classical field theory?

I am currently researching into classical field theories and have come across the idea of a position being considered a label in field theory, rather than a dynamic variable.

I am not sure why this is, and what it really means or implies in terms of a classical theory.

Answer

In a field theory, the values of the field at each position make up the set of dynamical variables. Therefore the role of the position is just to which dynamical variable (i.e., which location for the field value) you are talking about.

The role of the position variable is actually similar to the role of the spatial dimension index of a position variable in classical mechanics of point particles.

To make this similarity more clear, lets consider first the equation of motion for a simple harmonic oscillator, whose position is denoted by $x$. We get

$$m\ddot{x}_i=-kx_i.$$

Above, $m$ is the mass of the osillating object, $k$ is the spring constant, and $i$ is and index representing the spatial dimension, so that $x_i$ is the $i$th component of the position $\mathbf{x}$. Notice here that $i$ is not a dynamical variable in that its value does not change with time depending on the configuration of the system. Instead, the three dynamical variables are $x_1$, $x_2$, and $x_3$. The symbol "$i$" is merely a label being used to indicate which of the three components of $\mathbf{x}$ is being discussed.

Now lets compare this to classical field theory. Let's consider air pressure in some three dimensional region. Under the right conditions, the equation of motion for the pressure $p(\mathbf{x})$ is

$$\ddot{p}(\mathbf{x})=c^2 \nabla^2 p(\mathbf{x}),$$

where $c$ is the speed of sound. In the above equation $\mathbf{x}$ is not a dynamical variable. Instead, the dynamical variables are the values of $p(\mathbf{x})$. Notice that there are infinitely many of these variables: one for each value of the position $\mathbf{x}$. The role of $\mathbf{x}$ in this equation simply to indicate which of the dynamical variables $p(\mathbf{x})$ is being discussed. This is exactly analogous to the role of the spatial index $i$ in the above classical mechanics of particles example. Thus $\mathbf{x}$ serves the role of an "index" or "label" instead of a dynamical variable.