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.
 A: 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.
Classical Mechanics of 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.
Classical Field Theory
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.
A: The typical way to introduce field theory is by first considering system of finitely many d.o.f. and then "take a continuum limit". Imagine I have a bunch of $n$ one-dimensional oscillators sitting on a line. Then I can label them from $1$ to $n$ and write the amplitude of each as $\phi_i$, or $\phi(i)$, where $\phi$ is a number between $1$ and $n$.
Now I want to look at a string (a classical one, not a string of string theory), like a guitar string. I can't make out individual oscillators on it, but I can sure imagine the amplitude of each point on the string. So now I label my oscillators as $\phi(x)$, where $x$ is now any real number between 0 and the length of the string. Those oscillators $\phi(x)$ are now uncountably many, since there are uncountably many real numbers between 0 and the length of the string.
So now position has become the label for my "field" $\phi(x)$ that tells me what the amplitude of the string at the position $x$ along it is. This position itself is no longer dynamical - the string doesn't stretch or move in this scenario, it's fixed at the ends and just oscillates in the orthogonal direction I chose to call $\phi$.
Instead of such an amplitude, there are many more things I can measure at any point  in space - for instance, temperature. You can more a thermometer around and measure temperature at each point in space, and in principle there's no limit how fine your spatial resolution for this can be. So instead of modelling a temperature gradient with a measly finite number of $T_i$ that we have to adjust depending on the experimental setup, we choose to model temperature as a field $T(x)$ - the temperature at the spot $x$. Again, in thermodynamics, $T(x)$ is variable by $x$ is fixed, just "a label".
A: In classical point particles mechanics the dynamics of a system is determined by solving a system of coupled differential equations (Newton's law): if you have $N$ distinct particles then the solution is represented by the knowledge of position and velocity for each one of the particles at any time, namely the collection 
$$
t\in\,[a,b]\to(\mathbf{r}_i(t), \mathbf{v}_i(t)), \quad i=1,\ldots,N
$$
namely $6N$ different functions of the time. Here the index $i$ is discrete and counts each single particle we have at our disposal. As, we said, for each $i$ we have a map $f_i(t)$ (representing whatever physical quantity).
There exist some systems in nature whose dynamics can be described as if they were composed by infinitely many point particles extremely close to each other. The standard example is that of an oscillating string, whereby you want to calculate the displacements from the equilibrium position at any point in space and time; as such, if the above $f_i(t)$ represent the physics of one point particle, a string can be thought of many several particles very close to each other and hence occupying each single point $x$ of the real line: $f_i(t)\leadsto f(x,t)$, where now you can see that the discrete index $i$ has become the continuous variable $x$ (as the point particles composing the strings are present at any point in space). The equivalent of Newton's equations for the $f(x,t)$ will be any other differential equation whose solution will now be the function $f$.
Comparing the two examples we see that:


*

*point particle mechanics: the position is the solution of the differential equations. We have as many "positions" as particles, in particular countably many.

*field theory: the position is not the solutions of the equations, rather the dynamics is represented by any function $f(x,t)$ of the position. The spaces of functions of the positions where we search for the solutions are usually infinite-dimensional, hence we refer to this fact saying that the system itself has infinite degrees of freedom.
