Different definitions of configuration space

Question

Configuration space has been mentioned in many different areas of physics and from my personal experience I have found the definition has been different from topic to topic.

There are two examples to demonstrate my confusion. In standard lagrangian mechanics, the generalised coordinates span the configuration space. For example a particle in a newtonian setting has a configuration space $Q=\mathbb{R}^3$. The particles trajectory is then a particular curve in this configuration space. The configuration space for this example is finite dimensional.

Now consider classical field theory. We are told the configuration space for a field (with $n$ components) that depends on position and time, $\phi(t,x,y,z)$, is infinite dimensional with each point in configuration space representing a different function $\phi: \mathbb{R}^4 \rightarrow \mathbb{R}^n$, that is, a particular configuration is an element of configuration space. The feynman functional integral over configurations is a functional integrated over configuration space.

These seem like two different definitions to me. Following the first example then by analogy surely the configuration space would just be $\mathbb{R}^n$ and a particular configuration would be represented by a 'path' (function $f:\mathbb{R}^4$ to configuration space) in said space?

Answer 1

They are different definitions.

Take your first definition where $Q=\mathbb R^3$ for a Newtonian particle. According to that, the configuration space in classical field theory per your example is the space of functions $$ \mathbb R^3\to\mathbb R^n\,. $$ So for each point in time $t$, you get a function that depends on the coordinates $(x,y,z)\in\mathbb R^3$: $$ \phi(t,x,y,z)\,. $$

Of course in relativistic field theory splitting time from space like this is not always convenient, hence your second definition. Perhaps it's best to think of them like this: in the first case, the configuration space is the space of snapshots, while in the second case it is the space of histories.

Answer 2

For example a particle in a newtonian setting has a configuration space 𝑄=ℝ3. The particles trajectory is then a particular curve in this configuration space.

Right. A configuration is a point in $Q=\mathbb R^3$, and the particle evolves along a curve $\gamma:t \mapsto \gamma(t)\in Q$.

These seem like two different definitions to me. Following the first example then by analogy surely the configuration space would just be ℝ𝑛 and a particular configuration would be represented by a 'path' (function 𝑓:ℝ4 to configuration space) in said space?

This doesn't make sense to me. The configuration in point mechanics is an element of $Q=\mathbb R^n$, not a path. The "trajectory" in field theory is a path $\gamma: t \mapsto \gamma(t)\in Q$ as before, but now $\gamma(t)$ is a field configuration, i.e. a function from $\mathbb R^n \rightarrow \mathbb R$ (for an $\mathbb R$-valued field, obviously).