What is the difference between the meaning of “state space” and “configuration space”?

What is the difference between the meaning of "state space" and "configuration space"?

I'm only familiar with the first, and when I look up the second I can't tell the difference.

Configuration space = manifold of allowed position configurations. It is the same for classical mechanics and quantum mechanics; for $n$ distinguishable particles, $R^{3n}$ minus the set of coordinates where two particles occupy the same position.

State space = manifold of pure states of the system = manifold on which a deterministic dynamics is valid. Thus classically, state space = phase space; for $n$ distinguishable particles, $R^{6n}$. But in quantum mechanics, the state space is the projective space of rays in $L^2(\Omega)$, where $\Omega$ is the configuration space.

What yuggib described in his answer is not the state space but the space of mixed states (= limits of convex combinations of pure states). He also ignored that different particles cannot occupy the same position - which is relevant for topological issues.

In classical mechanics, a state is described by a probability distribution on the phase space, i.e. the space of all possible positions and momenta. In the simplest cases, the phase space is $\mathbb{R}^{2nd}$ - where $n$ is the number of particles and $d$ is the spatial dimension - endowed with a suitable symplectic structure.

The configuration space on the other hand is simply the space of possible positions. In the situation above, it is $\mathbb{R}^{nd}$.

In other words, the state space is the space of probabilities acting on the phase space; the configuration space is the space of all possible positions. Therefore they are two quite different objects.

• Wow, I really appreciate this but actually I'm even more confused now. I didn't expect probability to be involved at all in classical mechanics. How would probabilities be involved? Or did you mean quantum mechanics? – user541686 Feb 15 '16 at 10:46
• Also in classical mechanics probabilities come out, usually in the context of the so-called statistical mechanics. In general you may not have a perfect knowledge of the system also at a classical level, so you describe it probabilistically. Of course, the Dirac's delta probability distribution is an allowed state classically (and that means concentration on a single phase-space point/ trajectory as usual). – yuggib Feb 15 '16 at 12:18

I will explain these two spaces in the context of fluid dynamics. In fluid dynamics, flow velocity components can be expressed as the derivative of scalar stream function. Interestingly, the structure of the stream function is analogous to Hamilton's equation. This similarity was realized in the 80's which gave a new direction to the field.

Configuration space or real space is what you see in the real world. For example motion of flower or leaves in the pond. However, this (real) space can also be visualized from the perspe3ctive of phase space (space where dynamics live!). This is due to the similarity of the stream function and Hamilton's equation. The advantage of phase space/ state space over configurational space is that it gives you a lot of mathematical tools to handle nonlinearity. Also, it reveals the qualitative nature of the complexity.