If $|ψ\rangle$ is the state of a system of two indistinguishable particles, then we have an exchange operator P which switches the states of the two particles. Since the two particles are indistinguishable, the physical state cannot change under the action of the exchange operator, so we must have $P|ψ\rangle=λ|ψ\rangle$ where $|λ|=1$. Obviously switching the states of the two particles, and then switching them back, leaves the particles with their original states, so $P^2|ψ\rangle=(λ^2)|ψ\rangle=|ψ\rangle$, so $λ=±1$, and thus the state of the system must be either symmetric or anti-symmetric with respect to exchange.
Now I've heard that this reasoning does not hold for two dimensions, leading to the possibility of anyons, for which you can have $λ$ be something other than 1 or -1. How in the world is that possible? Where is the flaw or oversight in the above reasoning, that makes it exclude the 2D case? Where in the above proof are we assuming that space is three-dimensional?
Any help would be greatly appreciated.
Thank You in Advance.
EDIT: Let me present the proof in a step-by-step fashion, so the error can be more easily identified:
For any states $|ψ_1\rangle$ and $|ψ_2\rangle$, define $P|ψ_1\rangle|ψ_2\rangle$ to be $|ψ_2\rangle|ψ_1\rangle$
For identical particles, $P |\psi \rangle$ and $|\psi \rangle$ correspond to the same physical state (i.e. ray), so we must have $P |\psi \rangle = \lambda |\psi \rangle$ for some complex number $\lambda$.
Applying the definition of $P$ in step 1 twice, we have $P^2|ψ_1\rangle|ψ_2\rangle=PP|ψ_1\rangle|ψ_2\rangle=P|ψ_2\rangle|ψ_1\rangle = |ψ_1\rangle|ψ_2\rangle$, so for any two-particle state $|\psi\rangle$, we have $P^2|\psi \rangle = |\psi \rangle$.
Applying step 2 twice, we have $P^2|ψ\rangle = PP|ψ\rangle = P \lambda |ψ\rangle = \lambda P |ψ\rangle = \lambda^2 |ψ\rangle$
By steps 3 and 4, we have $\lambda^2 = 1$ and thus $\lambda = ±1$
I assume the problem is with step 3 somehow, but I'm not sure what the problem is, since it follows directly from the definition in step 1. Is the problem with the definition in step 1, then? But how can a definition be wrong?