I am trying to find a quantum field theory (a Yang-Mills theory) for the identical particles exchange interaction. For a system of $N$ identical particles one has the state $|x_1,x_2,\ldots,x_N\rangle$ that is invariant under permutations of states. The permutation operator $P$ is a discrete, not a continuous symmetry group. But is there a way turning the permutation operator $P$ into a continuous operator?
In a quantum field theory I have continuum states, i.e. the $\psi(x)$ (for fermions) function. Due to this fact I cannot define a proper permutation operator. However if I compute the $N$-point function (with $M$ ingoing and $N-M$ outgoing states)
$$\langle\psi_1 \psi_2 \ldots \psi^\dagger_{N-1}\psi^\dagger_N\rangle:=\int ~\mathrm d[\psi]\int ~\mathrm d[\psi^\dagger]e^{iS}(\psi_1 \psi_2 ... \psi^\dagger_{N-1}\psi^\dagger_N)$$
I could impose permutation symmetry in the functions $\psi_1,...\psi^\dagger_N$. If $N \mapsto \infty$ then I can define a function $P(x)$ (a local symmetry group) that fixes permuation symmetry on every point of spacetime.
How I can define a generalized permutation operator in continuum field?
Edit: I am trying to assume some additional symmetries in the quantum field vacuum, i.e. two fields of equal wave structure but on different spacetime points.
