Identical particle exchange as a Yang-Mills theory

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 $|x_1,x_2,ldots,x_Nrangle$ 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)

$$langlepsi_1 psi_2 ldots psi^dagger_{N-1}psi^dagger_Nrangle:=int ~mathrm d[psi]int ~mathrm d[psi^dagger]e^{iS}(psi_1 psi_2 … psi^dagger_{N-1}psi^dagger_N)$$

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.