Creation and Annihilation Operators for Identical Particles

General formalism

The state space of a system of N distinguishable particles

$$\varepsilon_N=\varepsilon_1(1)\otimes\varepsilon_1(2)\otimes..\otimes\varepsilon_1(N)$$

• The space of the completely symmetric states for bosons $\varepsilon_S(N)$

The projectors

$$S_N=\frac{1}{N!}\sum_\alpha P_\alpha$$

• The space of the completely antisymmetric states for fermions $\varepsilon_A(N)$

The projectors

$$A_N=\frac{1}{N!}\sum_\alpha\varepsilon_\alpha P_\alpha$$

• The $P_\alpha$ is the N! permutation operators for the N particles

• The $\varepsilon_\alpha$ is the parity of $P_\alpha$

• The subscript $\alpha$ distinguishes the different permutations of the N particles, and therefore take N! different values

Fock states for identical bosons

$$|n_i,n_j,..,n_l,..\rangle\\=cS_N|1:u_i;2:u_i;..;n_i:u_i;\ \ n_i+1:u_j;..n_i+n_j:u_j;..\rangle$$

• The subscripts i, j, k, l, ..denote different basis vectors {$|u_i\rangle$} of the state space $\varepsilon_1$ of a single particle