Second quantization

Summary

Second quantization is the description of systems of many identical particles, with a variable number of them, on the Fock space. It lifts operators on the one-particle Hilbert space to operators on Fock space. A unitary $U$ becomes $\Gamma(U)$, acting as $U$ on every particle. A self-adjoint $A$ becomes $d\Gamma(A)$, the sum of $A$ over all particles. For bounded $A$ and an orthonormal basis $(e_j)$, it reads $d\Gamma(A) = \sum_{j,k} \ip{e_j}{A e_k}\, a^*(e_j) a(e_k)$. Here $a^*$ and $a$ are the creation and annihilation operators. It is the bridge between many-body quantum mechanics and free quantum field theory.