Fock Space

$F_{\nu }(H):=\overline{{⨁}_{n=0}^{\infty }{S}_{\nu }{H}^{⨂n}}$ , where

• $H^{⨂n}$ represents an Quantum Hilbert Space with $n$ particles
• $S_{\nu }$ represents the symmetrization or antisymetrization of a tensor, i.e. $S_{\nu }(H)$ is the set of all symmetric ( anti symmetric ) tensors acting on an Hilbert Space $H$.
  • Symmetric tensors are needed for bosonic statistics
  • Anti symmetric tensors are needed for fermionic statistics
• $⨁$ describes the Direct Sum of Hilbert Spaces
• $\overline{V}$ represents the closure of a Hilbert Space $V$.

The fock space allows us to act on spaces with a varying number of particles through the use of annihilation operators and creation operators.