Fock

Fock space is a Hilbert space framework used in quantum mechanics to describe systems with an arbitrary number of indistinguishable particles. It is constructed as the direct sum of the n-particle Hilbert spaces: H^0 ⊕ H^1 ⊕ H^2 ⊕ ..., where H^0 is the vacuum state, representing zero particles. For bosons, each H^n is the n-fold symmetric tensor product of a single-particle space; for fermions, it is the n-fold antisymmetric tensor product. The resulting Fock basis can be labeled by occupation numbers n_k, with the total particle number N = Σ_k n_k.

Creation and annihilation operators act on Fock space to add or remove quanta of a given mode. They satisfy canonical commutation relations for bosons [a_k, a†_l] = δ_kl, [a_k, a_l] = [a†_k, a†_l] = 0, and canonical anti-commutation relations for fermions {a_k, a†_l} = δ_kl, {a_k, a_l} = {a†_k, a†_l} = 0. The vacuum state |0⟩ is annihilated by all a_k. Any Fock state can be built by applying creation operators to |0⟩.

Fock space is central to the formalism of second quantization in quantum field theory and many-body physics, where particle number is not fixed and interactions naturally couple states with different N. In mathematics, the construction is realized via symmetric or exterior algebras of a single-particle Hilbert space, yielding the bosonic or fermionic Fock space, respectively. The concept is named after Vladimir A. Fock, who introduced it in the 1930s; it forms a foundational tool in describing photons, phonons, electrons, and other indistinguishable quanta.