Focktiloja

Focktiloja, or Fock spaces in English, are a concept in quantum mechanics and quantum field theory that describe a Hilbert space capable of representing states with a variable number of indistinguishable particles. The term is named after the Russian physicist Vladimir Fock, and the Finnish plural form Focktiloja reflects the idea of multiple such spaces or sectors.

The standard construction starts from a single-particle Hilbert space H. The Fock space F(H) is the direct sum of the n-particle spaces, with the symmetry imposed by the particle statistics. For bosons, the n-particle space is the symmetric tensor product, while for fermions it is the antisymmetric tensor product. In mathematical terms, F(H) = ⊕_{n=0}^∞ H^{⊗ n}_s for bosons or H^{⊗ n}_a for fermions. This structure allows the description of states with any number of particles, from the vacuum (no particles) to arbitrarily many.

A common basis for F(H) is the occupation-number basis, written as |n1, n2, …>, where ni denotes the number of particles in each mode. The formalism of second quantization introduces creation and annihilation operators that add or remove particles in a given mode. These operators satisfy commutation relations [ai, aj†] = δij for bosons and anticommutation relations {ai, aj†} = δij for fermions, together with the action on the vacuum state |0>.

Focktiloja are central to many areas of physics, including quantum optics, condensed matter, and particle physics, because they provide a compact framework to describe systems where particle number is not fixed. They underpin calculations in many-body theory, field theories, and simulations of quantum systems. See also Fock space, second quantization, and occupation-number representation.