# Jib room

The **Fock space** (after the Russian physicist Vladimir Alexandrowitsch Fock ) is used in quantum physics , especially in quantum field theory , for the mathematical description of many-particle systems with a variable number of particles . Depending on whether the particles are bosons or fermions , one speaks of the **bosonic** or the **fermionic Fock space** . According to its structure, the Fock space is a quantum mechanical Hilbert space .

The base states (of a Fock space) with a fixed number of particles ( i.e. elements of or density operators above it, each with a magnitude of 1, or the eigenstates of the particle number operator ) are called Fock states . In this context one speaks of second quantization or occupation number representation.

Mathematically it is

- the bosonic Fock space the symmetrical tensor algebra over a one-particle Hilbert space, more precisely its completion with respect to the scalar product
- the fermionic Fock space the Graßmann algebra over the one-particle Hilbert space, more precisely its completion.

The suitably normalized symmetrized tensor product (in the bosonic case) or the wedge product (in the fermionic case) induce images

With

The illustrations

are called **creation** operators,

the adjoint operators to it

are called **annihilation operators** .

The canonical (anti) commutation relations apply to them

where the upper sign ( commutator ) applies in the bosonic case and the lower sign (anti- commutator ) in the fermionic case.

## literature

- Kehe Zhu:
*Analysis on Fock Spaces.*Graduate Texts in Mathematics, 263. Springer, New York, 2012. ISBN 978-1-4419-8800-3 - Ola Bratteli, Derek W. Robinson: Operator algebras and quantum statistical mechanics . 2. Equilibrium states, Models in quantum statistical mechanics ". Springer, Berlin / Heidelberg 1981, ISBN 3-540-10381-3 (English, 505 pages).