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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 ${\displaystyle {\mathcal {F}}_{+}({\mathcal {H}})}$ ${\displaystyle {\mathcal {H}},}$
• the fermionic Fock space the Graßmann algebra over the one-particle Hilbert space, more precisely its completion. ${\displaystyle {\mathcal {F}}_{-}({\mathcal {H}})}$

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

${\displaystyle a^{*}:{\mathcal {H}}\times {\mathcal {F}}_{\pm }({\mathcal {H}})\to {\mathcal {F}}_{\pm }({\mathcal {H}}),\quad (\psi ,\Phi )\mapsto a_{\psi }^{*}\Phi .}$

With ${\displaystyle \psi \in {\mathcal {H}},\Phi \in {\mathcal {F}}_{\pm }({\mathcal {H}}).}$

The illustrations

${\displaystyle a_{\psi }^{*}:\qquad {\mathcal {F}}_{\pm }({\mathcal {H}})\to {\mathcal {F}}_{\pm }({\mathcal {H}})}$

are called creation operators,

the adjoint operators to it

${\displaystyle a_{\psi }:\qquad {\mathcal {F}}_{\pm }({\mathcal {H}})\to {\mathcal {F}}_{\pm }({\mathcal {H}})}$

are called annihilation operators .

The canonical (anti) commutation relations apply to them

${\displaystyle a_{\psi }a_{\phi }\mp a_{\phi }a_{\psi }=0}$

${\displaystyle a_{\psi }^{*}a_{\phi }^{*}\mp a_{\phi }^{*}a_{\psi }^{*}=0}$

${\displaystyle a_{\psi }a_{\phi }^{*}\mp a_{\phi }^{*}a_{\psi }=\langle \psi ,\phi \rangle _{\mathcal {H}}\operatorname {id} _{{\mathcal {F}}_{\pm }({\mathcal {H}})},}$

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).