nLab Fock space



Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts


Quantum systems

quantum logic

quantum physics

quantum probability theoryobservables and states

quantum information

quantum computation


quantum algorithms:

quantum sensing

quantum communication

Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT



In physics the symmetric tensor algebra on a space of quantum states of some quantum mechanical system is called its bosonic Fock space. This is regarded in turn as the space of quantum states of arbitrarily many copies of the system. In particular if the original system describes some bosonic particle species, then its Fock space is the space of quantum states of arbitrarily many such particles.

Similarly the exterior algebra/Grassmann algebra is called the fermionic Fock space.

The process of passing from a given space of quantum states to its Fock space is also known as (or rather: is part of) what is called second quantization.

Fock spaces hence appear as spaces of quantum states of free fields in quantum field theory. In perturbative quantum field theory they are still used indirectly in for non-free field theories.

quantum probability theoryobservables and states


Named after Vladimir Aleksandrovich Fock.


A textbook account with an eye towards perturbative algebraic quantum field theory is in

See also

In linear type theory

The Fock space construction may be axiomatized as the exponential modality in linear type theory. This is discussed in the following articles.

Last revised on May 7, 2021 at 10:04:20. See the history of this page for a list of all contributions to it.