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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.
Named after Vladimir Aleksandrovich Fock.
A textbook account with an eye towards perturbative algebraic quantum field theory is in
See also
The Fock space construction may be axiomatized as the exponential modality in linear type theory. This is discussed in the following articles.
Richard Blute, Prakash Panangaden, R. A. G. Seely, Fock Space: A Model of Linear Exponential Types (1994) (web)
Marcelo Fiore, Differential Structure in Models of Multiplicative Biadditive Intuitionistic Linear Logic, Lecture Notes in Computer Science Volume 4583, 2007, pp 163-177 (pdf)
Jamie Vicary, A categorical framework for the quantum harmonic oscillator, International Journal of Theoretical Physics December 2008, Volume 47, Issue 12, pp 3408-3447 (arXiv:0706.0711)
(in the context of finite quantum mechanics in terms of dagger-compact categories)