nLab Fock space

Idea

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.

Related entries

• tensor algebra

• second quantization

• quantum many-body physics

quantum probability theory – observables and states

• states

  • classical state

  • quantum state

    • space of states (in geometric quantization)

    • state on a star-algebra, quasi-state

    • qbit, Bell state

      dimer, tensor network state

    • quantum state preparation

    • probability amplitude, quantum fluctuation

    • pure state

      • wave function

      • bra-ket

      • Bell state

    • quantum superposition, quantum interference

    • quantum entanglement

    • quantum measurement

      wave function collapse

      Born rule

      deferred measurement principle

      quantum reader monad

      measurement problem

    • superselection sector

    • mixed state, density matrix

      entanglement entropy

      holographic entanglement entropy

    • coherent quantum state

  • ground state, excited state

  • quasi-free state

  • Fock space, second quantization

  • vacuum, vacuum state

    • Hadamard state

    • vacuum diagram

    • vacuum expectation value, vacuum amplitude, vacuum fluctuation

    • vacuum energy

    • vacuum polarization

    • interacting vacuum

    • thermal vacuum, KMS state

    • vacuum stability

      • false vacuum, tachyon, Coleman-De Luccia instanton

    • theta vacuum

    • perturbative string theory vacuum

      • non-geometric string theory vacuum

      • landscape of string theory vacua

  • entangled state

  • tensor network state

    • matrix product state

    • tree tensor network state

• observables

  • quantum observable, beable

  • algebra of observables, star-algebra

  • Bohr topos

  • quantum operator (in geometric quantization)

  • quantum operation, quantum effect, effect algebra

  • in quantum field theory

    • local observable

    • polynomial observable

      • linear observable

        • field observable

    • regular observable

    • microcausal observable

    • normal-ordered product, time-ordered products, retarded product

    • Wick algebra

    • scattering amplitude

    • interacting field algebra of observables, Bogoliubov's formula

• GNS construction

• theorems

  • order-theoretic structure in quantum mechanics

    • Gleason's theorem

    • Alfsen-Shultz theorem

    • Harding-Döring-Hamhalter theorem

    • Kochen-Specker theorem

    • Nuiten's lemma

  • Wigner's theorem

  • no-cloning theorem

  • Bell's theorem

References

Named after Vladimir Aleksandrovich Fock.

General

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

• Michael Dütsch, appendix E of From classical field theory to perturbative quantum field theory

See also

• Wikipedia, Fock space

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.

• 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)