TheFock space is analgebraic construction used inquantum mechanics to construct thequantum states space of a variable or unknown number of identicalparticles from a single particleHilbert spaceH. It is named afterV. A. Fock who first introduced it in his 1932 paper "Konfigurationsraum und zweite Quantelung" ("Configuration space andsecond quantization").^[1][2]

Informally, a Fock space is the sum of a set of Hilbert spaces representing zero particle states, one particle states, two particle states, and so on. If the identical particles arebosons, then-particle states are vectors in asymmetrizedtensor product ofn single-particle Hilbert spacesH. If the identical particles arefermions, then-particle states are vectors in anantisymmetrized tensor product ofn single-particle Hilbert spacesH (seesymmetric algebra andexterior algebra respectively). A general state in Fock space is alinear combination ofn-particle states, one for eachn.

Technically, the Fock space is (theHilbert spacecompletion of) thedirect sum of the symmetric or antisymmetric tensors in thetensor powers of a single-particle Hilbert spaceH,$${\displaystyle F_{\nu }(H)=\overline{\underset{n=0}{\overset{\mathrm{\infty }}{⨁}}{S}_{\nu }{H}^{\otimes n}}.}$$

Here${\displaystyle S_{\nu }}$ is theoperator that symmetrizes orantisymmetrizes a tensor, depending on whether the Hilbert space describes particles obeyingbosonic${\displaystyle (\nu =+)}$ orfermionic${\displaystyle (\nu =-)}$ statistics, and the overline represents the completion of the space. The bosonic (resp. fermionic) Fock space can alternatively be constructed as (the Hilbert space completion of) thesymmetric tensors${\displaystyle F_{+}(H)=\overline{S^{\ast }H}}$ (resp.alternating tensors$F_{-}(H)=\overline{{\bigwedge }^{\ast }H}$). For every basis forH there is a natural basis of the Fock space, theFock states.

The Fock space is the (Hilbert)direct sum oftensor products of copies of a single-particle Hilbert space${\displaystyle H}$

$${\displaystyle F_{\nu }(H)=\underset{n=0}{\overset{\mathrm{\infty }}{⨁}}{S}_{\nu }{H}^{\otimes n}=\mathbb{C}\oplus H\oplus \left(S_{\nu }\left(H\otimes H\right)\right)\oplus \left(S_{\nu }\left(H\otimes H\otimes H\right)\right)\oplus \cdots }$$

Here${\displaystyle \mathbb{C}}$, thecomplex scalars, consists of the states corresponding to no particles,${\displaystyle H}$ the states of one particle,${\displaystyle S_{\nu }(H\otimes H)}$ the states of two identical particles etc.

A general state in${\displaystyle F_{\nu }(H)}$ is given by

$${\displaystyle |\mathrm{\Psi }{⟩}_{\nu }=|{\mathrm{\Psi }}_0{⟩}_{\nu }\oplus |{\mathrm{\Psi }}_1{⟩}_{\nu }\oplus |{\mathrm{\Psi }}_2{⟩}_{\nu }\oplus \cdots =a|0⟩\oplus \sum _i{a}_i|{\psi }_i⟩\oplus \sum _{ij}{a}_{ij}|{\psi }_i,{\psi }_j{⟩}_{\nu }\oplus \cdots }$$where

• ${\displaystyle |0⟩}$ is a vector of length 1 called the vacuum state and${\displaystyle a\in \mathbb{C}}$ is a complex coefficient,
• ${\displaystyle |{\psi }_i⟩\in H}$ is a state in the single particle Hilbert space and${\displaystyle a_i\in \mathbb{C}}$ is a complex coefficient,
• $|{\psi }_i,{\psi }_j{⟩}_{\nu }=a_{ij}|{\psi }_i⟩\otimes |{\psi }_j⟩+a_{ji}|{\psi }_j⟩\otimes |{\psi }_i⟩\in S_{\nu }(H\otimes H)$, and${\displaystyle a_{ij}=\nu a_{ji}\in \mathbb{C}}$ is a complex coefficient, etc.

The convergence of this infinite sum is important if${\displaystyle F_{\nu }(H)}$ is to be a Hilbert space. Technically we require${\displaystyle F_{\nu }(H)}$ to be the Hilbert space completion of the algebraic direct sum. It consists of all infinitetuples${\displaystyle |\mathrm{\Psi }{⟩}_{\nu }=(|{\mathrm{\Psi }}_0{⟩}_{\nu },|{\mathrm{\Psi }}_1{⟩}_{\nu },|{\mathrm{\Psi }}_2{⟩}_{\nu },\dots )}$ such that thenorm, defined by the inner product is finitewhere the${\displaystyle n}$ particle norm is defined by$${\displaystyle ⟨{\mathrm{\Psi }}_n|{\mathrm{\Psi }}_n{⟩}_{\nu }=\sum _{i_1,\dots i_n}\sum _{j_1,\dots j_n}{a}_{i_1,\dots ,i_n}^{\ast }{a}_{j_1,\dots ,j_n}⟨{\psi }_{i_1}|{\psi }_{j_1}⟩\cdots ⟨{\psi }_{i_n}|{\psi }_{j_n}⟩}$$i.e., the restriction of thenorm on the tensor product${\displaystyle H^{\otimes n}}$

For two general states$${\displaystyle |\mathrm{\Psi }{⟩}_{\nu }=|{\mathrm{\Psi }}_0{⟩}_{\nu }\oplus |{\mathrm{\Psi }}_1{⟩}_{\nu }\oplus |{\mathrm{\Psi }}_2{⟩}_{\nu }\oplus \cdots =a|0⟩\oplus \sum _i{a}_i|{\psi }_i⟩\oplus \sum _{ij}{a}_{ij}|{\psi }_i,{\psi }_j{⟩}_{\nu }\oplus \cdots ,}$$ and$${\displaystyle |\mathrm{\Phi }{⟩}_{\nu }=|{\mathrm{\Phi }}_0{⟩}_{\nu }\oplus |{\mathrm{\Phi }}_1{⟩}_{\nu }\oplus |{\mathrm{\Phi }}_2{⟩}_{\nu }\oplus \cdots =b|0⟩\oplus \sum _i{b}_i|{\varphi }_i⟩\oplus \sum _{ij}{b}_{ij}|{\varphi }_i,{\varphi }_j{⟩}_{\nu }\oplus \cdots }$$theinner product on${\displaystyle F_{\nu }(H)}$ is then defined as$${\displaystyle ⟨\mathrm{\Psi }|\mathrm{\Phi }{⟩}_{\nu }:=\sum _n⟨{\mathrm{\Psi }}_n|{\mathrm{\Phi }}_n{⟩}_{\nu }=a^{\ast }b+\sum _{ij}{a}_i^{\ast }{b}_j⟨{\psi }_i|{\varphi }_j⟩+\sum _{ijkl}{a}_{ij}^{\ast }{b}_{kl}⟨{\psi }_i|{\varphi }_k⟩⟨{\psi }_j|{\varphi }_l{⟩}_{\nu }+\cdots }$$where we use the inner products on each of the${\displaystyle n}$-particle Hilbert spaces. Note that, in particular the${\displaystyle n}$ particle subspaces are orthogonal for different${\displaystyle n}$.

Aproduct state of the Fock space is a state of the form

$${\displaystyle |\mathrm{\Psi }{⟩}_{\nu }=|{\varphi }_1,{\varphi }_2,\cdots ,{\varphi }_n{⟩}_{\nu }=|{\varphi }_1⟩\otimes |{\varphi }_2⟩\otimes \cdots \otimes |{\varphi }_n⟩}$$

which describes a collection of${\displaystyle n}$ particles, one of which has quantum state${\displaystyle {\varphi }_1}$, another${\displaystyle {\varphi }_2}$ and so on up to the${\displaystyle n}$th particle, where each${\displaystyle {\varphi }_i}$ isany state from the single particle Hilbert space${\displaystyle H}$. Here juxtaposition (writing the single particle kets side by side, without the${\displaystyle \otimes }$) is symmetric (resp. antisymmetric) multiplication in the symmetric (antisymmetric)tensor algebra. The general state in a Fock space is a linear combination of product states. A state that cannot be written as a convex sum of product states is called anentangled state.

When we speak ofone particle in state${\displaystyle {\varphi }_i}$, we must bear in mind that in quantum mechanics identical particles areindistinguishable. In the same Fock space, all particles are identical. (To describe many species of particles, we take the tensor product of as many different Fock spaces as there are species of particles under consideration). It is one of the most powerful features of this formalism that states are implicitly properly symmetrized. For instance, if the above state${\displaystyle |\mathrm{\Psi }{⟩}_{-}}$ is fermionic, it will be 0 if two (or more) of the${\displaystyle {\varphi }_i}$ are equal because the antisymmetric(exterior) product${\displaystyle |{\varphi }_i⟩|{\varphi }_i⟩=0}$. This is a mathematical formulation of thePauli exclusion principle that no two (or more) fermions can be in the same quantum state. In fact, whenever the terms in a formal product are linearly dependent; the product will be zero for antisymmetric tensors. Also, the product of orthonormal states is properly orthonormal by construction (although possibly 0 in the Fermi case when two states are equal).

A useful and convenient basis for a Fock space is theoccupancy number basis. Given a basis${\displaystyle \{|{\psi }_i⟩{\}}_{i=0,1,2,\dots }}$ of${\displaystyle H}$, we can denote the state with${\displaystyle n_0}$ particles in state${\displaystyle |{\psi }_0⟩}$,${\displaystyle n_1}$ particles in state${\displaystyle |{\psi }_1⟩}$, ...,${\displaystyle n_k}$ particles in state${\displaystyle |{\psi }_k⟩}$, and no particles in the remaining states, by defining

$${\displaystyle |n_0,n_1,\dots ,n_k{⟩}_{\nu }=|{\psi }_0{⟩}^{n_0}|{\psi }_1{⟩}^{n_1}\cdots |{\psi }_k{⟩}^{n_k},}$$

where each${\displaystyle n_i}$ takes the value 0 or 1 for fermionic particles and 0, 1, 2, ... for bosonic particles. Note that trailing zeroes may be dropped without changing the state. Such a state is called aFock state. When the${\displaystyle |{\psi }_i⟩}$ are understood as the steady states of a free field, the Fock states describe an assembly of non-interacting particles in definite numbers. The most general Fock state is a linear superposition of pure states.

Two operators of great importance are thecreation and annihilation operators, which upon acting on a Fock state add or respectively remove a particle in the ascribed quantum state. They are denoted${\displaystyle a^{†}(\varphi )}$ for creation and${\displaystyle a(\varphi )}$for annihilation respectively. To create ("add") a particle, the quantum state${\displaystyle |\varphi ⟩}$ is symmetric or exterior- multiplied with${\displaystyle |\varphi ⟩}$; and respectively to annihilate ("remove") a particle, an (even or odd)interior product is taken with${\displaystyle ⟨\varphi |}$, which is the adjoint of${\displaystyle a^{†}(\varphi )}$. It is often convenient to work with states of the basis of${\displaystyle H}$ so that these operators remove and add exactly one particle in the given basis state. These operators also serve as generators for more general operators acting on the Fock space, for instance thenumber operator giving the number of particles in a specific state${\displaystyle |{\varphi }_i⟩}$ is${\displaystyle a^{†}({\varphi }_i)a({\varphi }_i)}$.

Often the one particle space${\displaystyle H}$ is given as${\displaystyle L_2(X,\mu )}$, the space ofsquare-integrable functions on a space${\displaystyle X}$ withmeasure${\displaystyle \mu }$ (strictly speaking, theequivalence classes of square integrable functions where functions are equivalent if they differ on aset of measure zero). The typical example is thefree particle with${\displaystyle H=L_2({\mathbb{R}}^3,d^{3}x)}$ the space of square integrable functions on three-dimensional space. The Fock spaces then have a natural interpretation as symmetric or anti-symmetric square integrable functions as follows.

Let${\displaystyle X^0=\{\ast \}}$ and${\displaystyle X^1=X}$,${\displaystyle X^2=X\times X}$,${\displaystyle X^3=X\times X\times X}$, etc.Consider the space of tuples of points which is thedisjoint union

$${\displaystyle X^{\ast }=X^0⨿X^1⨿X^2⨿X^3⨿\cdots .}$$

It has a natural measure${\displaystyle {\mu }^{\ast }}$ such that${\displaystyle {\mu }^{\ast }(X^0)=1}$ and the restriction of${\displaystyle {\mu }^{\ast }}$ to${\displaystyle X^n}$ is${\displaystyle {\mu }^n}$.The even Fock space${\displaystyle F_{+}(L_2(X,\mu ))}$ can then be identified with the space of symmetric functions in${\displaystyle L_2(X^{\ast },{\mu }^{\ast })}$ whereas the odd Fock space${\displaystyle F_{-}(L_2(X,\mu ))}$ can be identified with the space of anti-symmetric functions. The identification follows directly from theisometric mapping$${\displaystyle L_2(X,\mu {)}^{\otimes n}\to L_2(X^n,{\mu }^n)}$$$${\displaystyle {\psi }_1(x)\otimes \cdots \otimes {\psi }_n(x)\mapsto {\psi }_1(x_1)\cdots {\psi }_n(x_n)}$$.

Given wave functions${\displaystyle {\psi }_1={\psi }_1(x),\dots ,{\psi }_n={\psi }_n(x)}$, theSlater determinant

is an antisymmetric function on${\displaystyle X^n}$. It can thus be naturally interpreted as an element of the${\displaystyle n}$-particle sector of the odd Fock space. The normalization is chosen such that${\displaystyle \Vert \mathrm{\Psi }\Vert =1}$ if the functions${\displaystyle {\psi }_1,\dots ,{\psi }_n}$ are orthonormal. There is a similar "Slater permanent" with the determinant replaced with thepermanent which gives elements of${\displaystyle n}$-sector of the even Fock space.

Define theSegal–Bargmann space${\displaystyle B_N}$^[3] of complexholomorphic functions square-integrable with respect to aGaussian measure:

where$${\displaystyle \Vert f{\Vert }_{{\mathcal{F}}^2({\mathbb{C}}^N)}:={\int }_{{\mathbb{C}}^N}|f(\mathbf{z}){|}^2{e}^{-\pi |\mathbf{z}{|}^2}d\mathbf{z}.}$$Then defining a space${\displaystyle B_{\mathrm{\infty }}}$ as the nested union of the spaces${\displaystyle B_N}$ over the integers${\displaystyle N\ge 0}$, Segal^[4] and Bargmann showed^[5][6] that${\displaystyle B_{\mathrm{\infty }}}$ is isomorphic to a bosonic Fock space. The monomial$${\displaystyle x_1^{n_1}...x_k^{n_k}}$$corresponds to the Fock state$${\displaystyle |n_0,n_1,\dots ,n_k{⟩}_{\nu }=|{\psi }_0{⟩}^{n_0}|{\psi }_1{⟩}^{n_1}\cdots |{\psi }_k{⟩}^{n_k}.}$$

• Feynman diagrams and Wick products associated with q-Fock space - noncommutative analysis, Edward G. Effros and Mihai Popa, Department of Mathematics, UCLA
• R. Geroch, Mathematical Physics, Chicago University Press, Chapter 21.

• Quantum mechanics
• Quantum field theory

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