Fock space

The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space H. It is named after V. A. Fock who first introduced it in his 1932 paper "Konfigurationsraum und zweite Quantelung".[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 are bosons, the n-particle states are vectors in a symmetrized tensor product of n single-particle Hilbert spaces H. If the identical particles are fermions, the n-particle states are vectors in an antisymmetrized tensor product of n single-particle Hilbert spaces H. A general state in Fock space is a linear combination of n-particle states, one for each n.

Technically, the Fock space is (the Hilbert space completion of) the direct sum of the symmetric or antisymmetric tensors in the tensor powers of a single-particle Hilbert space H,

\( {\displaystyle F_{\nu }(H)={\overline {\bigoplus _{n=0}^{\infty }S_{\nu }H^{\otimes n}}}~.} \)

Here \( S_{\nu } \) is the operator which symmetrizes or antisymmetrizes a tensor, depending on whether the Hilbert space describes particles obeying bosonic \( (\nu = +) \) or fermionic \( (\nu = -) \) 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) the symmetric tensors \( F_+(H) = \overline{S^*H} \) (resp. alternating tensors \( F_-(H) = \overline{{\bigwedge}^* H}) \). For every basis for H there is a natural basis of the Fock space, the Fock states.

Definition

The Fock space is the (Hilbert) direct sum of tensor products of copies of a single-particle Hilbert space H

\( F_\nu(H)=\bigoplus_{n=0}^{\infty}S_\nu H^{\otimes n} =\mathbb{C} \oplus H \oplus \left(S_\nu \left(H \otimes H\right)\right) \oplus \left(S_\nu \left( H \otimes H \otimes H\right)\right) \oplus \ldots \)

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

A typical state in \( F_\nu(H) \) is given by

where

\( |0\rangle \) is a vector of length 1, called the vacuum state and \( \,a_0 \in \mathbb{C} \) is a complex coefficient,
\( |\psi_1\rangle \in H \) is a state in the single particle Hilbert space, and \( {\displaystyle \,a_{1}\in \mathbb {C} } \) is a complex coefficient,
\( {\displaystyle |\psi _{2i}\,,\psi _{2j}\rangle _{\nu }={\frac {1}{2}}(|\psi _{2i}\rangle \otimes |\psi _{2j}\rangle +\nu \,|\psi _{2j}\rangle \otimes |\psi _{2i}\rangle )\in S_{\nu }(H\otimes H)} \) , and \( a_{ij} = \nu a_{ji} \in \mathbb{C} \) is a complex coefficient
etc.

The convergence of this infinite sum is important if \( F_\nu(H) \) is to be a Hilbert space. Technically we require \( F_\nu(H) \) to be the Hilbert space completion of the algebraic direct sum. It consists of all infinite tuples \( |\Psi\rangle_\nu = (|\Psi_0\rangle_\nu , |\Psi_1\rangle_\nu , |\Psi_2\rangle_\nu, \ldots) \) such that the norm, defined by the inner product is finite

\( {\displaystyle \||\Psi \rangle _{\nu }\|_{\nu }^{2}=\sum _{n=0}^{\infty }\langle \Psi _{n}|\Psi _{n}\rangle _{\nu }<\infty } \)

where the n particle norm is defined by

\( \langle \Psi _{n}|\Psi _{n}\rangle _{\nu }=\sum _{{i_{1},\ldots i_{n},j_{1},\ldots j_{n}}}a_{{i_{1},\ldots ,i_{n}}}^{*}a_{{j_{1},\ldots ,j_{n}}}\langle \psi _{{i_{1}}}|\psi _{{j_{1}}}\rangle \cdots \langle \psi _{{i_{n}}}|\psi _{{j_{n}}}\rangle \)

i.e. the restriction of the norm on the tensor product \( H^{\otimes n} \)

For two states

\( {\displaystyle |\Psi \rangle _{\nu }=|\Psi _{0}\rangle _{\nu }\oplus |\Psi _{1}\rangle _{\nu }\oplus |\Psi _{2}\rangle _{\nu }\oplus \ldots =a_{0}|0\rangle \oplus a_{1}|\psi _{1}\rangle \oplus \sum _{ij}a_{ij}|\psi _{2i},\psi _{2j}\rangle _{\nu }\oplus \ldots } \), and
\( {\displaystyle |\Phi \rangle _{\nu }=|\Phi _{0}\rangle _{\nu }\oplus |\Phi _{1}\rangle _{\nu }\oplus |\Phi _{2}\rangle _{\nu }\oplus \ldots =b_{0}|0\rangle \oplus b_{1}|\phi _{1}\rangle \oplus \sum _{ij}b_{ij}|\phi _{2i},\phi _{2j}\rangle _{\nu }\oplus \ldots } \)

the inner product on \( F_\nu(H) \) is then defined as

\( {\displaystyle \langle \Psi |\Phi \rangle _{\nu }:=\sum _{n}\langle \Psi _{n}|\Phi _{n}\rangle _{\nu }=a_{0}^{*}b_{0}+a_{1}^{*}b_{1}\langle \psi _{1}|\phi _{1}\rangle +\sum _{ijkl}a_{ij}^{*}b_{kl}\langle \psi _{2i}|\phi _{2k}\rangle \langle \psi _{2j}|\phi _{2l}\rangle _{\nu }+\ldots } \)

where we use the inner products on each of the n-particle Hilbert spaces. Note that, in particular the n particle subspaces are orthogonal for different n.

Product states, indistinguishable particles, and a useful basis for Fock space

A product state of the Fock space is a state of the form

which describes a collection of n particles, one of which has quantum state \( \phi_1\,, \) another \( \phi_2\, \) and so on up to the n {\displaystyle n} nth particle, where each \( \phi_i\, \) is any state from the single particle Hilbert space H. Here juxtaposition (writing the single particle kets side by side, without the \( \otimes \) ) 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 an entangled state.

When we speak of one particle in state \( \phi_i\,, \) it must be borne in mind that in quantum mechanics identical particles are indistinguishable. 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 \( |\Psi\rangle_- \) is fermionic, it will be 0 if two (or more) of the \( \phi_i\, \) are equal because the antisymmetric (exterior) product \( |\phi_i \rangle |\phi_i \rangle = 0 \) . This is a mathematical formulation of the Pauli 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 the occupancy number basis. Given a basis \( \{|\psi_i\rangle\}_{i = 0,1,2, \dots} \) of H, we can denote the state with \( n_{0} \) particles in state \( |\psi _{0}\rangle \) , \( n_{1} \) particles in state \( |\psi _{1}\rangle \) , ..., \( n_{k} \) particles in state \( |\psi_k\rangle \), and no particles in the remaining states, by defining

\( |n_{0},n_{1},\ldots ,n_{k}\rangle _{\nu }=|\psi _{0}\rangle ^{{n_{0}}}|\psi _{1}\rangle ^{{n_{1}}}\cdots |\psi _{k}\rangle ^{{n_{k}}}, \)

where each \( n_{i} \) 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 a Fock state. When the \( |\psi _{i}\rangle \) 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 the creation and annihilation operators, which upon acting on a Fock state add or respectively remove a particle in the ascribed quantum state. They are denoted \( a^{\dagger}(\phi)\, \) for creation and \( a(\phi)\, \) for annihilation respectively. To create ("add") a particle, the quantum state \( |\phi\rangle \) is symmetric or exterior- multiplied with \( |\phi\rangle \); and respectively to annihilate ("remove") a particle, an (even or odd) interior product is taken with \( \langle \phi | \) , which is the adjoint of \( a^\dagger(\phi)\, \) . It is often convenient to work with states of the basis of H {\displaystyle H} 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 the number operator giving the number of particles in a specific state \( |\phi _{i}\rangle \) is \( a^{\dagger}(\phi_i)a(\phi_i)\,. \)
Wave function interpretation

Often the one particle space H is given as \( L_2(X, \mu) \) , the space of square-integrable functions on a space X with measure \( \mu \) (strictly speaking, the equivalence classes of square integrable functions where functions are equivalent if they differ on a set of measure zero). The typical example is the free particle with \( H = L_2(\mathbb{R}^3, d^3x) \) 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 \( X^0 = \{*\} \) and \( X^1 = X, \) \( X^2 = X\times X \) , \( X^3 = X \times X \times X \) etc. Consider the space of tuples of points which is the disjoint union

\( X^* = X^0 \bigsqcup X^1 \bigsqcup X^2 \bigsqcup X^3 \bigsqcup \ldots. \)

It has a natural measure \( \mu ^{*} \) such that \( \mu^*(X^0) = 1 \) and the restriction of \( \mu ^{*} \) to \( X^{n} \) is \( \mu^n \) . The even Fock space \( F_+(L_2(X,\mu))\, \) can then be identified with the space of symmetric functions in\( L_2(X^*, \mu^*) \)whereas the odd Fock space\( F_-(L_2(X,\mu))\, \) can be identified with the space of anti-symmetric functions. The identification follows directly from the isometric mapping

\( L_2(X, \mu)^{\otimes n} \to L_2(X^n, \mu^n) \)
\( \psi_1(x)\otimes\cdots\otimes\psi_n(x) \mapsto \psi_1(x_1)\cdots \psi_n(x_n). \)

Given wave functions \( \psi_1 = \psi_1(x), \ldots , \psi_n = \psi_n(x) \) \) , the Slater determinant

\( \Psi(x_1, \ldots x_n) = \frac{1}{\sqrt{n!}}\left|\begin{matrix} \psi_1(x_1) & \ldots & \psi_n(x_1) \\ \vdots & & \vdots \\ \psi_1(x_n) & \dots & \psi_n(x_n) \\ \end{matrix} \right| \)

is an antisymmetric function on \( X^{n} \). It can thus be naturally interpreted as an element of the n-particle sector of the odd Fock space. The normalization is chosen such that \( \|\Psi\| = 1 \) if the functions \( \psi_1, \ldots, \psi_n \) are orthonormal. There is a similar "Slater permanent" with the determinant replaced with the permanent which gives elements of n {\displaystyle n} n-sector of the even

Fock space.
Relation to the Segal–Bargmann space

Define the Segal–Bargmann space space \( B_{N} \)[3] of complex holomorphic functions square-integrable with respect to a Gaussian measure:

\( {\displaystyle {\mathcal {F}}^{2}(\mathbb {C} ^{N})=\{f\colon \mathbb {C} ^{N}\to \mathbb {C} \mid \Vert f\Vert _{{\mathcal {F}}^{2}(\mathbb {C} ^{N})}<\infty \}}, \)

where

\( {\displaystyle \Vert f\Vert _{{\mathcal {F}}^{2}(\mathbb {C} ^{N})}:=\int _{\mathbb {C} ^{n}}\vert f(\mathbf {z} )\vert ^{2}e^{-\pi \vert \mathbf {z} \vert ^{2}}\,d\mathbf {z} }. \)

Then defining a space \( B_\infty \) as the nested union of the spaces \( B_{N} \) over the integers \( {\displaystyle N\geq 0} \), Segal [4] and Bargmann showed [5][6] that \( B_\infty \) is isomorphic to a bosonic Fock space. The monomial

\( {\displaystyle x_{1}^{n_{1}}...x_{k}^{n_{k}}} \)

corresponds to the Fock state

\( {\displaystyle |n_{0},n_{1},\ldots ,n_{k}\rangle _{\nu }=|\psi _{0}\rangle ^{n_{0}}|\psi _{1}\rangle ^{n_{1}}\cdots |\psi _{k}\rangle ^{n_{k}}.} \)