Fock space

 ( Quantum Mechanics ) 

The Fock space is an construction used in quantum mechanics to construct the space of a variable or unknown number of identical particles from a single particle Hilbert space . It is named after Vladimir Fock who first introduced it in his 1932 paper "Konfigurationsraum und zweite Quantelung" ("Configuration space and second quantization").M.C. Reed, Barry Simon, "Methods of Modern Mathematical Physics, Volume II", Academic Press 1975. Page 328.

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 -particle states are vectors in a Symmetric tensor tensor product of single-particle Hilbert spaces . If the identical particles are fermions, the -particle states are vectors in an antisymmetrized tensor product of single-particle Hilbert spaces (see symmetric algebra and exterior algebra respectively). A general state in Fock space is a linear combination of -particle states, one for each .

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 , $$F\_\backslash nu(H)=\backslash overline\{\backslash bigoplus\_\{n=0\}^\{\backslash infty\}S\_\backslash nu\; H^\{\backslash otimes\; n\}\}\; ~.$$

Here $S\_\backslash nu$ is the operator that symmetrizes or antisymmetrizes a tensor, depending on whether the Hilbert space describes particles obeying bosonic $(\backslash nu\; =\; +)$ or fermionic $(\backslash nu\; =\; -)$ statistics, and the overline represents the completion of the space. The bosonic (or fermionic) Fock space can alternatively be constructed as (the Hilbert space completion of) the $F\_+(H)\; =\; \backslash overline\{S^*H\}$ (or alternating tensors $F\_-(H)\; =\; \backslash overline\{\; \{\backslash bigwedge\}^*\; H\}$). For every basis of , there is a natural basis of the Fock space, the .

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

$$F\_\backslash nu(H)=\backslash bigoplus\_\{n=0\}^\{\backslash infty\}S\_\backslash nu\; H^\{\backslash otimes\; n\}\; =\; \backslash Complex\; \backslash oplus\; H\; \backslash oplus\; \backslash left(S\_\backslash nu\; \backslash left(H\; \backslash otimes\; H\backslash right)\backslash right)\; \backslash oplus\; \backslash left(S\_\backslash nu\; \backslash left(\; H\; \backslash otimes\; H\; \backslash otimes\; H\backslash right)\backslash right)\; \backslash oplus\; \backslash cdots$$

Here $\backslash Complex$, the Complex number, consists of the states corresponding to no particles, $H$ the states of one particle, $S\_\backslash nu\; (H\backslash otimes\; H)$ the states of two identical particles etc.

A general state in $F\_\backslash nu(H)$ is given by

$$|\backslash Psi\backslash rangle\_\backslash nu=\; |\backslash Psi\_0\backslash rangle\_\backslash nu\; \backslash oplus\; |\backslash Psi\_1\backslash rangle\_\backslash nu\; \backslash oplus\; |\backslash Psi\_2\backslash rangle\_\backslash nu\; \backslash oplus\; \backslash cdots\; =\; a\; |0\backslash rangle\; \backslash oplus\; \backslash sum\_i\; a\_i|\backslash psi\_i\backslash rangle\; \backslash oplus\; \backslash sum\_\{ij\}\; a\_\{ij\}|\backslash psi\_i,\; \backslash psi\_j\; \backslash rangle\_\backslash nu\; \backslash oplus\; \backslash cdots$$ where

• $|0\backslash rangle$ is a vector of length 1 called the vacuum state and $a\; \backslash in\; \backslash Complex$ is a complex coefficient,
• $|\backslash psi\_i\backslash rangle\; \backslash in\; H$ is a state in the single particle Hilbert space and $a\_i\; \backslash in\; \backslash Complex$ is a complex coefficient,
• $|\backslash psi\_i\; ,\; \backslash psi\_j\; \backslash rangle\_\backslash nu\; =\; a\_\{ij\}\; |\backslash psi\_i\backslash rangle\; \backslash otimes|\backslash psi\_j\backslash rangle\; +\; a\_\{ji\}\; |\backslash psi\_j\backslash rangle\backslash otimes|\backslash psi\_i\backslash rangle\; \backslash in\; S\_\backslash nu(H\; \backslash otimes\; H)$, and $a\_\{ij\}\; =\; \backslash nu\; a\_\{ji\}\; \backslash in\; \backslash Complex$ is a complex coefficient, etc.

The convergence of this infinite sum is important if $F\_\backslash nu(H)$ is to be a Hilbert space. Technically we require $F\_\backslash nu(H)$ to be the Hilbert space completion of the algebraic direct sum. It consists of all infinite $|\backslash Psi\backslash rangle\_\backslash nu\; =\; (|\backslash Psi\_0\backslash rangle\_\backslash nu\; ,\; |\backslash Psi\_1\backslash rangle\_\backslash nu\; ,\; |\backslash Psi\_2\backslash rangle\_\backslash nu,\; \backslash ldots)$ such that the norm, defined by the inner product is finite where the $n$ particle norm is defined by $$\backslash langle\; \backslash Psi\_n\; |\; \backslash Psi\_n\; \backslash rangle\_\backslash nu\; =\; \backslash sum\_\{i\_1,\backslash ldots\; i\_n\}\; \backslash sum\_\{j\_1,\; \backslash ldots\; j\_n\}\; a\_\{i\_1,\backslash ldots,\; i\_n\}^*\; a\_\{j\_1,\; \backslash ldots,\; j\_n\}\; \backslash langle\; \backslash psi\_\{i\_1\}|\; \backslash psi\_\{j\_1\}\; \backslash rangle\backslash cdots\; \backslash langle\; \backslash psi\_\{i\_n\}|\; \backslash psi\_\{j\_n\}\; \backslash rangle$$ i.e., the restriction of the norm on the tensor product $H^\{\backslash otimes\; n\}$

For two general states $$|\backslash Psi\backslash rangle\_\backslash nu=\; |\backslash Psi\_0\backslash rangle\_\backslash nu\; \backslash oplus\; |\backslash Psi\_1\backslash rangle\_\backslash nu\; \backslash oplus\; |\backslash Psi\_2\backslash rangle\_\backslash nu\; \backslash oplus\; \backslash cdots\; =\; a\; |0\backslash rangle\; \backslash oplus\; \backslash sum\_i\; a\_i|\backslash psi\_i\backslash rangle\; \backslash oplus\; \backslash sum\_\{ij\}\; a\_\{ij\}|\backslash psi\_i,\; \backslash psi\_j\; \backslash rangle\_\backslash nu\; \backslash oplus\; \backslash cdots,$$ and $$|\backslash Phi\backslash rangle\_\backslash nu=|\backslash Phi\_0\backslash rangle\_\backslash nu\; \backslash oplus\; |\backslash Phi\_1\backslash rangle\_\backslash nu\; \backslash oplus\; |\backslash Phi\_2\backslash rangle\_\backslash nu\; \backslash oplus\; \backslash cdots\; =\; b\; |0\backslash rangle\; \backslash oplus\; \backslash sum\_i\; b\_i\; |\backslash phi\_i\backslash rangle\; \backslash oplus\; \backslash sum\_\{ij\}\; b\_\{ij\}|\backslash phi\_i,\; \backslash phi\_j\; \backslash rangle\_\backslash nu\; \backslash oplus\; \backslash cdots$$ the inner product on $F\_\backslash nu(H)$ is then defined as $$\backslash langle\; \backslash Psi\; |\backslash Phi\backslash rangle\_\backslash nu\; :=\; \backslash sum\_n\; \backslash langle\; \backslash Psi\_n|\; \backslash Phi\_n\; \backslash rangle\_\backslash nu\; =\; a^*\; b\; +\; \backslash sum\_\{ij\}\; a\_i^*\; b\_j\backslash langle\backslash psi\_i\; |\; \backslash phi\_j\; \backslash rangle\; +\backslash sum\_\{ijkl\}a\_\{ij\}^*b\_\{kl\}\backslash langle\; \backslash psi\_i|\backslash phi\_k\backslash rangle\backslash langle\backslash psi\_j|\; \backslash phi\_l\; \backslash rangle\_\backslash nu\; +\; \backslash cdots$$ 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 $n$.

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Product states, indistinguishable particles, and a useful basis for Fock space A product state of the Fock space is a state of the form

$$|\backslash Psi\backslash rangle\_\backslash nu=|\backslash phi\_1,\backslash phi\_2,\backslash cdots,\backslash phi\_n\backslash rangle\_\backslash nu\; =\; |\backslash phi\_1\backslash rangle\; \backslash otimes\; |\backslash phi\_2\backslash rangle\; \backslash otimes\; \backslash cdots\; \backslash otimes\; |\backslash phi\_n\backslash rangle$$

which describes a collection of $n$ particles, one of which has quantum state $\backslash phi\_1$, another $\backslash phi\_2$ and so on up to the $n$th particle, where each $\backslash phi\_i$ is any state from the single particle Hilbert space $H$. Here juxtaposition (writing the single particle kets side by side, without the $\backslash otimes$) 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 $\backslash phi\_i$, we must bear 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 $|\backslash Psi\backslash rangle\_-$ is fermionic, it will be 0 if two (or more) of the $\backslash phi\_i$ are equal because the antisymmetric exterior product product $|\backslash phi\_i\; \backslash rangle\; |\backslash phi\_i\; \backslash rangle\; =\; 0$. 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 $\backslash \{|\backslash psi\_i\backslash rangle\backslash \}\_\{i\; =\; 0,1,2,\; \backslash dots\}$ of $H$, we can denote the state with $n\_0$ particles in state $|\backslash psi\_0\backslash rangle$, $n\_1$ particles in state $|\backslash psi\_1\backslash rangle$, ..., $n\_k$ particles in state $|\backslash psi\_k\backslash rangle$, and no particles in the remaining states, by defining

$$|n\_0,n\_1,\backslash ldots,n\_k\backslash rangle\_\backslash nu\; =\; |\backslash psi\_0\backslash rangle^\{n\_0\}|\backslash psi\_1\backslash rangle^\{n\_1\}\; \backslash cdots\; |\backslash psi\_k\backslash rangle^\{n\_k\},$$

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 $|\backslash psi\_i\backslash rangle$ 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 $a^\{\backslash dagger\}(\backslash phi)\backslash ,$ for creation and $a(\backslash phi)$for annihilation respectively. To create ("add") a particle, the quantum state $|\backslash phi\backslash rangle$ is symmetric or exterior- multiplied with $|\backslash phi\backslash rangle$; and respectively to annihilate ("remove") a particle, an (even or odd) interior product is taken with $\backslash langle\backslash phi|$, which is the adjoint of $a^\backslash dagger(\backslash phi)$. It is often convenient to work with states of the basis of $H$ 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 $|\backslash phi\_i\backslash rangle$ is $a^\{\backslash dagger\}(\backslash phi\_i)a(\backslash phi\_i)$.

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Wave function interpretation Often the one particle space $H$ is given as $L\_2(X,\; \backslash mu)$, the space of square-integrable functions on a space $X$ with measure $\backslash mu$ (strictly speaking, the equivalence classes of square integrable functions where functions are equivalent if they differ on a Null set). The typical example is the free particle with $H\; =\; L\_2(\backslash R^3,\; d^3x)$ 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 $X^0\; =\; \backslash \{*\backslash \}$ and $X^1\; =\; X$, $X^2\; =\; X\backslash times\; X$, $X^3\; =\; X\; \backslash times\; X\; \backslash times\; X$, etc. Consider the space of tuples of points which is the disjoint union

$$X^*\; =\; X^0\; \backslash amalg\; X^1\; \backslash amalg\; X^2\; \backslash amalg\; X^3\; \backslash amalg\; \backslash cdots\; .$$

It has a natural measure $\backslash mu^*$ such that $\backslash mu^*(X^0)\; =\; 1$ and the restriction of $\backslash mu^*$ to $X^n$ is $\backslash mu^n$. The even Fock space $F\_+(L\_2(X,\backslash mu))$ can then be identified with the space of symmetric functions in $L\_2(X^*,\; \backslash mu^*)$ whereas the odd Fock space $F\_-(L\_2(X,\backslash mu))$ can be identified with the space of anti-symmetric functions. The identification follows directly from the isometry mapping $$L\_2(X,\; \backslash mu)^\{\backslash otimes\; n\}\; \backslash to\; L\_2(X^n,\; \backslash mu^n)$$ $$\backslash psi\_1(x)\backslash otimes\backslash cdots\backslash otimes\backslash psi\_n(x)\; \backslash mapsto\; \backslash psi\_1(x\_1)\backslash cdots\; \backslash psi\_n(x\_n)$$.

Given wave functions $\backslash psi\_1\; =\; \backslash psi\_1(x),\; \backslash ldots\; ,\; \backslash psi\_n\; =\; \backslash psi\_n(x)$, the Slater determinant

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 $\backslash |\backslash Psi\backslash |\; =\; 1$ if the functions $\backslash psi\_1,\; \backslash ldots,\; \backslash psi\_n$ are orthonormal. There is a similar "Slater permanent" with the determinant replaced with the permanent which gives elements of $n$-sector of the even Fock space.

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Relation to the Segal–Bargmann space Define the Segal–Bargmann space $B\_N$ of complex holomorphic functions square-integrable with respect to a Gaussian measure:

where $$\backslash Vert\; f\backslash Vert\_\{\backslash mathcal\{F\}^2(\backslash Complex^N)\}\; :=\; \backslash int\_\{\backslash Complex^N\}\backslash vert\; f(\backslash mathbf\{z\})\backslash vert^2\; e^\{-\backslash pi\backslash vert\; \backslash mathbf\{z\}\backslash vert^2\}\backslash ,d\backslash mathbf\{z\}.$$ Then defining a space $B\_\backslash infty$ as the nested union of the spaces $B\_N$ over the integers $N\; \backslash ge\; 0$, Segal and Bargmann showed that $B\_\backslash infty$ is isomorphic to a bosonic Fock space. The monomial $$x\_1^\{n\_1\}...x\_k^\{n\_k\}$$ corresponds to the Fock state $$|n\_0,n\_1,\backslash ldots,n\_k\backslash rangle\_\backslash nu\; =\; |\backslash psi\_0\backslash rangle^\{n\_0\}|\backslash psi\_1\backslash rangle^\{n\_1\}\; \backslash cdots\; |\backslash psi\_k\backslash rangle^\{n\_k\}.$$

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See also

• Fock state
• Tensor algebra
• Holomorphic Fock space
• Creation and annihilation operators
• Slater determinant
• Wick's theorem
• Noncommutative geometry
• Grand canonical ensemble, thermal distribution over Fock space
• Schrödinger functional

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External links

• 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.

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