Fock space explained

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 . It is named after V. A. Fock who first introduced it in his 1932 paper "Konfigurationsraum und zweite Quantelung" ("Configuration space and second 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 are bosons, the -particle states are vectors in a symmetrized 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,$$F\_\backslash nu(H)=\backslash overline\; ~.$$

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 $F\_-(H)\; =\; \backslash overline$). For every basis for 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

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

Here

\Complex , the complex scalars, consists of the states corresponding to no particles, H the states of one particle, S _{\nu} (H ⊗ H) the states of two identical particles etc.

A general state in

F _{\nu(H)} is given by

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

| 0 \rangle is a vector of length 1 called the vacuum state and a \in \Complex is a complex coefficient,|\psi _{i\rangle} \in H is a state in the single particle Hilbert space and a _{i} \in \Complex is a complex coefficient,

• $|\backslash psi\_i,\; \backslash psi\_j\; \backslash rangle\_\backslash nu\; =\; a\_\; |\backslash psi\_i\backslash rangle\; \backslash otimes|\backslash psi\_j\backslash rangle\; +\; a\_\; |\backslash psi\_j\backslash rangle\backslash otimes|\backslash psi\_i\backslash rangle\; \backslash in\; S\_\backslash nu(H\; \backslash otimes\; H)$, and

a _{ij} = \nu a _{ji} \in \Complex 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 finitewhere the n particle norm is defined by$$\backslash langle\; \backslash Psi\_n\; |\; \backslash Psi\_n\; \backslash rangle\_\backslash nu\; =\; \backslash sum\_\; a\_^*\; a\_\; \backslash langle\; \backslash psi\_|\; \backslash psi\_\; \backslash rangle\backslash cdots\; \backslash langle\; \backslash psi\_|\; \backslash psi\_\; \backslash rangle$$i.e., the restriction of the norm on the tensor product H ^{⊗}

For two general states$$|\backslash Psi\backslash rangle\_\backslash nu=\; |\backslash Psi\_0\backslash rangle\_\backslash nu\; \backslash oplus\; |\backslash Psi\_1\backslash rangle\_\backslash nu\; \backslash oplus\; |\backslash Psi\_2\backslash rangle\_\backslash nu\; \backslash oplus\; \backslash cdots\; =\; a\; |0\backslash rangle\; \backslash oplus\; \backslash sum\_i\; a\_i|\backslash psi\_i\backslash rangle\; \backslash oplus\; \backslash sum\_\; a\_|\backslash psi\_i,\; \backslash psi\_j\; \backslash rangle\_\backslash nu\; \backslash oplus\; \backslash cdots,$$ and$$|\backslash Phi\backslash rangle\_\backslash nu=|\backslash Phi\_0\backslash rangle\_\backslash nu\; \backslash oplus\; |\backslash Phi\_1\backslash rangle\_\backslash nu\; \backslash oplus\; |\backslash Phi\_2\backslash rangle\_\backslash nu\; \backslash oplus\; \backslash cdots\; =\; b\; |0\backslash rangle\; \backslash oplus\; \backslash sum\_i\; b\_i\; |\backslash phi\_i\backslash rangle\; \backslash oplus\; \backslash sum\_\; b\_|\backslash phi\_i,\; \backslash phi\_j\; \backslash rangle\_\backslash nu\; \backslash oplus\; \backslash cdots$$the inner product on

F _{\nu(H)} is then defined as$$\backslash langle\; \backslash Psi\; |\backslash Phi\backslash rangle\_\backslash nu\; :=\; \backslash sum\_n\; \backslash langle\; \backslash Psi\_n|\; \backslash Phi\_n\; \backslash rangle\_\backslash nu\; =\; a^*\; b\; +\; \backslash sum\_\; a\_i^*\; b\_j\backslash langle\backslash psi\_i\; |\; \backslash phi\_j\; \backslash rangle\; +\backslash sum\_a\_^*b\_\backslash langle\; \backslash psi\_i|\backslash phi\_k\backslash rangle\backslash langle\backslash psi\_j|\; \backslash phi\_l\; \backslash rangle\_\backslash nu\; +\; \backslash cdots$$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 .

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 \phi _{1} , another \phi _{2} and so on up to the n th 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 ⊗ ) 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} , 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 |\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} of H , we can denote the state withn _{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

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

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 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 ^{3 ,} _{2 (\R} 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 x X , X ^{3} = X x X x X , etc.Consider the space of tuples of points which is the disjoint union

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

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$$L\_2(X,\; \backslash mu)^\; \backslash to\; L\_2(X^n,\; \backslash mu^n)$$$$\backslash psi\_1(x)\backslash otimes\backslash cdots\backslash otimes\backslash psi\_n(x)\; \backslash mapsto\; \backslash psi\_1(x\_1)\backslash cdots\; \backslash psi\_n(x\_n)$$.

Given wave functions

\psi _{1} = \psi _{1(x),} \ldots , \psi _{n} = \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 \|\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 -sector of the even Fock space.

Relation to the Segal–Bargmann space

B _{N} ^[3] of complex holomorphic functions square-integrable with respect to a Gaussian measure:

$$\backslash mathcal^2\backslash left(\backslash Complex^N\backslash right)\; =\; \backslash left\backslash ,$$where$$\backslash Vert\; f\backslash Vert\_\; :=\; \backslash int\_\backslash vert\; f(\backslash mathbf)\backslash vert^2\; e^\backslash ,d\backslash mathbf.$$Then defining a space

B _{infty} as the nested union of the spaces B _{N} over the integers N \ge 0 , Segal^[4] and Bargmann showed^{[5] [6]} that B _{infty} is isomorphic to a bosonic Fock space. The monomial$$x\_1^...x\_k^$$corresponds to the Fock state$$|n\_0,n\_1,\backslash ldots,n\_k\backslash rangle\_\backslash nu\; =\; |\backslash psi\_0\backslash rangle^|\backslash psi\_1\backslash rangle^\; \backslash cdots\; |\backslash psi\_k\backslash rangle^.$$

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

References

1. Fock . V. . Vladimir Fock. Konfigurationsraum und zweite Quantelung . Zeitschrift für Physik . Springer Science and Business Media LLC . 75 . 9–10 . 1932 . 1434-6001 . 10.1007/bf01344458 . 622–647 . 1932ZPhy...75..622F . 186238995 . de.
2. [Michael C. Reed|M.C. Reed]
3. Bargmann. V.. On a Hilbert space of analytic functions and associated integral transform I. Communications on Pure and Applied Mathematics . 1961. 14. 187–214. 10.1002/cpa.3160140303. 10338.dmlcz/143587. free.
4. I. E. . Segal . 1963 . Mathematical problems of relativistic physics . Chap. VI . Proceedings of the Summer Seminar, Boulder, Colorado, 1960, Vol. II .
5. Bargmann. V. Remarks on a Hilbert space of analytic functions . Proc. Natl. Acad. Sci.. 1962. 48. 2. 199–204. 10.1073/pnas.48.2.199. 16590920. 1962PNAS...48..199B . 220756. free.
6. Stochel. Jerzy B.. Representation of generalized annihilation and creation operators in Fock space. Universitatis Iagellonicae Acta Mathematica. 1997. 34. 135–148. 13 December 2012.

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.