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 $H$ ,

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

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

|\Psi\rangle_\nu= |\Psi_0\rangle_\nu \oplus |\Psi_1\rangle_\nu \oplus |\Psi_2\rangle_\nu \oplus \ldots = a_0 |0\rangle \oplus |\psi_1\rangle \oplus \sum_{ij} a_{ij}|\psi_{2i}, \psi_{2j} \rangle_\nu \oplus \ldots

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,

|\psi_{2i} \psi_{2j} \rangle_\nu = \frac{1}{2}(|\psi_{2i}\rangle \otimes|\psi_{2j}\rangle + (-1)^\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

\| |\Psi\rangle_\nu \|_\nu^2 = \sum_{n=1}^\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

|\Psi\rangle_\nu= |\Psi_0\rangle_\nu \oplus |\Psi_1\rangle_\nu \oplus |\Psi_2\rangle_\nu \oplus \ldots = a_0 |0\rangle \oplus |\psi_1\rangle \oplus \sum_{ij} a_{ij}|\psi_{2i}, \psi_{2j} \rangle_\nu \oplus \ldots

, and

|\Phi\rangle_\nu=|\Phi_0\rangle_\nu \oplus |\Phi_1\rangle_\nu \oplus |\Phi_2\rangle_\nu \oplus \ldots = b_0 |0\rangle \oplus |\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

\langle \Psi |\Phi\rangle_\nu:= \sum_n \langle \Psi_n| \Phi_n \rangle_\nu = a_0^* b_0 + \langle\psi_1 | \phi_1 \rangle +\sum_{ijkl}a_{ij}^*b_{kl}\langle \phi_{2i}|\psi_{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 $n$ .

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

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

|\Psi\rangle_\nu=|\phi_1,\phi_2,\cdots,\phi_n\rangle_\nu = |\phi_1\rangle|\phi_2\rangle \cdots |\phi_n\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 is symmetric respectively antisymmetric multiplication in the symmetric and antisymmetric tensor algebra. The general state in a Fock space is a linear combination of pure states. A state that cannot be written as a product of pure 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, 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 anti symmetric (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. 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 (respectively remove) a particle in the ascribed quantum state. They are denoted $a^{\dagger}(\phi)\,$ and $a(\phi)\,$ respectively, with the quantum state $|\phi\rangle$ the particle which is "added" by (symmetric or exterior) multiplication with $|\phi\rangle$ respectively "removed" by (even or odd) interior product 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 a basis 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 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 $n$ -particle section of the odd Fock space. The normalisation 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.

Some professors, such as Csaba Csaki of Cornell University fame, believe that this formula in fact has an error corresponding to the random variable $X$ distributed normally with mean $\mu$ and standard deviation $\frac{\sqrt{\mu}^2}{\psi_1!}$ . Applying this adjustment to the above Slater determinant, one determines the following necessary and sufficient condition for Slater permanency:

\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|

if and only if

P(\Psi(x_1, \ldots, x_n) = \bigsqcup_{i=0}^n X^i ) > \frac{\sqrt{\mu}^2}{\psi_1!} \cdot \Psi

This result is still being reviewed, however.

Relation to Segal-Bargmann space

Define the Segal–Bargmann space space $B_{n}$ ^[3] of complex holomorphic functions square-integrable with respect to a Gaussian measure:

\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

\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 amalgamation of spaces $B_{n}$ over the integers $n\geq 0$ , Segal ^[4] and Bargmann showed ^[5]^[6] that $B_\infty$ is isomorphic to a bosonic Fock space.

References

↑ V. Fock, Z. Phys. 75 (1932), 622-647
↑ M.C. Reed, B. Simon, "Methods of Modern Mathematical Physics, Volume II", Academic Press 1975. Page 328.
↑ Bargmann, V. (1961). "On a Hilbert space of analytic functions and associated integral transform I". Comm. Pure Math. Appl. 14: 187–214. doi:10.1002/cpa.3160140303.
↑ Segal, I. E. (1963). "Mathematical problems of relativistic physics". Proceedings of the Summer Seminar, Boulder, Colorado, 1960, Vol. II. Chap. VI.
↑ Bargmann, V (1962). "Remarks on a Hilbert space of analytic functions". Proc. Natl. Acad. Sci. 48: 199–204. Bibcode:1962PNAS...48..199B. doi:10.1073/pnas.48.2.199.
↑ Stochel, Jerzy B. (1997). "Representation of generalized annihilation and creation operators in Fock space" (PDF). UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA. 34: 135–148. Retrieved 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.

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