Fock space

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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 paper Konfigurationsraum und zweite Quantelung.[1][2]

Informally, a Fock space consists of a set of Hilbert spaces representing a zero particle state, a one particle state, a two particle state, and so on. If the identical particles are bosons, the n-particle state is a symmetrized tensor product of n single particle Hilbert spaces H. If the identical particles are fermions, the n-particle state is an antisymmetrized tensor product of n single particle Hilbert spaces H. A state in Fock space is a linear combination of states, where each state has a definite number of particles.

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[edit]

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}, a complex scalar, represents the states of no particles, H the state 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 = \lim_{M\to \infty}\sum_{i_1,\ldots i_n, j_1, \ldots j_n < M}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.

Example[edit]

An example of a pure state of the Fock space is

|\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 nth 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 the choice of 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 by

|n_0,n_1,\cdots,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. 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 pure state is the linear superposition of Fock 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 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[edit]

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.

Relation to Bargmann-Fock space[edit]

Define a space B_n[3] of complex holomorphic functions convergent 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 \ge 0 , Bargmann in 1961 showed [4] [5] that B_\infty is isomorphic to a bosonic Fock space.

See also[edit]

References[edit]

  1. ^ V. Fock, Z. Phys. 75 (1932), 622-647
  2. ^ M.C. Reed, B. Simon, "Methods of Modern Mathematical Physics, Volume II", Academic Press 1975. Page 328.
  3. ^ 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. 
  4. ^ Bargmann, V (1962). "Remarks on a Hilbert space of analytic functions". Proc. Nat. Acad. Sci. 48: 199–204. Bibcode:1962PNAS...48..199B. doi:10.1073/pnas.48.2.199. 
  5. ^ Stochel, Jerzy B. (1997). "Representation of generalized annihilation and creation operators in Fock space". UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA 34: 135–148. Retrieved 13 December 2012. 

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