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The '''Fock space''' is an [[algebra]]ic construction used in [[quantum mechanics]] to construct the [[quantum state]]s space of a variable or unknown number of identical [[Subatomic particle|particles]] from a single particle [[Hilbert space]] <math>H</math>. It is named after [[Vladimir Fock|V. A. Fock]] who first introduced it in his paper ''Konfigurationsraum und zweite Quantelung''.<ref>V. Fock, ''Z. Phys''. '''75''' (1932), 622-647</ref><ref>[[Michael C. Reed|M.C. Reed]], [[Barry Simon|B. Simon]], "Methods of Modern Mathematical Physics, Volume II", Academic Press 1975. Page 328.</ref>
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Informally, a Fock space consists of a set of Hilbert spaces representing a zero particle state, a one particle state,  
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a two particle state, and so on. If the identical particles are [[bosons]], the <math>n</math>-particle state is a [[Symmetric tensor|symmetrized]] [[tensor product]] of <math>n</math> single particle Hilbert spaces <math>H</math>. If the identical particles are [[fermions]], the <math>n</math>-particle state is an [[Antisymmetric tensor|antisymmetrized]] tensor product of <math>n</math> single particle Hilbert spaces <math>H</math>. A state in Fock space is a [[linear combination]] of states, where each state has a definite number of particles.
 
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Technically, the Fock space is (the [[Hilbert space]] [[completion (metric space)|completion]] of) the [[Direct sum of modules|direct sum]] of the symmetric or antisymmetric tensors in the  
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[[Tensor product of Hilbert spaces|tensor power]]s of a single-particle Hilbert space ''H'':


:<math>F_\nu(H)=\overline{\bigoplus_{n=0}^{\infty}S_\nu H^{\otimes n}}</math>
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Here <math>S_\nu</math> is the [[Operator (physics)|operator]] which symmetrizes or [[Antisymmetric tensor|antisymmetrizes a tensor]], depending on whether the Hilbert space describes particles obeying [[Bose–Einstein statistics|bosonic]] <math>(\nu = +)</math> or [[Fermi–Dirac statistics|fermion]]ic <math>(\nu = -)</math> 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 tensor]]s <math>F_+(H) = \overline{S^*H}</math>  (resp. [[alternating tensor]]s <math>F_-(H) = \overline{{\bigwedge}^* H}</math>). For every basis for <math>H</math> there is a natural basis of the Fock space, the [[Fock state]]s.
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Fock space is the (Hilbert) [[Direct sum of modules|direct sum]] of [[tensor product]]s of copies of a single-particle Hilbert space <math>H</math>
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:<math>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</math>
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Here <math>\mathbb{C}</math>, a complex scalar, represents the states of no particles, <math>H</math> the state of one particle, <math>S_\nu (H\otimes H)</math> the states of two identical particles etc.
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A typical state in <math>F_\nu(H)</math> is given by
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:<math>|\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 </math>
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where
:<math>|0\rangle</math> is a vector of length 1, called the vacuum state and <math>\,a_0 \in \mathbb{C}</math> is a complex coefficient,
:<math> |\psi_1\rangle \in H</math> is a state in the single particle Hilbert space,
:<math> |\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)</math>, and <math> a_{ij} = \nu a_{ji} \in \mathbb{C}</math> is a complex coefficient
:etc.


The convergence of this infinite sum is important if <math>F_\nu(H)</math> is to be a Hilbert space. Technically we require <math>F_\nu(H)</math> to be the Hilbert space completion of the algebraic direct sum. It consists of all infinite [[tuple]]s <math>|\Psi\rangle_\nu = (|\Psi_0\rangle_\nu , |\Psi_1\rangle_\nu ,  
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|\Psi_2\rangle_\nu, \ldots)</math> such that the [[Norm (mathematics)|norm]], defined by the inner product is finite
 
:<math>\| |\Psi\rangle_\nu \|_\nu^2 = \sum_{n=1}^\infty \langle \Psi_n |\Psi_n \rangle_\nu < \infty </math>
  <li></li>
where the <math>n</math> particle norm is defined by
 
:<math> \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  </math>
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i.e. the restriction of the [[Tensor product of Hilbert spaces|norm on the tensor product]] <math>H^{\otimes n}</math>
 
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For two states
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:<math>|\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</math>, and
:<math>|\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</math>
the [[inner product]] on <math>F_\nu(H)</math> is then defined as
:<math>\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 </math>
where we use the inner products on each of the <math>n</math>-particle Hilbert spaces. Note that, in particular the <math>n</math> particle subspaces are orthogonal for different <math>n</math>.


== Example ==
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An example of a [[quantum state#Pure quantum state|pure state]] of the Fock space is
 
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:<math>|\Psi\rangle_\nu=|\phi_1,\phi_2,\cdots,\phi_n\rangle_\nu = |\phi_1\rangle|\phi_2\rangle \cdots |\phi_n\rangle</math>
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which describes a collection of <math>n</math> particles, one of which has quantum state <math>\phi_1\,</math>, another <math>\phi_2\,</math> and so on up to the <math>n</math><sup>th</sup> particle, where each <math>\phi_i\,</math> is ''any'' state from the single particle Hilbert space <math>H</math>. 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]].
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When we speak of ''one particle in state <math>\phi_i\,</math>'', it must be borne in mind that in quantum mechanics identical particles are [[identical particles|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 <math>|\Psi\rangle_-</math> is fermionic, it will be 0 if two (or more) of the <math>\phi_i\,</math> are equal because the anti symmetric [[exterior product|(exterior)]] product <math>|\phi_i \rangle |\phi_i \rangle = 0 </math>. 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).
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A useful and convenient basis for a Fock space is the ''occupancy number basis''. Given the choice of a basis <math>\{|\psi_i\rangle\}_{i = 0,1,2, \dots}</math> of <math>H</math>, we can denote the state with
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<math>n_0</math> particles in state <math>|\psi_0\rangle</math>,
 
<math>n_1</math> particles in state <math>|\psi_1\rangle</math>, ...,
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<math>n_k</math> particles in state <math>|\psi_k\rangle</math> by
 
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:<math>|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}</math>
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where each <math>n_i</math> 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  <math>|\psi_i\rangle</math> 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.
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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 <math>a^{\dagger}(\phi)\,</math> and <math>a(\phi)\,</math> respectively, with the quantum state <math>|\phi\rangle</math> the particle which is "added" by multiplication with <math>|\phi\rangle</math> respectively "removed" by (even or odd) [[interior product]] with <math>\langle\phi|</math> which is the adjoint of <math>a^\dagger(\phi)\,</math>. It is often convenient to work with states of the basis of <math>H</math> 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 <math>|\phi_i\rangle</math> is <math>a^{\dagger}(\phi_i)a(\phi_i)\,</math>.
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Often the one particle space <math>H</math> is given as <math>L_2(X, \mu)</math>, the space of [[square-integrable function]]s on a space <math>X</math> with [[Measure (mathematics)|measure]] <math>\mu</math> (strictly speaking, the [[equivalence class]]es 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 <math> H = L_2(\mathbb{R}^3, d^3x)</math> 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
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<math>X^0 = \{*\}</math> and <math>X^1 = X</math>, <math>X^2 = X\times X </math>, <math>X^3 = X  \times X \times X</math> etc.
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:<math>X^* = X^0 \bigsqcup X^1 \bigsqcup X^2 \bigsqcup X^3 \bigsqcup \ldots</math>.
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It has a natural measure <math>\mu^*</math> such that <math>\mu^*(X^0) = 1</math> and the restriction of <math>\mu^*</math> to <math>X^n</math> is <math>\mu^n</math>.
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The even Fock space <math>F_+(L_2(X,\mu))\,</math> can then be identified with the space of symmetric functions in <math>L_2(X^*, \mu^*)</math> whereas odd Fock space <math>F_-(L_2(X,\mu))\,</math> can be identified with the space of anti-symmetric functions.  The identification follows directly from the [[isometry|isometric]] mapping
:<math> L_2(X, \mu)^{\otimes n} \to L_2(X^n, \mu^n) </math>
:<math> \psi_1(x)\otimes\cdots\otimes\psi_n(x) \mapsto \psi_1(x_1)\cdots \psi_n(x_n)</math>.


Given wave functions <math>\psi_1 = \psi_1(x), \ldots , \psi_n = \psi_n(x) </math>, the [[Slater determinant]]
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: <math>\Psi(x_1, \ldots x_n) = \frac{1}{\sqrt{n!}}\left|\begin{matrix}
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              \psi_1(x_1) & \ldots & \psi_n(x_1) \\
                \vdots      &        &  \vdots    \\
              \psi_1(x_n) & \dots  & \psi_n(x_n) \\
              \end{matrix} \right|
  </math>
is an antisymmetric function on <math>X^n</math>. It can thus be naturally interpreted as an element of <math>n</math>-particle section of the odd Fock space. The normalisation is chosen such that <math>\|\Psi\| = 1</math>  if the functions <math>\psi_1, \ldots, \psi_n</math> are orthonormal. There is a similar "Slater permanent" with the determinant replaced with the [[permanent]] which gives elements of <math>n</math>-sector of the even Fock space.


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Define a space <math>B_n</math><ref name=Bargmann1961>{{cite journal|last=Bargmann|first=V.|title=On a Hilbert space of analytic functions and associated integral transform I|journal=Comm. Pure Math. Appl.|year=1961|volume=14|pages=187–214}}</ref> of complex [[holomorphic]] functions convergent with respect to a [[Gaussian measure]]:
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:<math>\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\}</math>, where <math>\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}</math>
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Then defining a space <math>B_\infty</math> as the amalgamation of spaces <math>B_n</math> over the integers <math> n \ge 1 </math>, Bargmann in 1961 showed <ref name=Bargmann1962>{{cite journal|last=Bargmann|first=V|title=Remarks on a Hilbert space of analytic functions|journal=Proc. Nat. Acad. Sci.|year=1962|volume=48|pages=199–204}}</ref> <ref name=Stochel1997>{{cite journal|last=Stochel|first=Jerzy B.|title=Representation of generalized annihilation and creation operators in Fock space|journal=UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA|year=1997|volume=34|pages=135–148|url=http://www.emis.de/journals/UIAM/actamath/PDF/34-135-148.pdf|accessdate=13 December 2012}}</ref> that <math>B_\infty</math> is isomorphic to a bosonic Fock space.
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* [[Tensor algebra]]
 
* [[Boson]]
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* [[Fermion]]
 
* [[Holomorphic Fock space]]
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* [[Creation and annihilation operators]]
 
* [[Slater determinant]]
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* [[Wick's theorem]]
 
* [[Noncommutative geometry]]
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* [[Grand canonical ensemble]] - thermal distribution over Fock space.
 
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*[http://www.pnas.org/content/100/15/8629.full.pdf Feynman diagrams and Wick products associated with q-Fock space - noncommutative analysis], Edward G. Effros and Mihai Popa, Department of Mathematics, UCLA
 
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{{DEFAULTSORT:Fock Space}}
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