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 . It is named after V. A. Fock who first introduced it in his 1932 paper "Konfigurationsraum und zweite Quantelung". 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 , formula_1 Here formula_2 is the operator which symmetrizes or antisymmetrizes a tensor, depending on whether the Hilbert space describes particles obeying bosonic formula_3 or fermionic formula_4 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 formula_5 (resp. alternating tensors formula_6, and formula_7 is a complex coefficient, etc. The convergence of this infinite sum is important if formula_8 is to be a Hilbert space. Technically we require formula_8 to be the Hilbert space completion of the algebraic direct sum. It consists of all infinite tuples formula_10 such that the norm, defined by the inner product is finite formula_11 where the formula_12 particle norm is defined by formula_13 i.e. , the restriction of the norm on the tensor product formula_14 For two general states formula_15 and formula_16 the inner product on formula_8 is then defined as formula_18 where we use the inner products on each of the formula_12-particle Hilbert spaces. Note that, in particular the formula_12 particle subspaces are orthogonal for different formula_12. Product states, indistinguishable particles, and a useful basis for Fock space. A product state of the Fock space is a state of the form formula_22 which describes a collection of formula_12 particles, one of which has quantum state formula_24, another formula_25 and so on up to  the formula_12th particle, where each formula_27 is "any" state from the single particle Hilbert space formula_28. Here juxtaposition (writing the single particle kets side by side, without the formula_29) 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 formula_27", 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 formula_31 is fermionic, it will be 0 if two (or more) of the formula_27 are equal because the antisymmetric (exterior) product formula_33. 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 formula_34 of formula_28, we can denote the state with formula_36 particles in state formula_37, formula_38 particles in state formula_39, ..., formula_40 particles in state formula_41, and no particles in the remaining states, by defining formula_42 where each formula_43 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 formula_44 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 formula_45  for creation and formula_46for annihilation respectively. To create ("add") a particle, the quantum state formula_47 is symmetric or exterior- multiplied with formula_47; and respectively to annihilate ("remove") a particle, an (even or odd) interior product is taken with formula_49, which is the adjoint of formula_50. It is often convenient to work with states of the basis of formula_28 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 formula_52 is formula_53. Wave function interpretation. Often the one particle space formula_28 is given as formula_55, the space of square-integrable functions on a space formula_56 with measure formula_57 (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 formula_58 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 formula_59 and formula_60, formula_61, formula_62, etc. Consider the space  of tuples of points which is the disjoint union formula_63 It has a natural measure formula_64 such that formula_65 and the restriction of formula_64 to formula_67 is formula_68. The even Fock space formula_69 can then be identified with the space of symmetric functions in formula_70 whereas the odd Fock space formula_71 can be identified with the space of anti-symmetric functions. The identification follows directly from the isometric mapping formula_72 formula_73. Given wave functions formula_74, the Slater determinant formula_75 is an antisymmetric function on formula_67. It can thus be naturally interpreted as an element of the formula_12-particle sector of  the odd Fock space. The normalization is chosen such that formula_78 if the functions formula_79 are orthonormal. There is a similar "Slater permanent" with the determinant replaced with the permanent which gives elements of formula_12-sector of the even Fock space. Relation to the Segal–Bargmann space. Define the Segal–Bargmann space space formula_81 of complex holomorphic functions square-integrable with respect to a Gaussian measure: formula_82 where formula_83 Then defining a space formula_84 as the nested union of the spaces formula_81 over the integers formula_86, Segal and Bargmann showed that formula_84 is isomorphic to a bosonic Fock space. The  monomial formula_88 corresponds to the Fock state formula_89