Fock space
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The Fock space is an
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
ic construction used in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
to construct the
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
s space of a variable or unknown number of identical
particles In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from s ...
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 Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as ...
"). M.C. Reed, B. Simon, "Methods of Modern Mathematical Physics, Volume II", Academic Press 1975. Page 328. 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 In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer ...
, the -particle states are vectors in a symmetrized
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
of single-particle Hilbert spaces . If the identical particles are
fermions In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
, the -particle states are vectors in an antisymmetrized tensor product of single-particle Hilbert spaces (see
symmetric algebra In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ...
and
exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
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_\nu(H)=\overline ~. Here S_\nu is the operator which symmetrizes or antisymmetrizes a tensor, depending on whether the Hilbert space describes particles obeying
bosonic In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...
(\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 tensor In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments: :T(v_1,v_2,\ldots,v_r) = T(v_,v_,\ldots,v_) for every permutation ''σ'' of the symbols Alternatively, a symmetric tensor of orde ...
s F_+(H) = \overline (resp.
alternating tensor In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged. section §7. The index subset must generally either be all ...
s F_-(H) = \overline). For every basis for there is a natural basis of the Fock space, the
Fock state In quantum mechanics, a Fock state or number state is a quantum state that is an element of a Fock space with a well-defined number of particles (or quanta). These states are named after the Soviet physicist Vladimir Fock. Fock states play an im ...
s.


Definition

The Fock space is the (Hilbert) direct sum of
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
s of copies of a single-particle Hilbert space H F_\nu(H)=\bigoplus_^S_\nu H^ = \Complex \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 \cdots Here \Complex, 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 general 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 \cdots = a , 0\rangle \oplus \sum_i a_i, \psi_i\rangle \oplus \sum_ a_, \psi_i, \psi_j \rangle_\nu \oplus \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, * , \psi_i , \psi_j \rangle_\nu = a_ , \psi_i\rangle \otimes, \psi_j\rangle + a_ , \psi_j\rangle\otimes, \psi_i\rangle \in S_\nu(H \otimes H), and a_ = \nu a_ \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
tuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
s , \Psi\rangle_\nu = (, \Psi_0\rangle_\nu , , \Psi_1\rangle_\nu , , \Psi_2\rangle_\nu, \ldots) such that the
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
, defined by the inner product is finite \, , \Psi\rangle_\nu \, _\nu^2 = \sum_^\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_ a_^* a_ \langle \psi_, \psi_ \rangle\cdots \langle \psi_, \psi_ \rangle i.e., the restriction of the norm on the tensor product H^ For two general states , \Psi\rangle_\nu= , \Psi_0\rangle_\nu \oplus , \Psi_1\rangle_\nu \oplus , \Psi_2\rangle_\nu \oplus \cdots = a , 0\rangle \oplus \sum_i a_i, \psi_i\rangle \oplus \sum_ a_, \psi_i, \psi_j \rangle_\nu \oplus \cdots, and , \Phi\rangle_\nu=, \Phi_0\rangle_\nu \oplus , \Phi_1\rangle_\nu \oplus , \Phi_2\rangle_\nu \oplus \cdots = b , 0\rangle \oplus \sum_i b_i , \phi_i\rangle \oplus \sum_ b_, \phi_i, \phi_j \rangle_\nu \oplus \cdots the
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
on F_\nu(H) is then defined as \langle \Psi , \Phi\rangle_\nu := \sum_n \langle \Psi_n, \Phi_n \rangle_\nu = a^* b + \sum_ a_i^* b_j\langle\psi_i , \phi_j \rangle +\sum_a_^*b_\langle \psi_i, \phi_k\rangle\langle\psi_j, \phi_l \rangle_\nu + \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 In quantum mechanics, separable states are quantum states belonging to a composite space that can be factored into individual states belonging to separate subspaces. A state is said to be entangled if it is not separable. In general, determinin ...
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 \otimes , \phi_2\rangle \otimes \cdots \otimes , \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 (writing the single particle kets side by side, without the \otimes) is symmetric (resp. antisymmetric) multiplication in the symmetric (antisymmetric)
tensor algebra In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
. 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 Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of ...
. 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 In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulat ...
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 \_ 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^, \psi_1\rangle^ \cdots , \psi_k\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 In quantum mechanics, a Fock state or number state is a quantum state that is an element of a Fock space with a well-defined number of particles (or quanta). These states are named after the Soviet physicist Vladimir Fock. Fock states play an im ...
. 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 Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually d ...
, which upon acting on a Fock state add or respectively remove a particle in the ascribed quantum state. They are denoted a^(\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 In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, or inner derivation) is a degree −1 (anti)derivation on the exterior algebra of ...
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 In quantum mechanics, for systems where the total number of particles may not be preserved, the number operator is the observable that counts the number of particles. The number operator acts on Fock space. Let :, \Psi\rangle_\nu=, \phi_1,\phi_2 ...
giving the number of particles in a specific state , \phi_i\rangle is a^(\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 Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...
\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 In physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field-free" space. I ...
with H = L_2(\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 In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ( ...
X^* = X^0 \bigsqcup X^1 \bigsqcup X^2 \bigsqcup X^3 \bigsqcup \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, \mu)^ \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 In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two elect ...
\Psi(x_1, \ldots x_n) = \frac \begin \psi_1(x_1) & \cdots & \psi_n(x_1) \\ \vdots & \ddots & \vdots \\ \psi_1(x_n) & \cdots & \psi_n(x_n) \\ \end 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 Permanent may refer to: Art and entertainment * ''Permanent'' (film), a 2017 American film * ''Permanent'' (Joy Division album) * "Permanent" (song), by David Cook Other uses * Permanent (mathematics), a concept in linear algebra * Permanent (cy ...
which gives elements of n-sector of the even Fock space.


Relation to the Segal–Bargmann space

Define the
Segal–Bargmann space In mathematics, the Segal–Bargmann space (for Irving Segal and Valentine Bargmann), also known as the Bargmann space or Bargmann–Fock space, is the space of holomorphic functions ''F'' in ''n'' complex variables satisfying the square-integr ...
B_N of complex holomorphic functions square-integrable with respect to a
Gaussian measure In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space R''n'', closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are nam ...
: \mathcal^2\left(\Complex^N\right) = \left\, where \Vert f\Vert_ := \int_\vert f(\mathbf)\vert^2 e^\,d\mathbf. Then defining a space B_\infty as the nested union of the spaces B_N over the integers N \ge 0 , Segal and Bargmann showed 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,\ldots,n_k\rangle_\nu = , \psi_0\rangle^, \psi_1\rangle^ \cdots , \psi_k\rangle^.


See also

*
Fock state In quantum mechanics, a Fock state or number state is a quantum state that is an element of a Fock space with a well-defined number of particles (or quanta). These states are named after the Soviet physicist Vladimir Fock. Fock states play an im ...
*
Tensor algebra In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
* Holomorphic Fock space *
Creation and annihilation operators Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually d ...
*
Slater determinant In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two elect ...
*
Wick's theorem Wick's theorem is a method of reducing high-order derivatives to a combinatorics problem. It is named after Italian physicist Gian-Carlo Wick. It is used extensively in quantum field theory to reduce arbitrary products of creation and annihila ...
*
Noncommutative geometry Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some g ...
*
Grand canonical ensemble In statistical mechanics, the grand canonical ensemble (also known as the macrocanonical ensemble) is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that are in thermodynamic equilibriu ...
, thermal distribution over Fock space


References


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. {{DEFAULTSORT:Fock Space Quantum mechanics Quantum field theory