# Fock space

Fok space

In the simplest and most often used case, a Hilbert space consisting of infinite sequences of the form

$$\tag{1 } F = \{ f _ {0} , f _ {1} \dots f _ {n} ,\dots \} ,$$

where

$$f _ {0} \in \mathbf C ,\ \ f _ {1} \in L _ {2} ( \mathbf R ^ \nu , d ^ \nu x),\ \ f _ {n} \in L _ {2} ^ {s} (( \mathbf R ^ \nu ) ^ {n} ,\ ( d ^ \nu x) ^ {n} ),$$

or

$$f _ {n} \in L _ {2} ^ {a} (( \mathbf R ^ \nu ) ^ {n} ,\ ( d ^ \nu x) ^ {n} ),\ \ n = 2, 3 \dots \ \ \nu = 1, 2 \dots$$

in which

$$L _ {2} ^ {s} (( \mathbf R ^ \nu ) ^ {n} , ( d ^ \nu x) ^ {n} ) \ \ ( \textrm{ or } \ \ L _ {2} ^ {a} (( \mathbf R ^ \nu ) ^ {n} , ( d ^ \nu x) ^ {n} ))$$

denotes the Hilbert space of symmetric (respectively, anti-symmetric) functions in $n$ variables $x _ {1} \dots x _ {n} \in \mathbf R ^ \nu$, $n = 2, 3 ,\dots$. The scalar product of two sequences $F$ and $G$ of the form (1) is equal to

$$( F, G) = \ f _ {0} \overline{g}\; _ {0} + \sum _ {n = 1 } ^ \infty ( f _ {n} , g _ {n} ) _ {L _ {2} (( \mathbf R ^ \nu ) ^ {n} , ( d ^ \nu x) ^ {n} ) } .$$

In the case when the sequences $F$ consist of symmetric functions, one speaks of a symmetric (or boson) Fock space, and in the case of sequences of anti-symmetric functions the Fock space is called anti-symmetric (or fermion). Fock spaces were first introduced by V.A. Fock [V.A. Fok]  in this simplest case.

In the general case of an arbitrary Hilbert space $H$, the Fock space $\Gamma ^ {s} ( H)$( or $\Gamma ^ {a} ( H)$) constructed over $H$ is the symmetrized (or anti-symmetrized) tensor exponential of $H$, that is, the space

$$\tag{2 } \Gamma ^ \alpha ( H) \equiv \ \mathop{\rm Exp} _ \alpha H = \oplus \sum _ {n = 0 } ^ \infty ( H ^ {\otimes n } ) _ \alpha ,\ \ \alpha = s, a,$$

where the symbol $\oplus$ denotes the direct orthogonal sum of Hilbert spaces, $( H ^ {\otimes 0 } ) _ \alpha = \mathbf C ^ {1}$, $( H ^ {\otimes 1 } ) _ \alpha = H$, and $( H ^ {\otimes n } ) _ \alpha$, $n > 1$, is for $\alpha = s$ the symmetrized or for $\alpha = a$ the anti-symmetrized $n$- th tensor power of $H$. In the case $H = L _ {2} ( \mathbf R ^ \nu , d ^ \nu x)$ definition (2) is equivalent to the definition of a Fock space given at the beginning of the article, if one identifies the spaces $L _ {2} ^ \alpha (( \mathbf R ^ \nu ) ^ {n} , ( d ^ \nu x) ^ {n} )$ and $( L _ {2} ( \mathbf R ^ \nu , d ^ \nu x)) _ \alpha ^ {\otimes n }$ so that the tensor product

$$( f _ {1} \otimes \dots \otimes f _ {n} ) _ \alpha \ \in ( L _ {2} ( \mathbf R ^ \nu , d ^ \nu x)) _ \alpha ^ {\otimes n }$$

of the sequence of functions

$$f _ {1} \dots f _ {n} \in \ L _ {2} ( \mathbf R ^ \nu , d ^ \nu x)$$

corresponds to the function

$$\tag{3 } { \frac{1}{\sqrt n! } } \sum _ \sigma ( \pm 1) ^ { \mathop{\rm sign} \sigma } \prod _ {i = 1 } ^ { n } f ( x _ {\sigma ( i) } ) \in \ L _ {2} ^ \alpha (( \mathbf R ^ \nu ) ^ \nu , ( d ^ \nu x) ^ {n} ),$$

where the summation is taken over all permutations $\sigma$ of the indices $1 \dots n$; $\mathop{\rm sign} \sigma$ is the sign of $\sigma$, and the sign $+ 1$ or $- 1$ in (3) corresponds to the symmetric or anti-symmetric case.

In quantum mechanics, the Fock spaces $\Gamma ^ {s} ( H)$ or $\Gamma ^ {a} ( H)$ serve as the state spaces of quantum-mechanical systems consisting of an arbitrary (but finite) number of identical particles such that the state space of each separate particle is $H$. Here, depending on which of the Fock spaces — the symmetric $\Gamma ^ {s} ( H)$ or the anti-symmetric $\Gamma ^ {a} ( H)$— describes this system, the particles themselves are called bosons or fermions, respectively. For every $n = 1, 2 \dots$ the subspace $\Gamma _ {n} ^ \alpha ( H) \equiv ( H ^ {\otimes n } ) _ \alpha \subset \Gamma ^ \alpha ( H)$, $\alpha = s, a$, is called the $n$- particle subspace: The vectors in it describe those states in which there are exactly $n$ particles; the unit vector $\Omega \in ( H ^ {\otimes 0 } ) _ \alpha \subset \Gamma ^ \alpha ( H)$, $\alpha = s, a$( in the notation of (1): $\Omega = \{ 1, 0 \dots 0 ,\dots \}$), called the vacuum vector, describes the state of the system in which there are no particles.

In studying linear operators acting on the Fock spaces $\Gamma ^ {s} ( H)$ and $\Gamma ^ {a} ( H)$, one often applies a special formalism called the method of second quantization. It is based on introducing two families of linear operators on each of the spaces $\Gamma ^ \alpha ( H)$: the so-called annihilation operators $\{ {a _ \alpha ( f ) } : {f \in H } \}$, $\alpha = s, a$, and the family of operators adjoint to them $\{ {a _ \alpha ^ {*} ( f ) } : {f \in H } \}$, called creation operators. The annihilation operators are given as the closures of the operators acting on the vectors

$$\tag{4 } ( f _ {1} \otimes \dots \otimes f _ {n} ) _ \alpha \ \in \Gamma ^ \alpha ( H),\ \ \alpha = s, a,$$

where $( f _ {1} \otimes \dots \otimes f _ {n} ) _ \alpha$ are the symmetrized (for $\alpha = s$) or anti-symmetrized $( \alpha = a)$ tensor products of the sequences of vectors $f _ {1} \dots f _ {n} \in H$, $n = 1, 2 \dots$ according to the formulas

$$a _ \alpha ( f ) ( f _ {1} \otimes \dots \otimes f _ {n} ) _ \alpha = \ \sum _ {i = 1 } ^ { n } (- 1) ^ {g _ \alpha ( i) } ( f _ {i} , f ) \times$$

$$\times ( f _ {1} \otimes \dots \otimes f _ {i - 1 } \otimes f _ {i + 1 } \otimes \dots \otimes f _ {n} ) _ \alpha ,$$

$$\alpha = s , a,\ a _ \alpha ( f ) \Omega = 0,$$

where $g _ {s} ( i) = 0$ and $g _ {a} ( i) = i - 1$. The creation operators $a _ \alpha ^ {*} ( f )$ act on the vectors (3) according to the formulas

$$a _ \alpha ^ {*} ( f ) ( f _ {1} \otimes \dots \otimes f _ {n} ) _ \alpha = \ ( f \otimes f _ {1} \otimes \dots \otimes f _ {n} ) _ \alpha ,$$

$$a _ \alpha ^ {*} ( f ) \Omega = f.$$

Here for every $f \in H$, $a _ \alpha ( f )$: $\Gamma _ {n} ^ \alpha ( H) \rightarrow \Gamma _ {n - 1 } ^ \alpha ( H)$, $n = 1, 2 \dots$ and $a _ \alpha ^ {*} ( f ): \Gamma _ {n} ^ \alpha ( H) \rightarrow \Gamma _ {n + 1 } ^ {a} ( H)$, $n = 0, 1 \dots$ that is, states of the physical system with $n$ particles are mapped by the annihilation operators $a _ \alpha ( f )$ to states with $( n - 1)$ particles, and by the creation operators $a _ \alpha ^ {*} ( f )$ to states with $( n + 1)$ particles. Creation and annihilation operators occur in many cases of a similar system as "generators" in the collection of all operators (bounded and unbounded) acting on Fock spaces. The representation of such operators in the form of a sum (finite or infinite) of operators of the form

$$a _ \alpha ^ {*} ( f _ {1} ) \dots a _ \alpha ^ {*} ( f _ {n} ) a _ \alpha ( g _ {1} ) \dots a _ \alpha ( g _ {m} )$$

$$( f _ {1} \dots f _ {n} , g _ {1} \dots g _ {m} \in H,\ n, m = 0, 1 ,\dots)$$

— the so-called normal form of an operator — and methods of dealing with operators based on such a representation (computing functions of them, reducing operators to the "simplest" form, various examples of approximation, etc.) also constitute the content of the formalism of second quantization mentioned above (see ).

In the case of a symmetric Fock space over a real space $H$ there is a canonical isomorphism between that space and the Hilbert space of square-integrable functionals in a Gaussian linear random process $\{ {\xi _ {f} } : {f \in H } \}$ defined on $H$ such that

$$M _ {\xi _ {f} } = 0,\ \ M ( \xi _ {f _ {1} } \xi _ {f _ {2} } ) = \ ( f _ {1} , f _ {2} ) _ {H} ,\ \ f, f _ {1} , f _ {2} \in H.$$

This isomorphism, called the Itô–Segal–Wick mapping, is uniquely determined by the condition that for any orthonormal system of elements $f _ {1} \dots f _ {k} \in H$ and any collection of non-negative integers $n _ {1} \dots n _ {k}$ the vector

$$f _ {1} \otimes \dots \otimes f _ {1} \otimes \dots \otimes f _ {k} \otimes \dots \otimes f _ {k} \in \ \Gamma ^ {s} ( H)$$

( $n _ {1}$ times $f _ {1} \dots n _ {k}$ times $f _ {k}$) is mapped to the functional

$$\prod _ {i = 1 } ^ { k } H _ {n _ {i} } ( \xi _ {f _ {i} } ),$$

where the $H _ {n} ( \cdot )$, $n = 0, 1 \dots$ are the Hermite polynomials with leading coefficient one (see , ).

How to Cite This Entry:
Fock space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fock_space&oldid=46947
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article