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] [1] 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 [2]).

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 [3], [4]).

References

 [1] V. Fock, Z. Phys. , 75 (1932) pp. 622–647 [2] F.A. Berezin, "The method of second quantization" , Acad. Press (1966) (Translated from Russian) (Revised (augmented) second edition: Kluwer, 1989) [3] R.L. Dobrushin, R.A. Minlos, "Polynomials in linear random functions" Russian Math. Surveys , 32 : 2 (1971) pp. 71–127 Uspekhi Mat. Nauk , 32 : 2 (1977) pp. 67–122 [4] B. Simon, "The Euclidean (quantum) field theory" , Princeton Univ. Press (1974)

The number operator $\sum _ {f} a _ \alpha ^ {*} ( f ) a _ \alpha ( f )$, $\alpha = a , s$, where $f$ runs through an orthonormal basis of $H$, has as eigen spaces with eigen value $n$ the spaces $\Gamma _ {n} ^ \alpha ( H)$. It is interpreted as giving the number of particles.