A family of closed linear operators
, where
is some Hilbert space, acting on a Fock space constructed from
(i.e. on the symmetrization
or anti-symmetrization
of the space of tensors over
) such that on the vector
,
, consisting of the symmetrized
or anti-symmetrized
tensor product of a sequence of elements
,
in
, they are given by the formulas:
![](https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255014.png) | (1) |
in the symmetric case, and
![](https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255016.png) | (2) |
in the anti-symmetric case; the empty vector
,
(i.e. the unit vector in the subspace of constants in
) is mapped to zero by
. In these formulas
is the inner product in
. The operators
dual to the operators
are called creation operators; their action on the vectors
,
,
is given by the formulas
![](https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255029.png) | (3) |
and
As a consequence of these definitions, for each
the subspace
,
, the symmetrized or anti-symmetrized
-th tensor power of
, is mapped by
into
and by
into
.
In quantum physics, the Fock space
,
, is interpreted as the state space of a system consisting of an arbitrary (finite) number of identical quantum particles, the space
is the state space of a single particle, the subspace
corresponds to the states of the system with
particles, i.e. states in which there are just
particles. A state with
particles is mapped by
to a state with
particles ( "annihilation" of a particle), and by
to a state with
particles ( "creation" of a particle).
The operators
and
form irreducible families of operators satisfying the following permutation relations: In the symmetric case (the commutation relations)
![](https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255053.png) | (4) |
and in the anti-symmetric case (the anti-commutation relations)
![](https://www.encyclopediaofmath.org/legacyimages/a/a012/a012550/a01255056.png) | (5) |
where
is the identity operator in
or
. Besides the families of operators
and
,
, described here, there exist in the case of an infinite-dimensional space
also other irreducible representations of the commutation and anti-commutation relations (4) and (5), not equivalent to those given above. Sometimes they are also called creation and annihilation operators. In the case of a finite-dimensional space
, all irreducible representations of the commutation or anti-commutation relations are unitarily equivalent.
The operators
are in many connections convenient "generators" in the set of all linear operators acting in the space
,
, and the representation of such operators as the sum of arbitrary creation and annihilation operators (the normal form of an operator) is very useful in applications. The connection with this formalism bears the name method of second quantization, cf. [1].
In the particular, but for applications important, case in which
,
(or in a more general case
, where
is a measure space), the family of operators
defines two operator-valued generalized functions
and
such that
The introduction of
and
turns out to be convenient for the formalism of second quantization (e.g. it allows one directly to consider operators of the form
where
is a certain "sufficiently-good" function), without having to recourse to their decomposition as a series in the monomials
where
References
[1] | F.A. Berezin, "The method of second quantization" , Acad. Press (1966) (Translated from Russian) (Revised (augmented) second edition: Kluwer, 1989) |
[2] | R.L. Dobrushin, R.A. Minlos, Uspekhi Mat. Nauk , 32 : 2 (1977) pp. 67–122 |
[3] | L. Gårding, A. Wightman, Proc. Nat. Acad. Sci. U.S.A. , 40 : 7 (1954) pp. 617–626 |
References
[a1] | J. Glimm, A. Jaffe, "Quantum physics, a functional integral point of view" , Springer (1981) |
[a2] | N.N. Bogolyubov, A.A. Logunov, I.T. Todorov, "Introduction to axiomatic quantum field theory" , Benjamin (1975) (Translated from Russian) |
[a3] | J. de Boer, "Construction operator formalism in many particle systems" J. de Boer (ed.) G.E. Uhlenbeck (ed.) , Studies in statistical mechanics , North-Holland (1965) |
How to Cite This Entry:
Annihilation operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Annihilation_operators&oldid=45188
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article