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Density matrix

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of a state defined on the algebra of bounded linear operators acting on a Hilbert space

The positive nuclear operator such that

(1)

where . Conversely, any state , i.e. any linear positive normalized functional on , can be represented in the form (1), i.e. it has a density matrix , which is moreover unique.

The concept of a density matrix arose in statistical physics in defining a Gibbs quantum state. Let a quantum system occupying a finite volume in be described by the vectors of a certain Hilbert space , by the Hamiltonian and, possibly, by some set of mutually commuting "first integrals" , . A Gibbs state for such a system is a state on defined by the density matrix

(2)

where is a normalizing factor and , are real parameters.

In addition to the density matrix (2), the state of a system in quantum statistical physics may be defined by means of the so-called reduced density matrix. In the simplest case of a system of identical particles (bosons or fermions) described by the vectors of a Fock [Fok] space , the reduced density matrix of a state is the set of (in general, generalized) functions

(3)

where

and where , , are the creation operators and annihilation operators, respectively, acting in . If the creation and annihilation operators in are replaced by some other system of generators ( is a certain set of indices), then the reduced density matrix for a state is defined by analogy with (3) as the set of values of on all possible monomials of the form

The reduced density matrix is particularly convenient for determining the limiting Gibbs state in a -algebra of so-called quasi-local observables: (the bar denotes closure in the uniform topology).

References

[1] L.D. Landau, E.M. Lifshitz, "Statistical physics" , Pergamon (1980) (Translated from Russian)
[2] D. Ruelle, "Statistical mechanics: rigorous results" , Benjamin (1974)


Comments

The statement that any state has a representation (1) has been proved for finite-dimensional only.

The functions defined by (3) are the quantum analogues of distribution functions.

How to Cite This Entry:
Density matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Density_matrix&oldid=46627
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article