On the quantum moment problem. (English. Russian original) Zbl 0654.44006

Daletskij, A. Yu.

Theory Probab. Math. Stat. 32, 9-16 (1986); translation from Teor. Veroyatn. Mat. Stat. 32, 9-17 (1985).

Let G be a connected simply connected Lie group, let \({\mathcal G}\) be its complex Lie algebra with generators \(e_ 1,...,e_ n\), let \(\{U_ g\}\) be a cyclic representation of G on a Hilbert space H and let \(\Omega\) be a cyclic vector of this representation. The triple \((\{U_ g\|| H,\Omega)\) is a non-commutative measure on G and its moments are \(m_{k_ 1,...,k_ n}=(A_ 1^{k_ 1}...A_ n^{k_ n}\Omega,\Omega)\) where \(\{A_ x\}\) is the representation of \({\mathcal G}\) corresponding to \(\{U_ g\}\). By solving the non-commutative moment problem for \((\{U_ g\},H,\Omega)\) means find conditions on a sequence \(\{s_{k_ 1,...,k_ n}\}^{\infty}_{k_ 1,...,k_ n}=0\) of complex numbers in order that there exists a non-commutative measure on G whose moments are all finite and equal \(S_{k_ 1,...,k_ n}.\)
The main theorem is: Theorem: For systems satisfying the canonical commutation relations [cf. I. M. Gel’fand and N. Ja. Wilenkin, Generalized functions. IV. Some applications of harmonic analysis (1964; Zbl 0103.092) or A. S. Kholevo, Probabilistic and statistical aspects of quantum theory (1982; Zbl 0497.46053)] then conditions (A) and (B) below are sufficient for a matrix \((a_{n,m})\) in order that the moment problem be uniquely solvable. Condition (A): \((a_{n,m})\) is positive-definite. Condition (B): \(| a_{n,0}|\) and \(| a_{0,n}|\) are less or equal to \(CM^ nn!\) where C and M are positive constants.

Reviewer: R.W.Shonkwiler

MSC:

44A60	Moment problems
42A70	Trigonometric moment problems in one variable harmonic analysis
22E45	Representations of Lie and linear algebraic groups over real fields: analytic methods
81S05	Commutation relations and statistics as related to quantum mechanics (general)

Keywords:

canonical commutation relations; representation in Lie groups; simply connected Lie group; complex Lie algebra; cyclic representation; Hilbert space; non-commutative measure; non-commutative moment problem

Citations:

Zbl 0103.092; Zbl 0497.46053