Dirichlet forms and symmetric Markovian semigroups on CCR algebras with respect to quasi-free states

Abstract

Employing the construction method of Dirichlet forms on standard forms of von Neumann algebras developed in Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2000, Vol. 3, No. 1, pp. 1–14 (Ref. 1), we construct Dirichlet forms and associated symmetric Markovian semigroups on CCR algebras with respect to quasi-free states. More precisely, let A(h0) be the CCR algebra over a complex separable pre-Hilbert space h0 and let ω be a quasi-free state on A(h0). For any normalized admissible function f and complete orthonormal system (CONS) {gn}⊂h0, we construct a Dirichlet form and corresponding symmetric Markovian semigroup on the natural standard form associated to the GNS representation of (A(h0),ω). It turns out that the form is independent of admissible function f and CONS {gn} chosen. By analyzing the spectrum of the generator (Dirichlet operator) of the semigroup, we show that the semigroup is ergodic and tends to the equilibrium exponentially fast.

Remarks on the Structure of Dirichlet Forms on Standard Forms of Von Neumann Algebras

“…Also it may be possible to extend Theorem 2.1 to the case in which x is an unbounded operator affiliated with M [8,7].…”

“…One of the reasons is that the general structure of Dirichlet forms for non-tracial states is not well-understood compared to the tracial case [2,3,6,12]. For constructions of Dirichlet forms for non-tracial states, we refer to [8,9,11,18,20,21,25,23] and the references there in. In [23], we gave a general construction method of Dirichlet forms on standard forms of von Neumann algebras.…”

“…In [23], we gave a general construction method of Dirichlet forms on standard forms of von Neumann algebras. The method has been used to construct (translation invariant) symmetric Markovian semigroups for quantum spin systems [23], the CCR and CAR algebras with respect to quasi-free states [8,9] and quantum mechanical systems [7].…”

Continuity and generators of dynamical semigroups for infinite Bose systems

“…On von Neumann algebras in standard form a one-to-one correspondence between dynamical semigroups satisfying a certain symmetry condition and noncommutative Dirichlet forms has been established [13], and a method for constructing certain noncommutative Dirichlet forms has been given [30]. This has been applied to construct an example of a weak * -continuous quantum dynamical semigroup on representations of the CCR-algebra with respect to quasifree states [4]. However, these methods have not resulted in a rich supply of examples of dynamical semigroups on von Neumann algebras, in particular not on von Neumann algebras corresponding to representations of the CCR.…”

Decoherence in infinite quantum systems

“…In order to construct quantum dynamical semigroups on von Neumann algebras in standard form a one-to-one correspondence between dynamical semigroups satisfying a certain symmetry condition and noncommutative Dirichlet forms has been established [58], and a method for constructing certain noncommutative Dirichlet forms has been given [108]. This has been applied to construct an example of a weak* continuous quantum dynamical semigroup on representations of the CCR algebra with respect to quasifree states [43]. However, these methods have not resulted in a rich supply of examples of dynamical semigroups on von Neumann algebras, in particular not on von Neumann algebras corresponding to representations of the CCR.…”