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Journal of Mathematical Physics

Preliminary results in the Banach space formulation of master equation and subdynamics theory in quantum statistics

Maria Alberta Alberti, Paolo Cotta-Ramusino, G. Ramella

Published Jan 1, 1975• 4 citations
Publisher

Abstract

The Liouville−von Neumann equation and a useful decomposition of the resolvent of the generator of the time evolution operators are obtained in the formulation of quantum statistics by means of the pair of Banach spaces (τc, B), where τc is the space of the ’’trace−class’’ operators and B is the space of all bounded operators, defined on a Hilbert space.

DOI
10.1063/1.522405

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The Liouville−von Neumann equation and a useful decomposition of the resolvent of the generator of the time evolution operators are obtained in the formulation of quantum statistics by means of the pair of Banach spaces (τc, B), where τc is the space of the ’’trace−class’’ operators and B is the space of all bounded operators, defined on a Hilbert space.
We study some mathematical problems posed in nonequilibrium statistical mechanics and subdynamics theory developed by Prigogine and coworkers. We study in the superspace of the Hilbert–Schmidt operators the solution of the Liouville–von Neumann equation. The application in this frame of the spectral deformation methods yields expression of analytic continuations of a class of matrix elements of the resolvent of the Liouville–von Neumann operator, and this allows the analytic continuation of the collision and destruction operators. However, it is impossible to obtain simultaneously analytic continuation of the creation operator in this frame. The above results will be used in the second part of the article in order to study the pseudo-Markovian equation.
We study in the frame of the superspace of the Hilbert–Schmidt operators the contributions of some complex singularities of the analytic continuation of (Ψ(z)−z)−1 to the diagonal part of the solution of the Liouville–von Neumann equation. Under some conditions, the θ̄ operator of the pseudo-Markovian master equation can be explicitly constructed. It is necessary to specify the diagonal representation and the class of initial conditions having regularity properties. This Hilbertian structure does not allow the construction of a closed subspace which reduces the Liouville–von Neumann operator L, giving an exact irreversible subdynamics; more elaborated mathematical structures are therefore necessary. The above methods are illustrated in the case of the Lee model.
One of us (C.G) has developed a transformation theory in nonequilibrium statistical mechanics that generalizes the usual transformation theory of quantum mechanics in the framework of the resolvent formalism of the Liouville-von Neumann operator extensively used by the Brussels school. Prigogine, George, and Henin have shown that the evolution of a dissipative system can be decomposed into separate subdynamics in orthogonal subspaces. The π subspace, containing all thermodynamics, can be seen as an extension of the subspace of the invariants in the absence of dissipation. In this subspace a suitable contracted description of the system can be given through a dressing operator χ, which has the property that HR = χ−1H0 can be considered as the renormalized energy. A differential equation that determines χ has been proposed by Mandel. It is useful to see, even for a nondissipative system, what relation the transformation theory in the Liouville-von Neumann formalism has to the standard transformations used in quantum mechanics. In this paper we consider a system that can be completely diagonalized by the Bogoliubov transformation. It is shown that in this case the Mandel equation can be solved and that χ−1H0 = HD, where HD is the diagonalized Hamiltonian, obtained by the Bogoliubov transformation.
In this paper the results from various areas of mathematical research which are necessary for a consistent unification of the Dirac and von Neumann formulations of quantum mechanics are collected and presented as a single synthesis. For this purpose, direct integral decompositions of Hilbert space must be introduced into Dirac's formulation of spectral theory and representation theory; true unit vectors in the direct integral decomposition spaces replace unnormalizable vectors of infinite length. It then becomes clear that families of modified Dirac projection operators are simply related to the Radon-Nikodym derivative of von Neumann spectral measures. In terms of these mathematical preliminaries a second paper will present the more physical aspects of the resulting unified formulation of quantum mechanics.
For a system described in a phase space of generalized coordinates w and momenta J, the generalized master equation gives the time evolution of the reduced-density distribution function ρ(t, J) for the momenta. A generalization of the generalized master equation, having a similar non-Markoffian form, is derived for the full distribution function ρ(t, w, J). This equation is an alternate form of the Liouville equation. The derivation is an extension of a previous derivation of the generalized master equation from the Liouville equation utilizing projection operators in a Hilbert space. The time-evolution equation for the reduced distribution function ρr(t, wr, J), depending on the subset wr of the set of coordinates w, is derived. The approach to a stationary state for t → ∞ is discussed.