An algebraic extension of Dirac quantization: Examples

An extension of the Dirac procedure for the quantization of constrained systems is necessary to address certain issues that are left open in Dirac's original proposal. These issues play an important role especially in the context of non-linear, diffeomorphism invariant theories such as general relativity. Recently, an extension of the required type was proposed by one of us using algebraic quantization methods. In this paper, the key conceptual and technical aspects of the algebraic program are illustrated through a number of finite dimensional examples. The choice of examples and some of the analysis is motivated by certain peculiar problems endemic to quantum gravity. However, prior knowledge of general relativity is not assumed in the main discussion. Indeed, the methods introduced and conclusions arrived at are applicable to any system with first class constraints. In particular, they resolve certain technical issues which are present also in the reduced phase space approach to quantization of these systems.Comment: 43 pages, Latex, CGPG-94/6-1. (References added; particularly to earlier work by C.J.Isham using group theoretic ideas, in the introduction.

On the Skolem Problem for Continuous Linear Dynamical Systems

The Continuous Skolem Problem asks whether a real-valued function satisfying a linear differential equation has a zero in a given interval of real numbers. This is a fundamental reachability problem for continuous linear dynamical systems, such as linear hybrid automata and continuous-time Markov chains. Decidability of the problem is currently open---indeed decidability is open even for the sub-problem in which a zero is sought in a bounded interval. In this paper we show decidability of the bounded problem subject to Schanuel's Conjecture, a unifying conjecture in transcendental number theory. We furthermore analyse the unbounded problem in terms of the frequencies of the differential equation, that is, the imaginary parts of the characteristic roots. We show that the unbounded problem can be reduced to the bounded problem if there is at most one rationally linearly independent frequency, or if there are two rationally linearly independent frequencies and all characteristic roots are simple. We complete the picture by showing that decidability of the unbounded problem in the case of two (or more) rationally linearly independent frequencies would entail a major new effectiveness result in Diophantine approximation, namely computability of the Diophantine-approximation types of all real algebraic numbers.Comment: Full version of paper at ICALP'1

Passivity Enforcement via Perturbation of Hamiltonian Matrices

This paper presents a new technique for the passivity enforcement of linear time-invariant multiport systems in statespace form. This technique is based on a study of the spectral properties of related Hamiltonian matrices. The formulation is applicable in case the system input-output transfer function is in admittance, impedance, hybrid, or scattering form. A standard test for passivity is first performed by checking the existence of imaginary eigenvalues of the associated Hamiltonian matrix. In the presence of imaginary eigenvalues the system is not passive. In such a case, a new result based on first-order perturbation theory is presented for the precise characterization of the frequency bands where passivity violations occur. This characterization is then used for the design of an iterative perturbation scheme of the state matrices, aimed at the displacement of the imaginary eigenvalues of the Hamiltonian matrix. The result is an effective algorithm leading to the compensation of the passivity violations. This procedure is very efficient when the passivity violations are small, so that first-order perturbation is applicable. Several examples illustrate and validate the procedure