Dissipative forces and the algebra of operators in stochastic quantum mechanics

Davidson's construction of a Hilbert space and of quantum operators on the basis of the Fényes-Nelson stochastic mechanics is extended to the case in which a dissipative force linear in the velocity is present. The hamiltonian becomes a nonlinear operator but the position and linear momentum operators are the same as in ordinary quantum mechanics.     

Dissipative hamiltonian systems: A unifying principle

The concept of hamiltonian sysem is generalized to include a wide class of dissipative processes. Evolution of any observable is generated jointly by a hamiltonian, with an entropy-conserving Poisson bracket, and an entropy, with an energy-conserving dissipative bracket. This approach yields many of the standard kinetic equations, such as those representing particle collisions, three-wave interactions, and wave-particle resonances.     

An infinite-dimensional classical integrable system and the Heisenberg and Schrödinger representations

We present an infinite-dimensional classical integrable hamiltonian system on projective Hilbert space. We show that the equations of motion correspond to the Heisenberg ones of quantum mechanics when the hamiltonian operator is compact, and that the formulation of these equations as a classical Lax pair with parameter gives rise naturally to an infinite set of conversation laws. Further, an infinite-dimensional version of Moser's transformation for integrating classical systems is shown to relate the Heisenberg and Schrödinger pictures.     

Exact canonically conjugate momentum to the quadrupole tensor and a microscopic derivation of the nuclear collective hamiltonian

Two momenta conjugate to the mass quadrupole tensor are given. The first is a canonical momentum only in a subspace of the shell model space. A microscopic collective kinetic energy in terms of this momentum and the quadrupole tensor is then obtained and compared with that of Bohr's hamiltonian. The second momentum is, on the other hand, canonically conjugate to the quadrupole tensor in the entire state space.     

Vlasov equation on a symplectic leaf

The infinite dimensional phase space of the Vlasov equation is foliated by symplectic manifolds (leaves) which are invariant under the dynamics. By adopting a Lie transform representation, exp{W, }, for near-identity canonical transformations we obtain a local coordinate system on a leaf. The evolution equation defined by restricting the Vlasov equation to the leaf is approximately represented by the evolution of W. We derive the equation for ∂_tW and show that it is hamiltonian relative to the nondegenerate Kirillov-Kostant-Souriau symplectic structure.     

Vortices in He II,current algebras and quantum knots

A canonical quantization scheme is developed for vertices in superfluid He II, using Dirac's technique for constrained hamiltonian systems. Quantization introduces in the theory in natural way the structure of the infinite Lie algebra of incompressible flows. We argue that all the topological invariants of the vortex, considered as a knot, can be regarded as observables of the system. Finally unitary representations of measure preserving flows on R³ and current algebra are discussed.     

Difficulty with a kinematic concept of unstable particles: the SZ.-Nagy extension and the Matthews-Salam-Zwanziger representation

We discuss the possibility of describing unstable systems, or dissipative systems in general, by vectors in a Hilbert space, evolving in time according to some non-unitary group or semigroup of translations. If the states of the unstable or dissipative system are embedded in a larger Hilbert space containing decay products as well, so that the time evolution of the system as a whole becomes unitary, we show that the infinitesimal generator necessarily has all energies from minus to plus infinity in its spectrum. This result supplements and extends the well-known fact that a positive energy spectrum is incompatible with a decay law bounded by a decreasing exponential. As an example of both facts, we discuss Zwanziger's irreducible, nonunitary representation of the Poincaré group; and we find its minimal, unitary extension (the Sz.-Nagy construction). The answer provides a mathematically canonical approach to the Matthews-Salam theory of wave functions for unstable, elementary particles, where the spectrum difficulty was already recognized. We speculate on the possibility that the Matthews-Salam-Zwanziger representation might be a strong coupling approximation in the relativistic version of the Wigner-Weisskopf theory, but we have not shown the existence of a physically acceptable model where that is so.     

A single qubit Landau-Zener gate

We study the hamiltonian and dissipative dynamics of a system undergoing a sequence of level crossings. The resulting Landau-Zener effect makes a new implementation of a general single qubit gate possible. In the dissipative case, with a periodic bias, the level crossing counteracts the interlevel relaxation and drives the system toward a two dimensional attractor. This feature can in principle be used to implement quantum memory devices of new type.     

Power injected in dissipative systems and the fluctuation theorem

We consider three examples of dissipative dynamical systems involving many degrees of freedom, driven far from equilibrium by a constant or time dependent forcing. We study the statistical properties of the injected and dissipated power as well as the fluctuations of the total energy of these systems. The three systems under consideration are: a shell model of turbulence, a gas of hard spheres colliding inelastically and excited by a vibrating piston, and a Burridge-Knopoff spring-block model. Although they involve different types of forcing and dissipation, we show that the statistics of the injected power obey the “fluctuation theorem" demonstrated in the case of time reversible dissipative systems maintained at constant total energy, or in the case of some stochastic processes. Although this may be only a consequence of the theory of large deviations, this allows a possible definition of “temperature" for a dissipative system out of equilibrium. We consider how this “temperature" scales with the energy and the number of degrees of freedom in the different systems under consideration. Received 26 June 2000 and Received in final form 24 October 2000     

介观耗散电容耦合电路量子化中的正则变换

针对介观耗散电容耦合RLC电路提出一种一般的正则变换, 并证明了其正确性和合理性. 用这种正则变换研究了双回路介观耗散电容耦合电路的量子化问题, 得出的对角化哈密顿量比文献中多出一非线性项. 这种具有普遍性的一般正则变换可能对研究介观多回路耗散系统的量子涨落、量子噪声等性质具有重要的意义. 关键词： RLC电路')" href="#">RLC电路 量子化 正则变换     

Geometry of canonical variables in gravity theories

We show how to pass from an SL(2, C) covariant to an SU(2) covariant formulation of the theories of gravity. Our construction determines the canonical and gauge variables of the theory and establishes an appropriate framework for a hamiltonian picture.     

Infinite-dimensional symplectic capacities and a squeezing theorem for Hamiltonian PDE's

We study partial differential equations of hamiltonian form and treat them as infinite-dimensional hamiltonian systems in a functional phase-space ofx-dependent functions. In this phase space we construct an invariant symplectic capacity and prove a version of Gromov's (non)squeezing theorem. We give an interpretation of the theorem in terms of the energy transition to high frequencies problem.     

Quantum Langevin systems

It is shown that the canonical quantization of dissipative systems, following a Langevin equation, reduce the physical applicability to particular approximations.     

Application of the Dyson-type non-unitary representation of the self-consistent collective-coordinate method to simple models (I). SU(3) model

A reformulation of the self-consistent collective-coordinate (SCC) method has recently been given in a Dyson-type non-unitary representation. This method seems very useful to evaluate the coupling effect between the collective and non-collective degrees of freedom in nuclear many-body systems. In order to see how well it works, it is applied to an exactly solvable SU(3) model. The optimized boundary condition proposed by Matsuo and Matsuyanagi is used in solving the basic equations of the SCC method. The collective hamiltonian derived by the SCC method is quantized by using the canonical quantization procedure, in which we propose the Dyson ordering of operators. The resultant eigenstates of the quantized hamiltonian are compared with those of the exact calculation and the Dyson boson mapping.     

Roadmap to ultra-short record high-energy pulses out of laser oscillators

Highly-chirped dissipative solitons of the cubic-quintic Ginzburg-Landau equation found in this work may provide a roadmap to design passively mode-locked laser oscillators that generate pulses of extremely high energy. We provide a region in the space of the system parameters where high-energy dissipative solitons are found, along with their typical spectral and temporal features.     

Distribution of local entropy in the Hilbert space of bi-partite quantum systems: origin of Jaynes' principle

For a closed bi-partite quantum system partitioned into system proper and environment we interpret the microcanonical and the canonical condition as constraints for the interaction between those two subsystems. In both cases the possible pure-state trajectories are confined to certain regions in Hilbert space. We show that in a properly defined thermodynamical limit almost all states within those accessible regions represent states of some maximum local entropy. For the microcanonical condition this dominant state still depends on the initial state; for the canonical condition it coincides with that defined by Jaynes' principle. It is these states which thermodynamical systems should generically evolve into. Received 13 June 2002 / Received in final form 14 November 2002 Published online 4 February 2003 RID="a" ID="a"e-mail: jochen@theol.physik.uni-stuttgart.de     

Spatially inhomogeneous thermo field dynamics

Non-equilibrium thermo field dynamics is extended to treat spatially inhomogeneous systems. A canonical formalism of thermally dissipative semi-free fields describing spatially inhomogeneous situations is presented. With this formalism, a scheme of perturbative calculations is developed and the “on-shell” renormalization condition is discussed. We illustrate this scheme using a model of particles interacting with impurities and find that the self-consistent renormalization condition gives the kinetic equation for the averaged particle number density as well as the renormalized energy and the dissipative coefficient. It is also shown that this kinetic equation can be reduced to the Boltzmann equation in the gradient expansion approximation.     

Extending integrable hamiltonian systems from 2 to N dimensions

We describe a method of constructing N-dimensional integrable hamiltonian systems starting from two-dimensional ones. Several models are examined. Included are the two candidates for integrability discovered by Lakshmanan and Sahadevan for which we find the integrals of motion. Results for other N-dimensional integrable hamiltonian systems are also presented.