Stochastic Wess-Zumino-Witten model over a symplectic manifold

Over the path space of a symplectic manifold with end points in two Lagrangian submanifolds, we define a measure and a stochastic symplectic action in the simply connected case. We define a regularized Wess-Zumino-Witten Laplacian over the forms of finite degree over the path space. We perform a short time asymptotic near the critical points and find a limit Brownian harmonic oscillator: we can diagonalize it explicitly, and find the limit ground state of the Laplacian. We define a stochastic Witten complex, and its algebraic counterpart at the level of Chen forms.     

Excited states time evolution on a laser-ablated molybdenum plume

The dynamics of the excited states on a laser-ablated Mo plume was studied, both in air and in vacuum, by emission spectroscopy along the plume expansion axis. The emission related to ionized atoms occurs in the beginning of the plume expansion, near the metal surface, and is predominantly ultraviolet emission. In the middle of the plasma plume, it takes place the electron transitions between excited states of neutral atoms, and in the end of the plume, the visible emission is to transitions to the ground state of neutral molybdenum atoms. It was possible to determine plume parameters such as plasma expansion velocity of (5.0 ± 0.7) km/s at atmospheric pressure and (4.0 ± 0.7) km/s in vacuum, and the plasma duration that was (160 ± 14) ns at atmospheric pressure and (138 ± 18) ns in vacuum.     

Stochastic theory of metastable states and nucleation

We study on a model the role of fluctuations size in the nucleation of a first order phase transition. A bifurcation point exists between metastable and stable equilibrium solutions for a critical value of fluctuations size.     

Brownian motion on a manifold

The question of the existence and correct form of equations describing Brownian motion on a manifold cannot be answered by mathematics alone, but requires a study of the underlying physics. As in classical mechanics, manifolds enter through the transformation of variables needed to account for the presence of constraints. The constraints are either due to a physical agency that forces the motion to remain on a manifold, or they represent conserved quantities of the equation of motion themselves. Also the Brownian motion is described either by a Smoluchowski diffusion equation or by a Kramers equation. The four cases lead to the following conclusions, (i) Smoluchowski diffusion with a conserved quantity reduces to a diffusion equation on the manifold; (ii) The same is true for diffusion with a physical constraint in three dimensions, but in more dimensions it may happen thatno autonomous equation on the manifold results; (iii) A Kramers equation with a conserved quantity reduces to an equation on the manifold, but in general not of the form of a Kramers equation; (iv) The Kramers equation with a physical constraint reduces to an autonomous Kramers equation on the manifold only for a special shape of that constraint. Throughout, only a certain type of physical constraints has been envisaged, and global questions are ignored. Finally, the customary heuristic construction of a Fokker-Planck equation for a mechanical system on a manifold is demonstrated for the case of Brownian rotation of a rigid body, and its shortcomings are emphasized.     

Time evolution of quasi-stationary states

In this paper we study the time evolution of prepared states in some quantum mechanical models, and discuss the probability of decay and the rate of energy dissipation and their dependence on the form of the interaction. First we consider solvable models with divergent matrix elements for the operatorH ², whereH is the Hamiltonian of the system. We study two specific examples, one with well-defined eigenvalues and the other with renormalizable interaction. The time development of the initial state in the latter case depends on the cut-off parameter. In the second part of the paper, we show the possibility of existence of decaying states with long lifetime, where the amplitude of the initial state decreases like a Bessel function. In the third part, we determine the time development of a prepared state in a simple many-boson problem. Finally we study the problem of penetration of a wave packet through two phase-equivalent potential barriers, and we conclude that from the scattering phase shifts alone, it is not possible to determine the lifetime or the mode of decay of an unstable particle uniquely.     

Stochastic evolution of Yang-Mills connections on the noncommutative two-torus

We study quantum stochastic parallel transport processes where the noise terms arise from quantum Brownian motion in Fock space and the connection is chosen to minimize the Yang-Mills functional on a Heisenberg module over the smooth algebra of the noncommutative two-torus. Each such process yields a dilation of a quantum dynamical semigroup whose action on components of the connection induces a family of transformations of the moduli space. From a physical point of view, this describes a highly singular interaction between quantized Yang-Mills fields and the free boson field.     

A remark on time evolution of coherent states

The general condition for a coherent state to remain coherent at all times is considered.     

Stochastic evolution of Schwarzschild black hole in a radiation heat bath

The mass of a Schwarzschild black hole in equilibrium with black-body radiation is shown to undergo a random drift with a diffusion coefficient D ～ M^-3. This follows from the master equation for the radiation in a stochastically bistable system of a black hole in an isolating cavity.     

Difficulties for the evolution of pure states into mixed states

Motivated by Hawking's proposal that the quantum-mechanical density matrix ? obeys an equation more general than the Schrödinger equation, we study the general properties of evolution equations for ?. We argue that any more general equation for ? violates either locality or energy-momentum conservation.     

Algebras of local observables on a manifold

We propose a generalization of the Haag-Kastler axioms for local observables to Lorentzian manifolds. The framework is intended to resolve ambiguities in the construction of quantum field theories on manifolds. As an example we study linear scalar fields for globally hyperbolic manifolds.Supported by National Science Foundation PHY 77-21740.On leave from Department of Mathematics, SUNY at Buffalo, Buffalo, NY 14214, USA     

Nuclear structure on a Grassmann manifold

Products of particlelike representations of the homogeneous Lorentz group are used to construct the degrees of spin angular momentum of a composite system of protons and neutrons. If a canonical labeling system is adopted for each state, a shell structure emerges. Furthermore the use of the Dirac ring ensures that the spin is characterized by half-angles in accord with the neutron-rotation experiment. It is possible to construct a Clebsch-Gordan decomposition to reduce a state of complex angular momentum into simpler states which can be identified with and particles, multipole operators, etc. Finally, ground-state energy levels are calculated for all the even-even nuclei by using a differentiable manifold that is spin-graded and gauge-invariant by construction. It is shown that this manifold is Grassmann.     

Note on the time evolution of minimum uncertainty states

General conditions for a Hamiltonian to keep a minimum uncertainty state stable are obtained.     

Sensitivity of wave field evolution and manifold stability in chaotic systems

The sensitivity of a wave field's evolution to small perturbations is of fundamental interest. For chaotic systems, there are two distinct regimes of either exponential or Gaussian overlap decay in time. We develop a semiclassical approach for understanding both regimes and give a simple expression for the crossover time between the regimes. The wave field's evolution is considerably more stable than the exponential instability of chaotic trajectories seems to suggest. The resolution of this paradox lies in the collective behavior of the appropriate set of trajectories. Results are given for the standard map.     

Stochastic Schroedinger equation from optimal observable evolution

In this article, we consider a set of trial wave-functions denoted by |Q〉 and an associated set of operators A_α which generate transformations connecting those trial states. Using variational principles, we show that we can always obtain a quantum Monte-Carlo method where the quantum evolution of a system is replaced by jumps between density matrices of the form D = |Q_a〉〈Q_b|, and where the average evolutions of the moments of A_α up to a given order k, i.e., 〈A_α1〉,〈A_α1A_α2〉,…,〈A_α1?A_αk〉, are constrained to follow the exact Ehrenfest evolution at each time step along each stochastic trajectory. Then, a set of more and more elaborated stochastic approximations of a quantum problem is obtained which approach the exact solution when more and more constraints are imposed, i.e., when k increases. The Monte-Carlo process might even become exact if the Hamiltonian H applied on the trial state can be written as a polynomial of A_α. The formalism makes a natural connection between quantum jumps in Hilbert space and phase-space dynamics. We show that the derivation of stochastic Schroedinger equations can be greatly simplified by taking advantage of the existence of this hierarchy of approximations and its connection to the Ehrenfest theorem. Several examples are illustrated: the free wave-packet expansion, the Kerr oscillator, a generalized version of the Kerr oscillator, as well as interacting bosons or fermions.     

Stochastic differential equations anda posteriori states in quantum mechanics

In recent years a consistent theory describing measurements continuous in time in quantum mechanics has been developed. The result of such a measurement is atrajectoryfor one or more quantities observed with continuity in time. Applications are connected especially with detection theory in quantum optics. In such a theory of continuous measurements one can ask what is the state of the system given that a certain trajectory up to timet has been observed. The response to this question is the notion ofa posteriori states and afilteringequation governing the evolution of such states: this turns out to be a nonlinear stochastic differential equation for density matrices or for pure vectors. The driving noise appearing in such an equation is not an external one, but its probability law is determined by the system itself (it is the probability measure on the trajectory space given by the theory of continuous measurements).