STOCH-EQ · Stochastic differential equations in Hilbert spaces and application to collapse models
FP6 — Marie Curie Actions (Human Resources and Mobility)
- Duration
- 2007-01-01 → 2007-12-31
- EU contribution
- €40,000
- Participants
- 1
- Scheme
- ERG
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Results in brief
Final Activity Report Summary - STOCH-EQ (Stochastic differential equations in Hilbert spaces and application to collapse models)
1) Development of an independent research career: An independent research career has been started by the researcher at the Department of Theoretical Physics of the University of Trieste, in collaboration with Prof. GianCarlo Ghirardi from the same University. Other collaborations include Prof. D. Duerr (Munich) and S.L. Adler (Princeton). The researcher is supervising a Ph.D. student working on topics related to the ERG research project. 2) Research activity: A) The mathematical formalism of Collapse Models, i.e. stochastic differential equations, has been generalised in order to include also non white noises as the source of the reduction mechanism. This result is important in view of the possibility of identifying the collapsing field with some physical field already existing in nature. B) The possibility of extending the formalism of Collapse Models to include also relativistic quantum field theories has been analysed. A stimulating and fruitful debate is going on with Profs. Conway and Kochen from Princeton University. C) In a paper still in preparation, we are working on significantly improving recent proofs about the long time behavior of the solutions of certain stochastic differential equations. D) Stochastic differential equations have been applied to study the effect of the environment on quantum algorithms. It has been shown that such an approach has the advantaged (with respect to the other ones) that the environmental noise can be effectively treated as a linear gate acting on the state vector, more or less like any other standard quantum gate. 3) Participation to conferences: The above results have been publicised in several international meetings.
Data: CORDIS, © European Union
Project objective
Within Quantum Mechanics, stochastic differential equations find useful applications in the following research fields:- collapse models, to describe spontaneous localizations of the wave function;- decoherence theory, to mimic the effect of the environment on an open system;- the theory of continuous quantum measurement, to describe the action of a measuring device on a quantum system.In the past years, in particular during his experience as a Marie-Curie fellow in Germany, the researcher has started to study, both analytically and numerically, classes of stochastic equations which are of particular physical relevance; the time evolution of specific solutions (e.g. Gaussian solutions), which are of interest in all applications, have been analyzed, together with the reduction mechanism and its stability, and the localization probabilities; applications to experiments have also been considered.We now wish to pursue this line of research. In particular, we wish to focus on the following topics:Problem 1. Analysis of the general solution and its properties (in particular the asymptotic behaviour) of the stochastic differential equation for the free quantum particle subject to spontaneous localization in space.Problem 2. Analysis of the general solution and of the asymptotic behaviour of the stochastic differential equations for more complex systems, e.g. the harmonic oscillator and the hydrogen atom.Problem 3. If there is time left, we will tackle the problem of formulating collapse models which are relativistically invariant.Since stochastic differential equations are becoming an essential tool in the study of many physical phenomena (from non-equilibrium statistical mechanics, to biology, to mathematical finance, ...), the results of our analysis has the potentialities of being important also for research areas other than the one related to collapse models.
Original text from CORDIS.
Participants
- UNIVERSITÀ DEGLI STUDI DI TRIESTE · TRIESTECoordinatorItaly
Links
Data: CORDIS, © European Union