Quantum Dynamics: from Coarse Graining to a Tower of Scales via Multiresolution

Antonina N. Fedorova and Michael G. Zeitlin

IPME RAS, St.Petersburg, Russia

http://www.ipme.ru/zeitlin.html

http://mp.ipme.ru/zeitlin.html,

anton@math.ipme.ru

zeitlin@math.ipme.ru


We present a family of methods which can describe complex behaviour in quantum
ensembles.
We demonstrate the creation of nontrivial (meta) stable states (patterns),
localized, chaotic, entangled or decoherent, from the basic localized modes in
various collective models arising from the quantum hierarchy described by
Wigner-like equations.
The advantages of such an approach are as follows:
i). the natural realization of localized states in any proper functional
realization of (Hilbert) space of states,
ii). the representation of hidden symmetry of a chosen realization of the
functional model describes the (whole) spectrum of possible states via the
so-called multiresolution decomposition.
Effects we are interested in are as follows:
1. a hierarchy of internal/hidden scales (time, space, phase space);
2. non-perturbative multiscales: from slow to fast contributions, from the
coarser to the finer level of resolution/decomposition;
3. the coexistence of the levels of hierarchy of multiscale dynamics with
transitions between scales;
4. the realization of the key features of the complex quantum world such as the
existence of chaotic and/or entangled states with possible destruction in
"open/dissipative" regimes due to interactions with quantum/classical
environment and transition to decoherent states.
The numerical simulation demonstrates the formation of various (meta) stable  
patterns or orbits generated by internal hidden symmetry from generic
high-localized fundamental modes.
In addition, we can control the type of behaviour on the pure algebraic level
by means of properly reduced algebraic systems (generalized dispersion  
relations).