Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Theoretical Statistics and Mathematics Unit (TSMU-Delhi)


Sinha, Kalyan Bidhan (TSMU-Delhi; ISI)

Abstract (Summary of the Work)

The central theme of the present thesia is quantum stochastic dilation af semigroupe of completely panitive mapa on operator algebran. It is the sim of all mathemati- cal, or even all scientific theorics, to understand a given class of objects through a tanonical and simpler subclass of it. For example, abstract C"-algebras are studied through their conerete realisation as elgebra of operators, contractions on a Hilbert space by unitaries. Hilbert modules by the factorissble ones, to mention anly a few. In most af these caes, a general object of the relavant class is sociated with a canonical candidate of the simpler subelass, in which the former is "embedded ia some natural way" and abtained back by soma canonical aperatilon like restrietion or projection. Such an association of larger objects having simpler structure is known aa dilation. Typical examples include the Se Nagy's unitary dilation of contarctione and the Stinespring's dikstion of completely positive maps. On the other hand, in many physical thearies. A dilation coresponda to viewing a plysical phenomenon in an enlarged system contsining the original aystem as a subsystem. Let us now restrlet ourselves tn physical models whicı have some relevance to the marhematical theories developed in the thesis. It is customary to model the dynamics of a con servative physical systam by an sppropriate Hamiltonian mechaniam described by a group of unitaries ( in the Hilbert space framewock) or more alstractly a group of ausomorphisma ( in the operator algebrs framework), representing the reversibie time evolution of the system. However, in many real physical aystems, the evolution is irreversible and this is attributed to the interaction with the eavironment or the 8O-calied heat-bath. The evolution, when seen in the bigger system consisting of the original one as well as the the environment, will be reversible but due to our inability to observe or Jack of interest in the dynamics of the total system the phenomenon of irreversibillty or dissipativity in the system takem place.the context of dissipative systems arising in classical mechanics. it is often reasonable to model the environment by the space of a suitable Markov prooss itypically Browalas motlon) so that the behaviour of the total syatem is described by a stockastic differential flow aquation and the evalution within the original subaystam at a given time is obhtained by taking conditional expectation with respect to the it nation of the sbom stochastic proces upto that time point. This may be physically interpreted as wasking out environtmerstal ncions to recover the original evolution, by the pesielve (or Markov) semigroup asociated with the process carrying theoist. This is the cam of clasesieal stochastie dilstion The sarse klea extends to quantum mochanical systams, but tiete kre consider alke oureptual and techinical difficulties. ome af which are addressed and partisly sulved in this thesis. The time evolution of an irreversible quantum mechanical sys tem can be represented by a smigronp of linear mape acting on an appeopeinte perator algebra In clansical mechanics, this algebea ia taken to be some suitable ommutative algebra of functions on the atste space, which is naturally replaced by more general.


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Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.


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