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EEQT - mathematical formalism

There are two levels of EEQT - the ensemble level and the individual level. Let us consider first the ensemble level.
First of all, in EEQT, at that level, we use all the standard mathematical formalism of quantum theory, but we extend it adding an extra parameter tex2html_wrap_inline460 Thus all quantum operators A get an extra index tex2html_wrap_inline464 , quantum Hilbert space tex2html_wrap_inline466 is replaced by a family tex2html_wrap_inline468 quantum state vectors tex2html_wrap_inline470 are replaced by families tex2html_wrap_inline472 quantum Hamiltonian H is replaced by a family tex2html_wrap_inline476 etc.
The parameter tex2html_wrap_inline478 is used to distinguish between macroscopically different and non-superposable states of the universe. In the simplest possible model we are interested only in describing a "yes-no" experiment and we disregard any other parameter - in such a case tex2html_wrap_inline478 will have only two values 0 and 1. Thus, in this case, we will need two Hilbert spaces. This will be the case when we will deal with particle detectors. In a more realistic situation tex2html_wrap_inline478 will take values in a multi-dimensional, perhaps even infinite-dimensional manifold - like, for instance, in a phase space of a tensor field. But even that may prove to be insufficient. When, for instance, EEQT is used as an engine powering Everett-Wheeler many-world branching tree, in that case alpha will also have to have the corresponding dynamical branching tree structure, where the space in which the parameter tex2html_wrap_inline478 takes values, grows and becomes more and more complex together with the growing complexity of the branching structure.
An event is, in our mathematical model, represented by a change of tex2html_wrap_inline460 This change is discontinuous, is a branching. Depending on the situation this branching is accompanied by a more or less radical change of physical parameters. Sometimes, like in the case of a phase transition in Bose-Einstein condensate, we will need to change the nature of the underlying Hilbert space representation. In other cases, like in the case of a particle detector, the Hilbert spaces tex2html_wrap_inline492 and tex2html_wrap_inline494 will be indistinguishable copies of one standard quantum Hilbert space tex2html_wrap_inline496
A second important point is this: time evolution of an individual quantum system is described by piecewise continuous function tex2html_wrap_inline498 , tex2html_wrap_inline500 , a trajectory of a piecewise deterministic Markov process. The very concept and the theory of piecewise deterministic processes (in short: PDP) is not a part of the standard mathematical education, even for professional probabilists. But the point is that it is impossible to unerstand the essence of EEQT without having even a rough idea about PDPs.
Originally EEQT was described in terms of a master equation for a coupled, quantum+classical, system. Thus it was only applicable to ensambles - the question of how to describe individual systems was open. Then, after searching through the mathematical literature, we found that in his monographs [37, 38] dealing with stochastic control and optimization M. H. A. Davis, having in mind mainly queuing and insurance models, described a special class of piecewise deterministic processes that fitted perfectly the needs of quantum measurement theory, and that reproduced the master equation postulated originally by us in [16].
It took us another couple of years to show [26] that the special class of couplings between a classical and quantum system leads to a unique piecewise deterministic process with values on E-the pure state space of the total system. That process consists of random jumps, accompanied by changes of a classical state, interspersed by random periods of Schrödinger-type deterministic evolution. The process, although mildly nonlinear in quantum wave function tex2html_wrap_inline470 , after averaging, recovers the original linear master equation for statistical states.

It should be stressed that in EEQT the dynamics of the coupled total system which is being modelled is described not only by a Hamiltonian tex2html_wrap_inline466 , or better: not only by an tex2html_wrap_inline478 - parametrized family of Hamiltonians tex2html_wrap_inline476 , but also by a doubly parametrized family of operators tex2html_wrap_inline512 , where tex2html_wrap_inline514 is a linear operator from tex2html_wrap_inline516 to tex2html_wrap_inline518 . While Hamiltonians must be essentially self-adjoint, tex2html_wrap_inline514 need not be such - although in many cases, when information transfer and control is our concern (as in quantum computers), one wants them to be even positive operators (otherwise unnecessary entropy is created). This aspect of EEQT is rather difficult to accept for a newcomer, as the first question he will ask is "where do we take these operators from?" Our answer, elaborated in more details in FAQ-s - see Section 5 - amounts to this: we find the correct operators tex2html_wrap_inline522 the same way we find the correct Hamiltonians: by trial and error! Each new solved model is a lesson and, little by little, we learn more and more, and we aspire for more. As already said, more on this subject in our FAQs section.
It is to be noted that the time evolution of statistical ensembles is due to the presence of tex2html_wrap_inline512 's, non-automorphic. The system, as a whole, is open. This is necessary, as we like to emphasize: information (in this case: information gained by the classical part) must be paid for with dissipation! The appropriate mathematical formalism for discussing the ensemble level is that of completely positive semigroups, as discussed by Kossakowski et al. [27], Lindblad [28] and generalized so as to fit our purpose by Arveson [29] and Christensen [30].
A general form of the linear master equation describing statistical evolution of the coupled system is given by





The operators tex2html_wrap_inline522 can be allowed to depend explicitly on time. While the term with the Hamiltonian describes "dyna-mics", that is exchange of forces, of the system, the term with tex2html_wrap_inline522 describes its "bina-mics" - that is exchange of "bits of information" between the quantum and the classical subsytem.

As has been proven in [26] the above Liouville equation, provided the diagonal terms tex2html_wrap_inline530 vanish, can be considered as an average of a unique Markov process governing the behavior of an individual system. The real-time behavior of such an individual system is given by a PDP process realize by the following non-linear and non-local, EEQT algorithm:


The algorithm is non-linear, because it involves repeated normalizations. It is non-local because it needs repeated computing of the norms - they involve space-integrations. It is to be noted that PDP processes are more general than the popular diffusion processes. In fact, every diffusion process can be obtained as a limit of a family of PDP processes.

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Next: EEQT - a few Up: No Title Previous: Introduction: Is Quantum Theory

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