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Justification of the postulates

The above postulates are more or less "natural". They are in agreement with the existing ideas of non-unitarygif evolution. So, for instance, in [20] the authors considered the ionization model. They wrote: According to the usual procedure the ionization probability should be given by .
Even if our postulates are natural, it is worthwhile to notice that EEQT allows us to interpret them, to understand them and to derive them, in terms of classical Markov processes. First of all let us see that the above formula for can be understood in terms of an inhomogeneous Poisson decision process as follows.gif Assume the evolution starts with some quantum state , of norm one, as above. Define the positive function as




Then above happens to be nothing but the first-jump probability of the inhomogeneous Poisson process with intensity . It is instructive to see that this is indeed the case. To this end let us divide the interval into n subintervals of length . Denote The inhomogeneous Poisson process of intensity consists then of taking independent decisions jump--or--not--jump\ in each time subinterval. The probability for jumping in the k-th subinterval is assumed to be (that is why is called the intensity\ of the process). Thus the probability of not jumping up to time t is


Let us show that can be identified with given by Eq. (7). To this end notice that

Thus , given by Eq. (1) satisfies the same differential equation as given by Eq. (7). Because , it follows that , and so indeed is the first jump probability of the inhomogeneous Poisson process with intensity .

This observation is useful but rather trivial. It can not yet stand for a justification of the formula (1) - this for the simple reason that the jump process above, based upon a continuous observation of the variable and registering the time instant of its jump, is not a Markovian process. It would become Markovian if we know , but to know we must know . This leads us to consider pairs , where is the Hilbert space vector describing quantum state, and is the yes-no state of the counter. Then evolves deterministically according to the formula (2), the intensity function is computed on the spot, and the Poisson decision process described above is responsible for the jump of value of - in our case it corresponds to a "click" of the counter. The time of the click is a random variable , well defined and computable by the above prescription.
This prescription sheds some light onto the meaning of the quantum state vector . We see that codes in itself information that is being used by a decision mechanism working in an entirely classical way - the only randomness we need is that of a biased (by ) classical roulette. Until we ask why the bias is determined by this particular functional of the quantum state, until then we do not have to invoke more esoteric concepts of quantum probability -- whatever they mean. But, in fact, it is possible to understand somewhat more, still in pure classical terms. We will not need this extra knowledge in the rest of this paper, but we think it is worthwhile to sketch here at least the idea.
In the reasoning above we were interested only in what governs the time of the first jump, when the counter clicks. But in reality nothing ends with this click. A photon, for instance, when detected, is transformed into another form of energy. So, if we want to continue our process in time, after , we must feed it with an extra information: how is the quantum state transformed as the result of the jump. So, in general, we have a classical variable that can take finitely many, denumerably many, a continuum, or more, possible values, and to each ordered pair there corresponds an operator . The transition is called an event, and to each event there corresponds a transformation of the quantum state . In the case of a counter there is only one . In general, when there are several -s, we need to tell not only when to jump, but also where to jump. One obtains in this way a piecewise deterministic Markov process on pure states of the total system: (quantum object, classical monitoring device). It can be then shown [6,22] that this process, including the jump rate formula (5) follows uniquely from the simplest possible Liouville equation that couples the two systems.

next up previous
Next: The Time of Up: Time of Events Previous: Third Postulate -

Arkadiusz Jadczyk
Thu Feb 22 09:58:31 MET 1996