Decoherence theory

Because of our inability to analyze the interaction of microscopic QM systems and macroscopic measuring devices to a sufficient degree, we make use of a set of empirical rules that are known as measurement theory. Some day, measurement theory will become a proven set of theorems in QM,, as the proponents of the decoherence theory, among others, claim. Until such time, it is beneficial to introduce the measurement process, and the principles associated with it, separately from the dynamics described by the Schrbdinger equation. 

XV. Non-Poisson and Renewal Processes A Problem for Decoherence Theory... 

At this stage, we are confident that a clear connection between Levy statistics and critical random events is established. We have also seen that non-Poisson renewal yields a class of GME with infinite memory, from within a perspective resting on trajectories with jumps that act as memory erasers. The non-Poisson and renewal character of these processes has two major effects. The former will be discussed in detail in Section XV, and the latter will be discussed in Section XVI. The first problem has to do with decoherence theory. As we shall see, decoherence theory denotes an approach avoiding the use of wave function collapses, with the supposedly equivalent adoption of quantum densities becoming diagonal in the pointer basis set. In Section XV we shall see that the decoherence theory is inadequate to derive non-Poisson renewal processes from quantum mechanics. In Section XVI we shall show that the non-Poisson renewal properties, revealed by the BQD experiments, rule out modulation as a possible approach to complexity. 

XV. NON-POISSON AND RENEWAL PROCESSES A PROBLEM FOR DECOHERENCE THEORY... 

As discussed in Ref. 108, decoherence theory implies that the quantum superposition of two macroscopically distinct states, A) and B), is forbidden by the entanglement process... 

According to decoherence theory the human observer perceives only the system of interest. Thus, we have to evaluate first the total density matrix corresponding to the wave function of Eq. (259) and to make a contraction over the environmental degrees of freedom, to derive out of it the contracted density matrix out, p(f). As a result of this procedure we get... 

The function AB(f) should be evaluated using a rigorous quantum mechanical calculation. In earlier work [108] we noted that decoherence theory simplifies this task by assuming that the time-dependent quantum operator ,(t) is a stochastic classical variable, so that the function becomes a characteristic... 

We would like to attract the attention of the reader to the case when the environment is a source of anomalous diffusion. Paz et al. [116] studied the decoherence process generated by a supra-ohmic bath, but they did not find any problem with the adoption of the decoherence theory. It is convenient to devote some attention to the case when the fluctuation E, is a source of Levy diffusion [59]. If the fluctuation E, is an uncorrelated Levy process, the characteristic function again decays exponentially, and the only significant change is that the... 

In the BQD case the non-Poisson renewal condition is a property emerging from the interaction with a cooperative bath. We want to illustrate here another important case, although this is not of central importance for the main aim of this review. Research work done in the last 15 years proves also that the non-Poisson character of the system of interest, generated by internal dynamics, rather than by the interaction with an anomalous bath, creates problems in decoherence theory. Let us see why. 

In the Poisson case, the decoherence theory affords a more satisfactory justification for the correspondence principle [20]. Adopting the Wigner formalism, it is possible to express quantum mechanical problems in terms of the classical phase space, and the Wigner quasi-probability is expected to remain positive definite until the instant at which a quantum transition occurs, according to the estimate of Ref. 120, at the time... 

How does one use the decoherence theory to annihilate these quantum effects There are problems. In fact, in classical physics the adoption of environmental noise produces a departure from anomalous diffusion [125,126]. 

Thus, the assumption of the decoherence theory that there are no isolated systems, and that we have always to consider the influence of environmental fluctuations, would kill anomalous diffusion. Furthermore, the numerical results of Ref. 31 show that the quantum-induced transition from anomalous to ordinary diffusion is a quantum effect more robust than the localization phenomenon itself. This indicates that in the presence of a weak environmental fluctuation is now insufficient to reestablish the correspondence principle. 

We are now in the right position to reach a preliminary conclusion. Although the decoherence theory is an attractive and efficient way of defeating the emergence of quantum effects at a macroscopic level, the authors of Ref. 112 did not feel comfortable with it. The reason is that when the observer has the impression that a wave-function collapse occurs, actually the quantum mechanical coherence is becoming even more extended and macroscopic, since it spreads from the system to the environment, Eq. (256). 

In conclusion, the condition of ordinary statistical physics makes the decoherence theory a valuable perspective, as well as an attractive way of deriving classical from quantum physics. The argument that the Markov approximation itself is subtly related to introducing ingredients that are foreign to quantum mechanics [23] cannot convince the advocates of decoherence theory to abandon the certainties of quantum theory for the uncertainties for a search for a new physics. The only possible way of converting a philosophical debate into a scientific issue, as suggested by the results that we have concisely reviewed in this section, is to study the conditions of anomalous statistical mechanics. In the next sections we shall explore with more attention these conditions. 

The first reason that led Latora and Baranger to evaluate the time evolution of the Gibbs entropy by means of a bunch of trajectories moving in a phase space divided into many small cells is the following In the Hamiltonian case the density equation must obey the Liouville theorem, namely it is a unitary transformation, which maintains the Gibbs entropy constant. However, this difficulty can be bypassed without abandoning the density picture. In line with the advocates of decoherence theory, we modify the density equation in such a way as to mimic the influence of external, extremely weak fluctuations [141]. It has to be pointed out that from this point of view, there is no essential difference with the case where these fluctuations correspond to a modified form of quantum mechanics [115]. 

We have seen that decoherence theory, according to its advocates [128], makes the wave-function collapse assumption obsolete The environmental fluctuations are enough to destroy quantum mechanical coherence and generate statistical properties indistinguishable from those produced by genuine wave-function collapses. All this is unquestionable, and if a disagreement exists, it rests more on philosophy than on physical facts. Thus, there is apparently no need for a new theory. However, we have seen that all this implies the assumption that the environment produces white noise and that the system of interest, in the classical limit, produces ordinary diffusion. As we move from... 

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