Both classical and respectively quantum observables can be modeled as somewhat similar examples of random variables. In such a model the associated measurements preserve the values spectrum of an observable but change the corresponding probabilistic weights (probability density or respectively the wave function). Such a model ensures theoretical estimations for predicted errors specific to the mean values as well as to the fluctuations of both types of observables. The model stands out of ourdays prevalent opinion that measurement theories must be depicted in entirely different manners in the classical respectively quantum cases.
Both classical and respectively quantum observables can be modeled as somewhat similar examples of random variables. In such a model the associated measurements preserve the values spectrum of an observable but change the corresponding probabilistic weights (probability density or respectively the wave function). Such a model ensures theoretical estimations for predicted errors specific to the mean values as well as to the fluctuations of both types of observables. The model stands out of ourdays prevalent opinion that measurement theories must be depicted in entirely different manners in the classical respectively quantum cases.
Abstract In stark contrast to classical Newtonian mechanics, the standard interpretation of quantum mechanics associates the Schrödinger wavefunction, not in a determ- inistic way with definite predictions about a specific, individual experimental result, but only in a probabilistic manner, with |ψ(x, t )|2 acting as a probability density, in this case, for position measurements. Before analyzing this approach in more detail, we will find it profitable to briefly review some of the basic concepts of discrete and
Unsolved controversies about uncertainty relations and quantum measurements still persists nowadays. They originate around the shortcomings regarding the conventional interpretation of uncertainty relations. Here we show that the respective shortcomings disclose veridic and unavoidable facts which require the abandonment of the mentioned interpretation. So the primitive uncertainty relations appear as being either thought fictions or fluctuations formulae. Subsequently we reveal that the conventional approaches of quantum measurements are grounded on incorrect premises. We propose a new approach in which : (i) the quantum observables are considered as generalized stochastic variables, (ii) the view is focused only on the pre-existent state of the measured system, without any interest for the collapse of the respective state, (iii) a measurement is described as an input-output transformation which modify the probability density and current but preserve the expressions of the operators. The measuring uncertainties are evaluated as changes in the probabilistic estimators of observables. Related to different observables we do not find reasons of principle neither for uncertainties-connections nor for measuring compatibility/incompatibility.
Ever since the beginning of quantum mechanics, the measurement problem has been a subject of sometimes discontinued but nevertheless recurrent concern. As students, most physicists have raised questions about it, but the spectacular and renewed successes of quantum theory were a strong incentive for them to work out applications without further worrying about foundations. In each generation, however, some dissatisfaction reemerges, which leads to slow progress, and we may expect that the measurement problem will some day be cleared up rather than forgotten. The situation recalls statistical mechanics, the foundations of which still give rise to scientific discussions, in spite of more than one century’s achievements. Actually, the paradoxes of irreversibility and of Maxwell’s demon share some common features with quantum measurements, to wit, they involve probabilities in a crucial fashion and their explanation requires elucidating the role of the observer.
A probability of heads/tails for a coin toss or a number for a die is really a frequency in time with no consideration of space. Similarly, a uniform distribution of stationary particles in space with P(x) = 1/L (one dimension restricted to a length L) is a probability in space, with no time appearing. Some phenomena (not necessarily linked to probability) necessarily involve space x and time t, such as a water wave. It is possible to fix x and watch the height of the wave change strictly in t, or take a film such that one sees wave heights at different x points with no reference to time (as one does not have a single crest, but a steady stream of waves). Another observation one may make is to sit on the crest and see x change with t (phase velocity) without any consideration of the wave height changing. The above examples are common in everyday life. We suggest that there is another probability-wave phenomenon which is common (reflection-refraction), but not all details are directly observed for a specific reason, namely because x and t can become decoupled directly at the n1-n2 index of refraction region. Typically, in the classical world one does not decouple x and t. In terms of observation, if one had a math probability given by exp(ipx) for a moving object, with the probability being linked somehow to interaction, then one would not see any changes in time. exp(ipx) is then neither directly visible in space or time, but may exist and have physical consequences. (Its existence would have to be proved experimentally.) The reason that decoupling of x,t may occur is that one does not have a physical measurement, such as a wave height. A physical wave height is described by a periodic function (-wt+kx), for which one cannot decouple t and k because the wave has a physical phase velocity of w/k and the function represents a measurement at x,t even though x,t and t+constant/w and x+ constant/k yield the same functional value. t+constant/w and x+ constant/k must represent another physical wave with the same phase velocity, but based on a different starting x,t, not x=0, t=0. This is because x and t cannot be decoupled as one observes a physical crest. If one has a math function exp(-iEt+ipx), then again one would have x,t marking a certain function value and t+constant/E and x+constant/p marking the same function value. It is certainly possible to have the particle move with a velocity v. In this case, however, the function does not represent an observable in x,t, like the height of a physical wave. In principle, there is no reason why t and x could not be decoupled. This means that the same function value (which presumably has some meaning) could be associated with two different x points for the same t value (or t removed altogether because x and t are decoupled). The question then becomes: How does one reconcile this strange math approach to physical observables? For example, if exp(-iEt+ipx) is considered to be a probability (but not of a measurable object in coupled spacetime which is the usual approach), how does one link this probability to measurements in spacetime? We note that since probabilities may apply only to time or space and noted in the first paragraph, that a decoupled exp(-iEt+ipx) probability lends itself very well to such one variable probabilities. For example, if only space is considered exp(-iEt+ipx) is replaced with exp(ipx) which only pertains to space, just like P(x)=1/L only pertains to space. The latter, however, is based on physical measurements at single points in space, while exp(ipx) is based on the same math function value occurring for two x points (x and x+constant/p) with no time consideration. Normally, one would need a time value directly associated with x, but such is not required for a probability in an OR situation. Different times may be associated with two different events without one needing to one these time values as one is specifically calculating a probability.
It has already been suggested that quantum theory needs to be reformulated or modified in order to explain the measurement process and the successive collapse of the wave- function. However, there are also models of another type which keep quantum theory intact and instead modify the classical gravity by introducing stochasticity to it. These models suggest that there is a fluctuation in the background gravitational field which eventually results in the decoherence of the wavefunction. These fluctuations limit the precision with which one can measure the properties of a spacetime geometry with a quantum probe. Two similar models along this line have been suggested by Karolyhazy (K-model) and Diósi(D-model). They are based upon apparently different spacetime bounds. The results obtained for the coherence length are also somewhat different. In this article, we show that, given certain conditions apply, the minimal spacetime bounds in these two models are equivalent. We also derive the two-point correlation for the fluctuation potential in K-model which turns out to be non-white, unlike in D-model, where the corresponding correlation is white noise in time. In our opinion, this is the origin of discrepancy in the predictions of the two models. We argue that the noise correlation cannot be determined uniquely from a given spacetime bound.