Quantum mechanics over non-Archimedean number fields

Institute of Electronic Technology, Moscow. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 83, No. 3, pp. 406–418, June, 1990.     

p-Adic physics, non-well-founded reality and unconventional computing

We consider perspectives of application of coinductive and corecursive methods of non-well-founded mathematics to modern physics, especially to adelic and p-adic quantum mechanics. We also survey perspectives of relationship between modern physics and unconventional computing.     

Quantum rule for detection probability from Brownian motion in the space of classical fields

We obtain Born’s rule from the classical theory of random waves in combination with the use of thresholdtype detectors. We consider a model of classical random waves interacting with classical detectors and reproducing Born’s rule. We do not discuss complicated interpretational problems of quantum foundations. The reader can select between the “weak interpretation,” the classical mathematical simulation of the quantum measurement process, and the “strong interpretation,” the classical wave model of the real quantum (in fact, subquantum) phenomena.     

Born's rule from classical random fields

This Letter is an attempt to go beyond QM. In our approach density operators of QM can be represented as covariance operators of classical random fields. Born's rule can be obtained from measurement theory for classical random field under the assumption that the probability of detection of field is proportional to the power of this field.     

New experimental tests of the photon’s indivisibility

We consider the old foundational problem of quantum/classical optics ?C indivisibility of photon. Quantum theory predicts that in experiments on coincidence detection double clicks are impossible (up to noise); on the other hand, the known semiclassical and classical models predict a relatively high rate of coincidence. We present a model of the classical (random) wave type which predicts that in the same way as in quantum optics coincidence of clicks is a rare event. However, this model has a new prediction compared to quantum optics, namely, that the rate of double clicks depends substantially on the discrimination threshold of a detector. We propose to perform new detailed tests to check the discrimination threshold dependence predicted by our model. In experiments on coincidence detection performed to date, for example, the Grangier experiment does not contain statistical data on the threshold dependence.