Recently a new attempt to go beyond quantum mechanics (QM) was presented in the form of so called prequantum classical statistical field theory (PCSFT). Its main experimental prediction is violation of Borns rule which provides only an approximative
description of real probabilities. We expect that it will be possible to design numerous experiments demonstrating violation of Borns rule. Moreover, recently the first experimental evidence of violation was found in the triple slits interference experiment, see cite{WWW}. Although this experimental test was motivated by another prequantum model, it can be definitely considered as at least preliminary confirmation of the main prediction of PCSFT. In our approach quantum particles are just symbolic representations of prequantum random fields, e.g., electron-field or neutron-field; photon is associated with classical random electromagnetic field. Such prequantum fields fluctuate on time and space scales which are essentially finer than scales of QM, cf. `t Hoofts attempt to go beyond QM cite{H1}--cite{TH2}. In this paper we elaborate a detection model in the PCSFT-framework. In this model classical random fields (corresponding to quantum particles) interact with detectors inducing probabilities which match with Borns rule only approximately. Thus QM arises from PCSFT as an approximative theory. New tests of violation of Borns rule are proposed.
Monomial mappings, $xmapsto x^n$, are topologically transitive and ergodic with respect to Haar measure on the unit circle in the complex plane. In this paper we obtain an anologous result for monomial dynamical systems over $p-$adic numbers. The pro
cess is, however, not straightforward. The result will depend on the natural number $n$. Moreover, in the $p-$adic case we never have ergodicity on the unit circle, but on the circles around the point 1.
We show that the projection postulate plays a crucial role in the discussion on the so called quantum nonlocality, in particular in the EPR-argument. We stress that the original von Neumann projection postulate was crucially modified by extending it
to observables with degenerate spectra (the Luders postulate) and we show that this modification is highly questionable from a physical point of view, and it is the real source of quantum nonlocality. The use of the original von Neumann postulate eliminates this problem: instead of action at the distance-nonlocality, we obtain a classical measurement nonlocality. It seems that EPR did mistake in their 1935-paper: if one uses correctly von Neumann projection postulate, no ``elements of reality can be assigned to entangled systems. Our analysis of the EPR and projection postulate makes clearer Bohrs considerations in his reply to Einstein.
Modern development of quantum technologies based on quantum information theory stimulated analysis of proposed computational, cryptographic and teleportational schemes from the viewpoint of quantum foundations. It is evident that not all mathematical
calculations performed in complex Hilbert space can be directly realized in physical space. Recently by analyzing the original EPR paper we found that they argument was based on the misuse of the von Neumanns projection postulate. Opposite to von Neumann, Einstein, Podolsky and Rosen (EPR) applied this postulate to observables represented by operators with degenerate spectra. It was completely forbidden by von Neumanns axiomatics of QM. It is impossible to repeat the EPR considerations in the von Neumanns framework. In this note we analyze quantum teleportation by taking into account von Neumanns projection postulate. Our analysis shows that so called quantum teleportation is impossible in von Neumanns framework. On the other hand, our analysis implies that the main quantum algorithms are totally consistent with von Neumanns projection postulate.
We performed a comparative analysis of the arguments of Einstein, Podolsky and Rosen -- EPR, 1935 (against the completeness of QM) and the theoretical formalism of QM (due to von Neumann, 1932). We found that the EPR considerations do not match at al
l with the von Neumanns theory. Thus EPR did not criticize the real theoretical model of QM. The root of EPRs paradoxical conclusion on incompleteness of QM is the misuse of von Neumanns projection postulate. EPR applied this postulate to observables with degenerate spectra (which is totally forbidden by the axiomatics of QM).
In this paper we present a simple algorithm for representation of statistical data of any origin by complex probability amplitudes. Numerical simulation with Mathematica-6 is performed. The Blochs sphere is used for visualization of results of numeri
cal simulation. On the one hand, creation of such a quantum-like (QL) representation and its numerical approval is an important step in clarification of extremely complicated interrelation between classical and quantum randomness. On the other hand, it opens new possibilities for application the mathematical formalism of QM in other domains of science.
This paper is devoted to such a fundamental problem of quantum computing as quantum parallelism. It is well known that quantum parallelism is the basis of the ability of quantum computer to perform in polynomial time computations performed by classic
al computers for exponential time. Therefore better understanding of quantum parallelism is important both for theoretical and applied research, cf. e.g. David Deutsch cite{DD}. We present a realistic interpretation based on recently developed prequantum classical statistical field theory (PCSFT). In the PCSFT-approach to QM quantum states (mixed as well as pure) are labels of special ensembles of classical fields. Thus e.g. a single (!) ``electron in the pure state $psi$ can be identified with a special `` electron random field, say $Phi_psi(phi).$ Quantum computer operates with such random fields. By one computational step for e.g. a Boolean function $f(x_1,...,x_n)$ the initial random field $Phi_{psi_0}(phi)$ is transformed into the final random field $Phi_{psi_f}(phi)$ ``containing all values of $f.$ This is the objective of quantum computers ability to operate quickly with huge amounts of information -- in fact, with classical random fields.
In this paper we present quantum-like (QL) representation of the Shafir-Tversky statistical effect. We apply so called contextual approach. The Shafir-Tversky effect is considered as a consequence of combination of a number of incompatible contexts w
hich are involved e.g. in Prisoners Dilemma or in more general games inducing the disjunction effect. As a consequence, the law of total probability is violated for experimental data obtained by Shafir and Tversky (1992) as well as Tversky and Shafir (1992). Moreover, we can find a numerical measure of contextual incompatibility (so called coefficient of interference) as well as represent contexts which are involved in Prisoners Dilemma (PD) by probability amplitudes -- normalized vectors (``mental wave functions). We remark that statistical data from Shafir and Tversky (1992) and Tversky and Shafir (1992) experiments differ crucially from the point of view of mental interference. The second one exhibits the conventional trigonometric ($cos$-type) interference, but the first one exhibits so called hyperbolic ($cosh$-type) interference. We discuss QL processing of information by cognitive systems, in particular, QL decision making as well as classical and QL rationality.
We consider the following model of decision-making by cognitive systems. We present an algorithm -- quantum-like representation algorithm (QLRA) -- which provides a possibility to represent probabilistic data of any origin by complex probability ampl
itudes. Our conjecture is that cognitive systems developed the ability to use QLRA. They operate with complex probability amplitudes, mental wave functions. Since the mathematical formalism of QM describes as well (under some generalization) processing of such quantum-like (QL) mental states, the conventional quantum decision-making scheme can be used by the brain. We consider a modification of this scheme to describe decision-making in the presence of two ``incompatible mental variables. Such a QL decision-making can be used in situations like Prisoners Dilemma (PD) as well as others corresponding to so called disjunction effect in psychology and cognitive science.
