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AIP conference proceedings

Born’s rule for probability of quantum detection from classical random field

Andrei Yu. Khrennikov, Börje Nilsson, Louis Fishman, Anders Karlsson, Sven Nordebo

Published Jan 1, 2009• 0 citations
Publisher

Abstract

This paper is an attempt to reproduce predictions of quantum mechanics (QM) on probability of detection by starting with classical random field (e.g. random electromagnetic field). Thus it is an attempt to go beyond QM. The crucial point of our approach is that 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.

DOI
10.1063/1.3117102

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This paper is an attempt to reproduce predictions of quantum mechanics (QM) on probability of detection by starting with classical random field (e.g. random electromagnetic field). Thus it is an attempt to go beyond QM. The crucial point of our approach is that 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.
The very old problem of the statistical content of quantum mechanics (QM) is studied in a novel framework. The Born's rule (one of the basic postulates of QM) is derived from theory of classical random signals. We present a measurement scheme which transforms continuous signals into discrete clicks and reproduces the Born's rule. This is the scheme of threshold type detection. Calibration of detectors plays a crucial role. Creation of such a detection model provides a possibility to represent the basic quantum phenomena as classical field phenomena completed by measurements with threshold type and properly calibrated detectors. Finally, the wave-particle duality has been resolved in favor of the purely wave description. It is well known that coincidence probability plays an important role in the debate on the possibility to reduce quantum theory (especially quantum optics) to classical field theory. Our model matches well the predictions of quantum theory and experiment. It violates the prediction of the classical field model which does not take into account the presence of threshold type detectors with proper calibration.
We present a model of detectors operating with classical random fields and producing probabilities described by the formalism of quantum mechanics. The basic assumption of this measurement theory is that the probability of detection of field (= signal) is proportional to the power of this field. (Therefore we call this approach "power signal-field detection theory" – PFSDT.) Our approach has some similarities with stochastic electrodynamics. However, formalisms describing random fields – "quantum fluctuations" – are different. Moreover, our approach all particles are represented by random fields, e.g., electronic or neutronic field. Thus we have various types of "quantum fluctuations".
The very old problem of the statistical content of quantum mechanics is studied in a novel framework. We show that quantum mechanics (QM) can be considered as a measurement model of prequantum classical statistical field theory (PCSFT): a pure field theory reproducing quantum correlations. We present a measurement scheme for classical signals which reproduces the basic rule of QM, the Born's rule. This is the scheme of threshold type detection. Calibration of detectors plays a crucial role. The size of the detection threshold is determined by the mean energy of a prequantum signal. QM is interpreted as theory of measurements of classical random signals with threshold type detectors which are calibrated in a proper way. The essence of this paper is the creation of a rigorous mathematical model of threshold detection of classical (ergodic) stochastic processes.
We present quantum mechanics (QM) as theory of special classical random signals. On one hand, this approach provides a possibility to go beyond conventional QM: to create a finer description of micro processes than given by the QM-formalism. In fact, we present a model with hidden variables of the wave-type. On the other hand, our approach establishes coupling between quantum and classical information theories. We recall that quantum information theory has already been used for description of the entropy of Gaussian input signals for noisy channels. The entropy of a classical random input was invented as the entropy of the quantum density operator corresponding to the covariance operator of the input process.1 In this paper, we proceed the other way around: we apply classical signal theory to create a measurement model which reproduces quantum probabilities.
We present a purely wave model (based on classical random field) which reproduces quantum probabilities (given by the fundamental law of quantum mechanics, Born's rule) including probabilities for joint detection of a pair of quantum observables (e.g., spin or polarization projections). The crucial point of our approach is that the presence of detector's threshold and calibration procedure have to be treated not as simply experimental technicalities, but as the basic counterparts of the theoretical model. The presence of the background field (vacuum fluctuations) is also the key-element of our prequantum model. It is of the classical signal type and the methods of classical signal theory (including statistical radiophysics) are used for its development. We stress that our prequantum model is not objective, i.e., the values of observables (clicks of detectors) cannot be assigned in advance, i.e., before measurement. Hence, the dilemma, nonobjectivity or nonlocality, is resolved in favor of nonobjectivity (our model is local of the classical field type). In particular, we reproduce the probabilities for the EPR-experiment for photon polarization and, hence, violate CHSH inequality for classical random signals (measured by the threshold type and properly calibrated detectors acting in the presence of the background field).
Recently a new attempt to go beyond quantum mechanics (QM) was done in the form of so called prequantum classical statistical field theory (PCSFT). In this approach QM is considered as incomplete theory – in total accordance with Einstein [1]. PCSFT is (one of possible) completions of QM. Hidden variables are given by classical fields. (It is again in accordance with Einstein.) These prequantum fields fluctuate on time and space scales which are essentially finer than scales of QM, cf. 't Hooft's attempt to go beyond QM [2, 3]. However opposite to Einstein, PCSFT does not assign results of quantum observations directly to hidden variables. Bohr's idea that the whole experimental arrangement should be taken into account is basic for PCSFT-based measurement theory. This theory for measurements of momentum of quantum particles is presented in this paper.