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Phenomenological Detection Model

Andrei Yu. Khrennikov

Published Mar 19, 2014• 0 citations
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Abstract

In this chapter a measurement model for PCSFT will be presented [201]. We explore the basic idea of PCSFT: a quantum state (e.g., a pure state) is simply a label for an ensemble of classical fields- classical random field. We model the process of interaction of a detector with a random field. By using another terminology, we can speak about classical random signalsa and, in particular, about switching to time averages (see Section 3.1.5). The model developed in this chapter is of the phenomenological type. It is based on (heuristically justified) the coupling between the energy distribution of a random field and the probability distribution of clicks of a detector. This model will be concreted in Chapter 9 by considering the detectors of the threshold type. In the later model we shall be able to describe the process of the interaction of a random signal with a threshold-type detector.

DOI
10.1201/b15540-11

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In this chapter a measurement model for PCSFT will be presented [201]. We explore the basic idea of PCSFT: a quantum state (e.g., a pure state) is simply a label for an ensemble of classical fields- classical random field. We model the process of interaction of a detector with a random field. By using another terminology, we can speak about classical random signalsa and, in particular, about switching to time averages (see Section 3.1.5). The model developed in this chapter is of the phenomenological type. It is based on (heuristically justified) the coupling between the energy distribution of a random field and the probability distribution of clicks of a detector. This model will be concreted in Chapter 9 by considering the detectors of the threshold type. In the later model we shall be able to describe the process of the interaction of a random signal with a threshold-type detector.
All previous chapters of this book were devoted to the development of the theory of classical (prequantum) random fields reproducing quantum averages (including correlations for entangled systems) as averages of quadratic forms of fields. Since the basic model of prequantum fields is the Gaussian one, these fields are continuous. Hence, in the same way as in the classical signal theory, averages are calculated for variables with a continuous range of values. This model, PCSFT, is really a prequantum model. In accordance with the Bohr’s viewpoint, we consider QM as an operational formalism describing (predicting) results of measurements on microsystems. QM cannot describe intrinsic physical processes in the microworld, but onlymeasurements performed bymacroscopic classical devices. In contrast, PCSFT describes intrinsic processes in the microworld. However, it does not describe results of measurements on the level of individual events, clicks of detectors. As we pointed in Section 1.1.8, in principle one might be satisfied with creation of a prequantum model which reproduces only quantum probabilistic predictions, i.e., without establishing a direct connectionwith theory of measurement on the level of individual events. This approach would match with views of Schro¨dinger and the Bild concept in general. However, historical development of QM demonstrated that the Bild concept was not attractive for the majority of physicists. Since the experimental verification is considered as the basic counterpart of any physical theory, Bild-like prequantum theories are considered as metaphysical. Measurement theory connecting PCSFT with experiment on the level of individual events was developed in my paper [211] (see also Refs. [209, 210, 212, 213]). In this paper I elaborated a scheme of discrete measurements of classical random signals which reproduces the basic rule of QM, the Born’s rule. This is the scheme of threshold-type detection: such a detector clicks after it has “eaten” a special portion of energy
This paper is a contribution to the project "emergent quantum mechanics" unifying a variety of attempts to treat quantum mechanics (QMs) as emergent from other theories pretending on finer descriptions of quantum phenomena. More concretely it is about an attempt to model detection probabilities predicted by QM for single photon states by using classical random fields interacting with detectors of the threshold type. Continuous field model, prequantum classical statistical field theory (PCSFT), was developed in recent years and its predictions about probabilities and correlations match well with QM. The main problem is to develop the corresponding measurement theory which would describe the transition from continuous fields to discrete events, "clicks of detectors". Some success was achieved and the click-probabilities for quantum observables can be derived from PCSFT by modeling interaction of fields with the threshold type detectors. However, already for the coefficient of second-order coherence g2(0) calculations are too complicated and only an estimation of g2(0) was obtained. In this paper, we present results of numerical simulation based on PCSFT and modeling of interaction with threshold type detectors. The "prequantum random field" interacting with a detector is modeled as the Brownian motion in the space of classical fields (Wiener process in complex Hilbert space). Simulation for g2(0) shows that this coefficient approaches zero with increase of the number of detections.
We describe quantum theory by using a complex valued random signal, which is (coherently) averaged over a time window, and a photon click is defined as the energy of this averaged signal surpassing a detector threshold. The random signal is modeled as white noise but grows over time s (proportional to ). Eventually this random signal will lead to a detector "click" defined by a detector threshold condition, at which point the random signal is reset. Further we proceed by analysing correlations and finally by modeling the quantum state of an entangled pair by a complex valued correlation function.
The quantum measurement problem as was formulated by von Neumann in 1933 can be solved by going beyond the operational quantum formalism. In our "prequantum model" quantum systems are symbolic representations of classical random fields. The Schrödinger's dynamics is a special form of the linear dynamics of classical fields. Measurements are described as interactions of classical fields with detectors. Discontinuity, the "collapse of the wave function", has the trivial origin: usage of threshold type detectors. The von Neumann projection postulate can be interpreted as the formal mathematical encoding of the absence of coincidence detection in measurement on a single quantum system, e.g., photon's polarization measurement. Our model, prequantum classical statistical field theory (PCSFT), in combination with measurements by threshold detectors satisfies the quantum restriction on coincidence detections: the second order coherence is less than one (opposite to all known semiclassical and classical feld models). The basic rule of quantum probability, the Born's rule, is derived from properties of prequantum random felds interacting with threshold type detectors. Comparison with De Broglie's views to quantum mechanics as theory of physical waves with singularities is presented.
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.