Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1412.8036

Prequantum Classical Statistical Field Theory: Simulation of Probabilities of Photon Detection with the Aid of Classical Brownian Motion

Andrei Yu. Khrennikov

Published Dec 27, 2014• 0 citations
arXivHTMLPDF

Abstract

In this paper we present results of numerical simulation based on Prequantum Classical Statistical Field Theory (PCSFT), a model with hidden variables of the field-type reproducing probabilistic predictions of quantum mechanics (QM). PCSFT is combined with measurement theory based on detectors of the threshold type. The latter describes discrete events corresponding to the continuous fields model, PCSFT. Numerical modeling demonstrated that the classical Brownian motion (the Wiener process valued in complex Hilbert space) producing clicks when approaching the detection threshold gives probabilities of detection predcited by the formalism QM (as well as PCSFT). This numerical result is important, since the transition from PCSFT to the threshold detection has a complex mathematical structure (in the framework of classical random processes) and it was modeled only approximately. We also perform numerical simulation for the PCSFT-value of the coefficient of second order coherence. Our result matches well with the prediction of quantum theory. Thus, opposite to semiclassical theory, PCSFT cannot be rejected as a consequence of measurements of $g^{(2)}(0).$ Finally, we analyze the output of the recent experiment performed in NIST questioning the validity of some predictions of PCSFT.

DOI
10.48550/arxiv.1412.8036

Related papers

Emerging quantum mechanics: Coefficient of second-order coherence from classical random fields interacting with threshold type detectors

Andrei Yu. Khrennikov

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.

International Journal of Quantum Information•Nov 1, 2014•0 citations

Born's Rule from Measurements of Classical Random Signals under the Assumption of Ergodicity at the Subquantum Time Scale

Andrei Yu. Khrennikov

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.

Open Systems & Information Dynamics•Sep 1, 2012•0 citations

Prequantum Classical Statistical Field Theory—PCSFT

Andrei Yu. Khrennikov, Guillaume Adenier, Andrei Yu. Khrennikov +3 more

In this short note we present (without to go into mathematical details) the main notions of Prequantum Classical Statistical Field Theory (PCSFT): theory of classical fields which fluctuations produce averages approximatively coinciding with averages given by the quantum formalism. Thus the conventional quantum mechanics can be considered as a probabilistic approximation of a more detailed classical field model. Experimental predictions of PCSFT are discussed

AIP conference proceedings•Jan 1, 2007•4 citations

Measurement problem: from De Broglie to theory of classical random fields interacting with threshold detectors

Andrei Yu. Khrennikov

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.

arXiv:1301.0011•Journal of Physics Conference Series•Jun 10, 2013•1 citation

Prequantum Classical Statistical Field Theory (PCSFT)

Andrei Yu. Khrennikov

This chapter presents the basics of the mathematical formalism and methodologyof the prequantum classical statistical field theory (PCSFT). In theBild-conception framework, PCSFT gives an example of acausal theoretical model (CTM) beyond QM, considered as observationalmodel (OM). Generally CTM-OM correspondence is not as straightforwardas in Bell’s model with hidden variables. In PCSFT hidden variables are randomfields fluctuating at spatial and temporal scales which are essentiallyfiner than those approached by the present measurement technology. Thekey element of the PCSFT-QM correspondence is mapping of the complexcovariance operator of a subquantum random field to the density operator.For compound systems, the situation is more complicated. Here PCSFT providestwo descriptions of compound systems with random fields valued intensor vs. Cartesian product of the Hilbert spaces of subsystems. The lattermodel matches representation of compound systems in classical statisticalmechanics. Both approaches are used for measure-theoretic representationof the correlations violating the Bell inequalities.

Cambridge University Press eBooks•Nov 28, 2024•0 citations

Towards violation of Born’s rule: description of a simple experiment

Andrei Yu. Khrennikov, Andrei Yu. Khrennikov, Gregg S. Jaeger +2 more

Recently a new model with hidden variables of the wave type was elaborated, so called prequantum classical statistical field theory (PCSFT). Roughly speaking PCSFT is a classical signal theory applied to a special class of signals—“quantum systems”. PCSFT reproduces successfully all probabilistic predictions of QM, including correlations for entangled systems. This model peacefully coexists with all known no‐go theorems, including Bell’s theorem. In our approach QM is an approximate model. All probabilistic predictions of QM are only (quite good) approximations of “real physical averages”. The latter are averages with respect to fluctuations of prequantum fields. In particular, Born’s rule is only an approximate rule. More precise experiments should demonstrate its violation. We present a simple experiment which has to produce statistical data violating Born’s rule. Since the PCSFT‐presentation of this experiment may be difficult for experimenters, we reformulate consequences of PCSFT in terms of the conventional wave function. In general, deviation from Born’s rule is rather small. We found an experiment amplifying this deviation. We start with a toy example. Then we present a more realistic example based on Gaussian states with very small dispersion.

arXiv:1007.4677•AIP conference proceedings•Jan 1, 2011•0 citations
In this paper we present results of numerical simulation based on Prequantum Classical Statistical Field Theory (PCSFT), a model with hidden variables of the field-type reproducing probabilistic predictions of quantum mechanics (QM). PCSFT is combined with measurement theory based on detectors of the threshold type. The latter describes discrete events corresponding to the continuous fields model, PCSFT. Numerical modeling demonstrated that the classical Brownian motion (the Wiener process valued in complex Hilbert space) producing clicks when approaching the detection threshold gives probabilities of detection predcited by the formalism QM (as well as PCSFT). This numerical result is important, since the transition from PCSFT to the threshold detection has a complex mathematical structure (in the framework of classical random processes) and it was modeled only approximately. We also perform numerical simulation for the PCSFT-value of the coefficient of second order coherence. Our result matches well with the prediction of quantum theory. Thus, opposite to semiclassical theory, PCSFT cannot be rejected as a consequence of measurements of g(2)(0).g^{(2)}(0).g(2)(0). Finally, we analyze the output of the recent experiment performed in NIST questioning the validity of some predictions of PCSFT.
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.
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.
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.
This chapter presents the basics of the mathematical formalism and methodologyof the prequantum classical statistical field theory (PCSFT). In theBild-conception framework, PCSFT gives an example of acausal theoretical model (CTM) beyond QM, considered as observationalmodel (OM). Generally CTM-OM correspondence is not as straightforwardas in Bell’s model with hidden variables. In PCSFT hidden variables are randomfields fluctuating at spatial and temporal scales which are essentiallyfiner than those approached by the present measurement technology. Thekey element of the PCSFT-QM correspondence is mapping of the complexcovariance operator of a subquantum random field to the density operator.For compound systems, the situation is more complicated. Here PCSFT providestwo descriptions of compound systems with random fields valued intensor vs. Cartesian product of the Hilbert spaces of subsystems. The lattermodel matches representation of compound systems in classical statisticalmechanics. Both approaches are used for measure-theoretic representationof the correlations violating the Bell inequalities.
Recently a new model with hidden variables of the wave type was elaborated, so called prequantum classical statistical field theory (PCSFT). Roughly speaking PCSFT is a classical signal theory applied to a special class of signals—“quantum systems”. PCSFT reproduces successfully all probabilistic predictions of QM, including correlations for entangled systems. This model peacefully coexists with all known no‐go theorems, including Bell’s theorem. In our approach QM is an approximate model. All probabilistic predictions of QM are only (quite good) approximations of “real physical averages”. The latter are averages with respect to fluctuations of prequantum fields. In particular, Born’s rule is only an approximate rule. More precise experiments should demonstrate its violation. We present a simple experiment which has to produce statistical data violating Born’s rule. Since the PCSFT‐presentation of this experiment may be difficult for experimenters, we reformulate consequences of PCSFT in terms of the conventional wave function. In general, deviation from Born’s rule is rather small. We found an experiment amplifying this deviation. We start with a toy example. Then we present a more realistic example based on Gaussian states with very small dispersion.