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Journal of Mathematical Physics

Modeling quantum behavior with standard (nonquantum) probability theory

Robert F. Bordley

Published Oct 1, 1983• 7 citations
Publisher

Abstract

This paper argues that quantum behavior can be modeled using standard probability theory. To show this, such a model is constructed in which the Lagrangians associated with different paths are random. (This random Lagrangian formulation is equivalent to constrained entropy maximization.) We assume that the random error term varies as a harmonic oscillator over time. (We attribute this to certain properties of measuring devices.) The result is a formula which provides a good qualitative description of the n-slit interference experiment—indeed the formula is quite similar to the formulas of quantum mechanics. Hence standard probability theory models can describe interference effects so that a quantum probability theory is unnecessary.

DOI
10.1063/1.525622

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This paper argues that quantum behavior can be modeled using standard probability theory. To show this, such a model is constructed in which the Lagrangians associated with different paths are random. (This random Lagrangian formulation is equivalent to constrained entropy maximization.) We assume that the random error term varies as a harmonic oscillator over time. (We attribute this to certain properties of measuring devices.) The result is a formula which provides a good qualitative description of the n-slit interference experiment—indeed the formula is quite similar to the formulas of quantum mechanics. Hence standard probability theory models can describe interference effects so that a quantum probability theory is unnecessary.
Today it still remains a challenge whether quantum mechanics has an underlying statistical explanation or not. While there are and were a lot of models trying to explain quantum phenomena with statistical methods these all failed on certain levels. In this paper a method is proposed that is not only based on a classical statistical framework but it has an underlying physical model behind it and it can explain some of the basic characteristics of quantum mechanics. It will be shown that if look at the properties of a charged particle in a random electric field one can obtain the discrete energy values of the harmonic oscillator and the infinite potential well, and also gives a good qualitative description of the double-slit experiment and measurement theory. Also the side-effect of the model is the emergence of a constant with an action dimension.
In this thesis we are investigating a different formalism of non-relativistic quantum mechanics called the path integral formalism.It is a generalization of the classical least action principle.The introduction to this subject begins with the construction of the path integral in terms of the idea of probability amplitudes whose absolute square gives the probability of finding a system in a particular state.Then we show that if the Lagrangian is a quadratic form one needs only to calculate the classical action besides from a time-dependent normalization constant to find the explicit expression of the path integral.We look in to the subject of two kinds of slit-experiments: The square slit, the single-and the double-Gaussian slit.Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohm effect, which shows that the vector potential defined in classical electrodynamics have a physical meaning in quantum mechanics.It is also shown that the path integral formulation is equivalent to the Schrödinger description of quantum mechanics, by deriving the Schrödinger equation from the path integral.Further applications of the path integral are discussed.
The thesis is about the peculiar probabilistic structure of quantum mechanics. Typically, a random phenomenon is described using the measure-theoretic formulation of probability theory. Such a description can also be done using algebraic methods, which are capable to deal with non-commutative random variables. A central feature of quantum mechanics is the non-commutativity of operators used to describe physical observables. A method to construct a non-commutative probability theory starting from an ordinary measure-theoretic description of probability is proposed and applied to a series of model to reobtain the quantum behaviour od a particle. In particular, using ordinary probability theory, the kinematics of a point-like particle jumping at random over a discrete random space is studied. After the removal of the random space from the model, the position and velocity of the particle do not commute, when represented as operators on the same Hilbert space. Using a suitable random process for the space one is able to recover non-relativistic quantum mechanics in a suitable limit.
An analysis of quantum measurement is presented that relies on an information-theoretic description of quantum entanglement. In a consistent quantum information theory of entanglement, entropies (uncertainties) conditional on measurement outcomes can be negative, implying that measurement can be described via unitary, entropy-conserving, interactions, while still producing randomness in a measurement device. In such a framework, quantum measurement is not accompanied by a wave-function collapse, or a quantum jump. The theory is applied to the measurement of incompatible variables, giving rise to a stronger entropic uncertainty relation than heretofore known. It is also applied to standard quantum measurement situations such as the Stern-Gerlach and double-slit experiments to illustrate how randomness, inherent in the conventional quantum probabilities, arises in a unitary framework. Finally, the present view clarifies the relationship between classical and quantum concepts.
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 theoretical framework in which the interference pattern of the double-slit experiment emerges from a variational principle defined on an informational manifold rather than from postulated wave–particle duality. Within the Viscous Time Theory (VTT) framework, physical systems are described by identity-preserving trajectories that minimize an informational latency functional. The competition between two permissible trajectories under finite latency produces a coherent term analogous to an interference phase, without assuming a physical wave or pre-existing superposition. The resulting probability distribution reduces to the standard double-slit formula in the limit of uniform latency and recovers the disappearance of interference under which-way detection as a breakdown of coherent identity. The model introduces a gradient of informational awareness that predicts a localized collapse event associated with a tensor activation reflecting the transition to a single-path regime. We propose an experimental protocol combining single-photon interference with EEG recordings to test whether early variations in the awareness gradient correlate with the collapse of coherence. We further report a model-based validation using a synchronized double-slit and EEG-inspired signal protocol, in which analytically constructed waveforms—consistent with published spectral properties—are used to illustrate threshold-driven collapse behavior, finite-time collapse dynamics, and improved predictive performance of the VTT model compared to standard decoherence-based descriptions. The framework thus provides a testable and experimentally supported informational interpretation of quantum interference, suggesting that wave–particle transitions correspond to a reorganization of identity in viscous informational time rather than a change in physical ontology.