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arXiv:physics/0611043 · AIP conference proceedings

Interpreting Negative Probabilities in the Context of Double-Slit Interferometry

Ghenadie N. Mardari

Published Jan 1, 2007• 5 citations
arXivHTMLPDF

Abstract

Negative probabilities emerged at intermediate steps in various attempts to predict the distributions of quantum interference. There is no consensus on their meaning yet. It has been suggested (Khrennikov, 1998) that negative probabilities require the existence of unsuspected correlations between detection events. We evaluate this claim in light of several representative experiments. In our assessment, some of its implications are in good agreement with the data.

DOI
10.1063/1.2713478

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Negative probabilities emerged at intermediate steps in various attempts to predict the distributions of quantum interference. There is no consensus on their meaning yet. It has been suggested (Khrennikov, 1998) that negative probabilities require the existence of unsuspected correlations between detection events. We evaluate this claim in light of several representative experiments. In our assessment, some of its implications are in good agreement with the data.
In this paper we attempt to establish a theory of negative (quasi) probability distributions from fundamental principles and apply it to the study of the double-slit experiment in quantum mechanics. We do so in a way that preserves the main conceptual issues intact but allow for a clearer analysis, by representing the double-slit experiment in terms of the Mach-Zehnder interferometer, and show that the main features of quantum systems relevant to the double-slit are present also in the Mach-Zehnder. This converts the problem from a continuous to a discrete random variable representation. We then show that, for the Mach-Zehnder interferometer, negative probabilities do not exist that are consistent with interference and which-path information, contrary to what Feynman believed. However, consistent with Scully et al.'s experiment, if we reduce the amount of experimental information about the system and rely on counterfactual reasoning, a joint negative probability distribution can be constructed for the Mach-Zehnder experiment.
In this paper, we analyze the statistics of detection data in a general double-double-slit experiment. The two particles are detected at random times which are not equal in general and because we do not have any constraint on the distances of left and right screens from their slits and the ratio they can be detected in completely different timescales. As the detection of the first particle leads to collapse of the wave function, there is no a straightforward and agreed-upon method to study this problem in the orthodox formalism which lacks a clear prediction of these random events and therefore the quantum state afterward. This is not the case in the Bohmian framework which we implement in this paper, and we can predict the system up to the end of experiment. The main result is the joint distribution of detection data including the arrival time and position of the particles on left and right screens. As one of the main consequences, we see that although the joint spatial distribution can be affected by a change to the relative location of screens, the marginals on each side remain intact, compatible with signal locality. At the end, we see how this result is very sensitive to quantum equilibrium conditions.
We start from Feynman`s idea to use negative probabilities to describe the two slit experiment and other quantum interfernce experiments. Formally by using negative probability distributions we can explain the results of the two slit experiment on the basis of the pure corpuscular picture of quantum mechnanics. However, negative probabilities are absurd objects in the framework of the standard Kolmogorov theory of probability. We present a large class of non-Kolmogorovean probability models where negative probabilities are well defined on the frequency basis. These are models with probabilities which belong to the so-called field of ppp-adic numbers. However, these models are characterized by correlations between trails. Therefore, we predict correlations between particles in interference experiments. In fact, our predictions are similar to the predictions of the so-called nonergodic interpretation of quantum mechanics, which was proposed by V. Buonomano. We propose the concrete experiments (in particular, in the framework of the neutron interferometry) to verify our predictions on the correlations.
This work addresses the long-standing claim that interference phenomena—most notably in double-slit–type experiments—cannot be physically modeled without invoking wave propagation or particles following multiple trajectories. We show that this claim rests on an implicit restriction to local, dynamical descriptions acting between emission and detection. We introduce a minimal framework in which detection events are treated as discrete realizations occurring at an absorbing boundary, while the observed interference pattern arises from global structural admissibility constraints of the full experimental configuration. The model does not attempt to generate or predict individual detection events. Instead, it characterizes the statistical distribution of outcomes as a property of realization closure rather than of local evolution in transit. Within this approach, interference patterns are understood as emergent statistical signatures of global realization structure, not as evidence of propagating waves, hidden variables, or oscillatory dynamics in space. The framework preserves all empirical features of standard interference experiments while avoiding assumptions about intermediate physical processes that are neither observed nor required by theory. This work is consistent with relativistic and quantum no-go results concerning photon trajectories, localization, and in-flight observables. It complements earlier structural interpretations of light and realization by providing a concrete, physically explicit model demonstrating how stable interference statistics arise without propagation. The contribution clarifies the precise sense in which interference is modelable, and the sense in which it is not: individual events remain intrinsically unpredictable, while their global organization is physically intelligible at the level of realization structure. The model introduces no new forces, fields, or modifications of established equations, and is intended as a refinement of conceptual foundations rather than a replacement of existing formalisms.