Shedding light (and dark) on quantum probabilities

Inter-relation between quantum and classical probability models is one of the most fundamental problems of quantum foundations. Nowadays this problem also plays an important role in quantum technologies, in quantum cryptography and the theory of quantum random generators. In this letter, we compare the viewpoint of Richard Feynman that the behavior of quantum particles cannot be described by classical probability theory with the viewpoint that quantum–classical inter-relation is more complicated (cf, in particular, with the tomographic model of quantum mechanics developed in detail by Vladimir Man'ko). As a basic example, we consider the two-slit experiment, which played a crucial role in quantum foundational debates at the beginning of quantum mechanics (QM). In particular, its analysis led Niels Bohr to the formulation of the principle of complementarity. First, we demonstrate that in complete accordance with Feynman's viewpoint, the probabilities for the two-slit experiment have the non-Kolmogorovian structure, since they violate one of basic laws of classical probability theory, the law of total probability (the heart of the Bayesian analysis). However, then we show that these probabilities can be embedded in a natural way into the classical (Kolmogorov, 1933) probability model. To do this, one has to take into account the randomness of selection of different experimental contexts, the joint consideration of which led Feynman to a conclusion about the non-classicality of quantum probability. We compare this embedding of non-Kolmogorovian quantum probabilities into the Kolmogorov model with well-known embeddings of non-Euclidean geometries into Euclidean space (e.g., the Poincaré disk model for the Lobachvesky plane).

Recently, Nature magazine made a survey about how Physicists think of what quantum mechanics says about reality. The survey shows that 47% of respondents consider the wavefunction in the quantum mechanics representing simply a useful tool, and 36% respondents consider the wavefunction representing something real. In this article we show that the double slit experiments support the opinion of 47% of respondents. It is the standard understanding that the double slit experiment shows that the light is waves, not particles. The photoelectric experiment shows that the light is particles. The two experiments show the conceptually different natures of the light and thus led to the wave-particle duality of the light. To thoroughly study the real nature of the light in the double slit experiments, we utilizing the Photon-chamber, Shield, Blocker, Lenses and Metal tubes respectively. We observed the following phenomena in the same double slit experiments: the light is photons, not waves; and it is the photons that produce both the non-interference patterns in Zone-0, Zong-1 and Zone-2, and the interference patterns in Zone-3. We suggest that the observed phenomena: (1) invalidate the wave-particle duality; (2) show that the wavefunction is simply a useful mathematical tool to describe the wave phenomena in Zone-3; (3) show that the wavefunction cannot be utilized to describe the no-wave phenomena in Zone-0, Zong-1 and Zone-2; (4) support the particle reality of quantum mechanics.

“Hypervalency” and the chemical bond

The purpose of this work is to carry out a systematic, detailed analytical study, together with the direct interpretations, as they follow from the analytical expressions obtained, of three basic “experiments” which have been classical examples in our understanding of quantum mechanics. The experiments considered are: the single- and double-slit (-hole) experiments, and the final one, which, in particular, deals with the situation where a particle is “reflected off” the detection screen in the single-slit experiment. Special emphasis is put on pointing out some purely quantum mechanical results, which follow from the analytical expressions, as well as on making a comparison with classical physics. Needless to say, the analysis of such experiments has been extremely important in laying the foundations of quantum physics.

Many quantum phenomena, traditionally described by quantum electrodynamics, can be calculated within the framework of so-called semiclassical theory, in which atoms are described by the wave equations of quantum mechanics (Schrödinger, Dirac, etc.), while light is described by classical electrodynamics without quantization of the radiation. These phenomena include the photoelectric effect, the Compton effect, the Lamb shift, radiative effects, spontaneous emission, the Hanbury Brown and Twiss effect, etc. In this paper, I show that the double-slit experiment can also be calculated in detail in terms of semiclassical theory if we take into account the discrete (atomic) structure of matter. I show that the results of a semiclassical simulation of the “linear” double-slit experiment coincide with predictions of wave theory only for low-intensity light and for short exposure time, while stricter dependences exist for long-term exposure. The semiclassical theory is used for calculation of the “nonlinear” double-slit experiment with an intense laser beam in which multiphoton and tunnel excitation of atoms on a photographic plate occurs. The Born rule for light is derived from the semiclassical theory.

It is an often repeated claim in the literature that quantum mechanics is indeterministic and that it has put an end to the classical notion of causality. From the impossibility of determining the exact spatio-temporal trajectory of an atomic system, for instance, Heisenberg infers ‘the invalidity of the causal law’ in quantum mechanics [1]. What is tacitly assumed in such views is a chain of reasoning, which leads from determinism to causality. One form of determinism – predictive determinism – is the view that a sufficient knowledge of the laws of nature and appropriate boundary conditions will enable a superior intelligence to predict the future states of the physical world and to retrodict its past states with infinite precision. Laplace attributes this capacity to his famous demon: for the demon the physical world stretches out like the frames of a filmstrip. Each frame is caused by an earlier frame and in its turn causes a later frame. From the present frame the Laplacean demon is capable of predicting and retrodicting all other frames. Hence the demon identifies determinism and causality. ‘We ought to regard the present state of the universe as the effect of its antecedent state and as the cause of the state that is to follow’ [9]. Laplace assumes that these states are unique and can be determined with mathematical precision such that prediction and retrodiction become possible. The laws of physics are typically expressed in differential equations which describe the evolution of some physical parameter, P , as a function of time, t . As one state of a system, S1, evolves to another state, S2, where this temporal evolution is made precise by the employment of differential equations, it becomes easy to think of differential equations as precise mathematical representations of causal laws [10]. This is indeed how Einstein presented the matter: ‘The differential law is the only form which completely satisfies the modern physicist’s demand for causality’ [2]. Although Russell [11] had argued that the ‘law of causality (. . .) is the product of a bygone age’ he nevertheless admitted causal laws in the form of functional relations and differential equations into physics. This functional model of causality enjoyed great popularity amongst physicists. But the experimental results from quantum mechanics – like the double-slit experiments – seemed to threaten the Laplacean identification of determinism and causality. Physicists reacted to this threat in three different ways.

When both slits of the double slit experiment are open, closing one paradoxically increases the detection rate at some points on the detection screen. Feynman famously warned that temptation to understand such a puzzling feature only draws us into blind alleys. Nevertheless, we gain insight into this feature by drawing an analogy between the double slit experiment and a time reversed fire alarm. Much as closing the slit increases probability of a future detection, ruling out fire drill scenarios, having heard the fire alarm, increases probability of a past fire (using Bayesian inference). Classically, Bayesian inference is associated with computing probabilities of past events. We therefore identify this feature of the double slit experiment with a time reversed thermodynamic arrow. We believe that much of the enigma of quantum mechanics is simply due to some variation of time's arrow. In further support of this, we employ a plausible formulation of the thermodynamic arrow to derive an uncertainty in classical mechanics that is reminiscent of quantum uncertainty.

The [formula omitted]-Adic Schrödinger equation and the two-slit experiment in quantum mechanics

Summary form only given. Complementarity is one of the most basic principles of quantum mechanics. A quantum mechanical object, be it a photon or a massive particle, shows position (particle-like) or momentum (wave-like) attributes exclusively depending on the experimental situation. Over the years, Young's two-slit experiment has been emphasized as a good example of complementarity. We report two two-slit experiments in which complementarity is enforced by availability of which-path information at the measurement stage of the quantum, which is different from the traditional understanding. A type-II BBO crystal is pumped by a CW UV laser generating spontaneous parametric down-conversion (SPDC). Entangled signal and idler photons are then sent in different directions. A double slit is placed in the idler beam and the idler detectors are scanned in the transverse direction.

As is well known, the two–slit experiment is related, within Quantum Mechanics (Q.M.), to the dualistic character of radiation (for instance light), whose behavior is correctly represented as a wave-like phenomenon during its passage from the source, through the slits, up to the screen; and by a particle model (for instance, photons) when it impinges on the screen causing the extraction of electrons.KeywordsClassical LogicQuantum LogicClassical PhysicClassical ParticleLogical ConnectiveThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Quasi-set theory $\cal Q$ allows us to cope with certain collections of objects where the usual notion of identity is not applicable, in the sense that $x = x$ is not a formula, if $x$ is an arbitrary term. $\cal Q$ was partially motivated by the problem of non-individuality in quantum mechanics. In this paper I discuss the range of explanatory power of $\cal Q$ for quantum phenomena which demand some notion of indistinguishability among quantum objects. My main focus is on the double-slit experiment, a major physical phenomenon which was never modeled from a quasi-set-theoretic point of view. The double-slit experiment strongly motivates the concept of degrees of indistinguishability within a field-theoretic approach, and that notion is simply missing in $\cal Q$. Nevertheless, other physical situations may eventually demand a revision on quasi-set theory axioms, if someone intends to use it in the quantum realm for the purpose of a clear discussion about non-individuality. I use this opportunity to suggest another way to cope with identity in quantum theories.

A distinct method to show a quantum object behaving both as wave and as particle is proposed and described in some detail. We make a systematic analysis using the elementary methodology of quantum mechanics upon Young's two-slit interferometer and the Mach—Zehnder two-arm interferometer with the focus placed on how to measure the interference pattern (wave nature) and the which-way information (particle nature) of quantum objects. We design several schemes to simultaneously acquire the which-way information for an individual quantum object and the high-contrast interference pattern for an ensemble of these quantum objects by placing two sets of measurement instruments that are well separated in space and whose perturbation of each other is negligibly small within the interferometer at the same time. Yet, improper arrangement and cooperation of these two sets of measurement instruments in the interferometer would lead to failure of simultaneous observation of wave and particle behaviors. The internal freedoms of quantum objects could be harnessed to probe both the which-way information and the interference pattern for the center-of-mass motion. That quantum objects can behave beyond the wave—particle duality and the complementarity principle would stimulate new conceptual examination and exploration of quantum theory at a deeper level.

This chapter discuses two stock examples of quantum interference effects — the two-slit experiment and the Wigner's Stern–Gerlach experiment. It notes that each experiment shows the sort of quantum weirdness that any satisfactory formulation of quantum mechanics must ultimately predict and explain. It also enumerates points that will be discussed regarding Hugh Everett III's relative state theory and his solution to the quantum measurement problem.

The double-slit experiment has become a classic thought experiment, for its clarity in expressing the central puzzle of quantum mechanics -- wave-particle complementarity. Such wave-particle duality continues to be challenged and investigated in a broad range of entities with electrons, neutrons, helium atoms, C$_{60}$ fullerenes, Bose-Einstein condensates and biological molecules. All existing experiments are performed with the scale larger than angstrom. In this article, we present a double-slit scenario at fermi scale with new entities -- coherent photon products in heavy-ion collisions. Virtual photons from the electromagnetic fields of relativistic heavy ions can fluctuate to quark-antiquark pairs, scatter off a target nucleus and emerge as vector mesons. The two colliding nuclei can take turns to act as targets, forming a double-slit interference pattern. Furthermore, the `which-way' information can be partially solved by the violent strong interactions in the proposed scenario, which demonstrates a key concept of quantum mechanics -- complementary principle.

In response to Orion and Laitman’s [1] explanation of the classic double-slit experiment of quantum mechanics, we propose an alternate explanation of that experiment by treating physical degrees of freedom as a conserved physical quantity, instead of referring to “vague terms” used in previous explanations, [1], that are not broadly applicable. Explanation in [1] refers to properties of groups of particles, even though the double-slit experiment’s results should address only to a single particle. By using physical degrees of freedom and the application of Hamilton’s principle, we obtain a single particle explanation of the double-slit experiment in terms of properties and via methods which apply equally in a quantum and a classical regime.