Born Rule Research Articles

Bayesian 14C-rationality, Heisenberg Uncertainty, and Fourier Transform

Following some 30 years of radiocarbon research during which the mathematical principles of 14C-calibration have been on loan to Bayesian statistics, here they are returned to quantum physics. The return is based on recognition that 14C-calibration can be described as a Fourier transform. Following its introduction as such, there is need to reconceptualize the probabilistic 14C-analysis. The main change will be to replace the traditional (one-dimensional) concept of 14C-dating probability by a two-dimensional probability. This is entirely analogous to the definition of probability in quantum physics, where the squared amplitude of a wave function defined in Hilbert space provides a measurable probability of finding the corresponding particle at a certain point in time/space, the so-called Born rule. When adapted to the characteristics of 14C-calibration, as it turns out, the Fourier transform immediately accounts for practically all known so-called quantization properties of archaeological 14C-ages, such as clustering, age-shifting, and amplitude-distortion. This also applies to the frequently observed chronological lock-in properties of larger data sets, when analysed by Gaussian wiggle matching (on the 14C-scale) just as by Bayesian sequencing (on the calendar time-scale). Such domain-switching effects are typical for a Fourier transform. They can now be understood, and taken into account, by the application of concepts and interpretations that are central to quantum physics (e.g. wave diffraction, wave-particle duality, Heisenberg uncertainty, and the correspondence principle). What may sound complicated, at first glance, simplifies the construction of 14C-based chronologies. The new Fourier-based 14C-analysis supports chronological studies on previously unachievable geographic (continental) and temporal (Glacial-Holocene) scales; for example, by temporal sequencing of hundreds of archaeological sites, simultaneously, with minimal need for development of archaeological prior hypotheses, other than those based on the geo-archaeological law of stratigraphic superposition. As demonstrated in a variety of archaeological case studies, just one number, defined as a gauge-probability on a scale 0–100%, can be used to replace a stacked set of subjective Bayesian priors.

The Born rule as a statistics of quantum micro-events

We deduce the Born rule from a purely statistical take on quantum theory within minimalistic math-setup. No use is required of quantum postulates. One exploits only rudimentary quantum mathematics—a linear, not Hilbert’, vector space—and empirical notion of the Statistical Length of a state. Its statistical nature comes from the lab micro-events (detector-clicks) being formalized into the C -coefficients of quantum superpositions. We also comment that not only has the use not been made of quantum axioms (scalar-product, operators, interpretations , etc.), but that the involving thereof would be, in a sense, inconsistent when deriving the rule. In point of fact, the quadratic character of the statistical length, and even not (the ‘physics’ of) Born’s formula, represents a first step in constructing the mathematical structure we name the Hilbert space of quantum states.

Conditional probability and interferences in generalized measurements with or without definite causal order

In the context of generalized measurement theory, the Gleason-Busch theorem assures the unique form of the associated probability function. Recently, in Flatt et al. Phys. Rev. A 96, 062125 (2017), the case of subsequent measurements has been treated, with the derivation of the L\"uders rule and its generalization (Kraus update rule). Here we investigate the special case of subsequent measurements where an intermediate measurement is a composition of two measurements (a or b) and the case where the causal order is not defined (a and b or b and a). In both cases interference effects can arise. We show that the associated probability cannot be written univocally, and the distributive property on its arguments cannot be taken for granted. The two probability expressions correspond to the Born rule and the classical probability; they are related to the intrinsic possibility of obtaining definite results for the intermediate measurement. For indefinite causal order, a causal inequality is also deduced. The frontier between the two cases is investigated in the framework of generalized measurements with a toy model, a Mach-Zehnder interferometer with a movable beam splitter.

Elastic Properties and Nonlinear Elasticity of the Noncarbon Hexagonal Lattice Nanomaterials Based on the Multiscale Modeling

Abstract This study presents the elastic properties and nonlinear elasticity of the two-dimensional noncarbon nanomaterials of hexagonal lattice structures having molecular structure XY. Four nitride-based and two phosphide-based two-dimensional nanomaterials, having graphene-like hexagonal lattice structure, are considered in the present study. The four empirical parameters associated with the attractive and repulsive terms of the Tersoff–Brenner potential are calibrated for noncarbon nanomaterials and tested for elastic properties, nonlinear constitutive behavior, bending modulus, bending and torsional energy. The mathematical identities for the tangent constitutive matrix in terms of the interatomic potential function are derived through an atomistic–continuum coupled multiscale framework of the extended version of Cauchy–Born rule. The results obtained using newly calibrated empirical parameters for cohesive energy, bond length, elastic properties, and bending rigidity are compared with those reported in the literature through experimental investigations and quantum mechanical calculations. The continuum approximation is attained through the finite element method. Multiscale evaluations for elastic properties and nonlinear stretching of the nanosheets under in-plane loads are also compared with those obtained from atomistic simulations.

Refutation of the Quantum Theory Principles: Theorem

The theorem presented challenges the quantum mechanics and its relativistic theory generally posited as an ultimate unifying guideline of nature in fundamental and applied matters, refutes this theory, any bridges from it to the realm. We build the evidence on the rigorous statistical criteriaand arguments of compatibility at the interfaces not adduced previously against the theory. It calls in question the Born rule, particle-wave doublethink, probability sense of the quantum theory, any bridges from the theory to both Lagrangian and nonholonomic mechanics. The argumentation given to the matter of ambient noise impact at the interfaces by meaningful statistical methods paves the way towards the correct principles of causality, connectedness, robustness .

Derive the Born’s rule from environment-induced stochastic dynamics of wave functions in an open system

The lack of superposition of different position states or the emergence of classicality in macroscopic systems have been a puzzle for decades. Classicality exists in every measuring apparatus, and is the key for understanding what can be measured in quantum theory. Different theories have been proposed, including decoherence, einselection and the spontaneous wave-function collapse, with no consensus reached up to now. In this paper, we propose a stochastic dynamics for the wave-function in an open system (e.g. the measuring apparatus) that interacts with its environment. The trajectory of wave-function is random with a well-defined probability distribution. We show that the stochastic evolution results in the wave-function collapse and the Born's rule for specific system-environment interactions. While it reproduces the unitary evolution governed by the Schr\"{o}dinger equation when the interaction vanishes. Our results suggest that it is the way of system interacting with environment that determines whether quantum superposition dominates or classicality emerges.

Emergence of the Born rule in quantum optics

The Born rule provides a fundamental connection between theory and observation in quantum mechanics, yet its origin remains a mystery. We consider this problem within the context of quantum optics using only classical physics and the assumption of a quantum electrodynamic vacuum that is real rather than virtual. The connection to observation is made via classical intensity threshold detectors that are used as a simple, deterministic model of photon detection. By following standard experimental conventions of data analysis on discrete detection events, we show that this model is capable of reproducing several observed phenomena thought to be uniquely quantum in nature, thus providing greater elucidation of the quantum-classical boundary.

The Periodic Table needs negative orbitals in order to eliminate quantum weirdness: a new quantum chemistry mathematics

A consensus among quantum experts is that the quantum world is not properly understood. It is a mistake to think we can cure quantum weirdness by tinkering with superficial aspects of quantum mechanics (QM). We propose that nature uses (–ψ) as its wave function, whereas QM uses (+ψ). We propose therefore that the Periodical Table should be changed to negative orbitals (–ψ). Surprisingly, this change makes almost no difference to chemistry on a practical level. The Born rule takes the absolute square of an amplitude to obtain a probability to test in chemistry lab P=|–ψ|2=|+ψ|2. We propose a new math based on (–ψ) that is the mirror image of quantum mathematics. We call it the Theory of Elementary Waves (TEW). The negative sign is not an electrical charge. It has nothing to do with Coulomb’s law. Valence electrons are unchanged. Ions, covalent bonds, dipoles, metals, hydrogen bonding and the hydrogen 21 cm line are unchanged. The octet rule and rules for drawing dot structures of molecules do not change. Amino acids, sugars and DNA do not change their handedness. We cite abundant experimental evidence showing that TEW is correct and QM is wrong.

Ergodicity and Born's rule in an entangled two-qubit Bohmian system.

We study the Bohmian trajectories of a generic entangled two-qubit system, composed of coherent states of two harmonic oscillators with noncommensurable frequencies and focus on the relation between ergodicity and the dynamical approach to Born's rule for arbitrary distributions of initial conditions. We find that most Bohmian trajectories are ergodic and establish the same invariant ergodic limiting distributions of their points for any nonzero amount of entanglement. In the case of strong entanglement the distribution satisfying Born's rule is dominated by chaotic-ergodic trajectories. Therefore, P→|Ψ|^{2} for an arbitrary P_{0}. However, when the entanglement is weak the distribution satisfying Born's rule is dominated by ordered trajectories, which are not ergodic. In this case the ergodic trajectories do not, in general, lead to the distribution of Born's rule, therefore P=|Ψ|^{2} is guaranteed only if P_{0}=|Ψ_{0}|^{2}. Consequently, the existence of chaotic and ergodic Bohmian trajectories does not always lead to the dynamical establishment of Born's rule.

Discrete Wigner functions from informationally complete quantum measurements

Wigner functions provide a way to do quantum physics using quasiprobabilities, that is, "probability" distributions that can go negative. Informationally complete POVMs, a much younger subject than phase space formulations of quantum mechanics, are less familiar but provide wholly probabilistic representations of quantum theory. In this paper, we show that the Born Rule links these two classes of structure and discuss the art of interconverting between them. In particular, we demonstrate that the operator bases corresponding to minimal discrete Wigner functions (Wigner bases) are orthogonalizations of minimal informationally complete measurements (MICs). By not imposing a particular discrete phase space structure at the outset, we push Wigner functions to their limits in a suitably quantified sense, revealing a new way in which the symmetric informationally complete measurements (SICs) are significant. Finally, we speculate that astute choices of MICs from the orthogonalization preimages of Wigner bases may in general give quantum measurements conceptually underlying the associated quasiprobability representations.

Chaos in Bohmian Quantum Mechanics: A Short Review

This is a short review in the theory of chaos in Bohmian Quantum Mechanics based on our series of works in this field. Our first result is the development of a generic theoretical mechanism responsible for the generation of chaos in an arbitrary Bohmian system (in 2 and 3 dimensions). This mechanism allows us to explore the effect of chaos on Bohmian trajectories and study in detail (both analytically and numerically) the different kinds of Bohmian trajectories where, in general, chaos and order coexist. Finally we explore the effect of quantum entanglement on the evolution of the Bohmian trajectories and study chaos and ergodicity in qubit systems which are of great theoretical and practical interest. We find that the chaotic trajectories are also ergodic, i.e. they give the same final distribution of their points after a long time regardless of their initial conditions. In the case of strong entanglement most trajectories are chaotic and ergodic and an arbitrary initial distribution of particles will tends to Born's rule over the course of time. On the other hand, in the case of weak entanglement the distribution of Born's rule is dominated by ordered trajectories and consequently an arbitrary initial configuration of particles will not tend, in general, to Born's rule, unless it is initially satisfied. Our results shed light on a fundamental problem in Bohmian Mechanics, namely whether there is a dynamical approximation of Born's rule by an arbitrary initial distribution of Bohmian particles.

Apparatus contribution to observed nonclassicality

Any nonclassical effect noticed in measured statistics is usually attributed just to the light state being observed. According to Born's rule, quantum probabilities are given by the overlap between the system state and measurement states in a quite symmetrical way. We demonstrate that the nonclassicality of the measurement is a necessary condition to obtain nonclassical statistics. To this end, we present a detector characterization for two-observable joint detection processes based on detector tomography. This is particularized to the most common signatures of nonclassical light.

There are two solutions to the equations of Feynman’s Quantum Electrodynamics (QED): the newly discovered solution is free of quantum weirdness

No one previously noticed there is a second solution to the equations of Richard Feynman’s Quantum Electrodynamics (QED). It makes identical predictions in the lab. The new solution (Reverse-QED) is closer to nature: it is free of quantum weirdness. For example, it eliminates Schrödinger’s cat. This article is the first time the equations of R-QED have been published. The R-QED amplitude is the negative of Feynman’s amplitude. Because of the Born rule, both amplitude and negative amplitude, when squared, produce the same probability to be tested against empirical data. If you were to measure the distance from Los Angeles to New York City with R-QED’s accuracy, it would be exact to the breadth of a human hair. If reality corresponds to the newly discovered R-QED equations, but scientists use the old QED equations, the result would be predictions for the lab that are precisely accurate, but scientists would be unable to construct a coherent picture of the quantum world. R-QED is based on a different picture of how the quantum world is organized. Experiments, including a neutron interferometer experiment we review, show that particles follow waves backward. R-QED integrates in the same direction that the waves travel.

Modeling of progressive damage in concrete using multidimensional virtual internal bond method

SummaryDamage simulation via tensile and shear failure modes in quasi‐brittle material like concrete is computationally challenging. Multidimensional virtual internal bond (MVIB) method is one of the promising method developed to capture shear failure in materials like concrete. It assumes continuum as an assemblage of material points joined by normal and shear bonds possessing harmonic potential and bond density evolution function, which degrades with deformation to yield at macroscopic level, stress–strain law with inbuilt feature of failure via Cauchy–Born rule. The degradation of bond‐density in MVIB method originally based solely on compressive failure is extended to include tensile failure synonymous with tension and compression asymmetry in concrete. The modified MVIB model is applied to a variety of problems, and the effect of inclusion of both tensile and compressive failure is thoroughly assessed. Through the example of two notches in a square plate subjected to biaxial loading, a systematic study of damage evolution vis‐à‐vis biaxial loading ratio is presented. Capture of crack closure and formation of crack retarding protective shield are unique outcomes of the present work.

Mathematics of the classical and the quantum

Newtonian and Schrödinger dynamics can be formulated in a physically meaningful way within the same Hilbert space framework. This fact was recently used to discover an unexpected relation between classical and quantum motions that goes beyond the results provided by the Ehrenfest theorem. The Newtonian dynamics was shown to be the Schrödinger dynamics of states constrained to a submanifold of the space of states, identified with the classical phase space of the system. Quantum observables are identified with vector fields on the space of states. The commutators of observables are expressed through the curvature of the space. The resulting embedding of the Newtonian and Schrödinger dynamics into a unified geometric framework is rigid in the sense that the Schrödinger dynamics is a unique extension of the Newtonian one. Under the embedding, the normal distribution of measurement results associated with a classical measurement implies the Born rule for the probability of transition of quantum states. The mathematics of the discovered relationship between the classical and the quantum is reviewed and investigated here in detail and applied to the process of measurement of spin and position observables.

Quantum fractionalism: The Born rule as a consequence of the complex Pythagorean theorem

Everettian Quantum Mechanics, or the Many Worlds Interpretation, lacks an explanation for quantum probabilities. We show that the values given by the Born rule equal projection factors, describing the contraction of Lebesgue measures in orthogonal projections from the complex line of a quantum state to eigenspaces of an observable. Unit total probability corresponds to a complex Pythagorean theorem: the measure of a subset of the complex line is the sum of the measures of its projections on all eigenspaces.Postulating the existence of a continuum infinity of identical quantum universes, all with the same quasi-classical worlds, we show that projection factors give relative amounts of worlds. These appear as relative frequencies of results in quantum experiments, and play the role of probabilities in decisions and inference. This solves the probability problem of Everett's theory, allowing its preferred basis problem to be solved as well, and may help settle questions about the nature of probability.

Elastic in–plane properties of cellular materials: Discrete approach

Elastic properties of two–dimensional cellular lattice materials are studied using a discrete model. The model is based on a representation of the lattice as a set of interacting nodes. A potential of interaction between the nodes is calibrated such that to simulate elastic linking. The analytical homogenization based on Cauchy–Born rule allowed determining the elastic properties of the effective continuum corresponding to the specific lattice. On the other hand, the same properties were found from the numerical simulations based on particle dynamics. It is shown that both ways lead to the same values of elastic constants. This approach allows using particle–based numerical simulations to design various applications of lattice materials.

Fourier transform, quantum mechanics and quantum field theory on the manifold of general relativity

A proof is given for the Fourier transform for functions in a quantum mechanical Hilbert space on a non-compact manifold in general relativity. In the (configuration space) Newton–Wigner representation, we discuss the spectral decomposition of the canonical operators and give a proof of the Parseval–Plancherel relation and the Born rule for linear superposition. We then discuss the representations of pure quantum states and their dual vectors and construct the Fock space and the associated quantum field theory for Bose–Einstein and Fermi–Dirac statistics.

Psychological ‘double-slit experiment’ in decision making: Quantum versus classical

This paper is devoted to justification of the application of quantum probability theory to problems of cognition, psychology, and decision making. Such applications are heavily based on quantum-like representation of interference of events that is formalized with complex probability amplitudes (“mental wave functions”) and the Born rule for calculation of probability. In this paper, we present universal mathematical formalization of interference of events based on the calculus of intensities of interacting processes. Generally, this formalization leads to the nonlinear law of superposition of complex probability amplitudes with quantum linear superposition as a special important case. For intensities characterized by discrete occurrence of events, the formula for interference of intensities is transferred into the quantum-like formula for interference of probabilities. We illustrate the formalism by simple examples of possible applications of the calculus of intensities of processes to decision making and economics. We show that in special cases the classical (Kolmogorov) probabilistic model can give the same results as the quantum rule of summation of probabilities.

QBism Is Not So Simply Dismissed

QBism is one of the main candidates for an epistemic interpretation of quantum mechanics. According to QBism, the quantum state or the wavefunction represents the subjective degrees of belief of the agent assigning the state. But, although the quantum state is not part of the furniture of the world, quantum mechanics grasps the real via the Born rule which is a consistency condition for the probability assignments of the agent. In this paper, we evaluate the plausibility of recent criticism of QBism . We focus on the consequences of the subjective character of the quantum state, the issue of realism and the problem of the evolution of the quantum state in QBism. In particular, drawing upon Born's notion of invariance as the mark of the real, it is argued that there is no essential difference between Einstein's program of the "real" and QBists' realism. Also, it will be argued that QBism can account for the unitary evolution of the quantum state.