Statistical Interpretation Of Quantum Mechanics Research Articles

Is the Statistical Interpretation of Quantum Mechanics ψ-Ontic or ψ-Epistemic?

The ontological models framework distinguishes ψ-ontic from ψ-epistemic wave-functions. It is, in general, quite straightforward to categorize the wave-function of a certain quantum theory. Nevertheless, there has been a debate about the ontological status of the wave-function in the statistical interpretation of quantum mechanics: is it ψ-epistemic and incomplete or ψ-ontic and complete? I will argue that the wave-function in this interpretation is best regarded as ψ-ontic and incomplete.

A unifying nonlinear probabilistic epidemic model in space and time

Covid-19 epidemic dramatically relaunched the importance of mathematical modelling in supporting governments decisions to slow down the disease propagation. On the other hand, it remains a challenging task for mathematical modelling. The interplay between different models could be a key element in the modelling strategies. Here we propose a continuous space-time non-linear probabilistic model from which we can derive many of the existing models both deterministic and stochastic as for example SI, SIR, SIR stochastic, continuous-time stochastic models, discrete stochastic models, Fisher–Kolmogorov model. A partial analogy with the statistical interpretation of quantum mechanics provides an interpretation of the model. Epidemic forecasting is one of its possible applications; in principle, the model can be used in order to locate those regions of space where the infection probability is going to increase. The connection between non-linear probabilistic and non-linear deterministic models is analyzed. In particular, it is shown that the Fisher–Kolmogorov equation is connected to linear probabilistic models. On the other hand, a generalized version of the Fisher–Kolmogorov equation is derived from the non-linear probabilistic model and is shown to be characterized by a non-homogeneous time-dependent diffusion coefficient (anomalous diffusion) which encodes information about the non-linearity of the probabilistic model.

The Geometrical Meaning of Spinors Lights the Way to Make Sense of Quantum Mechanics

This paper aims at explaining that a key to understanding quantum mechanics (QM) is a perfect geometrical understanding of the spinor algebra that is used in its formulation. Spinors occur naturally in the representation theory of certain symmetry groups. The spinors that are relevant for QM are those of the homogeneous Lorentz group SO(3,1) in Minkowski space-time R4 and its subgroup SO(3) of the rotations of three-dimensional Euclidean space R3. In the three-dimensional rotation group, the spinors occur within its representation SU(2). We will provide the reader with a perfect intuitive insight about what is going on behind the scenes of the spinor algebra. We will then use the understanding that is acquired to derive the free-space Dirac equation from scratch, proving that it is a description of a statistical ensemble of spinning electrons in uniform motion, completely in the spirit of Ballentine’s statistical interpretation of QM. This is a mathematically rigorous proof. Developing this further, we allow for the presence of an electromagnetic field. We can consider the result as a reconstruction of QM based on the geometrical understanding of the spinor algebra. By discussing a number of problems in the interpretation of the conventional approach, we illustrate how this new approach leads to a better understanding of QM.

Classical dynamics, arrow of time, and genesis of the Heisenberg commutation relations

Based on the assumption that time evolves only in one direction and mechanical systems can be described by Lagrangeans, a dynamical C*-algebra is presented for non-relativistic particles at atomic scales. Without presupposing any quantization scheme, this algebra is inherently non-commutative and comprises a large set of dynamics. In contrast to other approaches, the generating elements of the algebra are not interpreted as observables, but as operations on the underlying system; they describe the impact of temporary perturbations caused by the surroundings. In accordance with the doctrine of Niels Bohr, the operations carry individual names of classical significance. Without stipulating from the outset their “quantization”, their concrete implementation in the quantum world emerges from the inherent structure of the algebra. In particular, the Heisenberg commutation relations for position and velocity measurements are derived from it. Interacting systems can be described within the algebraic setting by a rigorous version of the interaction picture. It is shown that Hilbert space representations of the algebra lead to the conventional formalism of quantum mechanics, where operations on states are described by time-ordered exponentials of interaction potentials. It is also discussed how the familiar statistical interpretation of quantum mechanics can be recovered from operations.

Two Faced Janus of Quantum Nonlocality.

This paper is a new step towards understanding why “quantum nonlocality” is a misleading concept. Metaphorically speaking, “quantum nonlocality” is Janus faced. One face is an apparent nonlocality of the Lüders projection and another face is Bell nonlocality (a wrong conclusion that the violation of Bell type inequalities implies the existence of mysterious instantaneous influences between distant physical systems). According to the Lüders projection postulate, a quantum measurement performed on one of the two distant entangled physical systems modifies their compound quantum state instantaneously. Therefore, if the quantum state is considered to be an attribute of the individual physical system and if one assumes that experimental outcomes are produced in a perfectly random way, one quickly arrives at the contradiction. It is a primary source of speculations about a spooky action at a distance. Bell nonlocality as defined above was explained and rejected by several authors; thus, we concentrate in this paper on the apparent nonlocality of the Lüders projection. As already pointed out by Einstein, the quantum paradoxes disappear if one adopts the purely statistical interpretation of quantum mechanics (QM). In the statistical interpretation of QM, if probabilities are considered to be objective properties of random experiments we show that the Lüders projection corresponds to the passage from joint probabilities describing all set of data to some marginal conditional probabilities describing some particular subsets of data. If one adopts a subjective interpretation of probabilities, such as QBism, then the Lüders projection corresponds to standard Bayesian updating of the probabilities. The latter represents degrees of beliefs of local agents about outcomes of individual measurements which are placed or which will be placed at distant locations. In both approaches, probability-transformation does not happen in the physical space, but only in the information space. Thus, all speculations about spooky interactions or spooky predictions at a distance are simply misleading. Coming back to Bell nonlocality, we recall that in a recent paper we demonstrated, using exclusively the quantum formalism, that CHSH inequalities may be violated for some quantum states only because of the incompatibility of quantum observables and Bohr’s complementarity. Finally, we explain that our criticism of quantum nonlocality is in the spirit of Hertz-Boltzmann methodology of scientific theories.

Heisenberg's uncertainty conditions for various higher order probability distribution functions based on Budiyono–Rohrlich statistical model of quantum mechanics

This report discusses the explicit form of Heisenberg's uncertainty conditions based on recently proposed Budiyono–Rohrlich quantum interpretation which is formulated in terms of modified classical statistical mechanics within the framework of Hamilton–Jacobi formalism. This new statistical interpretation of quantum mechanics is remarkably compatible with the Copenhagen interpretation. We consider various probability distribution functions in the form of symmetric and localized higher order Gaussian, Cauchy-Lorentz and t− distribution functions. The characteristics of all those higher order distribution functions and the related uncertainty conditions are discussed, and we show that all those conditions satisfy the Heisenberg–Kennard uncertainty relation, as expected.

EPR paradox, quantum nonlocality and physical reality

Eighty years ago Einstein, Podolsky and Rosen demonstrated that instantaneous reduction of wave function, believed to describe completely a pair of entangled physical systems, led to EPR paradox. The paradox disappears in statistical interpretation of quantum mechanics (QM) according to which a wave function describes only an ensemble of identically prepared physical systems. QM predicts strong correlations between outcomes of measurements performed on different members of EPR pairs in far-away locations. Searching for an intuitive explanation of these correlations John Bell analysed so called local realistic hidden variable models and proved that correlations consistent with these models satisfy Bell inequalities which are violated by some predictions of QM and by experimental data. Several different local models were constructed and inequalities proven. Some eminent physicists concluded that Nature is definitely nonlocal and that it is acting according to a law of nonlocal randomness. According to these law perfectly random, but strongly correlated events, can be produced at the same time in far away locations and a local and causal explanation of their occurrence cannot be given. We strongly disagree with this conclusion and we prove the contrary by analysing in detail some influential finite sample proofs of Bell and CHSH inequalities and so called Quantum Randi Challenges. We also show how one can win so called Bell's game without violating locality of Nature. Nonlocal randomness is inconsistent with local quantum field theory, with standard model in elementary particle physics and with causal laws and adaptive dynamics prevailing in the surrounding us world. The experimental violation of Bell-type inequalities does not prove the nonlocality of Nature but it only confirms a contextual character of quantum observables and gives a strong argument against counterfactual definiteness and against a point of view according to which experimental outcomes are produced in irreducible random way.

Towards Einstein's dream of a unified field theory: Reports from a journey on a long and winding road

With general arguments it is motivated that we are still in search of a hidden-variables theory or sub-quantum mechanics. Aspects of a solvable model for quantum measurements are discussed, and a solution of the quantum measurement problem within ordinary quantum theory. The related statistical interpretation of quantum mechanics is advocated. Soliton approaches for elementary particles are proposed and some classical looking aspects of the relativistic hydrogen atom are mentioned. The integration of these solitons with stochastic electrodynamics is sketched.

The Statistical Origins of Quantum Mechanics

It is shown that Schrödinger's equation may be derived from three postulates. The first is a kind of statistical metamorphosis of classical mechanics, a set of two relations which are obtained from the canonical equations of particle mechanics by replacing all observables by statistical averages. The second is a local conservation law of probability with a probability current which takes the form of a gradient. The third is a principle of maximal disorder as realized by the requirement of minimal Fisher information. The rule for calculating expectation values is obtained from a fourth postulate, the requirement of energy conservation in the mean. The fact that all these basic relations of quantum theory may be derived from premises which are statistical in character is interpreted as a strong argument in favor of the statistical interpretation of quantum mechanics. The structures of quantum theory and classical statistical theories are compared, and some fundamental differences are identified.

Can the Statistical Interpretation of Quantum Mechanics be Inferred from the Schrödinger Equation?—Bell and Gottfried

In his paper titled ‘Against “measurement” ’ [Physics World 3(8), 33–40 [1990]], Bell criticised arguments that use the concept of measurement to justify the statistical interpretation of quantum theory. Among these was the text of Gottfried [Quantum Mechanics (Benjamin, New York, [1966])]. Gottfried has replied to this criticism, claiming to show that, for systems with both continuous and discrete degrees of freedom, the statistical interpretation for the discrete variables is implied by requiring that the continuous variables are described classically. In the present paper, Gottfried’s argument is criticised. It is suggested that he takes over aspects of classical physics which are in conflict with the classical limit of the Schrodinger equation. He incorrectly assumes that, in the output from a Stern-Gerlach apparatus, the wave-function of any ion is restricted to one or another of the beams.

ENTANGLEMENT, LOCALITY, AND SEPARABILITY: A TREATISE ON DIFFERENT VIEWPOINTS

Within the framework of a statistical interpretation of quantum mechanics, entanglement (in a mathematical sense) manifests itself in the non-separability of the statistical operator ρ representing the ensemble in question. In experiments, on the other hand, entanglement can be detected, in the form of non-locality, by the violation of Bell's inequality Δ ≤ 2. How can these different viewpoints be reconciled? We first show that (non-)separability follows different laws to (non-)locality, and, moreover, it is much more difficult to characterize as long as the mostly employed operational rather than an ontic definition of separability is used. In consequence, (i) "entanglement" has two different meanings which may or may not be realized simultaneously on one and the same ensemble, and (ii) we have to disadvise the use of the common separability definition which is still employed by the majority of the physical community.

Mapping of the relativistic kinetic balance equations onto the Klein–Gordon and second-order Dirac equations

In previous work we have shown that the quantum potential can be derived from the classical kinetic equations both for particles with and without spin. Here, we extend these mappings to the relativistic case. The essence of the analysis consists of Fourier transforming the momentum coordinate of the distribution function. This procedure introduces a natural parameter η with units of angular momentum. In the non-relativistic case the ansatz of either separability, or separability and additivity, imposed on the probability distribution function produces mappings onto the Schrödinger equation and the Pauli equation, respectively. The former corresponds to an irrotational flow, the latter to a fluid with non-zero vorticity. In this work we show that the relativistic mappings lead to the Klein–Gordon equation in the irrotational case and to the second-order Dirac equation in the rotational case. These mappings are irreversible; an approximate inverse is the Wigner function. Taken together, these results provide a statistical interpretation of quantum mechanics.

GENERALIZATION OF CLASSICAL STATISTICAL MECHANICS TO QUANTUM MECHANICS AND STABLE PROPERTY OF CONDENSED MATTER

Classical statistical average values are generally generalized to average values of quantum mechanics. It is discovered that quantum mechanics is a direct generalization of classical statistical mechanics, and we generally deduce both a new general continuous eigenvalue equation and a general discrete eigenvalue equation in quantum mechanics, and discover that a eigenvalue of quantum mechanics is just an extreme value of an operator in possibility distribution, the eigenvalue f is just classical observable quantity. A general classical statistical uncertain relation is further given, and the general classical statistical uncertain relation is generally generalized to the quantum uncertainty principle; the two lost conditions in classical uncertain relation and quantum uncertainty principle, respectively, are found. We generally expound the relations among the uncertainty principle, singularity and condensed matter stability, discover that the quantum uncertainty principle prevents the appearance of singularity of the electromagnetic potential between nucleus and electrons, and give the failure conditions of the quantum uncertainty principle. Finally, we discover that the classical limit of quantum mechanics is classical statistical mechanics, the classical statistical mechanics may further be degenerated to classical mechanics and we discover that merely stating that the classical limit of quantum mechanics is classical mechanics is a mistake. As application examples, we deduce both the Schrödinger equation and the state superposition principle, and deduce that there exists a decoherent factor from a general mathematical representation of the state superposition principle; the consistent difficulty between statistical interpretation of quantum mechanics and determinant property of classical mechanics is overcome.

God's Action in the World: The Relevance of Quantum Mechanics

It has been suggested that God can act on the world by operating within the limits set by Heisenberg's uncertainty principle (HUP) without violating the laws of nature. This requires nature to be intrinsically indeterministic. However, according to the statistical interpretation the quantum mechanical wavefunction represents the average behavior of an ensemble of similar systems and not that of a single system. The HUP thus refers to a relation between the spreads of possible values of position and momentum and so is consistent with a fully deterministic world. This statistical interpretation of quantum mechanics is supported by reference to actual measurements, resolves the quantum paradoxes, and stimulates further research. If this interpretation is accepted, quantum mechanics is irrelevant to the question of God's action in the world.

Inferring the statistical interpretation of quantum mechanics from the classical limit

It is widely believed that the statistical interpretation of quantum mechanics cannot be inferred from the Schrodinger equation itself, and must be stated as an additional independent axiom. Here I propose that the situation is not so stark. For systems that have both continuous and discrete degrees of freedom (such as coordinates and spin respectively), the statistical interpretation for the discrete variables is implied by requiring that the system's gross motion can be classically described under circumstances specified by the Schrodinger equation. However, this is not a full-fledged derivation of the statistical interpretation because it does not apply to the continuous variables of classical mechanics.

Generalized coherent states for the Pöschl-Teller potential and a classical limit

Coherent states in the harmonic oscillator may be defined in several equivalent ways. One definition describes coherent states as special states satisfying a minimum-uncertainty requirement in position and momentum spaces. This definition is generalized for other potentials according to a method developed by Nieto et al. [Phys. Rev. D 20, 1321 (1979)] and is herein applied to the P\"oschl-Teller potential. A classical limit based on the harmonic-oscillator coherent-state classical limit is then developed and applied to the P\"oschl-Teller minimum-uncertainty states. In this limit, classical behavior may be obtained from these quantum states. Together with a completeness argument for the generalized coherent states, this result provides insight into quantum classical correspondence through the statistical interpretation of quantum mechanics.

REINTERPRETATION OF QUANTUM MECHANICS BASED ON THE STATISTICAL INTERPRETATION

I attempt to develop further the statistical interpretation of quantum mechanics proposed by Einstein and developed by Popper, Ballentine, etc. Two ideas are proposed in the present paper. One is to interpret momentum as a property of an ensemble of similarly prepared systems which is not satisfied by any one member of the ensemble of systems. Momentum is regarded as a statistical parameter like temperature in statistical mechanics. The other is the holistic assumption that a probability distribution is determined as a whole as most likely to be realized. This is the same as the chief assumption in statistical mechanics, and maximum likelihood in classical statistics. These ideas enable us to understand statistically (1) the formalism of quantum mechanics, (2) Heisenberg's uncertainty relations, and (3) the origin of quantum equations. They also explain violation of Bell's inequality and the interference of probabilities.

Position measurements on quantum mechanical particles in bound states

The statistical interpretation of quantum mechanics and its possible conflict with the conservation of energy in the case of measurements on bound systems is described. A semi-classical solution to the problem given by Heisenberg is quantitatively reconstructed and proved not to solve the problem. Finally an approach within the quantum theory of measurement is made and shown to yield promising results.

Quantum measurements and information theory

Max Born [1] (1926) has proposed the statistical interpretation of quantum mechanics and there is no doubt that it well describes the statistics of pointer readings. Is this knowledge sufficient for to understand the physical reality of microparticles, i.e. can a physical theory be complete if nothing behind the statistics of pointer readings is assumed? Such questions arose shortly after Max Born’s proposal in the Bohr-Einstein debate which culminated in the famous Einstein-Podolsky-Rosen paradox [2]. Later investigations have led to the Bell type inequalities [3]. The experimental tests by A. Aspect and his coworkers gave deep insight into the physical situation: IT somebody believes in the physical reality of accidental properties of a single microparticle described by some kind of hidden variables, then he has to assume the existence of actions at a distance.

Theory of Measurement: A Note on Conceptual Foundation of Quantum Mechanics

Theory of measurement is presented which is based on the statistical interpretation of quantum mechanics. The reduction of wave packet is ascribed to undue comparison of the state of an ensemble before measurement and the state of an individual system after measurement. By studying the evolution of statistical operators for subsystems and composite systems comprising the object, a measuring apparatus and an observer, one can definitely locate where the measurement is done, thus negating the principle of psycho-physical parallelism due to von Neumann