The symplectic structures (brackets, Hamilton's equations, and Lagrange's equations) for the Dirac electron and its classical model have exactly the same form. We give explicitly the Poisson brackets in the dynamical variables (x μ,p μ,v μ,S μv). The only difference is in the normalization of the Dirac velocities γμγμ=4 which has significant consequences.

In this paper, we show that the Kirchhoff equations are derived from the Schrödinger equation by assuming the wave function to be a polynomial like solution. These Kirchhoff equations describe the evolution of n point vortices in hydrodynamics. In two dimensions, Kirchhoff equations are used to demonstrate the solution to single particle Laughlin wave function as complex Hermite polynomials. We also show that the equation for optical vortices, a two dimentional system, is derived from Kirchhoff equation by (...) using paraxial wave approximation. These Kirchhoff equations satisfy a Poisson bracket relationship in phase space which is identical to the Heisenberg uncertainty relationship. Therefore, we conclude that being classical equations, the Kirchhoff equations, describe both a particle and a wave nature of single particle quantum mechanics in two dimensions. (shrink)

A quantum theory of the photon is developed in a natural manner. Newton-Wigner and Wightman demonstrated that the photon could not be strictly localized according to natural criteria. These investigations involved the identification of an elementary system with a uirrep of the Poincare group. We identify a particle with the localized measurement of the states satisfying the uirrep. In the case of zero mass and unit spin, the photon is identified with those components of the state that can be (...) localized. A c-number four-vector potential and Lorentz condition are derived from the relativistic wave equation. The Wightman localization is demonstrated for the three independent space components of the vector potential, and the photon is identified with these components. A position operator and probability density follow immediately from the localization. A consequence of the subjective definition of a photon is that the transformations of the vector potential are unitary, and hence the unitary scalar product can be obtained for the four-vector potential. A Hilbert space is defined for the three space components of the vector potential. A position operator and probability density are derived from the scalar product, which compare directly with those obtained from the localization and the non-relativistic theory. As the longitudinal and scalar polarizations do not contribute to the measured transition probability, they are considered virtual. Lastly, a conserved four-vector current is derived from the scalar product. The possibility of observing a strict localization of the photon in the laboratory is suggested. (shrink)

For a wave packet evolving quantum mechanically, the rates of change of the expectations and uncertainties of the position and momentum are exactly the same as if Wigner's function instantaneously obeyed a classical Liouville equation (whatever the Hamiltonian). This extension of Ehrenfest's theorem should be useful for dealing with the evolution and manipulation of quantum states.

We consider the relativistic statistical mechanics of an ensemble of N events with motion in space-time parametrized by an invariant “historical time” τ. We generalize the approach of Yang and Yao, based on the Wigner distribution functions and the Bogoliubov hypotheses to find approximate dynamical equations for the kinetic state of any nonequilibrium system, to the relativistic case, and obtain a manifestly covariant Boltzmann- type equation which is a relativistic generalization of the Boltzmann-Uehling-Uhlenbeck (BUU) equation for indistinguishable (...) particles. This equation is then used to prove the H-theorem for evolution in τ. In the equilibrium limit, the covariant forms of the standard statistical mechanical distributions are obtained. We introduce two-body interactions by means of the direct action potential V(q), where q is an invariant distance in the Minkowski space-time. The two- body correlations are taken to have the support in a relative O(2, 1)-invariant subregion of the full spacelike region. The expressions for the energy density and pressure are obtained and shown to have the same forms (in terms of an invariant distance parameter) as those of the nonrelativistic theory and to provide the correct nonrelativistic limit. (shrink)

In a recent paper in Astrophysics and Space Science Vol. 364 no. 11 (2019), S. Ershkov & D. Leschenko presented a new solving procedure for Euler-Poisson equations for solving momentum equations of the CR3BP near libration points for uniformly rotating planets having inclined orbits in the solar system with respect to the orbit of the Earth. The system of equations of the CR3BP has been explored with regard to the existence of an analytic way of presentation of the approximated (...) solution in the vicinity of libration points. A new and elegant ansatz has been suggested in their publication whereby, in solving, the momentum equation is reduced to a system two coupled Riccati ODEs. In this paper, we presented a numerical solution of such coupled Riccati ODEs using Mathematica software package. (shrink)

In this paper we show that the three main equations used by Bohm in his approach to quantum mechanics are already contained in the earlier paper by Moyal which forms the basis for what is known as the Wigner-Moyal approach. This shows, contrary to the usual perception, that there is a deep relation between the two approaches. We suggest the relevance of this result to the more general problem of constructing a quantum geometry.

Bohmian mechanics faces an underdetermination problem: when it comes to solving the measurement problem, alternatives to the Bohmian guidance equation work just as well as the official guidance equation. One way to argue that the guidance equation is superior to its rivals is to use a symmetry argument: of the candidate guidance equations, the official guidance equation is the simplest Galilean-invariant candidate. This symmetry argument---if it worked---would solve the underdetermination problem. But the argument does not work. (...) It fails because it rests on assumptions about how Galilean transformations (especially boosts) act on the wavefunction that are (in this context) unwarranted. My discussion has larger morals about the physical significance of certain mathematical results (like, for example, Wigner's theorem) in non-orthodox interpretations of quantum mechanics. (shrink)

We discuss the status of gravitational radiation in Newtonian theories. In order to do so, we consider various options for interpreting the Poisson equation as encoding propagating solutions, reflect on the extent to which limit considerations from general relativity can shed light on the Poisson equation’s conceptual status, and discuss various senses in which the Poisson equation counts as a dynamical equation.

Why are some models, like the harmonic oscillator, the Ising model, a few Hamiltonian equations in quantum mechanics, the poisson equation, or the Lokta-Volterra equations, repeatedly used within and across scientific domains, whereas theories allow for many more modeling possibilities? Some historians and philosophers of science have already proposed plausible explanations. For example, Kuhn and Cartwright point to a tendency toward conservatism in science, and Humphreys emphasizes the importance of the intractability of what he calls “templates.” This paper (...) investigates more systematically the reasons for this remarkable interdisciplinary recurrence. To this aim, the authors describe in more detail the phenomenon they focus on and review competing potentialexplanations. The authors disentangle the various assumptions underlying these explanations based on sensitivity to a computational constraints and assess its relationships with the other analyzed explanatons. (shrink)

Wigner's quantum mechanical formulation of the problem of biological replication is examined with special reference to DNA. His necessary condition for replication is that the number of independent equations in his formulation should not exceed the number of unknowns. Explicit hypotheses concerning the relevant collision matrix are proposed without assuming biotonic modifications of quantum mechanics. Schrödinger's description of the gene as a low-temperature solid, combined with the concept of template, is given mathematical expression which fails to satisfy Wigner's (...) necessary condition. The condition is satisfied by a further specialization of the collision matrix, which also implies metabolic stability of the replicating unit. Such metabolic stability is in fact a distinguishing feature of the DNA molecule. (shrink)

The Wigner distribution and its equation of motion in the scalar potential case are arrived at in an unusual way. This in turn suggests (a) a departure from the standard Wigner distribution treatment for a charged particle in a magnetic field and (b) a new approach to quantization of nonconservative systems. Suggestion (a) is found to be, like the standard treatment, in agreement with Schrödinger's equation but, unlike it, also satisfies local classical-type conservation laws and employs (...) a distribution which is gauge-invariant rather than merely gauge-covariant. Suggestion (b) gives a clear result only in the case of resistance proportional to velocity, when it agrees with the Schrödinger-Langevin equation; for other dissipative systems a fresh assumption is required, and a proposal in that direction is put forward. (shrink)

In this paper the quantum covariant relativistic dynamics of many bodies is reconsidered. It is emphasized that this is an event dynamics. The events are quantum statistically correlated by the global parameter τ. The derivation of an event Boltzmann equation emphasizes this. It is shown that this Boltzmann equation may be viewed as exact in a dilute event limit ignoring three event correlations. A quantum entropy principle is obtained for the marginal Wigner distribution function. By means of (...) event linking (concatenations) particle properties such as the equation of state may be obtained. We further reconsider the generalized quantum equilibrium ensemble theory and the free event case of the Fermi-Dirac and Bose-Einstein distributions, and some consequences. The ultra-relativistic limit differs from the non-covariant theory and is a test of this point of view. (shrink)

Using the time-dependent wave function we have studied the properties of the atomic transverse motion in an interferometer, and the cause of the non-classical behavior of atoms reported by Kurtsiefer, Pfau, and Mlynek [Nature 386, 150 (1997)]. The transverse wave function is derived from the solution of the two-dimensional Schrödinger's equation, written in the form of the Fresnel–Kirchhoff diffraction integral. It is assumed that the longitudinal motion is classical. Comparing data of the space distribution and of the transverse momentum (...) distribution in interferometers with one and two open slits, it follows that the atomic motion is influenced by the atomic matter wave and violates the laws of classical mechanics. However, the negative values of Wigner's function should not be taken as evidence that the atoms in an interferometer violate the classical statistical law of the addition of positive probabilities. This inference follows from the comparison of properties of Wigner's function and of the de Broglian probability density in phase space. (shrink)

The evolution of the equation of mass conservation in fluid mechanics is studied. Following early hydraulic approximations, and progress by Daniel and Johann Bernoulli, its first expression as a partial differential equation was achieved by d’Alembert, and soon given definitive form by Euler. Later reworkings by Lagrange, Laplace, Poisson and others advanced the subject, but all based their derivations on the conserved mass of a moving fluid particle. Later, Duhamel and Thomson gave a simpler derivation, by considering (...) mass flow into and out of a fixed portion of space. The later propagation of these derivations in nineteenth-century British textbooks and treatises is also examined, including Maxwell’s on the kinetic theory of gases. (shrink)

This paper expounds the relations between continuous symmetries and conserved quantities, i.e. Noether's ``first theorem'', in both the Lagrangian and Hamiltonian frameworks for classical mechanics. This illustrates one of mechanics' grand themes: exploiting a symmetry so as to reduce the number of variables needed to treat a problem. I emphasise that, for both frameworks, the theorem is underpinned by the idea of cyclic coordinates; and that the Hamiltonian theorem is more powerful. The Lagrangian theorem's main ``ingredient'', apart from cyclic coordinates, (...) is the rectification of vector fields afforded by the local existence and uniqueness of solutions to ordinary differential equations. For the Hamiltonian theorem, the main extra ingredients are the asymmetry of the Poisson bracket, and the fact that a vector field generates canonical transformations iff it is Hamiltonian. (shrink)

It has been almost eighty years since Paul Dirac delivered a lecture on the relationship between mathematics and physics and since 1960 that Eugene Wigner wrote about the unreasonable effectiveness of mathematics in the natural sciences. The field of cosmology and efforts to define a more comprehensive theory have changed significantly since the 1960s; thus, it is time to refocus on the issue. This paper expands on ideas addressed by these two great physicists, specifically, the ultimate effectiveness of mathematics (...) to describe nature. After illustrating how theoretical physic predicted discoveries, the concepts of established theories are summarized. Then, the “world equation” which combines all key physics theories is conceptually described. As the possible last piece of the puzzle, String Theory is briefly defined. And last, a cursory overview of an extreme proposal is presented — the world as a mathematical object. However, there is no fundamental reason to believe that math will allow mankind to completely comprehend nature. If math does not provide a comprehensive theory, nature will retain her secrets. (shrink)

Two hundred years ago, Friedrich Schleiermacher took critical issue with Immanuel Kant's intellectual notion of intuition as applied to human nature (Wellmon 2006). He found it necessary to modify—"hermeneutically," as he said—Kant's notion of anthropology by enabling it to include as human the new and strange human tribes Captain Cook found in the Pacific South Seas. A similar hermeneutic move is necessary if physics is to include the local contextual empirical syntheses of relativity and quantum physics. In this hermeneutical revision (...) the synthesis is formed around the notion of a Hilbert Vector Space as the universal grammar of physics, adding to it the dynamic of the Schrödinger equation, and representing empirical "observables" by projection operators that map the subspaces of definite measurable values. Among the set of observable projection operators, some pairs share the same subspace, commute with one another, and share a common laboratory setting. Other pairs do not share this property and are described as being mutually complementary. Complementary symmetries introduce into the discursive language of physics the commonsense notion of contextuality. The new synthesis, proposed by Eugene Wigner, John von Neumann, and (in his own way) Paul Dirac, brought physics into the community of common language and established it as a work of general human achievement. 1. (shrink)

The Quantum Measurement Problem is arguably one of the most debated issues in the philosophy of Quantum Mechanics, since it represents not only a technical difficulty for the standard formulation of the theory, but also a source of interpretational disputes concerning the meaning of the quantum postulates. Another conundrum intimately connected with the QMP is the Wigner friend paradox, a thought experiment underlining the incoherence between the two dynamical laws governing the behavior of quantum systems, i.e the Schrödinger (...) class='Hi'>equation and the projection rule. Thus, every alternative interpretation aiming to be considered a sound formulation of QM must provide an explanation to these puzzles associated with quantum measurements. It is the aim of the present essay to discuss them in the context of Relational Quantum Mechanics. In fact, it is shown here how this interpretative framework dissolves the QMP. More precisely, two variants of this issue are considered: on the one hand, I focus on the “the problem of outcomes” contained in Maudlin (1995)—in which the projection postulate is not mentioned—on the other hand, I take into account Rovelli’s reformulation of this problem proposed in Rovelli (2022), where the tension between the Schrödinger equation and the stochastic nature of the collapse rule is explicitly considered. Moreover, the relational explanation to the Wigner’s friend paradox is reviewed, taking also into account some interesting objections contra Rovelli’s theory contained in Laudisa (2019). I contend that answering these critical remarks leads to an improvement of our understanding of RQM. Finally, a possible objection against the relational solution to the QMP is presented and addressed. (shrink)

In this third paper in a series on stochastic electrodynamics (SED), the nonrelativistic dipole approximation harmonic oscillator-zero-point field system is subjected to an arbitrary classical electromagnetic radiation field. The ensemble-averaged phase-space distribution and the two independent ensemble-averaged Liouville or Fokker-Planck equations that it satisfies are derived in closed form without furtner approximation. One of these Liouville equations is shown to be exactly equivalent to the usual Schrödinger equation supplemented by small radiative corrections and an explicit radiation reaction (RR) vector (...) potential that is similar to the Crisp-Jaynes semiclassical theory (SCT) RR potential. The wave function in this SED Schrödinger equation is shown to have thea priori significance of position probability amplitude. The other Liouville equation has no counterpart in ordinary quantum mechanics, and is shown to restrict initial conditions such that (i) The Wigner-type phase-space distribution is always positive, (ii) in the absence of an applied field, the only allowed solution of both equations is the quantum ground state, and (iii) if a previously applied field is suddenly turned off, then spontaneous transitions occur, with no need for a triggering perturbation as in SCT, until the system is in the ground state. It is also shown that the oscillator energy is a fluctuating quantity that must take on a continuum of values, with average value equal to the quantum ground-state energy plus a contribution due to the applied classical field. (shrink)

We consider the superposition of macroscopically distinguishable states for a measuring process whose time evolution is described by the Schrödinger equation. We ask whether it is possible to observe interference effects due to the above mentioned superposition and how to observe them, taking into consideration an experiment performed by other authors. We find a necessary condition in order to be able to observe these effects. We also point out some very serious difficulties in observing them and analyse the connection (...) between some of these difficulties and the Wigner-Araki-Yanase theory. We try to explain why the whole problem seems to us to be far from having a solution and we suggest some paths we think worthy of further investigation. (shrink)

The recent discovery of an indeterministic system in classical mechanics, the Norton dome, has shown that answering the question whether classical mechanics is deterministic can be a complicated matter. In this paper I show that indeterministic systems similar to the Norton dome were already known in the nineteenth century: I discuss four nineteenth century authors who wrote about such systems, namely Poisson, Duhamel, Boussinesq and Bertrand. However, I argue that their discussion of such systems was very different from the (...) contemporary discussion about the Norton dome, because physicists in the nineteenth century conceived of determinism in essentially different ways: whereas in the contemporary literature on determinism in classical physics, determinism is usually taken to be a property of the equations of physics, in the nineteenth century determinism was primarily taken to be a presupposition of theories in physics, and as such it was not necessarily affected by the possible existence of systems such as the Norton dome. (shrink)

Dissipative systems are described by a Hamiltonian, combined with a “dynamical matrix” which generalizes the simplectic form of the equations of motion. Criteria for dissipation are given and the examples of a particle with friction and of the Lotka-Volterra model are presented. Quantization is first introduced by translating generalized Poisson brackets into commutators and anticommutators. Then a generalized Schrödinger equation expressed by a dynamical matrix is constructed and discussed.

We discuss some outstanding open questions regarding the validity and uniqueness of the standard second-order Newton-Einstein classical gravitational theory. On the observational side we discuss the degree to which the realm of validity of Newton's law of gravity can actually be extended to distances much larger than the solar system distance scales on which the law was originally established. On the theoretical side we identify some commonly accepted (but actually still open to question) assumptions which go into the formulation of (...) the standard second-order Einstein theory in the first place. In particular, we show that while the familiar second-order Poisson gravitational equation (and accordingly its second-order covariant Einstein generalization) may be sufficient to yield Newton's law of gravity they are not in fact necessary. The standard theory thus still awaits the identification of some principle which would then make it necessary too. We show that current observational information does not exclusively mandate the standard theory, and that the conformal invariant fourth-order theory of gravity considered recently by Mannheim and Kazanas is also able to meet the constraints of data, and in fact to do so without the need for any so far unobserved nonluminous or dark matter. (shrink)

One of the most unsettling problems in the history of philosophy examines how mathematics can be used to adequately represent the world. An influential thesis, stated by Eugene Wigner in his paper entitled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," claims that "the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve." Contrary to this view, this thesis delineates and (...) implements a strategy to show that the applicability of mathematics is very reasonable indeed. I distinguish three forms of the problem of the applicability of mathematics, and focus on one I call the problem of uncanny accuracy: Given that the construction and manipulation of mathematical representations is pervaded by uncertainty, error, approximation, and idealization, how can their apparently uncanny accuracy be explained? I argue that this question has found no satisfactory answer because our rational reconstruction of scientific practice has not involved tools rich enough to capture the logic of mathematical modelling. Thus, I characterize a general schema of mathematical analysis of real systems, focusing on the selection of modelling assumptions, on the construction of model equations, and on the extraction of information, in order to address contextually determinate questions on some behaviour of interest. A concept of selective accuracy is developed to explain the way in which qualitative and quantitative solutions should be utilized to understand systems. The qualitative methods rely on asymptotic methods and on sensitivity analysis, whereas the quantitative methods are best understood using backward error analysis. The basic underpinning of this perspective is readily understandable across scientific fields, and it thereby provides a view of mathematical tractability readily interpretable in the broader context of mathematical modelling. In addition, this perspective is used to discuss the nature of theories, the role of scaling, and the epistemological and semantic aspects of experimentation. In conclusion, we argue for a method of local and global conceptual analysis that goes beyond the reach of the tools standardly used to capture the logic of science; on their basis, the applicability of mathematics finds itself demystified. (shrink)

We investigate quantum features of three coupled dissipative nano-optomechanical oscillators. The Hamiltonian of the system is somewhat complicated due not only to the coupling of the optomechanical oscillators but to the dissipation in the system as well. In order to simplify the problem, a spatial unitary transformation approach and a matrix-diagonalization method are used. From such procedures, the Hamiltonian is eventually diagonalized. In other words, the complicated original Hamiltonian is transformed to a simple one which is associated to three independent (...) simple harmonic oscillators. By utilizing such a simplification of the Hamiltonian, complete solutions of the Schrödinger equation for the optomechanical system are obtained. We confirm that the probability density converges to the origin of the coordinate in a symmetric manner as the optomechanical energy dissipates. The wave functions that we have derived can be used as a basic tool for evaluating diverse quantum consequences of the system, such as quadrature fluctuations, entanglement entropy, energy evolution, transition probability, and the Wigner function. (shrink)

The use of the axial vector representing a three-dimensional rotation makes the rotation representation much more compact by extending the trigonometric functions to vectorial arguments. Similarly, the pure Lorentz transformations are compactly treated by generalizing a scalar rapidity to a vector quantity in spatial three-dimensional cases and extending hyperbolic functions to vectorial arguments. A calculation of the Wigner rotation simplified by using the extended functions illustrates the fact that the rapidity vector space obeys hyperbolic geometry. New representations bring a (...) Lorentz-invariant fundamental equation of motion corresponding to the Galilei-invariant equation of Newtonian mechanics. (shrink)

This is the first in a series of papers that present a new classical statistical treatment of the system of a charged harmonic oscillator (HO) immersed in an omnipresent stochastic zero-point (ZP) electromagnetic radiation field. This paper establishes the Gaussian statistical properties of this ZP field using Bourret's postulate that all statistical moments of the stochastic field plane waves at a given space-time point should agree with their corresponding quantized field vacuum expectations. This postulate is more than adequate to derive (...) the Planck spectrum classically via Boyer's and Theimer's methods, but it requires that the stochastic amplitude of each linearly polarized plane wave in the field contain two independent Gaussian random variables, not just a random phase as has sometimes been assumed. In the succeeding papers in the series, the total motion of a charged HO is described by a fully renormalized dipole-approximation Abraham-Lorentz equation. This leads without further approximation to the following major results concerning this stochastic electrodynamics (SED) of the HO: i) The ensemble-average Liouville equation for the oscillator-ZP field system in the presence of an arbitrary applied classical radiation field is exactly equivalent to the usual time-dependent Schrödinger equation supplemented by an explicit radiation reaction vector potential similar to that of the Crisp-Jaynes-Stroud theory; ii) this SED Schrödinger equation for the HO is incomplete, insmuch as there exists a companion equation that restricts initial conditions such that the corresponding Wigner phase-space distribution is always positive; iii) the wave function of the SED Schrödinger equation has thea priori significance of position probability amplitude; iv) first-order transition rates predicted for the HO by this theory agree with those predicted by quantum electrodynamics for resonance absorption and spontaneous emission, which occurs with no triggering necessary; and v) if SED is taken seriously, then the concepts of quantized energies and photons must be abandoned. (shrink)

The fact that the same equations or mathematical models reappear in the descriptions of what are otherwise disparate physical systems can be seen as yet another manifestation of Wigner's “unreasonable effectiveness of mathematics.” James Clerk Maxwell famously exploited such formal similarities in what he called the “method of physical analogy.” Both Maxwell and Hermann von Helmholtz appealed to the physical analogies between electromagnetism and hydrodynamics in their development of these theories. I argue that a closer historical examination of the (...) different ways in which Maxwell and Helmholtz each deployed this analogy gives further insight into debates about the representational and explanatory power of mathematical models. (shrink)

The connection between stochastic electrodynamics (SED) and the quantum theory of matter is further explored. The main result is that the Fokker-Planck-like equation of SED can be recast into the form of a Schrödinger equation with radiative corrections, when the system is close to a state of equilibrium. The phase-space distribution can be written as Wigner's pseudo-distribution plus corrections due to the nonlinearity of the external force and to radiative effects. The radiative corrections predicted by the theory, (...) namely the Lamb shift and the decay of excited atomic states, coincide with those predicted by QED. Moreover, the theory offers a clear physical interpretation of these phenomena as due to the coupling of the electric dipole of the system with the zero-point radiation field and to radiation reaction, respectively. (shrink)

We approach the relationship between classical and quantum theories in a new way, which allows both to be expressed in the same mathematical language, in terms of a matrix algebra in a phase space. This makes clear not only the similarities of the two theories, but also certain essential differences, and lays a foundation for understanding their relationship. We use the Wigner-Moyal transformation as a change of representation in phase space, and we avoid the problem of “negative probabilities” by (...) regarding the solutions of our equations as constants of the motion, rather than as statistical weight factors. We show a close relationship of our work to that of Prigogine and his group. We bring in a new nonnegative probability function, and we propose extensions of the theory to cover thermodynamic processes involving entropy changes, as well as the usual reversible processes. (shrink)

A macroscopic realization of the peculiar virtual particles is presented. The classical Helmholtz and the Schrödinger equations are differential equations of the same mathematical structure. The solutions with an imaginary wave number are called evanescent modes in the case of elastic and electromagnetic fields. In the case of non-relativistic quantum mechanical fields they are called tunneling solutions. The imaginary wave numbers point to strange consequences: The waves are non-local, they are not observable, and they are described as virtual particles. During (...) the last two decades QED calculations of the solutions with an imaginary wave number have been experimentally confirmed for phonons, photons, and electrons. The experimental proofs of the predictions of non-relativistic quantum mechanics and the Wigner phase time approach for the elastic, electromagnetic and Schrödinger fields will be presented in this article. The results are zero time in the barrier and an interaction time (i.e. a phase shift) at the barrier interfaces. The measured tunneling time scales approximately inversely with the particle energy. Actually, the tunneling time is given only by the barrier boundary interaction time, as zero time is spent inside a barrier. (shrink)

The significance of the de Broglie/Bohm hidden-particle position in the relativistic regime is addressed, seeking connection to the single-particle Newton–Wigner position. The effect of non-positive excursions of the ensemble density for extreme cases of positive-energy waves is easily computed using an integral of the equations of motion developed here for free spin-0 particles in 1 + 1 dimensions and is interpreted in terms of virtual-like pair creation and annihilation beneath the Compton wavelength. A Bohm-theoretic description of the acausal explosion (...) of a specific Newton–Wigner-localized state is presented in detail. The presence of virtual pairs found is interpreted as the Bohm picture of the spatial extension beyond single point particles proposed in the 1960s as to why space-like hyperplane dependence of the Newton–Wigner wavefunctions may be needed to achieve Lorentz covariance. For spin-1/2 particles the convective current is speculatively utilized for achieving parity with the spin-0 theory. The spin-0 improper quantum potential is generalized to an improper stress tensor for spin-1/2 particles. (shrink)

A model for the trapping of animals with a circular pitfall is formulated. The model's assumptions are: The animals move independently according to the same Brownian motions. The boundary of the pitfall acts as an absorbing or elastic barrier. Initially a fixed number of animals is independently homogeneously distributed over a finite study area, or the initial positions follow a homogeneous planar Poisson process. The model depends on three free parameters: the motility of the animals, their reaction to the (...) pitfall, the initial density.It appears that the catches in disjoint time intervals are multinomially or independently Poisson distributed. The parameters of these distributions are obtained by solving certain partial differential equations. (shrink)

We show that particle-antiparticle exchange and covariant motion reversal are two physically different aspects of the same mathematical transformation, either in the prequantal relativistic equation of motion of a charged point particle, in the general scheme of second quantization, or in the spinning wave equations of Dirac and of Petiau-Duffin-Kemmer. While, classically, charge reversal and rest mass reversal are equivalent operations, in the wave mechanical case mass reversal must be supplemented by exchange of the two adjoint equations, implying ψ (...) ⇄ $\bar \psi$ .Denoting by M the rest mass reversal, P the parity reversal, T the Racah time reversal, and Z the ψ ⇄ $\bar \psi$ exchange, the connection with the usual scheme of charge conjugation, parity reversal, and Wigner motion reversal, is with, of course. (shrink)

A solution of the sourceless Einstein's equation with an infinite value for the cosmological constant Λ is discussed by using Inönü–Wigner contractions of the de Sitter groups and spaces. When Λ→∞, spacetime becomes a four-dimensional cone, dual to Minkowski space by a spacetime inversion. This inversion relates the four-cone vertex to the infinity of Minkowski space, and the four-cone infinity to the Minkowski light-cone. The non-relativistic limit c→∞ is further considered, the kinematical group in this case being a (...) modified Galilei group in which the space and time translations are replaced by the non-relativistic limits of the corresponding proper conformal transformations. This group presents the same abstract Lie algebra as the Galilei group and can be named the conformal Galilei group. The results may be of interest to the early Universe Cosmology. (shrink)

The quantum mechanical description of the evolution of an unstable system defined initially as a state in a Hilbert space at a given time does not provide a semigroup (exponential) decay, law. The Wigner–Weisskopf survival amplitude, describing reversible quantum transitions, may be dominated by exponential type decay in pole approximation at times not too short or too long, but, in the two channel case, for example, the pole residues are not orthogonal, and the evolution does riot correspond to a (...) semigroup (experiments on the decay of the neutral K-meson system strongly support the semigroup evolution postulated, by Lee, Oehme and Yang, and Yang and Wu). The scattering theory of Lax and Phillips, originally developed for classical wave equations, has been recently extended to the description of the evolution of resonant states in the framework of quantum theory. The resulting evolution law of the unstable system is that of a semigroup, and the resonant state is a well-defined function in the Lax–Phillips Hilbert space. In this paper we apply this theory to a relativistically covariant quantum field theoretical form of the (soluble) Lee model. We construct the translation representations with the help of the wave operators, and show that the resulting Lax–Phillips S-matrix is an inner function (the Lax–Phillips theory is essentially a theory of translation invariant subspaces). In the special case that the S-matrix is a rational inner function, we obtain the resonant state explicitly and analyze its particle (V, N, θ) content. If there is an exponential bound, the general case differs only by a so-called trivial inner factor, which does not change the complex spectrum, but may affect the wave function of the resonant state. (shrink)

We apply the Galilean covariant formulation of quantum dynamics to derive the phase-space representation of the Pauli–Schrödinger equation for the density matrix of spin-1/2 particles in the presence of an electromagnetic field. The Liouville operator for the particle with spin follows from using the Wigner–Moyal transformation and a suitable Clifford algebra constructed on the phase space of a (4 + 1)-dimensional space–time with Galilean geometry. Connections with the algebraic formalism of thermofield dynamics are also investigated.

Dirac’s identification of the quantum analog of the Poisson bracket with the commutator is reviewed, as is the threat of self-inconsistent overdetermination of the quantization of classical dynamical variables which drove him to restrict the assumption of correspondence between quantum and classical Poisson brackets to embrace only the Cartesian components of the phase space vector. Dirac’s canonical commutation rule fails to determine the order of noncommuting factors within quantized classical dynamical variables, but does imply the quantum/classical correspondence of (...) Poisson brackets between any linear function of phase space and the sum of an arbitrary function of only configuration space with one of only momentum space. Since every linear function of phase space is itself such a sum, it is worth checking whether the assumption of quantum/classical correspondence of Poisson brackets for all such sums is still self-consistent. Not only is that so, but this slightly stronger canonical commutation rule also unambiguously determines the order of noncommuting factors within quantized dynamical variables in accord with the 1925 Born-Jordan quantization surmise, thus replicating the results of the Hamiltonian path integral, a fact first realized by E.H. Kerner. Born-Jordan quantization validates the generalized Ehrenfest theorem, but has no inverse, which disallows the disturbing features of the poorly physically motivated invertible Weyl quantization, i.e., its unique deterministic classical “shadow world” which can manifest negative densities in phase space. (shrink)

In this paper we unravel the connection between the quantum mechanical formalism and the Central limit theorem (CLT). We proceed to connect the results coming from this theorem with the derivations of the Schrödinger equation from the Liouville equation, presented by ourselves in other papers. In those papers we had used the concept of an infinitesimal parameter δx that raised some controversy. The status of this infinitesimal parameter is then elucidated in the framework of the CLT. Finally, we (...) use the formal apparatus developed in our previous papers and the results of the present one to advance an alternative objective interpretation of quantum mechanics in which its relations with the classical framework are made explicit. The relations between our approach and those using the Wigner–Moyal transformation are also addressed. (shrink)

Embodiment approaches to cognition and emotion have put forth the idea that the way we think and talk about affective events often recruits spatial information that stems, to some extent, from our bodily experiences. For example, metaphorical expressions such as “being someone’s right hand” or “leaving something bad behind” convey affectivity associated with the lateral and sagittal dimensions of space. Action tendencies associated with affect such as the directional fluency of hand movements (dominant right hand-side – positive; non-dominant left hand-side (...) – negative) and approach-avoidance behaviors (forward – positive; backwards – negative) might be mechanisms supporting such associations. Against this background, experimental research has investigated whether positive and negative words are freely allocated into space (e.g., close or far from one’s body) or resonate with congruent (vs. incongruent) predefined manual actions usually performed by joysticks or button presses (e.g., positive - right; negative - left, or vice versa). However, to date, it is unclear how the processing of affective concepts resonate with directional actions of the whole body that are performed freely within a context enabling lateral and sagittal movements simultaneously. Accordingly, 67 right-handed participants were to freely step on an 8-response pad (front, back, right, left, front-right, front-left, back-right, or back-left) after being presented in front of them valence-laden personal life-events submitted before the task (e.g., words or sentences such as “graduation” or “birth of a child”). The most revealing finding of this study indicates that approach-avoidance behaviors and space-valence associations across laterality are interwoven during whole body step actions: Positive events induced steps highly biased to front-right whereas negative events induced steps highly biased to back-left. (shrink)

The present discussion arose from the desire to explain, to an audience of non-physicists, the epistemology to which one is forced if one pursues the quantum mechanical theory of observation to its ultimate consequences. However, the conclusions will not be derived from the aforementioned theory but obtained on the basis of a rather general analysis of what we mean by real. Quantum theory will form the background but not the basis for the analysis. The concept of the real to be (...) arrived at shows considerable similarity to that of the idealist. As the title indicates, it is formulated as a dualism. It is quite possible that it will soon be rejected not only by the community of the philosophers but also by that of the scientists. If this should be the case, the attempt to derive an epistemology from physics will prove to have been premature. Naturally, the author hopes that this will not be the case because, quite apart from the quantum theoretical background, the concepts to be presented appear natural also as an outgrowth of common sense considerations. They have been arrived at by many who did not accept the epistemology of quantum mechanics. (shrink)