## No full-text available

To read the full-text of this research,

you can request a copy directly from the author.

In David Bohm’s causal/trajectory interpretation of quantum mechanics, a physical system is regarded as consisting of both a particle and a wavefunction, where the latter “pilots” the trajectory evolution of the former. In this paper, we show that it is possible to discard the pilot wave concept altogether, thus developing a complete mathematical formulation of time-dependent quantum mechanics directly in terms of real-valued trajectories alone. Moreover, by introducing a kinematic definition of the quantum potential, a generalized action extremization principle can be derived. The latter places very severe a priori restrictions on the set of allowable theoretical structures for a dynamical theory, though this set is shown to include both classical mechanics and quantum mechanics as members. Beneficial numerical ramifications of the above, “trajectories only” approach are also discussed, in the context of simple benchmark applications.

To read the full-text of this research,

you can request a copy directly from the author.

... Curiously, the particular choices made here can be interpreted as determining the precise form of the "interworld potential"�according to the "discrete" many-interacting-worlds (MIW) interpretation of quantum mechanics that has sprung up from the trajectorybased reformulation. 39 In the original, "continuous" MIW interpretation, however, 36,38 these are merely choices for the numerical discretization. ...

... Accordingly, in this work, we instead invoke the exact factorization formalism in combination with the interacting quantum trajectories approach presented above. 36,38 The exact factorization yields nuclear dynamics under the effect of a single time-dependent classical force accounting for the excited electronic states, thereby avoiding the aforementioned technical difficulty. ...

... One of the simplest choices is to take C to be the initial value of a given trajectory at time t = 0�that is, C = x 0 = x(t = 0). 36 Through probability conservation [i.e., eq 15], one then obtains the following relation for the density at any time t ...

We present a quantum dynamics method based on the propagation of interacting quantum trajectories to describe both adiabatic and nonadiabatic processes within the same formalism. The idea originates from the work of Poirier [Chem. Phys.2010,370, 4-14] and Schiff and Poirier [J. Chem. Phys.2012,136, 031102] on quantum dynamics without wavefunctions. It consists of determining the quantum force arising in the Bohmian hydrodynamic formulation of quantum dynamics using only information about quantum trajectories. The particular time-dependent propagation scheme proposed here results in very stable dynamics. Its performance is discussed by applying the method to analytical potentials in the adiabatic regime, and by combining it with the exact factorization method in the nonadiabatic regime.

... On the other hand, what if there were no need of a wavefunction at all in a quantum theory? In recent years, attempts have been made to formulate a complete standalone theory of quantum mechanics without wavefunctions [12,13,14,15,16,17,18,19,20,21,22,23]. In particular, in 2010, one of the authors (B. ...

... In particular, in 2010, one of the authors (B. Poirier) proposed a theoretical framework in which a quantum state is represented solely by an ensemble of real-valued probabilistic trajectories [14,17]. With the notable exception of spin, this nonrelativistic version of the trajectory-based theory turns out to be formally mathematically equivalent to the standard wave-based Schrödinger equation [12,13,14,17]-though it can be derived completely independently [14,17]. ...

... Poirier) proposed a theoretical framework in which a quantum state is represented solely by an ensemble of real-valued probabilistic trajectories [14,17]. With the notable exception of spin, this nonrelativistic version of the trajectory-based theory turns out to be formally mathematically equivalent to the standard wave-based Schrödinger equation [12,13,14,17]-though it can be derived completely independently [14,17]. More recently, a discrete version of the trajectory-based theory has also been proposed [20,21,22,23], which is not consistent with the Schrödinger equation except in the continuous limit. ...

We present novel aspects of a trajectory-based theory of massive spin-zero relativistic quantum particles. In this approach, the quantum trajectory ensemble is the fundamental entity. It satisfies its own action principle, leading to a dynamical partial differential equation (via the Euler-Lagrange procedure), as well as to conservation laws (via Noether’s theorem). In this paper, we focus on the derivation of the latter. In addition to the usual expected energy and momentum conservation laws, there is also a third law that emerges, associated with the conditions needed to maintain global simultaneity. We also show that the nonrelativistic limits of these conservation laws match those of the earlier, nonrelativistic quantum trajectory theory [J. Chem. Phys. 136, 031102 (2012)].

... As a promising alternative to the QTM pioneered by Wyatt, a fully wave function-free formulation of quantum mechanics has been developed [38]− [41], and first applications to atomic scattering have only very recently been published [42]. In this approach, the time-dependent quantum mechanical problem is recast into a dynamical problem of a parameterized density. ...

... It is worth to notice that previous work regarding the specific parametrization of the density used in this thesis involve its application to the description of model systems only [38]− [41]. The method implemented in this thesis gives accurate results, captures wellknown quantum effects and is a promising alternative to already existing, standard wave packet methods. ...

... Together with this theoretical background, we introduced the QTM, as the numerical implementation of the equation of motion of the quantum trajectories. As a promising alternative to the QTM pioneered by Wyatt, a fully wave function-free formulation of quantum mechanics has been proposed [38]− [41]. ...

In this thesis different trajectory-based methods for the study of quantum mechanical phenomena are developed. The first approach is based on a global expansion of the hydrodynamic fields in Chebyshev polynomials. The scheme is used for the study of one-dimensional vibrational dynamics of bound wave packets in harmonic and anharmonic potentials. Furthermore, a different methodology is developed, which, starting from a parametrization previously proposed for the density, allows the construction of effective interaction potentials between the pseudo-particles representing the density. Within this approach several model problems are studied and important quantum mechanical effects such as, zero point energy, tunneling, barrier scattering and over barrier reflection are founded to be correctly described by the ensemble of interacting trajectories. The same approximation is used for study the laser-driven atom ionization. A third approach considered in this work consists in the derivation of an approximate many-body quantum potential for cryogenic Ar and Kr matrices with an embedded Na impurity. To this end, a suitable ansatz for the ground state wave function of the solid is proposed. This allows to construct an approximate quantum potential which is employed in molecular dynamics simulations to obtain the absorption spectra of the Na impurity isolated in the rare gas matrix.

... In this paper, we present a quantum trajectory capture (QTC) technique that stems directly from a recent theoretical development in the exact formulation of quantum mechanics by one of the authors. 22,23 In this theory, a trajectory ensemble-rather than the usual wavefunction-is regarded as the fundamental quantum state entity. When applied in a time-independent 1D quantum reactive scattering context, the ensemble reduces to a single quantum trajectory-which necessarily always transmits from reactants to products, no matter how large the intervening barrier is. ...

... The quantum trajectory method outlined above has previously been applied in the context of abstraction reactions. [22][23][24] Here, as a proof-of-concept benchmark, we compute adiabaticchannel QTC probabilities and cross-sections for the Li + CaH(v = 0, j = 0) → LiH + Ca reaction, which we then compare with QM finite-difference 25,26 capture calculations performed previously by Tscherbul and Buchachenko. 27 ...

... Since QTC is essentially a scattering process, we start with a short discussion of a standard 1D quantum reactive scattering problem, from the perspective of the trajectoryensemble-based quantum theory. [22][23][24] Generalized boundary conditions as required for the capture process are introduced afterwards. ...

The Langevin capture model is often used to describe barrierless reactive collisions. At very low temperatures, quantum effects may alter this simple capture image and dramatically affect the reaction probability. In this paper, we use the trajectory-ensemble reformulation of quantum mechanics, as recently proposed by one of the authors (Poirier) to compute adiabatic-channel capture probabilities and cross-sections for the highly exothermic reaction Li + CaH(v = 0, j = 0) → LiH + Ca, at low and ultra-low temperatures. Each captured quantum trajectory takes full account of tunneling and quantum reflection along the radial collision coordinate. Our approach is found to be very fast and accurate, down to extremely low temperatures. Moreover, it provides an intuitive and practical procedure for determining the capture distance (i.e., where the capture probability is evaluated), which would otherwise be arbitrary.

... It resembles Bohmian mechanics in that quantum trajectories are indeed employed. However, unlike Bohmian mechanics, only trajectories are used-the wave being replaced with a trajectory ensemble, which thereby represents the quantum state [12][13][14][15][16][17][18][19][20]. The trajectory ensemble is continuous, with each individual member trajectory labeled by the parameter C. The time evolution of the trajectory ensemble, x(t, C), is governed by some partial differential equation (PDE) in (t, C) that replaces the usual timeindependent Schrödinger equation governing the Ψ(t, x) evolution. ...

... Note that Eq. (13) implies a specific parametrization for C-i.e., for any given trajectory, C is the initial x value, x 0 [14,21]. Likewise, our earlier specification of the natural time coordinate, λ = T , implies the following as the one remaining initial condition: ...

... From a numerical standpoint, the solution of Eq. (17) offers important advantages over both conventional, x-grid-based Crank-Nicholson propagation of Ψ, and traditional quantum trajectory methods [28]. Briefly, one has the simultaneous advantages of both a regular grid (in C) and probability-conserving trajectories (in x) [14]. There are nevertheless some nontrivial numerical issues, stemming from the fact that the true boundary conditions are unknownunlike for Ψ propagation, for which Dirichlet boundary conditions are in effect. ...

In the context of nonrelativistic quantum mechanics, Gaussian wavepacket solutions of the time-dependent Schrödinger equation provide useful physical insight. This is not the case for relativistic quantum mechanics, however, for which both the Klein-Gordon and Dirac wave equations result in strange and counterintuitive wavepacket behaviors, even for free-particle Gaussians. These behaviors include zitterbewegung and other interference effects. As a potential remedy, this paper explores a new trajectory-based formulation of quantum mechanics, in which the wavefunction plays no role [Phys. Rev. X, 4, 040002 (2014)]. Quantum states are represented as ensembles of trajectories, whose mutual interaction is the source of all quantum effects observed in nature—suggesting a "many interacting worlds" interpretation. It is shown that the relativistic generalization of the trajectory-based formulation results in well-behaved free-particle Gaussian wavepacket solutions. In particular, probability density is positive and well-localized everywhere, and its spatial integral is conserved over time—in any inertial frame. Finally, the ensemble-averaged wavepacket motion is along a straight line path through spacetime. In this manner, the pathologies of the wave-based relativistic quantum theory, as applied to wavepacket propagation, are avoided.

... The present theory departs from the well-known Bohmian interpretation of quantum mechanics due to deBroglie and Bohm [1][2][3][4][5], as well as the class of so-called trajectory-based models to be found in the literature. The latter are usually realized in the context of non-relativistic quantum mechanics [6][7][8][9][10][11][12][13], their primary relevance from the practical viewpoint being essentially restricted to the development of Lagrangian numerical solution methods [14][15][16][17][18][19][20], although significant examples actually occur for relativistic quantum systems [21] too. The GLP-approach is intended to apply in principle both to 1-and N -body non-relativistic quantum systems S N , formed by an arbitrary number N of like quantum point-particles of mass m which satisfy the N -body SE. ...

... Actually, as will be discussed in detail in the paper, these features place the new theory apart and distinguish it from the class of Bohmian-like theories usually considered in the literature [7][8][9][10][11][12][22][23][24]. The GLP-approach, in fact, is based on a new type of Lagrangian stochastic parametrization for the non-relativistic quantum wave function ψ ≡ ψ (r, t) obtained by means of the adoption of so-called generalized Lagrangian paths (GLPs) (see Ref. [25]). ...

... In the following (see in particular Sects. 7,8,9) we intend to show that this is indeed the case, providing for this purpose a number of examples holding for quantum N -body systems which include: (a) the free-particle case; (b) the elastic attractive/repulsive force; (c) the Coulomb-like and the power-law central force; (d) the Van der Waals force. ...

In this paper a new trajectory-based representation to non-relativistic quantum mechanics is formulated. This is ahieved by generalizing the notion of Lagrangian path (LP) which lies at the heart of the deBroglie-Bohm “ pilot-wave” interpretation. In particular, it is shown that each LP can be replaced with a statistical ensemble formed by an infinite family of stochastic curves, referred to as generalized Lagrangian paths (GLP). This permits the introduction of a new parametric representation of the Schrödinger equation, denoted as GLP-parametrization, and of the associated quantum hydrodynamic equations. The remarkable aspect of the GLP approach presented here is that it realizes at the same time also a new solution method for the N-body Schrödinger equation. As an application, Gaussian-like particular solutions for the quantum probability density function (PDF) are considered, which are proved to be dynamically consistent. For them, the Schrödinger equation is reduced to a single Hamilton–Jacobi evolution equation. Particular solutions of this type are explicitly constructed, which include the case of free particles occurring in 1- or N-body quantum systems as well as the dynamics in the presence of suitable potential forces. In all these cases the initial Gaussian PDFs are shown to be free of the spreading behavior usually ascribed to quantum wave-packets, in that they exhibit the characteristic feature of remaining at all times spatially-localized.

... The first of two recent but entirely independent developments in the foundations of quantum theory is a number of 'trajectory-only' formulations of quantum theory [1,2,3,4,5,6,7], though also see [8], intended to recover quantum mechanics without reference to a physical wavefunction. Two related approaches which however leads to an experimentally distinguishable theory are the real-ensemble formulations of [9,10]. ...

... The approach of Schiff and Poirier [5,6] is different in that their formulation does not rely on the introduction of a quantum potential in the usual form. Instead, they consider higher-order time derivatives of the variables to appear in the expressions for the Lagrangian and energy of their theory. ...

... Since this is to hold for any interval [t 0 , t 1 ], we have in general that p i = ∂ i S. This equation can be seen as determining S up to an arbitrary additive function of time. 5 It only restricts the momenta p i in that these must be the gradient of some function. Of course, S turns out to have all the properties of a generator of those canonical transformations that represent the time evolution of the system. ...

We illustrate how non-relativistic quantum mechanics may be recovered from a
dynamical Weyl geometry on configuration space and an `ensemble' of
trajectories (or `worlds'). The theory, which is free of a physical
wavefunction, is presented starting from a classical `many-systems' action to
which a curvature term is added. In this manner the equations of equilibrium
de~Broglie-Bohm theory are recovered. However, na\"ively the set of solution
precludes solutions with non-zero angular momentum (a version of a problem
raised by Wallstrom). This is remedied by a slight extension of the action,
leaving the equations of motion unchanged.

... The first question has been addressed by Holland, Poirier and Hall et al. in recent work [6,5,7]. Interestingly, these authors show that Bohm's quantum trajectories can be obtained without a guiding wave function from a reformulated theory that prescribes the equations of motion for the particles, along with a probability distribution for the resulting particle trajectories. ...

... Additional differences, for example how amenable each formulation is to numerical evaluation, will not be pursued in any detail here. The first formulation, supported by Bohm [1], Bell [3] and others [9] firmly assumes a pilot-wave, which is a solution of the Schrödinger equation, as part of reality; the two other formulations make no explicit reference to a wave function or Schrödinger equation [6,5,7]. ...

... To keep notations simple and following refs. [5,7], this section will consider quantum mechanics of a single particle without spin in one dimension. It is mostly straightforward to extend the discussion to non-relativistic quantum mechanics for multiple spin zero particles in three space dimensions. ...

After summarizing three versions of trajectory-based quantum mechanics, it is
argued that only the original formulation due to Bohm, which uses the
Schr\"odinger wave function to guide the particles, can be readily extended to
particles with spin. To extend the two wave function-free formulations, it is
argued that necessarily particle trajectories not only determine location, but
also spin. Since spin values are discrete, it is natural to revert to a
variation of Bohm's pilot wave formulation due originally to Bell. It is shown
that within this formulation with stochastic quantum trajectories, a wave
function free formulation can be obtained.

... The theory is based on ideas initially published as a preprint draft [5], which has been completely re-worked and enhanced, in particular by adding a logical framework to properly deal with propositions about physical systems in a multiplicity of worlds, and by providing the conceptual prerequisites for treating the collection of worlds as a continuous substance. After having finished and submitted an earlier version of this manuscript, I noticed that essentially the same theory, though with a stronger focus on formal aspects and less focus on ontological and epistemological matters, has independently been put forward by Poirier and Schiff [35,39]. Although already having been aware of, and having cited, these publications, I did not fully recognize how close their theory was to mine. ...

... In particular, the relation between objective reality and subjective experience in the presence of a multiplicity of worlds is addressed. My proposal is compared to Bohmian mechanics, to Tipler's formulation of quantum mechanics [43], to the MIW approach of Hall et al. [21], to Sebens' Newtonian QM [40], and to Poirier and Schiff's approach [35,39]. I will also respond to criticisms raised by [40,48] against the idea of a continuum of worlds. ...

... However, Madelung could not provide a consistent physical interpretation of this mathematical fact, so the hydrodynamical interpretation of quantum mechanics was abandoned. Recently, the hydrodynamic interpretation experienced a renaissance, and it was shown that the wavefunction can be completely removed from the theory, leaving only trajectories as the physically existing objects from where all observable values can be calculated [23,35,39]. However, these approaches leave it open as to how the fluid is interpreted physically. ...

A non-relativistic quantum mechanical theory is proposed that describes the
universe as a continuum of worlds whose mutual interference gives rise to
quantum phenomena. A logical framework is introduced to properly deal with
propositions about objects in a multiplicity of worlds. In this logical
framework, the continuum of worlds is treated similarly to the continuum of
time points; both "time" and "world" are considered as mutually independent
modes of existence. The theory combines elements of Bohmian mechanics and of
Everett's many-worlds interpretation; it has a clear ontology and a set of
precisely defined postulates from where the predictions of standard quantum
mechanics can be derived. Probability as given by the Born rule emerges as a
consequence of insufficient knowledge of observers about which world it is that
they live in. The theory describes a continuum of worlds rather than a single
world or a discrete set of worlds, so it is similar in spirit to many-worlds
interpretations based on Everett's approach, without being actually reducible
to these. In particular, there is no splitting of worlds, which is a typical
feature of Everett-type theories. Altogether, the theory explains 1) the
subjective occurrence of probabilities, 2) their quantitative value as given by
the Born rule, 3) the identification of observables as self-adjoint operators
on Hilbert space, and 4) the apparently random "collapse of the wavefunction"
caused by the measurement, while still being an objectively deterministic
theory.

... More recently, it been observed by Holland [8] and by Poirier and coworkers [9][10][11] that the evolution of such quantum systems can be formulated without reference even to a momentum potential S. Instead, nonlinear Euler-Lagrange equations are used to define trajectories of a continuum of fluid elements, in an essentially hydrodynamical picture. The trajectories are labelled by a continuous parameter, such as the initial position of each element, and the equations involve partial derivatives of * Electronic address: H.Wiseman@Griffith.edu.au up to fourth order with respect to this parameter. ...

... However, it may be recovered, in a nontrivial manner, by integrating the trajectories up to any given time [8]. This has proved a useful tool for making efficient and accurate numerical calculations in quantum chemistry [9,10]. Schiff and Poirier [11], while "drawing no definite conclusions", interpret their formulation as a "kind of "many worlds" theory", albeit they have a continuum of trajectories (i.e. ...

... Here the left-hand side is to be understood as an approximation of the right-hand side, obtained via a suitable smoothing of the empirical density P (q) in Eq. (8), analogous to the approximation of the quantum force r N (q) by r N (x; X) in Eq. (9). It is important to note that a good approximation of the force (which is essential to obtain QM in the large N limit) is not guaranteed by a good approximation in Eq. (19). ...

We investigate whether quantum theory can be understood as the continuum
limit of a mechanical theory, in which there is a huge, but finite, number of
classical 'worlds', and quantum effects arise solely from a universal
interaction between these worlds, without reference to any wave function. Here
a `world' means an entire universe with well-defined properties, determined by
the classical configuration of its particles and fields. In our approach each
world evolves deterministically; probabilities arise due to ignorance as to
which world a given observer occupies; and we argue that in the limit of
infinitely many worlds the wave function can be recovered (as a secondary
object) from the motion of these worlds. We introduce a simple model of such a
'many interacting worlds' approach and show that it can reproduce some generic
quantum phenomena---such as Ehrenfest's theorem, wavepacket spreading, barrier
tunneling and zero point energy---as a direct consequence of mutual repulsion
between worlds. Finally, we perform numerical simulations using our approach.
We demonstrate, first, that it can be used to calculate quantum ground states,
and second, that it is capable of reproducing, at least qualitatively, the
double-slit interference phenomenon.

... Nevertheless, exactly just such a theory was recently formulated for non-relativistic quantum mechanics. [20][21][22][23][24][25][26] For a number of reasons, it makes sense to try to extend the previous work to the relativistic case. As presented in this document, this goal is now also achieved-at least in the context of a single, spin-zero, massive, relativistic quantum particle, propagating on a flat Minkowski spacetime, with no external fields. ...

... The crucial development is the recent wavefunction-free reformulation of nonrelativistic quantum mechanics, alluded to above. [20][21][22][23][24][25][26] This approach is trajectory based, and in that sense reminiscent of Bohmian mechanics. Unlike the Bohm theory, however, here, the traditional wavefunction, Ψ(t, x), is entirely done away with, in favor of the trajectory ensemble, x(t, C) (where C labels individual trajectories) as the fundamental representation of a quantum state. ...

... This is perhaps most physically meaningful if one adopts a "many worlds"-type ontological interpretation of the multiple particle paths/trajectories, according to which each trajectory worldline literally represents a different world, as has been discussed in previous work. 22,24,25 The one particle is thus comprised of many "copies," distributed across all space. Locally, the structure of the orthogonal subspace described above ensures that each particle copy agrees with its nearest neighbors as to which events occur simultaneously. ...

In a recent paper [Bill Poirier, arXiv:1208.6260 [quant-ph]], a
trajectory-based formalism has been constructed to study the
relativistic dynamics of a single spin-zero quantum particle. Being a
generally covariant theory, this formalism introduces a new notion of
global simultaneity for accelerated quantum particles. In this talk, we
present several examples based on this formalism, including the time
evolution of a relativistic Gaussian wavepacket. Energy-momentum
conservation relations may also be discussed.

... The QTM of this paper, though also approximate, treats barrier tunneling and reflection interference much more accurately than these other approaches (though it might be improved by applying the latter in the perpendicular directions, which are treated classically here). It stems from a broader, and quite recent, theoretical development2728293031 that regards the trajectory ensemble itself as the fundamental quantum state entity, rather than the wavefunction. The exact quantum propagation is therefore described as a partial differential equation (PDE) directly on the trajectory ensemble itself, i.e. with no reference whatsoever to W. The present QTM is then derived by replacing the exact PDE with an approximate ODE, thus severing all intertrajectory communication, circumventing the node problem, and resulting in a much more classical-like time evolution (with all commensurate computational advantages, e.g. ...

... Alternatively, it can be shown293031 that the physical quantum state represented by the wavefunction W(x), can also be represented exactly using a single trajectory x(t). This quantum trajectory , moreover, evolves in accordance with the following fourthorder ordinary differential equation (ODE): ...

... Note that Eq. (2) would be the usual second-order classical ODE of Newton, but for the last term, which represents the ''quantum force.'' [11,293031 In general, any fourthorder ODE such as Eq. (2) admits a four-parameter family of solution trajectories, x(t), that can be specified using four appropriate initial conditions, e.g.: x 0 = x(t = 0); _ x 0 ¼ _ xðt ¼ 0Þ; € x 0 ¼ € xðt ¼ 0Þ; x v 0 ¼ x v ðt ¼ 0Þ. ...

A trajectory ensemble method is introduced that enables accurate computation of microcanonical quantum reactive scattering quantities, using a classical-like simulation scheme. Individual quantum trajectories are propagated independently, using a Newton-like ODE which treats quantum dynamical effects along the reaction coordinate exactly, and preserves the phase space volume element. The sampling of initial conditions resembles a classical microcanonical simulation, but modified so as to incorporate quantization in the perpendicular mode coordinates. The method is exact for one-dimensional or separable systems, and achieves ∼1% accuracy for the coupled multidimensional benchmark applications considered here, even in the deep tunneling regime.

... In classical mechanics, physical trajectories are obtained as the subset of dynamical paths that extremize the classical action. Until recently [2][3][4][5][6][7][8], it appears that no trajectory-based action extremization principle was known or even suspected in quantum mechanics -although the identification of the action S (in units of h) with the phase of the wavefunction goes back to the earliest days of the quantum theory [9][10][11], and still serves as the basis of modern semiclassical trajectory-based approximation methods [12][13][14][15][16][17][18]. ...

... The purpose of this paper is to evaluate whether or not BOMCA satisfies an action principle. This is motivated by the recent, startling discovery that the real-valued quantum trajectories of Bohmian mechanics do indeed satisfy a bona fide principle of least action [2][3][4][5][6][7][8], very similar to that of classical Lagrangian mechanics. The real-valued quantum action-extremizing trajectories turn out to be equivalent to the quantum trajectories of Bohmian mechanics. ...

... in addition to Equations (4) and (5). Whether working with the PDE or the ODE hierarchy, we take the primary defining feature of standard BOMCA to be that the complex trajectory velocity field is defined via Equations (1) and (4). ...

In a recent paper [B. Poirier, Chem. Phys. 370, 4 (2010)], a formulation of quantum mechanics was presented for which the usual wavefunction and Schrödinger equation are replaced with an ensemble of real-valued trajectories satisfying a principle of least action. It was found that the resultant quantum trajectories are those of Bohmian mechanics. In this paper, analogous ideas are applied to Bohmian Mechanics with Complex Action (BOMCA). The standard BOMCA trajectories as previously defined are found not to satisfy an action principle. However, an alternate set of complex equations of motion is derived that does exhibit this desirable property, and an approximate numerical implementation is presented. Exact analytical results are also presented, for Gaussian wavepacket propagation under quadratic potentials.

... Nevertheless, exactly just such a theory was recently formulated for non-relativistic quantum mechanics. [20][21][22][23][24][25][26] For a number of reasons, it makes sense to try to extend the previous work to the relativistic case. As presented in this document, this goal is now also achieved-at least in the context of a single, spin-zero, massive, relativistic quantum particle, propagating on a flat Minkowski spacetime, with no external fields. ...

... The crucial development is the recent wavefunction-free reformulation of nonrelativistic quantum mechanics, alluded to above. [20][21][22][23][24][25][26] This approach is trajectory based, and in that sense reminiscent of Bohmian mechanics. Unlike the Bohm theory, however, here, the traditional wavefunction, Ψ(t, x), is entirely done away with, in favor of the trajectory ensemble, x(t, C) (where C labels individual trajectories) as the fundamental representation of a quantum state. ...

... This is perhaps most physically meaningful if one adopts a "many worlds"-type ontological interpretation of the multiple particle paths/trajectories, according to which each trajectory worldline literally represents a different world, as has been discussed in previous work. 22,24,25 The one particle is thus comprised of many "copies," distributed across all space. Locally, the structure of the orthogonal subspace described above ensures that each particle copy agrees with its nearest neighbors as to which events occur simultaneously. ...

Recently, a self-contained trajectory-based formulation of non-relativistic
quantum mechanics was developed [Ann. Phys. 315, 505 (2005); Chem. Phys. 370, 4
(2010); J. Chem. Phys. 136, 031102 (2012)], that makes no use of wavefunctions
or complex amplitudes of any kind. Quantum states are represented as ensembles
of real-valued quantum trajectories that extremize a suitable action. Here, the
trajectory-based approach is developed into a viable, generally covariant,
relativistic quantum theory for single (spin-zero, massive) particles. Central
to this development is the introduction of a new notion of global simultaneity
for accelerated particles--together with basic postulates concerning
probability conservation and causality. The latter postulate is found to be
violated by the Klein-Gordon equation, leading to its well-known problems as a
single-particle theory. Various examples are considered, including the time
evolution of a relativistic Gaussian wavepacket.

... This is the idea behind the Bohmian formulation of quantum mechanics [166,167], where a wave is guiding test-particles. One difficulty of this approach of "particle position" is that the equation above can only be solved if Ψ(q, t) is known, i.e., if the problem is already solved [168,169]. Another approach to Bohmian mechanics is to solve both the set of Eqs. 6.11a and 6.11b on the hydrodynamical fields ρ(q, t) and the action S(q, t) along with the equations of motion of the trajectories. ...

... From this point of view, Bohmian quantum trajectories seem to be confined to illustrative purposes. Efforts were made, however, in recent years [162,168,169,[201][202][203][204] in order to dispense with the knowledge of the density, and solve the quantum problem using trajectories only. In the framework, it is not necessary to refer explicitly to any auxiliary equation on the density on top of the equations of motion of the trajectories anymore. ...

The dynamics of a quantum system of interacting particles rapidly becomes impossible to describe exactly when the number of particles increases. This is one of the main difficulties in the description of atomic nuclei, which may contain several hundred of nucleons. A simplified approach to the problem is to assume that some degrees of freedom contain more information than others. A classical approximation is to focus on one-body degrees of freedom: the dynamics of the system can be approximately described by a set of particles propagating in an effective mean-field. While the mean-field approximation has allowed many advances in the theoretical understanding of the properties of nuclei, it is still unable to describe certain of their properties, for example the effects of direct collisions between nucleons or the quantum fluctuations of one-body observables. The objective of the thesis is to account for these correlations beyond the mean-field approximation in order to improve the dynamical description of quantum correlated systems. One component of the thesis has been to study methods to treat collisions between particles by including the Born term beyond the mean-field. This term is particularly complex because of non-local effects in time, the so-called non-Markovian effects. Possible simplifications of this term have been studied for future applications. Two simplifying approaches have been proposed, one allowing to treat the collision term with master equations, the other allowing to get rid of time integrals while keeping the non-locality in time. The second part of the thesis was devoted to the improvement of the mean-field approximation in order to describe the quantum fluctuations. Based on existing phase space methods, a new method, called "Hybrid Phase Space Method" (HPS) has been proposed. This method is a combination of the mean-field theory with initial fluctuations and a theory where the two-body degrees of freedom are propagated explicitly. This new approach has been successfully tested for the description of an ensemble of fermions on a lattice, i.e. the Fermi-Hubbard model, and has given much better results than the phase-space approaches previously used to describe correlated systems, in particular in a weak coupling case. If this new approximation gives interesting results, it remains numerically rather heavy and empirical. This led to a detailed study of the Wigner-Weyl and Bohm formalisms in order to explore phase-space methods in a more systematic way. The notion of trajectory in quantum mechanics has been systematically investigated. The conclusion of this study, where illustrations have been made on the tunneling effect, is that it is necessary that the trajectories interfere with each other in the course of time to reproduce the quantum effects.

... We suggested that it might be possible to reproduce quantum phenomena without a universal wavefunction Ψ(q) (except to define initial conditions). In its place we postulated an enormous, but countable, ensemble X = x j : j of points x j in configuration space (similar ideas have been proposed earlier by a number of authors [32][33][34][35], but they considered a continuum of worlds, which, in our view, leads to some of the same conceptual issues that Everett's interpretation faces-see also [36]). Each point is a world-particle, just as Bohmian mechanics postulates, and the dynamics is intended to reproduce a deterministic Bohmian trajectory for each world-particle. ...

... Dealing with nodes is a problem in many quantum simulation methods based on Bohmian mechanics [41]. Nodes should not be a problem for interpretations involving a continuum of worlds [32][33][34][35], as they are formulated to be exactly equivalent to quantum mechanics. However, as remarked in Section 3, our view is that these interpretations do not solve the conceptual problems of the Everettian many-worlds interpretation. ...

“Locality” is a fraught word, even within the restricted context of Bell’s theorem. As one of us has argued elsewhere, that is partly because Bell himself used the word with different meanings at different stages in his career. The original, weaker, meaning for locality was in his 1964 theorem: that the choice of setting by one party could never affect the outcome of a measurement performed by a distant second party. The epitome of a quantum theory violating this weak notion of locality (and hence exhibiting a strong form of nonlocality) is Bohmian mechanics. Recently, a new approach to quantum mechanics, inspired by Bohmian mechanics, has been proposed: Many Interacting Worlds. While it is conceptually clear how the interaction between worlds can enable this strong nonlocality, technical problems in the theory have thus far prevented a proof by simulation. Here we report significant progress in tackling one of the most basic difficulties that needs to be overcome: correctly modelling wavefunctions with nodes.

... The present proposal is based on two recent but entirely independent developments in the foundations of quantum theory. The first is a number of 'trajectory-only' formulations of quantum theory [120,56,130,20,98,117,118], though also see ref. [60], intended to recover quantum mechanics without reference to a physical wavefunction. Two related approaches which however lead to an experimentally distinguishable theory are the realensemble formulations of refs. ...

... The approach of Schiff and Poirier [98,117] is different in that their formulation does not rely on the introduction of a quantum potential in the usual form. Instead, they consider higher-order time derivatives of the variables to appear in the expressions for the Lagrangian and energy of their theory. ...

] In this thesis we investigate a solution to the `problem of time' in canonical quantum gravity by splitting spacetime into surfaces of constant mean curvature parameterised by York time. We argue that there are reasons to consider York time a viable candidate for a physically meaningful notion of time. We investigate a number York-time Hamiltonian-reduced cosmological models and explore some technical aspects, such as the non-canonical Poisson structure. We develop York-time Hamiltonian-reduced cosmological perturbation theory by solving the Hamiltonian constraint perturbatively around a homogeneous background for the physical (non-vanishing) Hamiltonian that is the momentum conjugate to the York time parameter. We proceed to canonically quantise the cosmological models and the perturbation theory and discuss a number of conceptual and technical points, such as volume eigenfunctions and the absence of a momentum representation due to the non-standard commutator structure. We propose an alternative, wavefunction-free method of quantisation based on an ensemble of trajectories and a dynamical configuration-space geometry and discuss its application to gravity. We conclude by placing the York-time theories explored in this thesis in the wider context of a search for a satisfactory theory of quantum gravity.

... A similar proposal was made in [6][7][8][9][10][11], where the idea of many interacting classical worlds (MIW) was introduced to explain quantum mechanics. This also posits that the quantum state refers to an ensemble of real, existing, systems which interact with each other, only those were posited to be near copies of our universe that all simultaneously exist 1 . ...

... The elements could be particles or events or subsystems and the relations could be relative position, relative distance, causal relations, etc. We proceed by defining the view of the i'th element, 1 If the ontology posited by the [6][7][8][9][10][11] papers may seem extravagant, their proposal had the virtue of a simple form for the inter-ensemble interactions. This inspired me to seek to use such a simple dynamics in the real ensemble idea. ...

Quantum mechanics is derived from the principle that the universe contain as
much variety as possible, in the sense of maximizing the distinctiveness of
each subsystem.
The quantum state of a microscopic system is defined to correspond to an
ensemble of subsystems of the universe with identical constituents and similar
preparations and environments. A new kind of interaction is posited amongst
such similar subsystems which acts to increase their distinctiveness, by
extremizing the variety. In the limit of large numbers of similar subsystems
this interaction is shown to give rise to Bohm's quantum potential. As a result
the probability distribution for the ensemble is governed by the Schroedinger
equation.
The measurement problem is naturally and simply solved. Microscopic systems
appear statistical because they are members of large ensembles of similar
systems which interact non-locally. Macroscopic systems are unique, and are not
members of any ensembles of similar systems. Consequently their collective
coordinates may evolve deterministically.
This proposal could be tested by constructing quantum devices from entangled
states of a modest number of quits which, by its combinatorial complexity, can
be expected to have no natural copies.

... Numerically, the prospect of stable, synthetic quantum trajectory calculations for many-D molecular applications will be fully explored, as the benefits here could prove profound. 4,7,11 Our formalism offers flexibility for restricting action extremization to trajectory ensembles of a desired form (e.g., reduced dimensions), thereby providing useful variational approximations. Our exact TDQM equations are PDEs, not single-trajectory ODEs-the entire ensemble must be determined at once. ...

... The other two constants determine which particular TIQM state the trajectory is associated with. A one-to-one correspondence thus exists between trajectory solutions [of Eq. (4)] and (scattering) TIQM states.7 ...

We present a self-contained formulation of spin-free nonrelativistic quantum
mechanics that makes no use of wavefunctions or complex amplitudes of any kind.
Quantum states are represented as ensembles of real-valued quantum
trajectories, obtained by extremizing an action and satisfying energy
conservation. The theory applies for arbitrary configuration spaces and system
dimensionalities. Various beneficial ramifications - theoretical,
computational, and interpretational - are discussed.

... The first one is based on the adoption of deterministic Lagrangian trajectories g L (s), s ∈ I , or Lagrangian-Paths (LP). This is analogous to the customary literature approach previously adopted in the context of the Bohmian representation of non-relativistic QM [26][27][28][29][30][31]. ...

The logical structure of quantum gravity (QG) is addressed in the framework of the so-called manifestly covariant approach. This permits to display its close analogy with the logics of quantum mechanics (QM). More precisely, in QG the conventional 2-way principle of non-contradiction (2-way PNC) holding in Classical Mechanics is shown to be replaced by a 3-way principle (3-way PNC). The third state of logical truth corresponds to quantum indeterminacy/undecidability, i.e., the occurrence of quantum observables with infinite standard deviation. The same principle coincides, incidentally, with the earlier one shown to hold in Part I, in analogous circumstances, for QM. However, this conclusion is found to apply only provided a well-defined manifestly-covariant theory of the gravitational field is adopted both at the classical and quantum levels. Such a choice is crucial. In fact it makes possible the canonical quantization of the underlying unconstrained Hamiltonian structure of general relativity, according to an approach recently developed by Cremaschini and Tessarotto (2015–2021). Remarkably, in the semiclassical limit of the theory, Classical Logic is proved to be correctly restored, together with the validity of the conventional 2-way principle.

... Based on Ref. [5], one can show that a stochastic-trajectory formulation of NRQM, which is ontologically equivalent to NRQM, can be achieved in the framework of the so-called Generalized Lagrangian Path (GLP) representation, i.e., the parametrization of the quantum wave-function and hence of the SE itself in terms of suitable stochastic GLP trajectories. The GLP-representation builds on the Bohmian representation of NRQM, well-known in the literature [3,[37][38][39][40][41], i.e., based on the notion of Lagrangian Path (LP). Hence, while the latter is based on the LP-representation, namely in terms of the LP, i.e., a unique (namely deterministic) solution of the initial-value problem (13), the GLP-representation relies, instead, on the notion of the so-called generalized Lagrangian path (GLP), i.e., a suitable family of stochastic configuration-space trajectories. ...

One of the most challenging and fascinating issue in mathematical and theoretical physics concerns the possibility of identifying the logic underlying the so-called quantum universe, i.e., Quantum Mechanics and Quantum Gravity. Besides the sheer difficulty of the problem, inherent in the actual formulation of Quantum Mechanics—and especially of Quantum Gravity—to be used for such a task, a crucial aspect lies in the identification of the appropriate axiomatic logical proposition calculus to be associated to such theories. In this paper the issue of the validity of the conventional principle of non-contradiction (PNC) is called into question and is investigated in the context of non-relativistic Quantum Mechanics. In the same framework a modified form of the principle, denoted as 3-way PNC is shown to apply, which relates the axioms of quantum logic with the physical requirements placed by the Heisenberg Indeterminacy Principle.

... There are some similarities between the present model and the work of Madelung [25], and also various works on many-interactingworlds [26,27,28,29,30,31] for a single quantum particles. ...

A local theory of relativistic quantum physics in space-time, which makes all of the same empirical predictions as the conventional delocalized theory in configuration space, is presented and interpreted. Each physical system is characterized by a set of indexed piece-wise single-particle wavefunctions in space-time, each with with its own coefficient, and these 'wave-fields' replace entangled states in higher-dimensional spaces. Each wavefunction of a fundamental system describes the motion of a portion of a conserved fluid in space-time, with the fluid decomposing into many classical point particles, each following a world-line and recording a local memory. Local interactions between two systems take the form of local boundary conditions between the differently indexed pieces of those systems' wave-fields, with new indexes encoding each orthogonal outcome of the interaction. The general machinery is introduced, including the local mechanisms for entanglement and interference. The experience of collapse, Born rule probability, and environmental decoherence are discussed. A number of illustrative examples are given, including a Von Neumann measurement, and a test of Bell's theorem.

... The second, "quantum mechanics without wavefunctions" approach is also useful, as it enables the quantum wave to be discarded entirely [23][24][25][26][27][28][29][30][31][32]. Instead, the ensemble of quantum trajectories is used to propagate all quantum information on its own, exactly. ...

We re-examine the (inverse) Fermi accelerator problem by resorting to a quantum trajectory description of the dynamics. Quantum trajectories are generated from the time-independent Schrödinger equation solutions, using a unipolar treatment for the (light) confined particle and a bipolar treatment for the (heavy) movable wall. Analytic results are presented for the exact coupled two-dimensional problem, as well as for the adiabatic and mixed quantum-classical approximations.

... Throughout recent years, apart from our own model, several approaches to a quantum mechanics without wavefunctions have been proposed [1][2][3][4][5]. These refer to "many classical worlds" that provide Bohm-type trajectories with certain repulsion effects. ...

In the quest for an understanding of nonlocality with respect to an appropriate ontology, we propose a "cosmological solution". We assume that from the beginning of the universe each point in space has been the location of a scalar field representing a zero-point vacuum energy that nonlocally vibrates at a vast range of different frequencies across the whole universe. A quantum, then, is a nonequilibrium steady state in the form of a "bouncer" coupled resonantly to one of those (particle type dependent) frequencies, in remote analogy to the bouncing oil drops on an oscillating oil bath as in Couder's experiments. A major difference to the latter analogy is given by the nonlocal nature of the vacuum oscillations. We show with the examples of double- and $n$-slit interference that the assumed nonlocality of the distribution functions alone suffices to derive the de Broglie-Bohm guiding equation for $N$ particles with otherwise purely classical means. In our model, no influences from configuration space are required, as everything can be described in 3-space. Importantly, the setting up of an experimental arrangement limits and shapes the forward and osmotic contributions and is described as vacuum landscaping.

... For the tasks indicated above, in close similarity with non-relativistic quantum mechanics (see [22,23]), two choices are in principle available. The first one is based on the introduction of deterministic Lagrangian trajectories {g(s), s ∈ I}, or Lagrangian-Paths (LP), analogous to those adopted in the context of the Bohmian representation of non-relativistic quantum mechanics [24][25][26][27][28][29][30][31]. ...

A trajectory-based representation for the quantum theory of the gravitational field is formulated. This is achieved in terms of a covariant Generalized Lagrangian-Path (GLP) approach which relies on a suitable statistical representation of Bohmian Lagrangian trajectories, referred to here as GLP-representation. The result is established in the framework of the manifestly-covariant quantum gravity theory (CQG-theory) proposed recently and the related CQG-wave equation advancing in proper-time the quantum state associated with massive gravitons. Generally non-stationary analytical solutions for the CQG-wave equation with non-vanishing cosmological constant are determined in such a framework, which exhibit Gaussian-like probability densities that are non-dispersive in proper-time. As a remarkable outcome of the theory achieved by implementing these analytical solutions, the existence of an emergent gravity phenomenon is proven to hold. Accordingly, it is shown that a mean-field background space-time metric tensor can be expressed in terms of a suitable statistical average of stochastic fluctuations of the quantum gravitational field whose quantum-wave dynamics is described by GLP trajectories.

... Several discrete and continuous versions of this approach have been proposed in the literature [6][7][8][9][10][11][12][13]. Due the possible interpretation of Q (i) t , i = 1, . . . ...

Recently the Many-Interacting-Worlds (MIW) approach to a quantum theory without wave functions was proposed. This approach leads quite naturally to numerical integrators of the Schr\"odinger equation. It has been suggested that such integrators may feature advantages over fixed-grid methods for higher numbers of degrees of freedom. However, as yet, little is known about concrete MIW models for more than one spatial dimension and/or more than one particle. In this work we develop the MIW approach further to treat arbitrary degrees of freedom, and provide a systematic study of a corresponding numerical implementation for computing one-particle ground and excited states in one dimension, and ground states in two spatial dimensions. With this step towards the treatment of higher degrees of freedom we hope to stimulate their further study.

... Here it is used to provide alternative ways to inspect or probe the quantum systems. Also, the alternative formulation might lead to possibly more efficient methods to perform computer simulations of quantum systems [15,16]. However, Bohm's formulation has not gained general acceptance (yet) and the Copenhagen interpretation remains to be favored by the major-addressing both perturbative and non-perturbative phenomena. ...

The formulation of quantum mechanics developed by Bohm, which can generate well-defined trajectories for the underlying particles in the theory, can equally well be applied to relativistic Quantum Field Theories to generate dynamics for the underlying fields. However, it does not produce trajectories for the particles associated with these fields. Bell has shown that an extension of Bohm's approach can be used to provide dynamics for the fermionic occupation numbers in a relativistic Quantum Field Theory. In the present paper, Bell's formulation is adopted and elaborated on, with a full account of all technical detail required to apply his approach to a bosonic quantum field theory on a lattice. This allows an explicit computation of (stochastic) trajectories for massive and massless particles in this theory. Also particle creation and annihilation, and their impact on particle propagation, is illustrated using this model.

... Similarly, the Gaussian-based timedependent variational principle [46,47] yields classical-like equations of motion. Alternatively, Schiff and Poirier [48] build an effective Lagrangian method that contains higherorder derivatives, which in turn yields classical-looking equations with extra degrees of freedom [49]. Quantum Statistical Potentials (QSPs) [50][51][52] and empirical potentials for molecular systems [53] are purely classical in their form, with effective potentials; many of these methods have been reviewed elsewhere [54]. ...

Effective classical dynamics provide a potentially powerful avenue for modeling large-scale dynamical quantum systems. We have examined the accuracy of a Hamiltonian-based approach that employs effective momentum-dependent potentials (MDPs) within a molecular-dynamics framework through studies of atomic ground states, excited states, ionization energies, and scattering properties of continuum states. Working exclusively with the Kirschbaum-Wilets (KW) formulation with empirical MDPs [C. L. Kirschbaum and L. Wilets, Phys. Rev. A 21, 834 (1980)], optimization leads to very accurate ground-state energies for several elements (e.g., N, F, Ne, Al, S, Ar, and Ca) relative to Hartree-Fock values. The KW MDP parameters obtained are found to be correlated, thereby revealing some degree of transferability in the empirically determined parameters. We have studied excited-state orbits of electron-ion pair to analyze the consequences of the MDP on the classical Coulomb catastrophe. From the optimized ground-state energies, we find that the experimental first-and second-ionization energies are fairly well predicted. Finally, electron-ion scattering was examined by comparing the predicted momentum transfer cross section to a semiclassical phase-shift calculation; optimizing the MDP parameters for the scattering process yielded rather poor results, suggesting a limitation of the use of the KW MDPs for plasmas.

... Here I examine the problem of explaining the symmetry dichotomy within two interpretations of quantum mechanics which clarify the connection between particles and the wave function by including particles following definite trajectories through space in addition to, or in lieu of, the wave function: (1) Bohmian mechanics and (2) a hydrodynamic interpretation that posits a multitude of quantum worlds interacting with one another, which I have called "Newtonian quantum mechanics" (Hall et al. , 2014 have called this kind of approach "many interacting worlds"). Versions of this second interpretation have recently been put forward by Tipler (2006); Poirier (2010); Schiff & Poirier (2012); Boström (2012); Boström (2015); Hall et al. (2014); Sebens (2015); it builds on the hydrodynamic approach to quantum mechanics (see Madelung, 1927;Wyatt, 2005;Holland, 2005). Bohmian mechanics and Newtonian quantum mechanics are often called "interpretations" of quantum mechanics, but should really be thought of as distinct physical theories which seek to explain the same body of data (those experiments whose statistics are successfully predicted by the standard methods of non-relativistic quantum mechanics). ...

I address the problem of explaining why wave functions for identical particles must be either symmetric or antisymmetric (the symmetry dichotomy) within two interpretations of quantum mechanics which include particles following definite trajectories in addition to, or in lieu of, the wave function: Bohmian mechanics and Newtonian quantum mechanics (a.k.a. many interacting worlds). In both cases I argue that, if the interpretation is formulated properly, the symmetry dichotomy can be derived and need not be postulated.

... It follows Pauli exclusion principle [13][14][15][16][17]. Even though Pauli exclusion principle is mainly for wavefunctions [18][19][20][21][22][23][24][25][26][27][28][29][30], the physical mass waves are used as above to discuss the principle. ...

The paper "Unified Field Theory and the Configuration of Particles" opened a new chapter of physics. One of the predictions of the paper is that a proton has an octahedron shape. As Physics progresses, it focuses more on invisible particles and the unreachable grand universe as visible matter is studied theoretically and experimentally. The shape of invisible proton has great impact on the topology of atom. Electron orbits, electron binding energy, Madelung Rules, and Zeeman splitting, are associated with proton’s octahedron shape and three nuclear structural axes. An element will be chemically stable if the outmost s and p clouds have eight electrons which make atom a symmetrical cubic.

... It follows Pauli exclusion principle [13][14][15][16][17]. Even though Pauli exclusion principle is mainly for wavefunctions [18][19][20][21][22][23][24][25][26][27][28][29][30], the physical mass waves are used as above to discuss the principle. ...

The paper "Unified Field Theory and the Configuration of Particles" opened a new chapter of physics. One of the predictions of the paper is that a proton has an octahedron shape. As Physics progresses, it focuses more on invisible particles and the unreachable grand universe as visible matter is studied theoretically and experimentally. The shape of invisible proton has great impact on the topology of atom. Electron orbits, electron binding energy, Madelung Rules, and Zeeman splitting, are associated with proton’s octahedron shape and three nuclear structural axes. An element will be chemically stable if the outmost s and p clouds have eight electrons which make atom a symmetrical cubic.

... Also, PWT could be compatible with modications such as nonlinear Schrödinger equations − should they ever become necessary. Interestingly, in the recent decades, the benets of PWT for a numerical implementation and visualisation of quantum processes have also been realised [51], [17], [38], [34]. ...

Pilot wave theory (PWT), also called de Broglie-Bohm theory or Bohmian Mechanics, is a deterministic nonlocal `hidden variables' quantum theory without fundamental uncertainty. It is in agreement with all experimental facts about nonrelativistic quantum mechanics (QM) and furthermore
explains its mathematical structure. But in general, PWT describes a nonequilibrium state, admitting new physics beyond standard QM.
This essay is concerned with the problem how to generalise the PWT approach to quantum field theories (QFTs). First, we briefy state the formulation of nonrelativistic PWT and review its major results. We work out the parts of its structure that it shares with the QFT case. Next, we come to the main part: We show how PW QFTs can be constructed both for field and particle ontologies. In this context, we discuss some of the existing models as well as general issues: most importantly the status of Lorentz invariance in the context of quantum nonlocality. The essay concludes with a more speculative outlook in which the potential of PWT for open QFT questions as well as quantum nonequilibrium physics are considered.

... Our interest in studying the explicit model (1.1) is that rigorous investigation of its limiting behavior becomes feasible. Both Hall et al. (2014) and Sebens (2014) noted the ontological difficulty of a continuum of worlds, a feature of an earlier but closely related hydrodynamical approach due to Holland (2005), Poirier (2010) and Schiff and Poirier (2012). ...

From its beginning, there have been attempts by physicists to formulate
quantum mechanics without requiring the use of wave functions. An interesting
recent approach takes the point of view that quantum effects arise solely from
the interaction of finitely many classical "worlds." The wave function is then
recovered (as a secondary object) from observations of particles in these
worlds, without knowing the world from which any particular observation
originates. Hall, Deckert and Wiseman [Physical Review X 4 (2014) 041013] have
introduced an explicit many-interacting-worlds harmonic oscillator model to
provide support for this approach. In this note we provide a proof of their
claim that the particle configuration is asymptotically Gaussian, thus matching
the ground-state solution of Schrodinger's equation when the number of worlds
goes to infinity.

... It follows Pauli exclusion principle [13][14][15][16][17]. Even though Pauli exclusion principle is mainly for wavefunctions [18][19][20][21][22][23][24][25][26][27][28][29][30], the physical mass waves are used as above to discuss the principle. ...

The paper "Unified Field Theory and the Configuration of Particles" opened a new chapter of physics. One of the predictions of the paper is that a proton has an octahedron shape. As Physics progresses, it focuses more on invisible particles and the unreachable grand universe as visible matter is studied theoretically and experimentally. The shape of invisible proton has great impact on the topology of atom. Electron orbits, electron binding energy, Madelung Rules, and Zeeman splitting, are associated with proton's octahedron shape and three nuclear structural axes. An element will be chemically stable if the outmost s and p clouds have eight electrons which make atom a symmetrical cubic.

... Newtonian QM is somewhat similar to Böstrom's (2012) metaworld theory 1 and the proposal in Tipler (2006). Other ideas about how to remove the wave function are explored in Poirier (2010); Schiff & Poirier (2012), including an intimation of many worlds. ...

Here I explore a novel no-collapse interpretation of quantum mechanics that combines aspects of two familiar and well-developed alternatives, Bohmian mechanics and the many-worlds interpretation. Despite reproducing the empirical predictions of quantum mechanics, the theory looks surprisingly classical. All there is at the fundamental level are particles interacting via Newtonian forces. There is no wave function. However, there are many worlds. © 2015 by the Philosophy of Science Association. All rights reserved.

Since the 1950s mathematical physicists have been working on the construction of a formal mathematical foundation for relativistic quantum theory. In the literature the view that the axiomatization of the subject is primarily a mathematical problem has been prevalent. This view, however, implicitly asserts that said axiomatization can be achieved without readdressing the basic concepts of quantum theory-an assertion that becomes more implausible the longer the debate on the conceptual foundations of quantum mechanics itself continues. In this work we suggest a new approach to the above problem, which views the non-relativistic theory from a purely statistical perspective: to generalize the quantum-mechanical Born rule for particle position probability to the general-relativistic setting. The advantages of this approach are that one obtains a statistical theory from the onset and that it is independent of any particular dynamical models and the symmetries of Minkowski spacetime. Here we develop the smooth 1-body generalization, based on prior contributions mainly due to C. Eckart and J. Ehlers. This generalization respects the general principle of relativity and exposes the assumptions of spacelikeness of the hypersurface and global hyperbolicity of the spacetime as obsolete. We discuss two distinct formulations of the theory, which, borrowing terminology from the non-relativistic analog, we term the Lagrangian and Eulerian pictures. Though the development of the former one is the main contribution of this work, under these general conditions neither one of the two has received such a comprehensive treatment in the literature before. The Lagrangian picture also opens up a potentially viable path towards the many-body generalization. We further provide a simple example in which the number of bodies is not conserved. Readers interested in the theory of the general-relativistic continuity equation will also find this work to be of value.

Among the numerous concepts of time in quantum scattering, Smith's dwell time (Smith, 1960 [7]) and Eisenbud & Wigner's time delay (Wigner, 1955 [12]) are the most well established. The dwell time represents the amount of time spent by the particle inside a given coordinate range (typically a potential barrier interaction region), while the time delay measures the excess time spent in the interaction region because of the potential. In this paper, we use the exact trajectory-ensemble reformulation of quantum mechanics, recently proposed by one of the authors (Poirier), to study how tunneling and reflection unfold over time, in a one-dimensional rectangular potential barrier. Among other dynamical details, the quantum trajectory approach provides an extremely robust, accurate, and straightforward method for directly computing the dwell time and time delay, from a single quantum trajectory. The resultant numerical method is highly efficient, and in the case of the time delay, completely obviates the traditional need to energy-differentiate the scattering phase shift. In particular, the trajectory variables provide a simple expression for the time delay that disentangles the contribution of the self-interference delay. More generally, quantum trajectories provide interesting physical insight into the tunneling process.

Bohmian mechanics is an alternative to standard quantum mechanics that does not suffer from the measurement problem. While it agrees with standard quantum mechanics concerning its experimental predictions, it offers novel types of approximations not suggested by the latter. Of particular interest are semi-classical approximations, where part of the system is treated classically. Bohmian semi-classical approximations have been explored before for systems without electromagnetic interactions. Here, the Rabi model is considered as a simple model involving light-matter interaction. This model describes a single mode electromagnetic field interacting with a two-level atom. As is well-known, the quantum treatment and the semi-classical treatment (where the field is treated classically rather than quantum mechanically) give qualitatively different results. We analyze the Rabi model using a different semi-classical approximation based on Bohmian mechanics. In this approximation, the back-reaction from the two-level atom onto the classical field is mediated by the Bohmian configuration of the two-level atom. We find that the Bohmian semi-classical approximation gives results comparable to the usual mean field one for the transition between ground and first excited state. Both semi-classical approximations tend to reproduce the collapse of the population inversion, but fail to reproduce the revival, which is characteristic of the full quantum description. Also an example of a higher excited state is presented where the Bohmian approximation does not perform so well.

A recent article has treated the question of how to generalize the Born rule from non-relativistic quantum theory to curved spacetimes (Lienert and Tumulka, Lett. Math. Phys. 110, 753 (2019)). The supposed generalization originated in prior works on 'hypersurface Bohm-Dirac models' as well as approaches to relativistic quantum theory developed by Bohm and Hiley. In this comment, we raise three objections to the rule and the broader theory in which it is embedded. In particular, to address the underlying assertion that the Born rule is naturally formulated on a spacelike hypersurface, we provide an analytic example showing that a spacelike hypersurface need not remain spacelike under proper time evolution -- even in the absence of curvature. We finish by proposing an alternative `curved Born rule' for the one-body case, which overcomes these objections, and in this instance indeed generalizes the one Lienert and Tumulka attempted to justify. Our approach can be generalized to the many-body case, and we expect it to be also of relevance for the general case of a varying number of bodies.

The quantum dynamics of vibrational predissociation of the Ar ⋯Br2 triatomic molecule is described within a trajectory-based framework. The Br2 stretching mode is mapped into a set of classical (coupled) harmonic oscillators, associated to each vibrational state of the diatomic molecule. The time evolution of the molecular wave packet along the dissociation coordinate is described within the hydrodynamical formulation of quantum mechanics, specifically using the interacting trajectory representation. The relatively small number of interacting trajectories required to attain numerical convergence (N=100), makes the present model very appealing in comparison with other trajectory-based methods. The underlying parameterisation of the density was found to represent accurately the evolution of the projection of the molecular wave packet along the van der Waals mode, from the ground vibrational state into the continuum. The computed lifetime of the predissociating level and the population dynamics are in very good agreement with those observed experimentally.

Confined systems often exhibit unusual behavior regarding their structure, stability, reactivity, bonding, interactions, and dynamics. Quantization is a direct consequence of confinement. Confinement modifies the electronic energy levels, orbitals, electronic shell filling, etc. of a system, thereby affecting its reactivity as well as various response properties as compared to the cases of corresponding unconfined systems. Confinement may force two rare gas atoms to form a partly covalent bond. Gas storage is facilitated through confinement and unprecedented optoelectronic properties are observed in certain cases. Some slow reactions get highly accelerated in an appropriate confined environment. In the current Feature Article we analyze these aspects with a special emphasis on the work done by our research group.

A methodology of quantum dynamics based on interacting trajectories without reference to any wave function is applied to ultrashort laser ionization of a model hydrogen atom. The pulses are chosen to be so short that the relative phase between the carrier wave and the pulse envelope becomes important. As main results we show that the trajectory-only approach is capable of correctly describing the large amplitude motion and energetics of the laser driven electron and of reproducing carrier-envelope effects onto the photoelectron spectra. It also provides an intuitive picture of the dynamical quantum processes involved.

In this paper a trajectory-based relativistic quantum wave equation is established for extended charged spinless particles subject to the action of the electromagnetic (EM) radiation-reaction (RR) interaction. The quantization pertains the particle dynamics, in which both the external and self EM fields are treated classically. The new equation proposed here is referred to as the RR quantum wave equation. This is shown to be an evolution equation for a complex scalar quantum wave function and to be realized by a first-order PDE with respect to a quantum proper time s. The latter is uniquely prescribed by representing the RR quantum wave equation in terms of the corresponding quantum hydrodynamic equations and introducing a parametrization in terms of Lagrangian paths associated with the quantum fluid velocity. Besides the explicit proper time dependence, the theory developed here exhibits a number of additional notable features. First, the wave equation is variational and is consistent with the principle of manifest covariance. Second, it permits the definition of a strictly positive 4-scalar quantum probability density on the Minkowski space-time, in terms of which a flow-invariant probability measure is established. Third, the wave equation is non-local, due to the characteristic EM RR retarded interaction. Fourth, the RR wave equation recovers the Schrödinger equation in the non-relativistic limit and the customary Klein-Gordon wave equation when the EM RR is negligible or null. Finally, the consistency with the classical RR Hamilton-Jacobi equation is established in the semi-classical limit. © 2015, Società Italiana di Fisica and Springer-Verlag Berlin Heidelberg.

The complex quantum Hamilton–Jacobi equation for the complex action is approximately solved by propagating individual Bohmian trajectories in real space. Equations of motion for the complex action and its spatial derivatives are derived through use of the derivative propagation method. We transform these equations into the arbitrary Lagrangian–Eulerian version with the grid velocity matching the flow velocity of the probability fluid. Setting higher-order derivatives equal to zero, we obtain a truncated system of equations of motion describing the rate of change in the complex action and its spatial derivatives transported along approximate Bohmian trajectories. A set of test trajectories is propagated to determine appropriate initial positions for transmitted trajectories. Computational results for transmitted wave packets and transmission probabilities are presented and analyzed for a one-dimensional Eckart barrier and a two-dimensional system involving either a thick or thin Eckart barrier along the reaction coordinate coupled to a harmonic oscillator.

The paper "Unified field theory" (UFT) unified four fundamental forces with help of the Torque model. UFT gives a new definition of Physics: “A natural science that involves the study of motion of space-time-energy-force to explain and predict the motion, interaction and configuration of matter.” One of important pieces of matter is the atom. Unfortunately, the configuration of an atom cannot be visually observed. Two of the important accepted theories are the Pauli Exclusion Principle and the Schrodinger equations. In these two theories, the electron configuration is studied. Contrary to the top down approach, UFT theory starts from structure of Proton and Neutron using bottom up approach instead. Interestingly, electron orbits, electron binding energy, Madelung Rules, Zeeman splitting and crystal structure of the metals, are associated with proton’s octahedron shape and three nuclear structural axes. An element will be chemically stable if the outmost s and p orbits have eight electrons which make atom a symmetrical cubic. Most importantly, the predictions of atomic configurations in this paper can be validated by characteristics of chemical elements which make the UFT claims credible. UFT comes a long way from space-time-energy-force to the atom. The conclusions of UFT are more precise and clearer than the existing theories that have no proper explanation regarding many rules, such as eight outer electrons make element chemically stable and the exception on Madelung's rules. Regardless of the imperfections of the existing atomic theories, many particle Physics theories have no choice but to build on top of atomic theories, mainly Pauli Exclusion Principle and Schrodinger equations. Physics starts to look for answer via ambiguous mathematical equations as the proper clues are missing. Physics issues are different from mathematical issues, as they are Physical. Pauli Exclusion works well in electron configuration under specific physical condition and it is not a general Physics principal. Schrodinger’s mathematical equations are interpreted differently in UFT. UFT is more physical as it built itself mainly on concept of Space, Time, Energy and Force, in the other word, UFT is Physics itself. Theory of Everything (ToE), the final theory of the Physics, can be simply another name for UFT. This paper connects an additional dot to draw UFT closer to ToE.

A fast and robust time-independent method to calculate thermal rate constants in the deep resonant tunneling regime for scattering reactions is presented. The method is based on the calculation of the cumulative reaction probability which, once integrated, gives the thermal rate constant. We tested our method with both continuous (single and double Eckart barriers) and discontinuous first derivative potentials (single and double rectangular barriers). Our results show that the presented method is robust enough to deal with extreme resonating conditions such as multiple barrier potentials. Finally, the calculation of the thermal rate constant for double Eckart potentials with several quasi-bound states and the comparison with the time-independent log-derivative method are reported. An implementation of the method using the Mathematica Suite is included in the Supporting Information. © 2013 Wiley Periodicals, Inc.

If the classical structure of space-time is assumed to define an a priori scenario for the formulation of quantum theory (QT), the coordinate representation of the solutions \(\psi (\vec x,t)(\psi (\vec x_1 , \ldots ,\vec x_N ,t))\) of the Schroedinger equation of a quantum system containing one (N) massive scalar particle has a preferred status. Let us consider all of the solutions admitting a multipolar expansion of the probability density function \(\rho (\vec x,t) = \left| {(\psi (\vec x,t)} \right|^2\) (and more generally of the Wigner function) around a space-time trajectory \(\vec x_c (t)\)
to be properly selected. For every normalized solution \(\left( {\smallint d^3 x\rho (\vec x,t) = 1} \right)\) there is a privileged trajectory implying the vanishing of the dipole moment of the multipolar expansion: it is given by the expectation value of the position operator
\(\left\langle {\psi (t)\left| {\hat \vec x} \right|\psi (t)} \right\rangle = \vec x_c (t)\). Then, the special subset of solutions \(\psi (\vec x,t)\) which satisfy Ehrenfest’s Theorem (named thereby Ehrenfest monopole wave functions (EMWF)), have the important property that this privileged classical trajectory \(\vec x_c (t)\) is determined by a closed Newtonian equation of motion where the effective force is the Newtonian force plus non-Newtonian terms (of order ħ
2 or higher) depending on the higher multipoles of the probability distribution ρ. Note that the superposition of two EMWFs is not an EMWF, a result to be strongly hoped for, given the possible unwanted implications concerning classical spatial perception. These results can be extended to N-particle systems in such a way that, when N classical trajectories with all the dipole moments vanishing and satisfying Ehrenfest theorem are associated with the normalized wave functions of the N-body system, we get a natural transition from the 3N-dimensional configuration space to the space-time. Moreover, these results can be extended to relativistic quantum mechanics. Consequently, in suitable states of N quantum particle which are EMWF, we get the “emergence” of corresponding “classical particles” following Newton-like trajectories in space-time. Note that all this holds true in the standard framework of quantum mechanics, i.e. assuming, in particular, the validity of Born’s rule and the individual system interpretation of the wave function (no ensemble interpretation). These results are valid without any approximation (like ħ → 0, big quantum numbers, etc.). Moreover, we do not commit ourselves to any specific ontological interpretation of quantum theory (such as, e.g., the Bohmian one). We will argue that, in substantial agreement with Bohr’s viewpoint, the macroscopic description of the preparation, certain intermediate steps and the detection of the final outcome of experiments involving massive particles are dominated by these classical “effective” trajectories. This approach can be applied to the point of view of de-coherence in the case of a diagonal reduced density matrix ρ
red (an improper mixture) depending on the position variables of a massive particle and of a pointer. When both the particle and the pointer wave functions appearing in ρ
red are EMWF, the expectation value of the particle and pointer position variables becomes a statistical average on a classical ensemble. In these cases an improper quantum mixture becomes a classical statistical one, thus providing a particular answer to an open problem of de-coherence about the emergence of classicality.

This chapter provides a comprehensive overview of the Bohmian formulation of
quantum mechanics. It starts with a historical review of the difficulties found
by Louis de Broglie, David Bohm, and John S. Bell to convince the scientific
community about the validity and utility of Bohmian mechanics. Then, a formal
explanation of Bohmian mechanics for nonrelativistic, single-particle quantum
systems is presented. The generalization to many-particle systems, where the
exchange interaction and the spin play an important role, is also presented.
After that, the measurement process in Bohmian mechanics is discussed. It is
emphasized that Bohmian mechanics exactly reproduces the mean value and
temporal and spatial correlations obtained from the standard, that is the
Copenhagen or orthodox, formulation. The ontological characteristics of Bohmian
mechanics provide a description of measurements as another type of interaction
without the need for introducing the wave function collapse. Several solved
problems are presented at the end of the chapter, giving additional
mathematical support to some particular issues. A detailed description of
computational algorithms to obtain Bohmian trajectories from the numerical
solution of the Schrodinger or the Hamilton-Jacobi equations are presented in
an appendix. The motivation of this chapter is twofold: first, as a didactic
introduction to Bohmian formalism, which is used in the subsequent chapters,
and second, as a self-contained summary for any newcomer interested in using
Bohmian mechanics in his or her daily research activity.

We analyze the attosecond electron dynamics in hydrogen molecular ion driven by an external intense laser field using the Bohmian trajectories. To this end, we employ a one-dimensional model of the molecular ion in which the motion of the protons is frozen. The Bohmian trajectories clearly visualize the electron transfer between the two protons in the field and, in particular, confirm the recently predicted attosecond transient localization of the electron at one of the protons and the related multiple bunches of the ionization current within a half cycle of the laser field. Further analysis based on the quantum trajectories shows that the electron dynamics in the molecular ion can be understood via the phase difference accumulated between the Coulomb wells at the two protons.

This paper explores the quantum fluid dynamical (QFD) representation of the time-dependent Schrödinger equation for the motion of a wave packet in a high dimensional space. A novel alternating direction technique is utilized to single out each of the many dimensions in the QFD equations. This technique is used to solve the continuity equation for the density and the equation for the convection of the flux for the quantum particle. The ability of the present scheme to efficiently and accurately describe the dynamics of a quantum particle is demonstrated in four dimensions where analytical results are known. We also apply the technique to the photodissociation of NOCl and NO2 where the systems are reduced to two coordinates by freezing the angular variable at its equilibrium value.

In this paper we establish three variational principles that provide new foundations for Nelson’s stochastic mechanics in the case of nonrelativistic particles without spin. The resulting variational picture is much richer and of a different nature with respect to the one previously considered in the literature. We first develop two stochastic variational principles whose Hamilton–Jacobi‐like equations are precisely the two coupled partial differential equations that are obtained from the Schrödinger equation (Madelung equations). The two problems are zero‐sum, noncooperative, stochastic differential games that are familiar in the control theory literature. They are solved here by means of a new, absolutely elementary method based on Lagrange functionals. For both games the saddle‐point equilibrium solution is given by the Nelson’s process and the optimal controls for the two competing players are precisely Nelson’s current velocity v and osmotic velocity u, respectively. The first variational principle includes as special cases both the Guerra–Morato variational principle [Phys. Rev. D 27, 1774 (1983)] and Schrödinger original variational derivation of the time‐independent equation.

Diffraction and interference of matter waves are key phenomena in quantum mechanics. Here we present some results on particle diffraction in a wide variety of situations, ranging from simple slit experiments to more complicated cases such as atom scattering by corrugated metal surfaces and metal surfaces with simple and isolated adsorbates. The principal novelty of our study is the use of the so-called Bohmian formalism of quantum trajectories. These trajectories are able to satisfactorily reproduce the main features of the experimental results and, more importantly, they provide a causal intuitive interpretation of the underlying dynamics. In particular, we will focus our attention on: (a) a revision of the concepts of near and far field in undulatory optics; (b) the transition to the classical limit, where it is found that although the quantum and classical diffraction patterns tend to be quite similar, some quantum features are maintained even when the quantum potential goes to zero; and (c) a qualitative description of the scattering of atoms by metal surfaces in the presence of a single adsorbate.

Nine formulations of nonrelativistic quantum mechanics are reviewed. These are the wavefunction, matrix, path integral, phase space, density matrix, second quantization, variational, pilot wave, and Hamilton–Jacobi formulations. Also mentioned are the many-worlds and transactional interpretations. The various formulations differ dramatically in mathematical and conceptual overview, yet each one makes identical predictions for all experimental results. © 2002 American Association of Physics Teachers.

The method of quantum trajectories proposed by de Broglie and Bohm is applied to the study of atom diffraction by surfaces. As an example, a realistic model for the scattering of He off corrugated Cu is consid-ered. In this way, the final angular distribution of trajectories is obtained by box counting, which is in excellent agreement with the results calculated by standard S matrix methods of scattering theory. More interestingly, the accumulation of quantum trajectories at the different diffraction peaks is explained in terms of the corre-sponding quantum potential. This nonlocal potential ''guides'' the trajectories causing a transition from a distribution near the surface, which reproduces its shape, to the final diffraction pattern observed in the asymptotic region, far from the diffracting object. These two regimes are homologous to the Fresnel and Fraunhofer regions described in undulatory optics. Finally, the turning points of the quantum trajectories provide a better description of the surface electronic density than the corresponding classical ones, usually employed for this task.

This is the first part of what will be a two-part review of distribution functions in physics. Here we deal with fundamentals and the second part will deal with applications. We discuss in detail the properties of the distribution function defined earlier by one of us (EPW) and we derive some new results. Next, we treat various other distribution functions. Among the latter we emphasize the so-called P distribution, as well as the generalized P distribution, because of their importance in quantum optics.

In previous articles [J. Chem. Phys. 121, 4501 (2004); J. Chem. Phys. 124, 034115 (2006); J. Chem. Phys. 124, 034116 (2006); J. Phys. Chem. A 111, 10400 (2007); J. Chem. Phys. 128, 164115 (2008)] an exact quantum, bipolar wave decomposition, psi=psi(+)+psi(-), was presented for one-dimensional stationary state and time-dependent wavepacket dynamics calculations, such that the components psi(+/-) approach their semiclassical WKB analogs in the large action limit. The corresponding bipolar quantum trajectories are classical-like and well behaved, even when psi has many nodes or is wildly oscillatory. In this paper, both the stationary state and wavepacket dynamics theories are generalized for multidimensional systems and applied to several benchmark problems, including collinear H+H(2).

The usual interpretation of the quantum theory is self-consistent, but it involves an assumption that cannot be tested experimentally, viz., that the most complete possible specification of an individual system is in terms of a wave function that determines only probable results of actual measurement processes. The only way of investigating the truth of this assumption is by trying to find some other interpretation of the quantum theory in terms of at present "hidden" variables, which in principle determine the precise behavior of an individual system, but which are in practice averaged over in measurements of the types that can now be carried out. In this paper and in a subsequent paper, an interpretation of the quantum theory in terms of just such "hidden" variables is suggested. It is shown that as long as the mathematical theory retains its present general form, this suggested interpretation leads to precisely the same results for all physical processes as does the usual interpretation. Nevertheless, the suggested interpretation provides a broader conceptual framework than the usual interpretation, because it makes possible a precise and continuous description of all processes, even at the quantum level. This broader conceptual framework allows more general mathematical formulations of the theory than those allowed by the usual interpretation. Now, the usual mathematical formulation seems to lead to insoluble difficulties when it is extrapolated into the domain of distances of the order of 10-13 cm or less. It is therefore entirely possible that the interpretation suggested here may be needed for the resolution of these difficulties. In any case, the mere possibility of such an interpretation proves that it is not necessary for us to give up a precise, rational, and objective description of individual systems at a quantum level of accuracy.

Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies

A potential barrier of the kind studied by Fowler and others may be represented by the analytic function V (Eq. (1)). The Schrödinger equation associated to this potential is soluble in terms of hypergeometric functions, and the coefficient of reflection for electrons approaching the barrier with energy W is calculable (Eq. (15)). The approximate formula, 1-ρ=exp{-∫4πh(2m(V-W))12dx} is shown to agree very well with the exact formula when the width of the barrier is great compared to the de Broglie wave-length of the incident electron, and W<Vmax.

Despite its enormous practical success, quantum theory is so contrary to intuition that, even after 45 years, the experts themselves still do not all agree what to make of it. The area of disagreement centers primarily around the problem of describing observations. Formally, the result of a measurement is a superposition of vectors, each representing the quantity being observed as having one of its possible values. The question that has to be answered is how this superposition can be reconciled with the fact that in practice we only observe one value. How is the measuring instrument prodded into making up its mind which value it has observed? Could the solution to the dilemma of indeterminism be a universe in which all possible outcomes of an experiment actually occur?

DOI:https://doi.org/10.1103/RevModPhys.29.454

The Statistical Interpretation of quantum theory is formulated for the purpose of providing a sound interpretation using a minimum of assumptions. Several arguments are advanced in favor of considering the quantum state description to apply only to an ensemble of similarily prepared systems, rather than supposing, as is often done, that it exhaustively represents an individual physical system. Most of the problems associated with the quantum theory of measurement are artifacts of the attempt to maintain the latter interpretation. The introduction of hidden variables to determine the outcome of individual events is fully compatible with the statistical predictions of quantum theory. However, a theorem due to Bell seems to require that any such hidden-variable theory which reproduces all of quantum mechanics exactly (i.e., not merely in some limiting case) must possess a rather pathological character with respect to correlated, but spacially separated, systems.

An adaptive grid approach to a computational study of the scattering of a wavepacket from a repulsive Eckart barrier is described. The grids move in an arbitrary Lagrangian–Eulerian (ALE) framework and a hybrid of the moving path transform of the Schrödinger equation and the hydrodynamic equations are used for the equations of motion. Boundary grid points follow Lagrangian trajectories and interior grid points follow non-Lagrangian paths. For the hydrodynamic equations the interior grid points are equally spaced between the evolving Lagrangian boundaries. For the moving path transform of the Schrödinger equation interior grid distribution is determined by the principle of equidistribution, and by using a grid smoothing technique these grid points trace a path that continuously adapts to reflect the dynamics of the wavepacket. The moving grid technique is robust and allows accurate computations to be obtained with a small number of grid points for wavepacket propagation times exceeding 5 ps.

Although Bohmian mechanics has attracted considerable interest as a causal interpretation of quantum mechanics, it also possesses intrinsic heuristic value, arising from calculational tools and physical insights that are unavailable in ``standard'' quantum mechanics. We illustrate by examining the behavior of Gaussian harmonic oscillator wave packets from the Bohmian perspective. By utilizing familiar classical concepts and techniques, we obtain a physically transparent picture of packet behavior. This example provides, at a level accessible to students, a concrete illustration of Bohmian mechanics as a heuristic device that can enhance both understanding and discovery.

A novel method for integrating the time-dependent Schrödinger equation is presented. Hydrodynamic quantum trajectories are used to adaptively define the boundaries and boundary conditions of a fixed grid. The result is a significant reduction in the number of grid points needed to perform accurate calculations. The Eckart barrier, along with uphill and downhill ramp potentials, was used to evaluate the method. Excellent agreement with fixed boundary grids was obtained for each example. By moving only the boundary points, stability was increased to the level of the full fixed grid.

The de Broglie-Bohm causual (hydrodynamic) formulation of quantum mechanics is computationally implemented in the Lagrangian (moving with the fluid) viewpoint. The quantum potential and force are accurately evaluated with a moving weighted least squares algorithm. The quantum trajectory method is then applied to barrier tunneling on smooth potential surfaces. Analysis of the tunneling mechanism leads to a novel and accurate approximation: shortly after the wave packet is launched, completely neglect all quantum terms in the dynamical equations for motion along the tunneling coordinate.

The demonstrations of von Neumann and others, that quantum mechanics does not permit a hidden variable interpretation, are reconsidered. It is shown that their essential axioms are unreasonable. It is urged that in further examination of this problem an interesting axiom would be that mutually distant systems are independent of one another.

The quantum hydrodynamic equations associated with the de Broglie–Bohm formulation of quantum mechanics are solved using a meshless method based on a moving least squares approach. An arbitrary Lagrangian–Eulerian frame of reference is used which significantly improves the accuracy and stability of the method when compared to an approach based on a purely Lagrangian frame of reference. A regridding algorithm is implemented which adds and deletes points when necessary in order to maintain accurate and stable calculations. It is shown that unitarity in the time evolution of the quantum wave packet is significantly improved by propagating using averaged fields. As nodes in the reflected wave packet start to form, the quantum potential and force become very large and numerical instabilities occur. By introducing artificial viscosity into the equations of motion, these instabilities can be avoided and the stable propagation of the wave packet for very long times becomes possible. Results are presented for the scattering of a wave packet from a repulsive Eckart barrier. © 2003 American Institute of Physics.

Rather general expressions are derived which represent the semiclassical time‐dependent propagator as an integral over initial conditions for classical trajectories. These allow one to propagate time‐dependent wave functions without searching for special trajectories that satisfy two‐time boundary conditions. In many circumstances, the integral expressions are free of singularities and provide globally valid uniform asymptotic approximations. In special cases, the expressions for the propagators are related to existing semiclassical wave function propagation techniques. More generally, the present expressions suggest a large class of other, potentially useful methods. The behavior of the integral expressions in certain limiting cases is analyzed to obtain simple formulas for the Maslov index that may be used to compute the Van Vleck propagator in a variety of representations.

The origin of quantum interference characteristic of bound nonlinear systems is investigated within the Bohmian formulation of time-dependent quantum mechanics. By contrast to time-dependent semiclassical theory, whereby interference is a consequence of phase mismatch between distinct classical trajectories, the Bohmian, fully quantum mechanical expression for expectation values has a quasiclassical appearance that does not involve phase factors or cross terms. Numerical calculations reveal that quantum interference in the Bohmian formulation manifests itself directly as sharp spatial/temporal variations of the density surrounding kinky trajectories. These effects are most dramatic in regions where the underlying classical motion exhibits focal points or caustics, and crossing of the Bohmian trajectories is prevented through extremely strong and rapidly varying quantum mechanical forces. These features of Bohmian dynamics, which constitute the hallmark of quantum interference and are ubiquitous in bound nonlinear systems, represent a major source of instability, making the integration of the Bohmian equations extremely demanding in such situations. © 2003 American Institute of Physics.

The quantum trajectory method (QTM) was recently developed to solve the hydrodynamic equations of motion in the Lagrangian, moving-with-the-fluid, picture. In this approach, trajectories are integrated for N fluid elements (particles) moving under the influence of both the force from the potential surface and from the quantum potential. In this study, distributed approximating functionals (DAFs) are used on a uniform grid to compute the necessary derivatives in the equations of motion. Transformations between the physical grid where the particle coordinates are defined and the uniform grid are handled through a Jacobian, which is also computed using DAFs. A difficult problem associated with computing derivatives on finite grids is the edge problem. This is handled effectively by using DAFs within a least squares approach to extrapolate from the known function region into the neighboring regions. The QTM–DAF is then applied to wave packet transmission through a one-dimensional Eckart potential. Emphasis is placed upon computation of the transmitted density and wave function. A problem that develops when part of the wave packet reflects back into the reactant region is avoided in this study by introducing a potential ramp to sweep the reflected particles away from the barrier region. © 2000 American Institute of Physics.

Numerical solutions of the quantum time-dependent integro-differential Schrödinger equation in a coherent state Husimi representation are investigated. Discretization leads to propagation on a grid of nonorthogonal coherent states without the need to invert an overlap matrix, with the further advantage of a sparse Hamiltonian matrix. Applications are made to the evolution of a Gaussian wave packet in a Morse potential. Propagation on a static rectangular grid is fast and accurate. Results are also presented for a moving rectangular grid, guided at its center by a mean classical path, and for a classically guided moving grid of individual coherent states taken from a Monte Carlo ensemble. © 2000 American Institute of Physics.

A hydrodynamic approach is developed to describe nonadiabatic nuclear dynamics. We derive a hierarchy of hydrodynamic equations which are equivalent to the exact quantum Liouville equation for coupled electronic states. It is shown how the interplay between electronic populations and coherences translates into the coupled dynamics of the corresponding hydrodynamic fields. For the particular case of pure quantum states, the hydrodynamic hierarchy terminates such that the dynamics may be described in terms of the local densities and momentum fields associated with each of the electronic states. © 2001 American Institute of Physics.

A new method is proposed for computing the time evolution of quantum mechanical wave packets. Equations of motion for the real-valued functions C and S in the complex action = C(r,t)+iS(r,t)/ℏ, with ψ(r,t) = exp(), involve gradients and curvatures of C and S. In previous implementations of the hydrodynamic formulation, various time-consuming fitting techniques of limited accuracy were used to evaluate these derivatives around each fluid element in an evolving ensemble. In this study, equations of motion are developed for the spatial derivatives themselves and a small set of these are integrated along quantum trajectories concurrently with the equations for C and S. Significantly, quantum effects can be included at various orders of approximation, no spatial fitting is involved, there are no basis set expansions, and single quantum trajectories (rather than correlated ensembles) may be propagated, one at a time. Excellent results are obtained when the derivative propagation method is applied to anharmonic potentials involving barrier transmission. © 2003 American Institute of Physics.

Recently, the quantum trajectory method (QTM) has been utilized in solving several quantum mechanical wave packet scattering problems including barrier transmission and electronic nonadiabatic dynamics. By propagating the real-valued action and amplitude functions in the Lagrangian frame, only a fraction of the grid points needed for Eulerian fixed-grid methods are used while still obtaining accurate solutions. Difficulties arise, however, near wave functionnodes and in regions of sharp oscillatory features, and because of this many quantum mechanical problems have not yet been amenable to solution with the QTM. This study proposes a hybrid of both the Lagrangian and Eulerian techniques in what is termed the arbitrary Lagrangian–Eulerian method (ALE). In the ALE method, an additional equation of motion governing the momentum of the grid points is coupled into the quantum hydrodynamicequations. These new “quasi-” Bohmian trajectories can be dynamically adapted to the emergent features of the time evolving hydrodynamic fields and are non-Lagrangian. In this study it is shown that the ALE method applied to an uphill ramp potential that was previously unsolvable by the current Lagrangian QTM not only yields stable transmission probabilities with accuracies comparable to that of a high resolution Eulerian method, but does so with a small number of grid points and for extremely long propagation times. To determine the grid point positions at each new time, an equidistribution method is used that is constructed similar to the stiffness matrix of a classical spring system in equilibrium. Each “smart” spring is dependent on a local function M(x) called the monitor function which can sense gradients or curvatures of the fields surrounding its position. To constrain grid points from having zero separation and possible overlap, a new system of equations is derived that includes a minimum separation parameter which prevents this from occurring.

It is shown that the quantum force in the Bohmian formulation of quantum mechanics can be related to the stability properties of the given trajectory. In turn, the evolution of the stability properties is governed by higher order derivatives of the quantum potential, leading to an infinite hierarchy of coupled differential equations whose solution specifies completely all aspects of the dynamics. Neglecting derivatives of the quantum potential beyond a certain order allows truncation of the hierarchy, leading to approximate Bohmian trajectories. Use of the method in conjunction with Bohmian initial value formulations [J. Chem. Phys. 2003, 119, 60] gives rise to simple position-space representations of observables or time correlation functions. These are analogous to approximate quasiclassical expressions based on the Wigner or Husimi phase space density but involve lower dimensional integrals with smoother integrands and avoid the costly evaluation of phase space transforms. The lowest-order version of the truncated hierarchy can capture large corrections to classical mechanical treatments and yields (with fewer trajectories) results that are somewhat more accurate than those based on quasiclassical phase space treatments.

The semiclassical (SC) initial value representation (IVR) provides a potentially practical way for adding quantum mechanical effects to classical molecular dynamics (MD) simulations of the dynamics of complex molecular systems (i.e., those with many degrees of freedom). It does this by replacing the nonlinear boundary value problem of semiclassical theory by an average over the initial conditions of classical trajectories. This paper reviews the background and rebirth of interest in such approaches and surveys a variety of their recent applications. Special focus is on the ability to treat the dynamics of complex systems, and in this regard, the forward−backward (FB) version of the approach is especially promising. Several examples of the FB-IVR applied to model problems of many degrees of freedom show it to be capable of describing quantum effects quite well and also how these effects are quenched when some of the degrees of freedom are averaged over (“decoherence”).

A new procedure for developing a coarse-grained representation of the free particle propagator in Cartesian, cylindrical, and spherical polar coordinates is presented. The approach departs from a standard basis representation of the propagator and the state function to which it is applied. Instead, distributed approximating functions (DAFs), developed recently in the context of propagating wave packets in 1-D on an infinite line, are used to create a coarse-grained, highly banded matrix which produces arbitrarily accurate results for the free propagation of wave packets. The new DAF formalism can be used with nonuniform grid spacings. The banded, discretized matrix DAF representation of <x\exp(-iK-tau/h)\x'> can be employed in any wave packet propagation scheme which makes use of the free propagator. A major feature of the DAF expression for the effective free propagator is that the modulus of the x(j),x(j), element is proportional to the Gaussian exp(-sigma(2)(0)(x(j) - x(f)2/2(sigma(4)(0) + h2-tau(2)/m2)). The occurrence of a tau-dependent width is a manifestation of the fundamental spreading of a wave packet as it evolves through time, and it is the minimum possible because the DAF representation of the free propagator is based on evolving the Gaussian generator of the Hermite polynomials. This suggests that the DAFs yield the most highly banded effective free propagator possible. The second major feature of the DAF representation of the free propagator is that it can be used for real time dynamics based on Feynman path integrals. This holds the possibility that the real time dynamics for multidimensional systems could be done by Monte Carlo methods with a Gaussian as the importance sampling function.

We review various methods of deriving expressions for quantum-mechanical quantities in the limit when hslash is small (in comparison with the relevant classical action functions). To start with we treat one-dimensional problems and discuss the derivation of WKB connection formulae (and their reversibility), reflection coefficients, phase shifts, bound state criteria and resonance formulae, employing first the complex method in which the classical turning points are avoided, and secondly the method of comparison equations with the aid of which uniform approximations are derived, which are valid right through the turningpoint regions. The special problems associated with radial equations are also considered. Next we examine semiclassical potential scattering, both for its own sake and also as an example of the three-stage approximation method which must generally be employed when dealing with eigenfunction expansions under semiclassical conditions, when they converge very slowly. Finally, we discuss the derivation of semiclassical expressions for Green functions and energy level densities in very general cases, employing Feynman's path-integral technique and emphasizing the limitations of the results obtained. Throughout the article we stress the fact that all the expressions obtained involve quantities characterizing the families of orbits in the corresponding purely classical problems, while the analytic forms of the quantal expressions depend on the topological properties of these families.
This review was completed in February 1972.

Correlations of linear polarizations of pairs of photons have been measured with time-varying analyzers. The analyzer in each leg of the apparatus is an acousto-optical switch followed by two linear polarizers. The switches operate at incommensurate frequencies near 50 MHz. Each analyzer amounts to a polarizer which jumps between two orientations in a time short compared with the photon transit time. The results are in good agreement with quantum mechanical predictions but violate Bell's inequalities by 5 standard deviations.

In this article, we develop a series of hierarchical mode-coupling equations for the momentum cumulants and moments of the density matrix for a mixed quantum system. Working in the Lagrange representation, we show how these can be used to compute quantum trajectories for dissipative and nondissipative systems. This approach is complementary to the de Broglie–Bohm approach in that the moments evolve along hydrodynamic/Lagrangian paths. In the limit of no dissipation, the paths are the Bohmian paths. However, the “quantum force” in our case is represented in terms of momentum fluctuations and an osmotic pressure. Representative calculations for the relaxation of a harmonic system are presented to illustrate the rapid convergence of the cumulant expansion in the presence of a dissipative environment. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002

The Van Vleck formula is an approximate, semiclassical expression for the quantum propagator. It is the starting point for the derivation of the Gutzwiller trace formula, and through this, a variety of other expansions representing eigenvalues, wave functions, and matrix elements in terms of classical periodic orbits. These are currently among the best and most promising theoretical tools for understanding the asymptotic behavior of quantum systems whose classical analogs are chaotic. Nevertheless, there are currently several questions remaining about the meaning and validity of the Van Vleck formula, such as those involving its behavior for long times. This article surveys an important aspect of the Van Vleck formula, namely, the relationship between it and phase space geometry, as revealed by Maslov's theory of wave asymptotics. The geometrical constructions involved are developed with a minimum of mathematical formalism.

Es wird gezeigt, da man die Schrdingersche Gleichung des Einelektronen-problems in die Form der hydrodynamischen Gleichungen transformieren kann.

The purpose of this paper was to justify the fact that deterministic corpuscular description of a free particle can be made reconciled with its dual probabilistic wave description in complex space. It is found that the known wave-particle duality can be best manifested in complex space by showing that the wave motion associated with a material particle is just the phenomenon of projection of its complex motion into real space. To verify this new interpretation of matter wave, the equation of motion for a particle moving in complex space is derived first, then it is solved to reveal how the interaction between the real and imaginary motion can produce the particle’s wave motion observed in real space. The derived complex equation of motion for a “free” particle indicates that a so-called free particle is only free from classical potential, but not free from the complex quantum potential. Due to the action of this complex quantum potential, a free particle may move either right or left in a classical way retaining its corpuscular property, or may oscillate between the two directions producing a non-local wave motion. A propagation criterion is derived in this paper to determine when a particle follows a classical corpuscular motion and when it follows a quantum wave motion. Based on this new interpretation, the internal mechanism producing polarization of matter wave and the formation of interference fringes can all be understood from the particle’s motion in complex space, and the reason why wave function can be served as a probability density function also becomes clear.

A method is presented for the construction of asymptotic formulas for the large eigenvalues and the corresponding eigenfunctions of boundary value problems for partial differential equations. It is an adaptation to bounded domains of the method previously devised to deduce the corrected Bohr-Sommerfeld quantum conditions.When applied to the reduced wave equation in various domains for which the exact solutions are known, it yields precisely the asymptotic forms of those solutions. In addition it has been applied to an arbitrary convex plane domain for which the exact solutions are not known. Two types of solutions have been found, called the “whispering gallery” and “bouncing ball” modes. Applications have also been made to the Schrödinger equation.

A discussion of aspects of probability relevant to the differing interpretations of quantum theory is given, followed by an account of so-called orthodox interpretations of quantum theory that stresses their flexibility and subtlety as well as their problems. An account of ensemble interpretations is then presented, with discussion of the approaches of Einstein and Ballentine, and of later developments, including those interpretations usually called “stochastic”. A general study of ensemble interpretations follows, including investigation of PIV (premeasurement initial values) and minimal interpretations, an account of recent developments, and an introduction to unsharp measurements. Finally, application is made to particular problems, EPR, Schrödinger's cat, the quantum Zeno “paradox”, and Bell's theorem.

A basic aspect of the recently proposed approach to quantum mechanics is that no use of any axiomatic interpretation of the wave function is made. In particular, the quantum potential turns out to be an intrinsic potential energy of the particle, which, similarly to the relativistic rest energy, is never vanishing. This is related to the tunnel effect, a consequence of the fact that the conjugate momentum field is real even in the classically forbidden regions. The quantum stationary Hamilton–Jacobi equation is defined only if the ratio ψD/ψ of two real linearly independent solutions of the Schrödinger equation, and therefore of the trivializing map, is a local homeomorphism of the extended real line into itself, a consequence of the Möbius symmetry of the Schwarzian derivative. In this respect we prove a basic theorem relating the request of continuity at spatial infinity of ψD/ψ, a consequence of the q↔q−1 duality of the Schwarzian derivative, to the existence of solutions of the corresponding Schrödinger equation. As a result, while in the conventional approach one needs the Schrödinger equation with the condition, consequence of the axiomatic interpretation of the wave function, the equivalence principle by itself implies a dynamical equation that does not need any assumption and reproduces both the tunnel effect and energy quantization.

A justification is given for the use of non-spreading or frozen gaussian packets in dynamics calculations. In this work an initial wavefunction or quantum density operator is expanded in a complete set of grussian wavepackets. It is demonstrated that the time evolution of this wavepacket expansion for the quantum wavefunction or density is correctly given within the approximations employed by the classical propagation of the avarage position and momentum of each gaussian packet, holding the shape of these individual gaussians fixed. The semiclassical approximation is employed for the quantum propagator and the stationary phase approximation for certain integrals is utilized in this derivation. This analysis demonstrates that the divergence of the classical trajectories associated with the individual gaussian packets accounts for the changes in shape of the quantum wavefunction or density, as has been suggested on intuitive grounds by Heller. The method should be exact for quadratic potentials and this is verified by explicitly applying it for the harmonic oscillator example.