De Broglie–Bohm theory

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On a closer view, though, one must admit that these empty branches do not actually disappear. As the wavefunction is taken to describe a really existing field, all their branches really exist and will evolve forever by the Schrödinger dynamics, no matter how many of them will become empty in the course of the evolution. Every branch of the global wavefunction potentially describes a complete world which is, according to Bohm's ontology, only a possible world that would be the actual world if only it were filled with particles, and which is in every respect identical to a corresponding world in Everett's theory. Only one branch at a time is occupied by particles, thereby representing the actual world, while all other branches, though really existing as part of a really existing wavefunction, are empty and thus contain some sort of "zombie worlds" with planets, oceans, trees, cities, cars and people who talk like us and behave like us, but who do not actually exist. Now, if the Everettian theory may be accused of ontological extravagance, then Bohmian mechanics could be accused of ontological wastefulness. On top of the ontology of empty branches comes the additional ontology of particle positions that are, on account of the quantum equilibrium hypothesis, forever unknown to the observer. Yet, the actual configuration is never needed for the calculation of the statistical predictions in experimental reality, for these can be obtained by mere wavefunction algebra. From this perspective, Bohmian mechanics may appear as a wasteful and redundant theory. I think it is considerations like these that are the biggest obstacle in the way of a general acceptance of Bohmian mechanics.

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it registers as spin-down. Thus, for the de Broglie–Bohm theory, the particle's spin is not an intrinsic property of the particle; instead spin is, so to speak, in the wavefunction of the particle in relation to the particular device being used to measure the spin. This is an illustration of what is sometimes referred to as contextuality and is related to naive realism about operators. Interpretationally, measurement results are a deterministic property of the system and its environment, which includes information about the experimental setup including the context of co-measured observables; in no sense does the system itself possess the property being measured, as would have been the case in classical physics.

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existed prior to the measurement, then one does reach contradictions. De Broglie–Bohm theory deals with this by noting that spin is not a feature of the particle, but rather that of the wavefunction. As such, it only has a definite outcome once the experimental apparatus is chosen. Once that is taken into account, the impossibility theorems become irrelevant. There are also objections to this theory based on what it says about particular situations usually involving eigenstates of an operator. For example, the ground state of hydrogen is a real wavefunction. According to the guiding equation, this means that the electron is at rest when in this state. Nevertheless, it is distributed according to

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position of the particle. Such initial position is not knowable or controllable by the experimenter, so there is an appearance of randomness in the pattern of detection. In Bohm's 1952 papers he used the wavefunction to construct a quantum potential that, when included in Newton's equations, gave the trajectories of the particles streaming through the two slits. In effect the wavefunction interferes with itself and guides the particles by the quantum potential in such a way that the particles avoid the regions in which the interference is destructive and are attracted to the regions in which the interference is constructive, resulting in the interference pattern on the detector screen.

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dynamics. The motivation for de Broglie–Bohm theory is to describe a system of particles. This implies that the goal of the theory is to describe the positions of those particles at all times. Other observables do not have this compelling ontological status. Having definite positions explains having definite results such as flashes on a detector screen. Other observables would not lead to that conclusion, but there need not be any problem in defining a mathematical theory for other observables; see Hyman et al. for an exploration of the fact that a probability density and probability current can be defined for any set of commuting operators.

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standard quantum mechanics, an additional axiom of the theory. The de Broglie–Bohm theory, by contrast, requires no such measurement axioms (and measurement as such is not a dynamically distinct or special sub-category of physical processes in the theory). In particular, the usual operators-as-observables formalism is, for de Broglie–Bohm theory, a theorem. A major point of the analysis is that many of the measurements of the observables do not correspond to properties of the particles; they are (as in the case of spin discussed above) measurements of the wavefunction.

4627: 1872:. In it, a beam of particles (such as electrons) travels through a barrier that has two slits. If a detector screen is on the side beyond the barrier, the pattern of detected particles shows interference fringes characteristic of waves arriving at the screen from two sources (the two slits); however, the interference pattern is made up of individual dots corresponding to particles that had arrived on the screen. The system seems to exhibit the behaviour of both waves (interference patterns) and particles (dots on the screen). 16640: 5030:{\displaystyle {\begin{aligned}{\frac {d\mathbf {Q} _{k}}{dt}}(t)&={\frac {\hbar }{m_{k}}}\operatorname {Im} \left({\frac {(\psi ,D_{k}\psi )}{(\psi ,\psi )}}\right)(\mathbf {Q} _{1},\ldots ,\mathbf {Q} _{N},t),\\i\hbar {\frac {\partial }{\partial t}}\psi &=\left(-\sum _{k=1}^{N}{\frac {\hbar ^{2}}{2m_{k}}}D_{k}^{2}+V-\sum _{k=1}^{N}\mu _{k}{\frac {\mathbf {S} _{k}}{\hbar s_{k}}}\cdot \mathbf {B} (\mathbf {q} _{k})\right)\psi ,\end{aligned}}} 12728:"While the testable predictions of Bohmian mechanics are isomorphic to standard Copenhagen quantum mechanics, its underlying hidden variables have to be, in principle, unobservable. If one could observe them, one would be able to take advantage of that and signal faster than light, which – according to the special theory of relativity – leads to physical temporal paradoxes." J. Kofler and A. Zeiliinger, "Quantum Information and Randomness", 1380: 6620: 6401: 3140:

increase of entropy. Similarly in the de Broglie–Bohm theory, there are anomalous initial conditions that would produce measurement statistics in violation of the Born rule (conflicting the predictions of standard quantum theory), but the typicality theorem shows that absent some specific reason to believe one of those special initial conditions was in fact realized, the Born rule behavior is what one should expect.

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could be directly affected by the condition of the particles the quantum theory can be understood completely in terms of the assumption that the quantum field has no sources or other forms of dependence on the particles". P. Holland considers this lack of reciprocal action of particles and wave function to be one "mong the many nonclassical properties exhibited by this theory". Holland later called this a merely

1853: 6585:, these additional elements or parameters could be called 'hidden' variables." Bohm and Hiley later stated that they found Bohm's choice of the term "hidden variables" to be too restrictive. In particular, they argued that a particle is not actually hidden but rather "is what is most directly manifested in an observation its properties cannot be observed with arbitrary precision (within the limits set by 4569:

conditions that violate the no-signal theorems of quantum theory. Just as special relativity is a limiting case of general relativity when the spacetime curvature vanishes, so, too is statistical no-entanglement signaling quantum theory with the Born rule a limiting case of the post-quantum action-reaction Lagrangian when the reaction is set to zero and the final boundary condition is integrated out.

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of relativity the concept of instantaneousness does not have an invariant meaning. Thus, to define particle trajectories, one needs an additional rule that defines which space-time points should be considered instantaneous. The simplest way to achieve this is to introduce a preferred foliation of space-time by hand, such that each hypersurface of the foliation defines a hypersurface of equal time.

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one of the photons was determined using weak measurements and postselection, can be understood in terms of a nonlocal connection between that photon's trajectory and the other photon's polarization. However, not only the De Broglie–Bohm interpretation, but also many other interpretations of quantum mechanics that do not include such trajectories are consistent with such experimental evidence.

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audience might have been lost in the technical details and de Broglie's mild manner left the impression that Pauli's objection was valid. He was eventually persuaded to abandon this theory nonetheless because he was "discouraged by criticisms which roused". De Broglie's theory already applies to multiple spin-less particles, but lacks an adequate theory of measurement as no one understood

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become such that the wavefunction of the system appears to be in a superposition of distinct states. But if the environment has recorded the results of the experiment, then using the actual Bohmian configuration of the environment to condition on, the conditional wavefunction collapses to just one alternative, the one corresponding with the measurement results.

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other particles in the universe. In some situations, such as in experimental systems, we can represent the system itself in terms of a de Broglie–Bohm theory in which the wavefunction of the system is obtained by conditioning on the environment of the system. Thus, the system can be analyzed with Schrödinger's equation and the guiding equation, with an initial

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particle has a "complex and subtle inner structure" that provides the capacity to react to the information provided by the wavefunction by the quantum potential. Also, unlike in classical mechanics, physical properties (e.g., mass, charge) are spread out over the wavefunction in de Broglie–Bohm theory, not localized at the position of the particle.

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was an independent origination of the pilot wave theory, and extended it to incorporate a consistent theory of measurement, and to address a criticism of Pauli that de Broglie did not properly respond to; it is taken to be deterministic (though Bohm hinted in the original papers that there should be disturbances to this, in the way

9341:. These results have been disputed: experiments fail to reproduce aspects of the double-slit experiments. High precision measurements in the tunneling case point to a different origin of the unpredictable crossing: rather than initial position uncertainty or environmental noise, interactions at the barrier seem to be involved. 5831:, which they refer to as "Bell-type quantum field theories". The basic idea is that configuration space becomes the (disjoint) space of all possible configurations of any number of particles. For part of the time, the system evolves deterministically under the guiding equation with a fixed number of particles. But under a 6768:

knowledge of the particle's trajectory that accounts for the uncertainty relation. What one can know about a particle at any given time is described by the wavefunction. Since the uncertainty relation can be derived from the wavefunction in other interpretations of quantum mechanics, it can be likewise derived (in the

6836:(see p. 14 in Bell), in which he showed that, if they are to agree with the empirical predictions of quantum mechanics, all such "hidden-variable" completions of quantum mechanics must either be nonlocal (as the Bohm interpretation is) or give up the assumption that experiments produce unique results (see 5819:. Modern approaches to SED, like those proposed by the group around late Gerhard Grössing, among others, consider wave and particle-like quantum effects as well-coordinated emergent systems. These emergent systems are the result of speculated and calculated sub-quantum interactions with the zero-point field. 7163:& Wallace, the de Broglie–Bohm particles play no role in the solution of the measurement problem. For these authors, the "result assumption" (see above) is inconsistent with the view that there is no measurement problem in the predictable outcome (i.e. single-outcome) case. They also say that a standard 10084:

In 1992, Englert, Scully, Sussman, and Walther proposed experiments that would show particles taking paths that differ from the Bohm trajectories. They described the Bohm trajectories as "surrealistic"; their proposal was later referred to as ESSW after the last names of the authors. In 2016, Mahler

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and Basil Hiley. Bohm is clear that this theory is non-deterministic (the work with Hiley includes a stochastic theory). As such, this theory is not strictly speaking a formulation of de Broglie–Bohm theory, but it deserves mention here because the term "Bohm Interpretation" is ambiguous between this

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pointed out that it was not compatible with a semi-classical technique Fermi had previously adopted in the case of inelastic scattering. Contrary to a popular legend, de Broglie actually gave the correct rebuttal that the particular technique could not be generalized for Pauli's purpose, although the

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theory, if the (never collapsing) wavefunction is taken to be physically real, then it is natural to interpret the theory as having the same many worlds as Everett's theory. In the Everettian view the role of the Bohmian particle is to act as a "pointer", tagging, or selecting, just one branch of the

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In the history of de Broglie–Bohm theory, the proponents have often had to deal with claims that this theory is impossible. Such arguments are generally based on inappropriate analysis of operators as observables. If one believes that spin measurements are indeed measuring the spin of a particle that

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In the standard quantum formalism, measuring observables is generally thought of as measuring operators on the Hilbert space. For example, measuring position is considered to be a measurement of the position operator. This relationship between physical measurements and Hilbert space operators is, for

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In de Broglie–Bohm theory, the results of a spin experiment cannot be analyzed without some knowledge of the experimental setup. It is possible to modify the setup so that the trajectory of the particle is unaffected, but that the particle with one setup registers as spin-up, while in the other setup

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has extended de Broglie–Bohm theory to include signal nonlocality that would allow entanglement to be used as a stand-alone communication channel without a secondary classical "key" signal to "unlock" the message encoded in the entanglement. This violates orthodox quantum theory but has the virtue of

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The relation between nonlocality and preferred foliation can be better understood as follows. In de Broglie–Bohm theory, nonlocality manifests as the fact that the velocity and acceleration of one particle depends on the instantaneous positions of all other particles. On the other hand, in the theory

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In the formulation of the de Broglie–Bohm theory, there is only a wavefunction for the entire universe (which always evolves by the Schrödinger equation). Here, the "universe" is simply the system limited by the same boundary conditions used to solve the Schrödinger equation. However, once the theory

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The wavefunction itself, and not the particles, determines the dynamical evolution of the system: the particles do not act back onto the wave function. As Bohm and Hiley worded it, "the Schrödinger equation for the quantum field does not have sources, nor does it have any other way by which the field

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If this experiment is modified so that one slit is closed, no interference pattern is observed. Thus, the state of both slits affects the final results. It can also be arranged to have a minimally invasive detector at one of the slits to detect which slit the particle went through. When that is done,

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But why then had Born not told me of this "pilot wave"? If only to point out what was wrong with it? Why did von Neumann not consider it? More extraordinarily, why did people go on producing "impossibility" proofs, after 1952, and as recently as 1978?... Why is the pilot wave picture ignored in text

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The theory was historically developed in the 1920s by de Broglie, who, in 1927, was persuaded to abandon it in favour of the then-mainstream Copenhagen interpretation. David Bohm, dissatisfied with the prevailing orthodoxy, rediscovered de Broglie's pilot-wave theory in 1952. Bohm's suggestions were

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in their support. At first sight, the empty branches do not appear problematic but on the contrary very helpful as they enable the theory to explain unique outcomes of measurements. Also, they seem to explain why there is an effective "collapse of the wavefunction", as in ordinary quantum mechanics.

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The de Broglie–Bohm theory describes the physics in the Bell test experiments as follows: to understand the evolution of the particles, we need to set up a wave equation for both particles; the orientation of the apparatus affects the wavefunction. The particles in the experiment follow the guidance

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In de Broglie–Bohm theory, there is always a matter of fact about the position and momentum of a particle. Each particle has a well-defined trajectory, as well as a wavefunction. Observers have limited knowledge as to what this trajectory is (and thus of the position and momentum). It is the lack of

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of space-time. While this is in conflict with the standard interpretation of relativity, the preferred foliation, if unobservable, does not lead to any empirical conflicts with relativity. In 2013, Dürr et al. suggested that the required foliation could be covariantly determined by the wavefunction.

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of particles as in classical mechanics but defined by non-Newtonian mechanics. At every moment of time there exists not only a wavefunction, but also a well-defined configuration of the whole universe (i.e., the system as defined by the boundary conditions used in solving the Schrödinger equation).

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To explain the behavior when the particle is detected to go through one slit, one needs to appreciate the role of the conditional wavefunction and how it results in the collapse of the wavefunction; this is explained below. The basic idea is that the environment registering the detection effectively

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Even though this latter relation is frequently presented as an axiom of the theory, Bohm presented it as derivable from statistical-mechanical arguments in the original papers of 1952. This argument was further supported by the work of Bohm in 1953 and was substantiated by Vigier and Bohm's paper of

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I just received your long letter of 20th November, and I also have studied more thoroughly the details of your paper. I do not see any longer the possibility of any logical contradiction as long as your results agree completely with those of the usual wave mechanics and as long as no means is given

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After publishing a popular textbook on Quantum Mechanics that adhered entirely to the Copenhagen orthodoxy, Bohm was persuaded by Einstein to take a critical look at von Neumann's theorem. The result was 'A Suggested Interpretation of the Quantum Theory in Terms of "Hidden Variables" I and II' . It

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Generalized particle trajectories can be extrapolated from numerous weak measurements on an ensemble of equally prepared systems, and such trajectories coincide with the de Broglie–Bohm trajectories. In particular, an experiment with two entangled photons, in which a set of Bohmian trajectories for

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Unlike de Broglie–Bohm theory, Valentini's theory the wavefunction evolution also depends on the ontological variables. This introduces an instability, a feedback loop that pushes the hidden variables out of "sub-quantal heat death". The resulting theory becomes nonlinear and non-unitary. Valentini

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Experiments on hydrodynamical analogs of quantum mechanics beginning with the work of Couder and Fort (2006) have purported to show that macroscopic classical pilot-waves can exhibit characteristics previously thought to be restricted to the quantum realm. Hydrodynamic pilot-wave analogs have been

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had actually submitted a preprint with a similar proposal but, not convinced, had withdrawn it before publication. According to Holland, failure to appreciate key points of the de Broglie–Bohm theory has led to confusion, the key point being "that the trajectories of a many-body quantum system are

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A fifth derivation, given by Dürr et al. is appropriate for generalization to quantum field theory and the Dirac equation. The idea is that a velocity field can also be understood as a first-order differential operator acting on functions. Thus, if we know how it acts on functions, we know what it

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A fourth derivation was given by Dürr et al. In their derivation, they derive the velocity field by demanding the appropriate transformation properties given by the various symmetries that Schrödinger's equation satisfies, once the wavefunction is suitably transformed. The guiding equation is what

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Many authors have expressed critical views of de Broglie–Bohm theory by comparing it to Everett's many-worlds approach. Many (but not all) proponents of de Broglie–Bohm theory (such as Bohm and Bell) interpret the universal wavefunction as physically real. According to some supporters of Everett's

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in the early 2000s attempted to use the Bohm "particles" as an adaptive mesh that follows the actual trajectory of a quantum state in time and space. In the "quantum trajectory" method, one samples the quantum wavefunction with a mesh of quadrature points. One then evolves the quadrature points in

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of a particle directly; instead, the component in one direction is measured; the outcome from a single particle may be 1, meaning that the particle is aligned with the measuring apparatus, or −1, meaning that it is aligned the opposite way. An ensemble of particles prepared by a polarizer to be in

1375:{\displaystyle m_{k}{\frac {dq^{k}}{dt}}(t)=\hbar \nabla _{k}\operatorname {Im} \ln \psi (q,t)=\hbar \operatorname {Im} \left({\frac {\nabla _{k}\psi }{\psi }}\right)(q,t)={\frac {m_{k}\mathbf {j} _{k}}{\psi ^{*}\psi }}=\operatorname {Re} \left({\frac {\mathbf {\hat {P}} _{k}\Psi }{\Psi }}\right),} 7016:

The second issue with Bohmian mechanics may, at first sight, appear rather harmless, but which on a closer look develops considerable destructive power: the issue of empty branches. These are the components of the post-measurement state that do not guide any particles because they do not have the

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It requires a special setup for the conditional wavefunction of a system to obey a quantum evolution. When a system interacts with its environment, such as through a measurement, the conditional wavefunction of the system evolves in a different way. The evolution of the universal wavefunction can

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De Broglie–Bohm theory is a theory that applies primarily to the whole universe. That is, there is a single wavefunction governing the motion of all of the particles in the universe according to the guiding equation. Theoretically, the motion of one particle depends on the positions of all of the

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Hrvoje Nikolić introduces a purely deterministic de Broglie–Bohm theory of particle creation and destruction, according to which particle trajectories are continuous, but particle detectors behave as if particles have been created or destroyed even when a true creation or destruction of particles

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is typical. There are anomalous initial conditions that would give rise to violations of the second law; however in the absence of some very detailed evidence supporting the realization of one of those conditions, it would be quite unreasonable to expect anything but the actually observed uniform

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In de Broglie–Bohm theory, the wavefunction is defined at both slits, but each particle has a well-defined trajectory that passes through exactly one of the slits. The final position of the particle on the detector screen and the slit through which the particle passes is determined by the initial

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The Bohmian, of course, cannot accept this argument. For her, it is decidedly the particle configuration in three-dimensional space and not the wave function on the abstract configuration space that constitutes a world (or rather, the world). Instead, she will accuse the Everettian of not having

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Below are some highlights of the results that arise out of an analysis of de Broglie–Bohm theory. Experimental results agree with all of quantum mechanics' standard predictions insofar as it has them. But while standard quantum mechanics is limited to discussing the results of "measurements", de

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but the dynamics are different. In classical mechanics, the accelerations of the particles are imparted directly by forces, which exist in physical three-dimensional space. In de Broglie–Bohm theory, the quantum "field exerts a new kind of "quantum-mechanical" force". Bohm hypothesized that each

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that test Bell's inequality using an EPR-type setup. Aspect's results show experimentally that Bell's inequality is in fact violated, meaning that the relevant quantum-mechanical predictions are correct. In these Bell test experiments, entangled pairs of particles are created; the particles are

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Operators as observables leads many to believe that many operators are equivalent. De Broglie–Bohm theory, from this perspective, chooses the position observable as a favored observable rather than, say, the momentum observable. Again, the link to the position observable is a consequence of the

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state 1 will all measure polarized in state 1 in a subsequent apparatus. A polarized ensemble sent through a polarizer set at angle to the first pass will result in some values of 1 and some of −1 with a probability that depends on the relative alignment. For a full explanation of this, see the

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De Broglie–Bohm theory gives the almost results as (non-relativisitic) quantum mechanics. It treats the wavefunction as a fundamental object in the theory, as the wavefunction describes how the particles move. This means that no experiment can distinguish between the two theories. This section

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in his transactional interpretation except the beables that exist between the von Neumann strong projection operator measurements. Sutherland's Lagrangian includes two-way action-reaction between pilot wave and beables. Therefore, it is a post-quantum non-statistical theory with final boundary

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Bohm showed explicitly how parameters could indeed be introduced, into nonrelativistic wave mechanics, with the help of which the indeterministic description could be transformed into a deterministic one. More importantly, in my opinion, the subjectivity of the orthodox version, the necessary

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is no longer a probability density in space, but a probability density in space-time. He uses this generalized probabilistic interpretation to formulate a relativistic-covariant version of de Broglie–Bohm theory without introducing a preferred foliation of space-time. His work also covers the

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In contrast to the usual interpretation, this alternative interpretation permits us to conceive of each individual system as being in a precisely definable state, whose changes with time are determined by definite laws, analogous to (but not identical with) the classical equations of motion.

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The de Broglie–Bohm theory makes the same (empirically correct) predictions for the Bell test experiments as ordinary quantum mechanics. It is able to do this because it is manifestly nonlocal. It is often criticized or rejected based on this; Bell's attitude was: "It is a merit of the de

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has advanced a statistical variant of this approach that uses Bayesian sampling technique to sample the quantum density and compute the quantum potential on a structureless mesh of points. This technique was recently used to estimate quantum effects in the heat capacity of small clusters

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This results in an infinite force on the sample particles forcing them to move away from the node and often crossing the path of other sample points (which violates single-valuedness). Various schemes have been developed to overcome this; however, no general solution has yet emerged.

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emphasized the importance of deriving a well-defined description of light in terms of particle trajectories in the framework of either the Bohmian mechanics or the Nelson stochastic mechanics. The same year, Ghose worked out Bohmian photon trajectories for specific cases. Subsequent

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The fact that the conditional wavefunction of a subsystem does not always evolve by the Schrödinger equation is related to the fact that the usual collapse rule of standard quantum theory emerges from the Bohmian formalism when one considers conditional wavefunctions of subsystems.

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of de Broglie–Bohm theory (that an observer becomes aware of configurations of particles of ordinary objects by means of correlations between such configurations and the configuration of the particles in the observer's brain) is unreasonable. This conclusion has been challenged by

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over configuration space, and the potential in Schrödinger's equation becomes a local self-adjoint operator acting on that space. The field equations for the de Broglie–Bohm theory in the relativistic case with spin can also be given for curved space-times with torsion.

6300:" distributions may in principle be observed and exploited, for which the statistical predictions of quantum theory are violated. It is controversially argued that quantum theory is merely a special case of a much wider nonlinear physics, a physics in which non-local ( 5619: 6879:

Bohm's formulation of de Broglie–Bohm theory in a classical-looking version has the merits that the emergence of classical behavior seems to follow immediately for any situation in which the quantum potential is negligible, as noted by Bohm in 1952. Modern methods of

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local beables (in Bell's sense) in her theory, that is, the ontological variables that refer to localized entities in three-dimensional space or four-dimensional spacetime. The many worlds of her theory thus merely appear as a grotesque consequence of this omission.

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All of non-relativistic quantum mechanics can be fully accounted for in this theory. Recent studies have used this formalism to compute the evolution of many-body quantum systems, with a considerable increase in speed as compared to other quantum-based methods.

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has been devised to test characteristics of the trajectory predictions. Experimental realization of this concept disagreed with the Bohm predictions. where they differed from standard quantum mechanics. These conclusions have been the subject of debate.

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There remain difficulties using the Bohmian approach, mostly associated with the formation of singularities in the quantum potential due to nodes in the quantum wavefunction. In general, nodes forming due to interference effects lead to the case where

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way for systems that seem to follow their own Schrödinger's equation. As this is an effective description of the system, it is a matter of choice as to what to define the experimental system to include, and this will affect when "collapse" occurs.

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interpretation of Bohmian mechanics, in which the cosmos is an Aristotelian substance composed of material particles and a substantial form. The wave function is assigned a dispositional role in choreographing the trajectories of the particles.

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In Bohm's original papers, he discusses how de Broglie–Bohm theory results in the usual measurement results of quantum mechanics. The main idea is that this is true if the positions of the particles satisfy the statistical distribution given by

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books? Should it not be taught, not as the only way, but as an antidote to the prevailing complacency? To show us that vagueness, subjectivity, and indeterminism, are not forced on us by experimental facts, but by deliberate theoretical choice?

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Quantum-mechanical probabilities are regarded (like their counterparts in classical statistical mechanics) as only a practical necessity and not as an inherent lack of complete determination in the properties of matter at the quantum level.

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to measure the values of your hidden parameters both in the measuring apparatus and in the observe system. As far as the whole matter stands now, your 'extra wave-mechanical predictions' are still a check, which cannot be cashed.

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and Bohm's causal interpretation of quantum mechanics was later extended by Bohm, Vigier, Hiley, Valentini and others to include stochastic properties. Bohm and other physicists, including Valentini, view the Born rule linking

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It is usually overlooked that Bohm's theory contains the same "many worlds" of dynamically separate branches as the Everett interpretation (now regarded as "empty" wave components), since it is based on precisely the same ...

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Nikolić has proposed a Lorentz-covariant formulation of the Bohmian interpretation of many-particle wavefunctions. He has developed a generalized relativistic-invariant probabilistic interpretation of quantum theory, in which

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were the first to perform numeric computations on the basis of the quantum potential to deduce ensembles of particle trajectories. Their work renewed the interests of physicists in the Bohm interpretation of quantum physics.

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Kocsis, Sacha; Braverman, Boris; Ravets, Sylvain; Stevens, Martin J.; Mirin, Richard P.; Shalm, L. Krister; Steinberg, Aephraim M. (2011). "Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer".

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said of Bohm's theory: "There was a time when you couldn't even talk about it because it was heretical. It probably still is the kiss of death for a physics career to be actually working on Bohm, but maybe that's changing."

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of the universal wavefunction never occurs in de Broglie–Bohm theory. Its entire evolution is governed by Schrödinger's equation, and the particles' evolutions are governed by the guiding equation. Collapse only occurs in a

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time according to the Bohm equations of motion. At each time step, one then re-synthesizes the wavefunction from the points, recomputes the quantum forces, and continues the calculation. (QuickTime movies of this for H + H

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claimed to duplicate the double slit experiment, tunneling, quantized orbits, and numerous other quantum phenomena which have led to a resurgence in interest in pilot wave theories. The analogs have been compared to the

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Grössing, G.; Fussy, S.; Mesa Pascasio, J.; Schwabl, H. (2012). "An explanation of interference effects in the double slit experiment: Classical trajectories plus ballistic diffusion caused by zero-point fluctuations".

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at UT Austin.) This approach has been adapted, extended, and used by a number of researchers in the chemical physics community as a way to compute semi-classical and quasi-classical molecular dynamics. A 2007 issue of

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Unlike the universal wavefunction, the conditional wavefunction of a subsystem does not always evolve by the Schrödinger equation, but in many situations it does. For instance, if the universal wavefunction factors as

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Abstract: "The quantum theory of de Broglie and Bohm solves the measurement problem, but the hypothetical corpuscles play no role in the argument. The solution finds a more natural home in the Everett interpretation."

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Preserving the density under the time evolution is another method of derivation. This is the method that Bell cites. It is this method that generalizes to many possible alternative theories. The starting point is the

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distributed according to the Born rule (that is, a distribution "out of quantum equilibrium") and evolving under the de Broglie–Bohm dynamics is overwhelmingly likely to evolve dynamically into a state distributed as

8203: 13563:(1994) discusses "the hegemony of the Copenhagen interpretation of quantum mechanics" over theories like Bohmian mechanics as an example of how the acceptance of scientific theories may be guided by social aspects. 3049:

For a given experiment, one can postulate this as being true and verify it experimentally. But, as argued by Dürr et al., one needs to argue that this distribution for subsystems is typical. The authors argue that

2206: 1754: 6825:, the authors describe a thought experiment that one could perform on a pair of particles that have interacted, the results of which they interpreted as indicating that quantum mechanics is an incomplete theory. 6680:

The Heisenberg's uncertainty principle states that when two complementary measurements are made, there is a limit to the product of their accuracy. As an example, if one measures the position with an accuracy of

5835:, particles may be created and annihilated. The distribution of creation events is dictated by the wavefunction. The wavefunction itself is evolving at all times over the full multi-particle configuration space. 5533: 5794: 8274: 7852: 13652:

Solvay Conference, 1928, Electrons et Photons: Rapports et Descussions du Cinquieme Conseil de Physique tenu a Bruxelles du 24 au 29 October 1927 sous les auspices de l'Institut International Physique Solvay

12806:

Golshani, M., and O. Akhavan. "Bohmian prediction about a two double-slit experiment and its disagreement with standard quantum mechanics." Journal of Physics A: Mathematical and General 34.25 (2001): 5259.

9817: 2565:

particles in the universe. As explained below, in most experimental situations, the influence of all of those particles can be encapsulated into an effective wavefunction for a subsystem of the universe.

9249:

that the probability follows the Born rule. The term "Bohmian mechanics" is also often used to include most of the further extensions past the spin-less version of Bohm. While de Broglie–Bohm theory has

945:

Measurements are a particular case of quantum processes described by the theory—for which it yields the same quantum predictions as other interpretations of quantum mechanics. The theory does not have a

2830: 2535:{\displaystyle {\frac {d\mathbf {Q} _{k}}{dt}}(t)={\frac {\hbar }{m_{k}}}\operatorname {Im} \left({\frac {\nabla _{k}\psi }{\psi }}\right)(\mathbf {Q} _{1},\mathbf {Q} _{2},\ldots ,\mathbf {Q} _{N},t).} 954:

in de Broglie–Bohm theory is not a postulate. Rather, in this theory, the link between the probability density and the wave function has the status of a theorem, a result of a separate postulate, the "

15826: 11265: 9748: 4501:

Chris Dewdney and G. Horton have proposed a relativistically covariant, wave-functional formulation of Bohm's quantum field theory and have extended it to a form that allows the inclusion of gravity.

7191:

De Broglie–Bohm theory has been derived many times and in many ways. Below are six derivations, all of which are very different and lead to different ways of understanding and extending this theory.

7035:

determines the observed result of a given experiment is called the "result assumption"); the other branches are designated "empty" and implicitly assumed by Bohm to be devoid of conscious observers.

6855:

separated, traveling to remote measuring apparatus. The orientation of the measuring apparatus can be changed while the particles are in flight, demonstrating the apparent nonlocality of the effect.

4477:

Initially, it had been considered impossible to set out a description of photon trajectories in the de Broglie–Bohm theory in view of the difficulties of describing bosons relativistically. In 1996,

4323: 5360: 4016:(this is what standard quantum theory would regard as the wavefunction of subsystem (I)). If, in addition, the Hamiltonian does not contain an interaction term between subsystems (I) and (II), then 3575: 7616:. This equation describes a probability flow along a current. We take the velocity field associated with this current as the velocity field whose integral curves yield the motion of the particle. 9609: 4632: 3258: 2112:. While the particle positions themselves are in real space, the velocity field and wavefunction are on configuration space, which is how particles are entangled with each other in this theory. 7394: 13346: 7266: 6983: 3864: 13417: 8985: 8478: 6353:. This is a statement of observer ignorance: the initial positions are represented by a statistical distribution so deterministic trajectories will result in a statistical distribution. 7427: 6763: 5911: 4465:

Thus, Dürr et al. (1999) showed that it is possible to formally restore Lorentz invariance for the Bohm–Dirac theory by introducing additional structure. This approach still requires a

3848: 1938: 9541: 9489: 9159:

complicates this hope, as it demonstrates that there can be no local hidden-variable theory that is compatible with the predictions of quantum mechanics. The Bohmian interpretation is

8774: 4560:'s retrocasual weak measurements to explain many-particle entanglement in a special relativistic way without the need for configuration space. The basic idea was already published by 9643: 8894: 8309: 6581:

De Broglie–Bohm theory is often referred to as a "hidden-variable" theory. Bohm used this description in his original papers on the subject, writing: "From the point of view of the

8394: 7775: 1831:, and this consequently holds for all times. Such a state is named quantum equilibrium. With quantum equilibrium, this theory agrees with the results of standard quantum mechanics. 7432: 6178:

in a lecture about the De Broglie–Bohm theory. Valentini argues quantum theory is a special equilibrium case of a wider physics and that it may be possible to observe and exploit

5093: 2658: 12480: 5530:— the wavefunction defined on the multidimensional configuration space; e.g. a system consisting of two spin-1/2 particles and one spin-1 particle has a wavefunction of the form 1435: 14372: 12816:

Brida, G.; Cagliero, E.; Falzetta, G.; Genovese, M.; Gramegna, M.; Novero, C. (2002). "A first experimental test of de Broglie-Bohm theory against standard quantum mechanics".

8811: 7614: 2822: 2110: 9568: 5710: 5393: 5323: 5148: 4612: 4424: 4397: 4041: 4014: 3806: 3658: 3384: 2650: 2601: 2343: 2198: 2078: 2045: 1809: 1068: 7229: 5677: 3770: 3730: 3611: 3348: 3091:

of the positions of the particles. The authors then prove that the vast majority of possible initial configurations will give rise to statistics obeying the Born rule (i.e.,

6086: 6055: 10072: 10041: 9930: 7092:

is an attempt to demonstrate that the wavefunction alone is sufficient to account for all our observations. When we see the particle detectors flash or hear the click of a

6783:. Thus, with succeeding observations, the initial conditions become more and more restricted. This formalism is consistent with the normal use of the Schrödinger equation. 3312: 10010: 9958: 9899: 6122: 5290: 4370: 3285: 2141: 1402: 9437: 6567: 6497: 6351: 6243: 4540: 3194: 3126: 3085: 3044: 3007: 9847: 9691: 9266:

as primary and has the guiding equation as its icon. They are mathematically equivalent in so far as the Hamilton-Jacobi formulation applies, i.e., spin-less particles.

8473: 1474: 6844:). In particular, Bell proved that any local theory with unique results must make empirical predictions satisfying a statistical constraint called "Bell's inequality". 5266: 1996: 9982: 9871: 7619:

A method applicable for particles without spin is to do a polar decomposition of the wavefunction and transform Schrödinger's equation into two coupled equations: the

6725: 6702: 5639: 1117: 17: 8924: 6253:. It can be shown that, once an equilibrium has been reached, the system remains in such equilibrium over the course of its further evolution: this follows from the 9109:

Coulomb) but because all are acted upon by an entity – mathematically described by the wavefunction or functions of it – that lies beyond them". This entity is the

9054:

at the 1927 Solvay Conference, after close collaboration with Schrödinger, who developed his wave equation for de Broglie's theory. At the end of the presentation,

8851: 8681: 8641: 7138: 7112:). No particle (in the Bohm sense of having a defined position and velocity) exists according to that theory. For this reason Everett sometimes referred to his own 6275: 6022: 5902: 5528: 5201: 4253: 4061: 3631: 2770: 2161: 1019: 9232:

The trajectories of the Bohm model that would result for particular experimental arrangements were termed "surreal" by some. Still in 2016, mathematical physicist

9229:

began to defend the theory. In "Speakable and Unspeakable in Quantum Mechanics" , several of the papers refer to hidden-variables theories (which include Bohm's).

4447:. Various extensions of "Bohm-like" mechanics exist that attempt to resolve this problem. Bohm himself in 1953 presented an extension of the theory satisfying the 3687: 6867:

provides an analysis of exactly what kind of nonlocality is present and how it is compatible with relativity. Bell has shown that the nonlocality does not allow

8831: 8444: 8424: 8349: 8329: 6207: 5502: 5225: 5172: 5117: 4343: 2790: 2621: 2563: 2363: 1961: 1829: 1088: 1039: 992: 14537:

Andersen, Anders; Madsen, Jacob; Reichelt, Christian; Rosenlund Ahl, Sonja; Lautrup, Benny; Ellegaard, Clive; Levinsen, Mogens T.; Bohr, Tomas (6 July 2015).

7012:'s many-worlds. In particular, the unreal many-worlds interpretation of Hawking and Weinberg is similar to the Bohmian concept of unreal empty branch worlds: 3396: 942:: the velocity of any one particle depends on the value of the guiding equation, which depends on the configuration of all the particles under consideration. 11262: 9003:

affiliations. The de Broglie–Bohm theory was widely deemed unacceptable by mainstream theorists, mostly because of its explicit non-locality. On the theory,

3205:

is formulated, it is convenient to introduce a notion of wavefunction also for subsystems of the universe. Let us write the wavefunction of the universe as

7635: 7276: 4225:{\displaystyle \psi (t,q^{\text{I}},q^{\text{II}})=\psi ^{\text{I}}(t,q^{\text{I}})\psi ^{\text{II}}(t,q^{\text{II}})+\phi (t,q^{\text{I}},q^{\text{II}}),} 7980:{\displaystyle -{\frac {\partial \rho (\mathbf {x} ,t)}{\partial t}}=\nabla \cdot \left(\rho (\mathbf {x} ,t){\frac {\nabla S(\mathbf {x} ,t)}{m}}\right)} 5715: 5399: 7508: 15112: 14011:

Pauli, W. (1953). "Remarques sur le probleme des parametres caches dans la mecanique quantique et sur la theorie de l'onde pilote". In A. George (Ed.),

6989:

These methods, as does Bohm's Hamilton–Jacobi formulation, do not apply to situations in which the full dynamics of spin need to be taken into account.

15869: 15727: 7140:

is a real field, then the associated particle is superfluous, since, as we have endeavored to illustrate, the pure wave theory is itself satisfactory.

1653:{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi (q,t)=-\sum _{i=1}^{N}{\frac {\hbar ^{2}}{2m_{i}}}\nabla _{i}^{2}\psi (q,t)+V(q)\psi (q,t).} 14132: 13384: 12869:

Struyve, W.; De Baere, W. (2001). "Comments on some recently proposed experiments that should distinguish Bohmian mechanics from quantum mechanics".

14742:

De Broglie, Louis (1956). "Une tentative d'interprétation causale et non linéaire de la mécanique ondulatoire: (la théorie de la double solution)".

14520: 11507:

Dewdney, Chris; Horton, George (2010). "A relativistic hidden-variable interpretation for the massive vector field based on energy-momentum flows".

10551: 2051:

particles moving in 3 dimensions. In the first instance, configuration space and real space are the same, while in the second, real space is still

915:, an actual configuration of particles exists, even when unobserved. The evolution over time of the configuration of all particles is defined by a 49: 14826:

Haisch, Bernard; Rueda, Alfonso (2000). "On the relation between a zero-point-field-induced inertial effect and the Einstein-de Broglie formula".

10281:

Kocsis, Sacha; Braverman, Boris; Ravets, Sylvain; Stevens, Martin J.; Mirin, Richard P.; Shalm, L. Krister; Steinberg, Aephraim M. (3 June 2011).

4498:

experiments yielded trajectories that coincide with the predicted trajectories. The significance of these experimental findings is controversial.

16437: 15823:, August 2010 international conference on de Broglie-Bohm theory. Site contains slides for all the talks – the latest cutting-edge deBB research. 13421: 2306:{\displaystyle {\frac {d\mathbf {Q} }{dt}}(t)={\frac {\hbar }{m}}\operatorname {Im} \left({\frac {\nabla \psi }{\psi }}\right)(\mathbf {Q} ,t).} 14585: 13443:

Valentini, Antony (2010). "De Broglie-Bohm Pilot-Wave Theory: Many Worlds in Denial?". In Saunders, Simon; Barrett, Jon; Kent, Adrian (eds.).

10085:

et al. verified the ESSW predictions. However they propose the surealistic effect is a consequence the nonlocality inherent in Bohm's theory.

7995: 16664: 16070: 8360:

This leads to viewing the quantum theory as particles moving under the classical force modified by a quantum force. However, unlike standard

4462:

and references therein. Another approach is given by Dürr et al., who use Bohm–Dirac models and a Lorentz-invariant foliation of space-time.

3009:. And that distribution is guaranteed to be true for all time by the guiding equation if the initial distribution of the particles satisfies 385: 9071:) believed to prove that all hidden-variable theories are impossible. This sealed the fate of de Broglie's theory for the next two decades. 9039:

De Broglie–Bohm theory has a history of different formulations and names. In this section, each stage is given a name and a main reference.

1663: 14892:

Mahler, Dylan H.; Rozema, Lee; Fisher, Kent; Vermeyden, Lydia; Resch, Kevin J.; Wiseman, Howard M.; Steinberg, Aephraim (5 February 2016).

13495: 13690: 12588:

Dürr, Detlef; Goldstein, Sheldon; Zanghì, Nino (2003). "Quantum Equilibrium and the Role of Operators as Observables in Quantum Theory".

11792:

Sutherland, Roderick (2015). "Lagrangian Description for Particle Interpretations of Quantum Mechanics -- Entangled Many-Particle Case".

8212: 6641: 6422: 886: 12794: 9178:, did not consider Bohm's interpretation to be a satisfactory answer to the quantum nonlocality question, calling it "too cheap", while 3386:

the actual configuration of subsystem (I) and of the rest of the universe. For simplicity, we consider here only the spinless case. The

2603:. The equation represents a quantized version of the total energy of a classical system evolving under a real-valued potential function 7780: 7144:

In the Everettian view, then, the Bohm particles are superfluous entities, similar to, and equally as unnecessary as, for example, the

16608: 14421: 9348:

A comparison by Bush (2015) among the walking droplet system, de Broglie's double-solution pilot-wave theory and its extension to SED

9024:

Since the 1990s, there has been renewed interest in formulating extensions to de Broglie–Bohm theory, attempting to reconcile it with

8657:. It is based on three basic postulates and an additional fourth postulate that links the wavefunction to measurement probabilities: 6863:

of the wavefunction. It is the wavefunction that carries the faster-than-light effect of changing the orientation of the apparatus.

15806: 7008:

Kim Joris Boström has proposed a non-relativistic quantum mechanical theory that combines elements of de Broglie-Bohm mechanics and

2947:{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi =-\sum _{k=1}^{N}{\frac {\hbar ^{2}}{2m_{k}}}\nabla _{k}^{2}\psi +V\psi .} 1041:. The configuration space is different for different versions of pilot-wave theory. For example, this may be the space of positions 15919: 15539: 14192:

Hiley, B. J.; E Callaghan, R.; Maroney, O. (2000). "Quantum trajectories, real, surreal or an approximation to a deeper process?".

11454:

Dewdney, Chris; Horton, George (2004). "A relativistically covariant version of Bohm's quantum field theory for the scalar field".

9754: 3733: 137: 15783:, the homepage of the international research network on Bohmian Mechanics that was started by D. Dürr, S. Goldstein and N. Zanghì. 11401:

Dewdney, Chris; Horton, George (2002). "Relativistically invariant extension of the de Broglie Bohm theory of quantum mechanics".

9245:

Bohmian mechanics is the same theory, but with an emphasis on the notion of current flow, which is determined on the basis of the

16620: 4547: 593: 16304: 15862: 14996:

Barbosa, G. D.; N. Pinto-Neto (2004). "A Bohmian Interpretation for Noncommutative Scalar Field Theory and Quantum Mechanics".

10749: 9697: 9211: 3135:

initial condition will, with overwhelmingly high probability, evolve into a higher-entropy state: behavior consistent with the

3128:) for measurement outcomes. In summary, in a universe governed by the de Broglie–Bohm dynamics, Born rule behavior is typical. 366: 7172:, who argues that the entirety of such objections arises from a failure to interpret de Broglie–Bohm theory on its own terms. 6997: 5614:{\displaystyle \psi :\mathbb {R} ^{9}\times \mathbb {R} \to \mathbb {C} ^{2}\otimes \mathbb {C} ^{2}\otimes \mathbb {C} ^{3},} 4258: 3613:

satisfies a guiding equation identical to the one presented in the formulation of the theory, with the universal wavefunction

16238: 15693: 15679: 15525: 15335: 15163: 14810: 14109: 13478: 13358: 13037: 12518: 11864: 10534: 10507: 10441: 7152:. This argument is sometimes called the "redundancy argument", since the superfluous particles are redundant in the sense of 5329: 4482: 3501: 1856:

The Bohmian trajectories for an electron going through the two-slit experiment. A similar pattern was also extrapolated from

549: 16161: 6133: 904: 472: 6313:

Broglie–Bohm theory governs the dynamics of a system without the intervention of outside observers (p. 117 in Bell).

3087:, by virtue of its equivariance under the dynamical evolution of the system, is the appropriate measure of typicality for 15259: 11990:

Duerr, Detlef; Goldstein, Sheldon; Tumulka, Roderich; Zanghi, Nino (2004). "Bohmian Mechanics and Quantum Field Theory".

9575: 6907: 5828: 3208: 14942: 14492: 12114:

Dürr, D.; Goldstein, S.; Taylor, J.; Tumulka, R.; Zanghì, N. (2007). "Quantum Mechanics in Multiply-Connected Spaces".

10109: 6129: 3979:{\displaystyle \psi (t,q^{\text{I}},q^{\text{II}})=\psi ^{\text{I}}(t,q^{\text{I}})\psi ^{\text{II}}(t,q^{\text{II}}),} 15322:(Demonstrates incompleteness of the Bohm interpretation in the face of fractal, differentiable-nowhere wavefunctions.) 13190:

Valentini, Antony; Westman, Hans (2012). "Combining Bohm and Everett: Axiomatics for a Standalone Quantum Mechanics".

7357: 15892: 15855: 15710: 15665: 15644: 14691: 13543: 13400: 13303: 12712: 10766:

Dürr, Detlef; Goldstein, Sheldon; Zanghí, Nino (2003). "Quantum Equilibrium and the Origin of Absolute Uncertainty".

7238: 6993: 6667: 6448: 132: 10552:"Hamiltonian theory of wave and particle in quantum mechanics II: Hamilton-Jacobi theory and particle back-reaction" 10469:

H. R. Brown, C. Dewdney and G. Horton: "Bohm particles and their detection in the light of neutron interferometry",

8604:{\displaystyle (v(f))(q)=\operatorname {Re} {\frac {\left(\psi ,{\frac {i}{\hbar }}\psi \right)}{(\psi ,\psi )}}(q)} 6936: 6649: 6430: 4451:

for a single particle. However, this was not extensible to the many-particle case because it used an absolute time.

16002: 9087: 9082:, which is incorrectly considered as a basis for the density current derivation of the de Broglie–Bohm theory. The 6787: 221: 8933: 6296:

and form a "quantum equilibrium" that is analogous to thermal equilibrium in classical dynamics, such that other "

16575: 16284: 16279: 15987: 15751: 13920:

Holland, Peter (2005). "What's wrong with Einstein's 1927 hidden-variable interpretation of quantum mechanics?".

13867:

Holland, Peter (2005). "What's wrong with Einstein's 1927 hidden-variable interpretation of quantum mechanics?".

6137: 4618:

in spin space to reduce the complex vectors to complex numbers. The Schrödinger equation is modified by adding a

879: 328: 308: 176: 7399: 6730: 3811: 1900: 16587: 16259: 15794: 14598:

Oil droplets guided by "pilot waves" have failed to reproduce the results of the quantum double-slit experiment

12061:

Duerr, Detlef; Goldstein, Sheldon; Tumulka, Roderich; Zanghi, Nino (2005). "Bell-Type Quantum Field Theories".

9495: 9443: 9246: 8689: 6645: 6426: 6277:. It is less straightforward to demonstrate whether and how such an equilibrium is reached in the first place. 955: 568: 298: 10268: 16565: 16342: 16264: 16136: 16032: 15902: 15135: 10701:

Towler, M. D.; Russell, N. J.; Valentini, A. (2012). "Timescales for dynamical relaxation to the Born rule".

7486:{\displaystyle \mathbf {v} ={\frac {\hbar }{m}}\operatorname {Im} \left({\frac {\nabla \psi }{\psi }}\right)} 5505: 4372:. Then, again, the conditional wavefunction of subsystem (I) is (up to an irrelevant scalar factor) equal to 2742:{\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +V\psi .} 1941: 608: 346: 246: 11775: 9615: 8859: 8279: 6779:, successive observations of the particles' positions refine the experimenter's knowledge of the particles' 5851:

in mathematical parlance), one simply notes that all of the elements of these equations make sense, such as

16669: 16299: 16233: 16228: 16199: 15897: 15773: 15724: 15077:

Bohm, David (1952). "A Suggested Interpretation of the Quantum Theory in Terms of "Hidden Variables", II".

10099: 9329: 9203: 8367: 7736: 7624: 7195: 6371: 6316:

The basis for agreement with standard quantum mechanics is that the particles are distributed according to

6254: 6210: 3690: 3136: 544: 539: 510: 361: 142: 15328:

The Quantum Theory of Motion: An Account of the de Broglie–Bohm Causal Interpretation of Quantum Mechanics

14129: 11162:

Cufaro Petroni, Nicola; Vigier, Jean-Pierre (2001). "Remarks on Observed Superluminal Light Propagation".

10648:

Dürr, D.; Goldstein, S.; Zanghì, N. (1992). "Quantum Equilibrium and the Origin of Absolute Uncertainty".

10526:

The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics

9079: 5045: 1477: 16367: 16274: 15618: 15043:

Bohm, David (1952). "A Suggested Interpretation of the Quantum Theory in Terms of "Hidden Variables" I".

14517: 13632: 10174:

Bohm, David (1952). "A Suggested Interpretation of the Quantum Theory in Terms of 'Hidden Variables' I".

6837: 6797: 4561: 4552:

Roderick I. Sutherland at the University in Sydney has a Lagrangian formalism for the pilot wave and its

1411: 958:", which is additional to the basic principles governing the wave function. There are several equivalent 578: 323: 313: 14657:

Rozenman, Georgi Gary; Bondar, Denys I; Schleich, Wolfgang P; Shemer, Lev; Arie, Ady A (10 March 2023).

14296: 14224:

A. Fine: "On the interpretation of Bohmian mechanics", in: J. T. Cushing, A. Fine, S. Goldstein (Eds.):

12276:

F. R. Benard Guedes, N. J. Popławski (2024). "General-relativistic wave-particle duality with torsion".

11203: 10819:

Passon, Oliver (2006). "What you always wanted to know about Bohmian mechanics but were afraid to ask".

10574: 8653:

A further derivation has been given by Peter R. Holland, on which he bases his quantum-physics textbook

16643: 16405: 16213: 15268:

Hall, Michael J. W. (2004). "Incompleteness of trajectory-based interpretations of quantum mechanics".

13392: 13224: 8779: 7576: 7160: 7120:

Our main criticism of this view is on the grounds of simplicity – if one desires to hold the view that

7113: 7089: 6868: 6841: 5804: 2795: 2083: 872: 524: 12313:

S. K. Wong (1972). "Heisenberg equations of motion for spin-1/2 wave equation in general relativity".

9553: 6170: 5686: 5369: 5299: 5124: 4588: 4481:

presented a relativistic quantum-mechanical description of spin-0 and spin-1 bosons starting from the

4402: 4375: 4019: 3992: 3775: 3636: 3353: 2626: 2577: 2319: 2174: 2054: 2021: 1762: 1044: 16427: 16294: 16218: 16179: 16116: 16095: 16052: 15945: 14051:

Philippidis, C.; Dewdney, C.; Hiley, B. J. (2007). "Quantum interference and the quantum potential".

11934:

Grössing, G.; Fussy, S.; Mesa Pascasio, J.; Schwabl, H. (2012). "The Quantum as an Emergent System".

9255: 9210:, focusing on Bohm's sympathy with communists as exemplified by his refusal to give testimony to the 9175: 9091: 9063:

at the time. An analysis of de Broglie's presentation is given in Bacciagaluppi et al. Also, in 1932

7205: 6791: 6582: 5650: 3739: 3700: 3580: 3317: 2574:

The one-particle Schrödinger equation governs the time evolution of a complex-valued wavefunction on

1869: 1660:

This completes the specification of the theory for any quantum theory with Hamilton operator of type

529: 490: 443: 418: 341: 201: 15377: 15300: 15104: 14858: 13776: 12651:

Hyman, Ross; Caldwell, Shane A; Dalton, Edward (2004). "Bohmian mechanics with discrete operators".

12620: 11582: 10943:

Nikolic, H. (2005). "Bohmian particle trajectories in relativistic fermionic quantum field theory".

8999:

not then widely received, partly due to reasons unrelated to their content, such as Bohm's youthful

8660:

A physical system consists in a spatiotemporally propagating wave and a point particle guided by it.

6884:

are relevant to an analysis of this limit. See Allori et al. for steps towards a rigorous analysis.

6064: 6031: 4399:, and if the Hamiltonian does not contain an interaction term between subsystems (I) and (II), then 16494: 16474: 16464: 16454: 16410: 15970: 15594: 15070: 13160: 13053:

Allori, V.; Dürr, D.; Goldstein, S.; Zanghì, N. (2002). "Seven Steps Towards the Classical World".

10912: 10246: 10104: 10047: 10016: 9905: 9144: 8648:

This formulation allows for stochastic theories such as the creation and annihilation of particles.

6630: 6411: 6285: 5997:{\displaystyle u^{i}={\frac {e_{\mu }^{i}{\bar {\psi }}\gamma ^{\mu }\psi }{{\bar {\psi }}\psi }},} 3989:

then the conditional wavefunction of subsystem (I) is (up to an irrelevant scalar factor) equal to

3290: 613: 15212: 13470: 13228: 12022: 10880:

Nikolic, H. (2004). "Bohmian particle trajectories in relativistic bosonic quantum field theory".

9988: 9936: 9877: 6095: 5273: 4348: 3263: 2545:

The main fact to notice is that this velocity field depends on the actual positions of all of the

2124: 1385: 16189: 16100: 15765: 14346: 14151:

Englert, Berthold-Georg; Scully, Marian O.; Süssmann, Georg; Walther, Herbert (1 December 1992).

12903: 12345: 9415: 9132: 6634: 6535: 6515: 6465: 6415: 6319: 6297: 6215: 6179: 6165: 4508: 3162: 3094: 3053: 3012: 2975: 1841: 782: 500: 408: 16603: 16151: 14539:"Double-slit experiment with single wave-driven particles and its relation to quantum mechanics" 13510: 13374: 12707:, edition published in the Taylor & Francis e-library 2009 (first edition Routledge, 1993), 10996:

Dürr, D.; Goldstein, S.; Münch-Berndl, K.; Zanghì, N. (1999). "Hypersurface Bohm–Dirac Models".

10482:

J. Anandan, "The Quantum Measurement Problem and the Possible Role of the Gravitational Field",

10384: 9823: 9667: 9148: 9147:

description that would resolve or eliminate many of the paradoxes of quantum mechanics, such as

8449: 8331:

is the classical potential that appears in Schrödinger's equation, and the other term involving

6250: 1444: 920: 458: 356: 122: 16514: 16289: 16269: 16194: 16047: 15589: 15372: 15295: 15207: 14853: 12615: 12533:

Albert, D. Z., 1992, Quantum Mechanics and Experience, Cambridge, MA: Harvard University Press.

12017: 11577: 10907: 10241: 9306: 7045: 7028: 6775:

To put the statement differently, the particles' positions are only known statistically. As in

6602: 5230: 1966: 1865: 787: 505: 333: 303: 266: 16090: 14113: 9967: 9856: 6707: 6684: 5624: 4581:, the wavefunction becomes complex-vector-valued. The value space is called spin space; for a 1093: 413: 16539: 16037: 16017: 14043:

Scale in Conscious Experience: Is the Brain Too Important to be Left to Specialists to Study?

13687: 13380: 12742:

Mahler, DH; Rozema, L; Fisher, K; Vermeyden, L; Resch, KJ; Wiseman, HM; Steinberg, A (2016).

11776:"Bohmian mechanics in relativistic quantum mechanics, quantum field theory and string theory" 10524: 10497: 9164: 9136: 6917: 6902: 6851: 6815: 6589:)". However, others nevertheless treat the term "hidden variable" as a suitable description. 6586: 6510: 6387:

outlines the ideas as to how the standard quantum formalism arises out of quantum mechanics.

4485:, setting out Bohmian trajectories for massive bosons and for massless bosons (and therefore 4043:

does satisfy a Schrödinger equation. More generally, assume that the universal wave function

939: 256: 241: 15705:, Cambridge University Press, 1993 (re-printed 2000, transferred to digital printing 2004), 13538:, Cambridge University Press, 1993 (re-printed 2000, transferred to digital printing 2004), 12716: 8899: 8209:

The Hamilton–Jacobi equation is the equation derived from a Newtonian system with potential

6864: 5484:

is the covariant derivative, involving the vector potential, ascribed to the coordinates of

1476:

is the standard complex-valued wavefunction from quantum theory, which evolves according to

16570: 16499: 16444: 16174: 15982: 15940: 15581: 15483: 15436: 15364: 15287: 15199: 15174: 15086: 15052: 15015: 14976: 14845: 14706: 14455: 14387: 14311: 14060: 14046: 13939: 13886: 13833: 13784: 13740: 13586: 13458: 13253: 13119: 13072: 12991: 12884: 12835: 12670: 12607: 12562: 12437: 12392: 12239: 12186: 12133: 12080: 12009: 11953: 11900: 11811: 11741: 11688: 11635: 11569: 11516: 11473: 11420: 11357: 11292: 11228: 11128: 11072: 11015: 10962: 10899: 10838: 10785: 10720: 10667: 10566: 10294: 10233: 10183: 9251: 9029: 8836: 8666: 8614: 7199: 7123: 6992:

The properties of trajectories in the de Broglie–Bohm theory differ significantly from the

6366: 6260: 6149: 6007: 5880: 5848: 5513: 5179: 4238: 4046: 3616: 2755: 2146: 997: 935: 573: 485: 211: 168: 15508:(Describes a Bohmian resolution to the dilemma posed by non-differentiable wavefunctions.) 7627:. This is the method used by Bohm in 1952. The decomposition and equations are as follows: 3663: 3287:

denotes the configuration variables associated to some subsystem (I) of the universe, and

817: 8: 16555: 16524: 16469: 16449: 16357: 16314: 16169: 16080: 16007: 15997: 15909: 12629: 12543:

Daumer, M.; Dürr, D.; Goldstein, S.; Zanghì, N. (1997). "Naive Realism About Operators".

11965: 11550:

Nikolić, Hrvoje (2005). "Relativistic Quantum Mechanics and the Bohmian Interpretation".

10921: 10114: 9263: 9152: 9060: 8361: 7620: 7503: 6803: 6776: 5860: 5827:

In Dürr et al., the authors describe an extension of de Broglie–Bohm theory for handling

5204: 3131:

The situation is thus analogous to the situation in classical statistical physics. A low-

2003: 1405: 947: 672: 480: 398: 226: 206: 158: 15585: 15543: 15487: 15440: 15368: 15291: 15203: 15090: 15056: 15019: 14980: 14849: 14718: 14710: 14467: 14459: 14391: 14315: 14064: 13943: 13890: 13837: 13788: 13744: 13590: 13462: 13257: 13123: 13076: 12995: 12962: 12904:"Reply to Comment on Experimental realization of a first test of de Broglie–Bohm theory" 12888: 12839: 12674: 12611: 12566: 12441: 12396: 12243: 12190: 12137: 12084: 12013: 11957: 11904: 11815: 11745: 11692: 11639: 11573: 11520: 11477: 11424: 11361: 11296: 11232: 11132: 11076: 11019: 10966: 10903: 10842: 10789: 10724: 10671: 10570: 10298: 10237: 10187: 5151: 3488:{\displaystyle \psi ^{\text{I}}(t,q^{\text{I}})=\psi (t,q^{\text{I}},Q^{\text{II}}(t)).} 16615: 16484: 16382: 16075: 16022: 15914: 15803: 15607: 15571: 15499: 15473: 15452: 15426: 15403: 15390: 15354: 15313: 15277: 15241: 15189: 15031: 15005: 14871: 14835: 14758: 14611: 14413: 14272: 14239: 14193: 14076: 13955: 13929: 13902: 13876: 13849: 13823: 13800: 13756: 13730: 13610: 13574: 13448: 13269: 13243: 13191: 13088: 13062: 12874: 12851: 12825: 12768: 12743: 12686: 12660: 12633: 12597: 12552: 12461: 12427: 12285: 12255: 12229: 12202: 12176: 12149: 12123: 12096: 12070: 12043: 11999: 11969: 11943: 11916: 11890: 11827: 11801: 11757: 11731: 11704: 11678: 11651: 11625: 11595: 11559: 11532: 11489: 11463: 11436: 11410: 11347: 11316: 11244: 11218: 11179: 11144: 11093: 11062: 11050: 11031: 11005: 10978: 10952: 10925: 10889: 10862: 10828: 10801: 10775: 10745:. A video of the electron density in a 2D box evolving under this process is available 10736: 10710: 10683: 10657: 10396: 10326: 10259: 10223: 10094: 9298: 9083: 9025: 8816: 8429: 8409: 8334: 8314: 7149: 7145: 7082: 6780: 6192: 6153: 5832: 5487: 5210: 5157: 5102: 4490: 4444: 4328: 3143:

It is in this qualified sense that the Born rule is, for the de Broglie–Bohm theory, a

2775: 2606: 2548: 2348: 1946: 1814: 1073: 1024: 977: 393: 318: 251: 163: 15637:

Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy

15495: 15309: 14988: 14867: 14538: 12919: 12847: 12682: 12383:

Valentini, A (1991). "Signal-Locality, Uncertainty and the Subquantum H-Theorem. II".

12145: 11485: 11432: 11240: 11049:

Dürr, Detlef; Goldstein, Sheldon; Norsen, Travis; Struyve, Ward; Zanghì, Nino (2014).

10345: 10282: 6798:

Quantum entanglement, Einstein–Podolsky–Rosen paradox, Bell's theorem, and nonlocality

950:", due to the fact that the particles have a definite configuration at all times. The 16625: 16534: 16504: 16432: 16395: 16390: 16372: 16337: 16327: 16131: 16027: 15992: 15975: 15878: 15820: 15706: 15689: 15675: 15661: 15640: 15521: 15345:

Nikolic, H. (2005). "Relativistic quantum mechanics and the Bohmian interpretation".

15331: 15233: 15225: 15159: 15035: 14923: 14893: 14806: 14773: 14639: 14631: 14566: 14558: 14327: 14277: 14259: 14174: 14105: 14093: 13959: 13906: 13804: 13760: 13614: 13602: 13539: 13474: 13396: 13354: 13299: 13135: 13033: 12923: 12855: 12773: 12708: 12514: 12453: 12449: 12404: 12259: 12206: 12153: 12092: 12035: 11973: 11920: 11860: 11831: 11536: 11493: 11383: 11375: 11308: 11261:

Sacha Kocsis, Sylvain Ravets, Boris Braverman, Krister Shalm, Aephraim M. Steinberg:

11183: 11148: 11098: 10740: 10623: 10530: 10503: 10437: 10414: 10410: 10365: 10318: 10310: 10136: 9259: 9233: 9226: 9179: 9156: 9110: 9090:, differ philosophically from the de Broglie–Bohm mechanics and are the basis of the 9051: 9008: 9004: 8352: 7726:{\displaystyle \psi (\mathbf {x} ,t)=R(\mathbf {x} ,t)e^{iS(\mathbf {x} ,t)/\hbar }.} 7347:{\displaystyle \psi (\mathbf {x} ,t)=Ae^{i(\mathbf {k} \cdot \mathbf {x} -\omega t)}} 6833: 6829: 6811: 6807: 6301: 5812: 5808: 5477:{\textstyle D_{k}=\nabla _{k}-{\frac {ie_{k}}{\hbar }}\mathbf {A} (\mathbf {q} _{k})} 3088: 1438: 908: 827: 802: 742: 737: 637: 603: 583: 181: 40: 15503: 15456: 15394: 15317: 14612:"Predictability in a hydrodynamic pilot-wave system: Resolution of walker tunneling" 14417: 14080: 13853: 13107: 13092: 13084: 12465: 12100: 11761: 11708: 11655: 11616:

Nikolic, H (2010). "QFT as pilot-wave theory of particle creation and destruction".

11599: 11440: 11320: 11248: 11119:

Ghose, Partha (1996). "Relativistic quantum mechanics of spin-0 and spin-1 bosons".

11035: 10982: 10866: 10805: 10746: 10687: 10604:"A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. I" 10330: 10214:

Antony Valentini; Hans Westman (2005). "Dynamical origin of quantum probabilities".

16352: 16347: 16204: 16085: 15786: 15698: 15611: 15599: 15564:

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

15559: 15491: 15444: 15382: 15305: 15245: 15217: 15127: 15094: 15060: 15023: 14984: 14913: 14905: 14875: 14863: 14798: 14722: 14714: 14670: 14623: 14550: 14471: 14463: 14403: 14395: 14319: 14267: 14251: 14164: 14068: 13947: 13894: 13841: 13792: 13748: 13594: 13466: 13328: 13261: 13127: 13080: 12999: 12957: 12915: 12843: 12763: 12755: 12690: 12678: 12625: 12570: 12445: 12400: 12322: 12295: 12247: 12194: 12141: 12088: 12047: 12027: 11961: 11908: 11852: 11819: 11749: 11696: 11643: 11587: 11524: 11481: 11428: 11365: 11300: 11236: 11171: 11136: 11088: 11080: 11055:

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

11023: 10970: 10929: 10917: 10854: 10846: 10793: 10728: 10703:

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

10675: 10615: 10406: 10357: 10302: 10283:"Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer" 10263: 10251: 10191: 9097: 9064: 9047: 8396:, which is a symptom of this being a first-order theory, not a second-order theory. 7566:{\displaystyle -{\frac {\partial \rho }{\partial t}}=\nabla \cdot (\rho v^{\psi })} 7176: 7169: 7164: 7153: 7078: 7009: 6280: 6185: 6175: 6141: 6125: 5293: 4495: 1857: 924: 832: 822: 812: 712: 692: 677: 647: 515: 403: 15514: 15221: 14610:

Tadrist, Loïc; Gilet, Tristan; Schlagheck, Peter; Bush, John W. M. (9 July 2020).

14323: 13818:

Tsekov, Roumen (2012). "Bohmian Mechanics versus Madelung Quantum Hydrodynamics".

13688:"Pilot wave theory, Bohmian metaphysics, and the foundations of quantum mechanics" 13351:

Understanding Quantum Mechanics: The World According to Modern Quantum Foundations

13273: 12942: 12637: 12031: 11669:

Nikolic, H. (2009). "Time in relativistic and nonrelativistic quantum mechanics".

10603: 7273:

The guiding equation can be derived in a similar fashion. We assume a plane wave:

7179:, in a wider Hamiltonian framework, theories can be formulated in which particles 6859:

Broglie–Bohm version to bring this out so explicitly that it cannot be ignored."

6304:) signalling is possible, and in which the uncertainty principle can be violated. 16582: 16509: 16489: 16459: 16422: 16417: 16322: 16146: 15810: 15798: 15731: 15448: 14947: 14792: 14524: 14136: 13694: 13413: 12983: 11846: 11269: 10753: 10212:

Publications of D. Bohm in 1952 and 1953 and of J.-P. Vigier in 1954 as cited in

9174:, who had suggested that Bohm search for a realist alternative to the prevailing 9171: 9118: 9101: 7085: 6913: 6893: 6381: 5866:

For a de Broglie–Bohm theory on curved space with spin, the spin space becomes a

5096: 4543:

extension of the Bohmian interpretation to a quantization of fields and strings.

857: 727: 707: 453: 293: 15831: 15173:

Dürr, Detlef; Sheldon Goldstein; Roderich Tumulka; Nino Zanghì (December 2004).

15172: 14627: 14098:

Continuity and change: charting David Bohm's evolving ideas on quantum mechanics

12979:"Can Quantum Mechanical Description of Physical Reality Be Considered Complete?" 12371: 11370: 11335: 7059:

Pilot-wave theories are parallel-universe theories in a state of chronic denial.

5874:

In a general spacetime with curvature and torsion, the guiding equation for the

16560: 16529: 16519: 16141: 16121: 15950: 15760: 15027: 14675: 14658: 14554: 14255: 13287: 13030:

Quantum Non-Locality and Relativity: Metaphysical Intimations of Modern Physics

12299: 12251: 12198: 10361: 9400: 9312:

gave an in-depth analysis of possible interpretations of Bohm's model of 1952.

9186:, who had been unconvinced by de Broglie in 1927, conceded to Bohm as follows: 9183: 9075: 9055: 9033: 7093: 6910:

was dedicated to Prof. Wyatt and his work on "computational Bohmian dynamics".

6362: 6089: 5642: 5363: 4619: 4578: 4565: 4557: 4455: 4448: 4443:

Pilot-wave theory is explicitly nonlocal, which is in ostensible conflict with

3694: 792: 752: 732: 702: 682: 632: 598: 448: 438: 231: 15841: 15755: 15386: 15131: 14802: 14212:

Larder et al. (2019) Fast nonadiabatic dynamics of many-body quantum systems

13951: 13898: 13845: 13752: 13265: 12341: 11912: 11856: 11823: 11753: 11700: 11647: 11591: 11528: 11175: 10974: 16658: 16479: 16332: 16223: 16126: 16042: 16012: 15965: 15254: 15229: 14927: 14777: 14635: 14562: 14263: 14178: 13675: 13606: 13316: 12927: 12457: 11379: 11027: 10627: 10418: 10369: 10314: 7052: 7036: 6790:, noting that this article describes the principle from the viewpoint of the 6058: 5875: 5867: 5680: 4615: 2002:

The de Broglie–Bohm theory works on particle positions and trajectories like

852: 847: 777: 747: 717: 588: 534: 261: 236: 114:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle } 15804:"Pilot waves, Bohmian metaphysics, and the foundations of quantum mechanics" 15741: 14659:"Observation of Bohm trajectories and quantum potentials of classical waves" 13676:"Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference" 13332: 12759: 11304: 10306: 8643:

is the local Hermitian inner product on the value space of the wavefunction.

3808:

is given by the square modulus of the (normalized) conditional wavefunction

16362: 15960: 15955: 15632: 15603: 15535: 15237: 15098: 15064: 14909: 14643: 14570: 14331: 14281: 13139: 13004: 12777: 12481:"Cosmological Data Hint at a Level of Physics Underlying Quantum Mechanics" 12039: 11387: 11312: 11102: 11084: 10732: 10619: 10322: 10255: 10195: 9337: 9316: 6847: 6769: 6025: 5859:. Thus, we use equations that have the same form as above. Topological and 4478: 1119:. The configuration evolves (for spin=0) according to the guiding equation 912: 842: 837: 772: 757: 722: 216: 14169: 14152: 11722:

Nikolic, H. (2011). "Making nonlocal reality compatible with relativity".

10850: 9105:

correlated not because the particles exert a direct force on one another (

7063:

This conclusion has been challenged by Detlef Dürr and Justin Lazarovici:

1090:

particles, or, in case of field theory, the space of field configurations

16400: 15814: 15653: 15576: 15478: 15431: 15417:

Sanz, A. S.; F. Borondo (2007). "A Bohmian view on quantum decoherence".

15408: 15359: 15282: 15194: 14408: 14213: 14198: 13934: 13881: 13718: 13248: 13067: 12879: 12830: 12665: 12602: 12557: 12128: 12075: 12004: 11564: 11468: 11415: 11223: 11010: 10957: 10894: 10833: 10780: 10662: 10401: 10228: 9309: 9218: 9199: 9068: 7232: 7109: 6881: 6822: 6245:

as representing not a basic law, but a result of a system having reached

6145: 807: 762: 697: 652: 14967:

Albert, David Z. (May 1994). "Bohm's Alternative to Quantum Mechanics".

14918: 14727: 14476: 13629:

The Undivided Universe – An Ontological Interpretation of Quantum Theory

13131: 12370:, Stochastic Processes, Physics and Geometry II. World Scientific, 1995 8853:.The fourth postulate is subsidiary yet consistent with the first three: 8198:{\displaystyle {\frac {\partial S(\mathbf {x} ,t)}{\partial t}}=-\left.} 6361:

According to ordinary quantum theory, it is not possible to measure the

4553: 15658:

The Undivided Universe: An Ontological Interpretation of Quantum Theory

15649: 15156:

The Undivided Universe: An ontological interpretation of quantum theory

15010: 14072: 13796: 13598: 13229:"Solving the measurement problem: de Broglie–Bohm loses out to Everett" 12705:

The Undivided Universe: An Ontological Interpretation of Quantum Theory

12574: 12326: 11140: 10797: 10679: 10499:

The undivided universe: an ontological interpretation of quantum theory

10458:

The undivided universe: An ontological interpretation of quantum theory

9273: 9207: 5816: 2118:

to this theory include spin and more complicated configuration spaces.

1749:{\textstyle H=\sum {\frac {1}{2m_{i}}}{\hat {p}}_{i}^{2}+V({\hat {q}})} 928: 797: 767: 687: 662: 657: 642: 15827:"Observing the Trajectories of a Single Photon Using Weak Measurement" 14840: 14399: 11263:"Observing the trajectories of a single photon using weak measurement" 10858: 9125:

in Bell's work and is the basis for 'The Quantum Theory of Motion' .

6569:, and no contradiction to experimental results is possible to detect. 3314:

denotes the remaining configuration variables. Denote respectively by

16254: 15935: 15774:"Videos answering frequently asked questions about Bohmian Mechanics" 14297:"Single-Particle Diffraction and Interference at a Macroscopic Scale" 13721:(2010). "Von Neumann's 'No Hidden Variables' Proof: A Re-Appraisal". 13561:

Quantum Mechanics: Historical Contingency and the Copenhagen Hegemony

12902:

Brida, G; Cagliero, E; Genovese, M; Gramegna, M (28 September 2004).

11845:

Pena, Luis de la; Cetto, Ana Maria; Valdes-Hernandez, Andrea (2014).

9344:

Another classical analog has been reported in surface gravity waves.

9294:. The main reference is "The Undivided Universe" (Bohm, Hiley 1993). 9195:

He subsequently described Bohm's theory as "artificial metaphysics".

9000: 6293: 5856: 4466: 3148: 2966: 2015:

lack of back reaction, due to the incompleteness of the description.

951: 288: 15847: 15791: 15780: 15562:; Hans Westman (2005). "Dynamical Origin of Quantum Probabilities". 14126:

A story without an ending: the quantum physics controversy 1950–1970

13496:"Hamiltonian Theory of Wave and Particle in Quantum Mechanics I, II" 12978: 12275: 12167:

Fabbri, Luca (2022). "de Broglie-Bohm formulation of Dirac fields".

6619: 6400: 5863:

may apply in supplementing the evolution of Schrödinger's equation.

5789:{\displaystyle (\phi ,\psi )=\sum _{s=1}^{d}\phi _{s}^{*}\psi _{s}.} 4548:

Quantum potential § Relativistic and field-theoretic extensions

2957:

This is the same wavefunction as in conventional quantum mechanics.

14759:"Interpretation of quantum mechanics by the double solution theory" 14536: 12290: 12234: 12181: 11806: 11352: 8269:{\displaystyle V-{\frac {\hbar ^{2}}{2m}}{\frac {\nabla ^{2}R}{R}}} 7003: 5852: 4582: 2163:

represents the complex-valued wavefunction on configuration space.

667: 13828: 13735: 13453: 13196: 12432: 11948: 11895: 11736: 11683: 11630: 11067: 10715: 7116:

as the "pure wave theory". Of Bohm's 1952 approach, Everett said:

15842:

De Broglie–Bohm theory inspired visualization of atomic orbitals.

15516:

And Yet It Moves: Strange Systems and Subtle Questions in Physics

14128:, Science & Education, vol. 12, pp. 573–586, 2003, 13108:"The Short Story of My Life and My Career in Quantum Propagation" 11933: 11879: 11333: 9067:

published a paper, that was widely (and erroneously, as shown by

7847:{\displaystyle \rho (\mathbf {x} ,t)=|\psi (\mathbf {x} ,t)|^{2}} 7108:(which we think of as a "particle", but is actually just another 6499:

distribution for the particles in the system (see the section on

5905: 3144: 3132: 15686:

Bohmian Mechanics: The Physics and Mathematics of Quantum Theory

13774:

Madelung, E. (1927). "Quantentheorie in hydrodynamischer Form".

9170:

Bohm's paper was largely ignored or panned by other physicists.

9128:

This stage applies to multiple particles, and is deterministic.

7496:

Notice that this derivation does not use Schrödinger's equation.

6500: 15402:

Passon, Oliver (2004). "Why isn't every physicist a Bohmian?".

14794:

The Quantum Dice: An Introduction to Stochastic Electrodynamics

14240:"Cosmic Hylomorphism: a powerist ontology of quantum mechanics" 14037:

Statement on that they were in fact the first in: B. J. Hiley:

10995: 9160: 9140: 5807:(SED) is an extension of the de Broglie–Bohm interpretation of 4486: 3154:

It can also be shown that a distribution of particles which is

919:. The evolution of the wave function over time is given by the 15821:"21st-century directions in de Broglie-Bohm theory and beyond" 15464:

Sanz, A. S. (2005). "A Bohmian approach to quantum fractals".

9135:. Bohm originally hoped that hidden variables could provide a 6382:

Measurements, the quantum formalism, and observer independence

3199: 1852: 15792:

Bohmian Mechanics Group at University of Innsbruck (G. Grübl)

13447:. Vol. 2010. Oxford University Press. pp. 476–509. 12901: 12815: 9812:{\displaystyle \omega ^{2}\approx \omega _{c}^{2}+c^{2}k^{2}} 9182:

considered it a "superfluous 'ideological superstructure' ".

970:

De Broglie–Bohm theory is based on the following postulates:

14656: 14150: 11281: 10280: 10213: 2018:

In what follows below, the setup for one particle moving in

14891: 14347:"Fluid mechanics suggests alternative to quantum orthodoxy" 12908:

Journal of Physics B: Atomic, Molecular and Optical Physics

12818:

Journal of Physics B: Atomic, Molecular and Optical Physics

12741: 12060: 11989: 11202:

Ghose, Partha; Majumdar, A. S.; Guhab, S.; Sau, J. (2001).

6140:. This relativistic wave-particle duality follows from the 3850:(in the terminology of Dürr et al. this fact is called the 3577:

satisfies the guiding equation that also the configuration

15768:

from the original on 11 December 2021 – via YouTube.

14609: 14586:"Famous Experiment Dooms Alternative to Quantum Weirdness" 14102:

Brazilian Studies in the Philosophy and History of Science

12220:

Fabbri, Luca (2023). "Dirac Theory in Hydrodynamic Form".

12113: 11848:

The Emerging Quantum: The Physics Behind Quantum Mechanics

11272:

19th Australian Institute of Physics (AIP) Congress, 2010.

11048: 9121:

disturbs Newtonian mechanics). This stage is known as the

3147:

rather than (as in ordinary quantum theory) an additional

14104:, Boston Studies in the Philosophy of Science, Springer, 13168: 13052: 12542: 12366:

See for ex. Detlef Dürr, Sheldon Goldstein, Nino Zanghí:

11844: 9743:{\displaystyle \omega _{F}^{2}\approx \sigma k^{3}/\rho } 9297:

This stage covers work by Bohm and in collaboration with

6132:

Dirac equation, then the particle moves according to the

2752:

For many particles, the equation is the same except that

14894:"Experimental nonlocal and surreal Bohmian trajectories" 12744:"Experimental nonlocal and surreal Bohmian trajectories" 7104:, which is responding in turn to the passage of another 4318:{\displaystyle \phi (t,q^{\text{I}},Q^{\text{II}}(t))=0} 15832:"Bohmian trajectories are no longer 'hidden variables'" 14191: 14050: 12795:

Quantum weirdness may hide an orderly reality after all

11336:"How (not) to understand weak measurements of velocity" 5355:{\displaystyle \mathbf {B} =\nabla \times \mathbf {A} } 3570:{\displaystyle Q(t)=(Q^{\text{I}}(t),Q^{\text{II}}(t))} 1884:

separates the two wave packets in configuration space.

10700: 9258:

as a primary focus and backdrop, with the icon of the

6772:

sense mentioned above) on the de Broglie–Bohm theory.

5402: 1666: 15558: 15113:"A new theory of the relationship of mind and matter" 14446:

Bush, John W. M. (2015). "Pilot-Wave Hydrodynamics".

13972:(Letter of 12 May 1952 from Einstein to Max Born, in 12976: 11201: 10050: 10019: 9991: 9970: 9939: 9908: 9880: 9859: 9826: 9757: 9700: 9670: 9618: 9578: 9556: 9498: 9446: 9418: 9016:

reference to the "observer", could be eliminated. ...

8936: 8902: 8862: 8839: 8819: 8782: 8692: 8669: 8617: 8481: 8452: 8432: 8412: 8370: 8364:, the initial velocity field is already specified by 8337: 8317: 8282: 8215: 7998: 7865: 7783: 7739: 7638: 7579: 7511: 7435: 7402: 7360: 7279: 7241: 7208: 7126: 6939: 6733: 6710: 6687: 6609: 6538: 6468: 6322: 6263: 6218: 6195: 6098: 6067: 6034: 6010: 5914: 5883: 5718: 5689: 5653: 5627: 5536: 5516: 5490: 5372: 5332: 5302: 5276: 5233: 5213: 5182: 5160: 5127: 5105: 5048: 4630: 4591: 4511: 4458:

extensions of Bohmian theory arose in the 1990s; see

4405: 4378: 4351: 4331: 4261: 4241: 4072: 4049: 4022: 3995: 3867: 3814: 3778: 3742: 3703: 3666: 3639: 3619: 3583: 3504: 3399: 3356: 3320: 3293: 3266: 3211: 3165: 3097: 3056: 3015: 2978: 2833: 2798: 2778: 2758: 2661: 2629: 2609: 2580: 2551: 2374: 2351: 2322: 2209: 2177: 2149: 2127: 2086: 2057: 2024: 1969: 1949: 1903: 1817: 1765: 1485: 1447: 1414: 1388: 1125: 1096: 1076: 1047: 1027: 1000: 980: 52: 15836: 14995: 14026:

Infinite Potential: The Life and Times of David Bohm

13557:

Infinite Potential: The Life and Times of David Bohm

11161: 9278:

Bohm developed his original ideas, calling them the

9274:

Causal interpretation and ontological interpretation

6786:

For the derivation of the uncertainty relation, see

6249:

during the course of the time development under the

15787:

Workgroup Bohmian Mechanics at LMU Munich (D. Dürr)

13998:Pauli to Bohm, 3 December 1951, in Wolfgang Pauli, 13641:

Bohmian Mechanics and Quantum Theory – An Appraisal

13389:

The Many-Worlds Interpretation of Quantum Mechanics

12789: 12787: 12650: 12587: 11340:

Studies in History and Philosophy of Science Part A

10765: 10647: 10346:"Experiment and the foundations of quantum physics" 9604:{\displaystyle mc^{2}\leftrightarrow \hbar \omega } 8663:The wave is described mathematically by a solution 6128:. If the wave function propagates according to the 3253:{\displaystyle \psi (t,q^{\text{I}},q^{\text{II}})} 1840:that drive a process of asymptotic relaxation from 15513: 14226:Bohmian mechanics and quantum theory: an appraisal 13661:Louis be Broglie, in the foreword to David Bohm's 13639:(1994). J. Cushing, A. Fine, S. Goldstein (eds.), 10436:. Routledge & Kegan Paul and D. Van Nostrand. 10066: 10035: 10004: 9976: 9952: 9924: 9893: 9865: 9841: 9811: 9742: 9685: 9637: 9603: 9562: 9535: 9483: 9431: 8979: 8918: 8888: 8845: 8825: 8805: 8768: 8686:The particle motion is described by a solution to 8675: 8635: 8603: 8467: 8438: 8418: 8388: 8343: 8323: 8303: 8268: 8197: 7979: 7846: 7769: 7725: 7608: 7565: 7485: 7421: 7388: 7346: 7260: 7223: 7132: 6977: 6757: 6719: 6696: 6561: 6491: 6356: 6345: 6269: 6237: 6201: 6116: 6080: 6049: 6016: 5996: 5896: 5847:To extend de Broglie–Bohm theory to curved space ( 5788: 5704: 5671: 5633: 5613: 5522: 5496: 5476: 5387: 5354: 5317: 5284: 5260: 5219: 5195: 5166: 5142: 5111: 5087: 5029: 4606: 4534: 4418: 4391: 4364: 4337: 4317: 4247: 4224: 4055: 4035: 4008: 3978: 3842: 3800: 3764: 3724: 3681: 3652: 3625: 3605: 3569: 3487: 3378: 3342: 3306: 3279: 3252: 3188: 3120: 3079: 3038: 3001: 2946: 2816: 2784: 2764: 2741: 2644: 2615: 2595: 2557: 2534: 2357: 2337: 2305: 2192: 2155: 2135: 2104: 2072: 2039: 1990: 1955: 1932: 1897:The de Broglie–Bohm theory describes a pilot wave 1823: 1803: 1748: 1652: 1468: 1429: 1396: 1374: 1111: 1082: 1062: 1033: 1013: 986: 113: 14002:, Vol IV – Part I, , (Berlin, 1996), pp. 436–441. 13445:Many Worlds? Everett, Quantum Theory, and Reality 13418:The Redundancy Argument Against Bohmian Mechanics 13296:Dennett's Philosophy: a comprehensive assessment. 7429:for the particle's actual velocity, we have that 7055:has expressed the same point more "acerbically": 7031:(the assumption that this branch indicates which 1421: 1344: 1021:, which is an element of the configuration space 16656: 14887: 14885: 13189: 12784: 12418:Valentini, Antony (2009). "Beyond the quantum". 8896:to find the particle in the differential volume 7389:{\displaystyle i\mathbf {k} =\nabla \psi /\psi } 7004:Similarities with the many-worlds interpretation 6802:De Broglie–Bohm theory highlighted the issue of 14790: 13219: 13217: 13215: 13213: 13211: 13209: 13207: 12868: 9323: 9155:and the collapse of the wavefunction. However, 7261:{\displaystyle \mathbf {p} =\hbar \mathbf {k} } 7000:from the unraveling of an open quantum system. 6390: 15813:, lecture course on de Broglie-Bohm theory by 15466:Journal of Physics A: Mathematical and General 15416: 15270:Journal of Physics A: Mathematical and General 15153: 14493:"Fluid Tests Hint at Concrete Quantum Reality" 14041:, in: Joseph S. King, Karl H. Pribram (eds.): 13345: 13294:In D. Ross, A. Brook, and D. Thompson (Eds.), 12873:. Vaxjo: Vaxjo University Press. p. 355. 12871:Quantum Theory: Reconsideration of Foundations 12653:Journal of Physics A: Mathematical and General 12511:Speakable and Unspeakable in Quantum Mechanics 12063:Journal of Physics A: Mathematical and General 11456:Journal of Physics A: Mathematical and General 11403:Journal of Physics A: Mathematical and General 10486:, March 1999, Volume 29, Issue 3, pp. 333–348. 9217:In 1979, Chris Philippidis, Chris Dewdney and 9131:The de Broglie–Bohm theory is an example of a 6978:{\displaystyle R^{-1}\nabla ^{2}R\to \infty .} 6292:argues that the laws of quantum mechanics are 1759:The configuration is distributed according to 15863: 15670:Detlef Dürr, Sheldon Goldstein, Nino Zanghì: 14882: 14146: 14144: 13321:British Journal for the Philosophy of Science 12411: 11611: 11609: 11506: 11453: 11400: 11114: 11112: 11051:"Can Bohmian mechanics be made relativistic?" 6257:associated with the Schrödinger evolution of 5799: 4614:. The guiding equation is modified by taking 880: 14100:, In: Décio Krause, Antonio Videira (eds.): 14045:, Psychology Press, 1995, pp. 318 ff., 13707:Mathematische Grundlagen der Quantenmechanik 13223: 13204: 12315:International Journal of Theoretical Physics 11671:International Journal of Quantum Information 10079: 8980:{\displaystyle |\psi (\mathbf {x} (t))|^{2}} 8426:, the equation to satisfy for all functions 6887: 6871:. Maudlin has shown this in greater detail. 5815:(ZPF) playing a central role as the guiding 2960: 108: 82: 14825: 6648:. Unsourced material may be challenged and 6522: 6501:the conditional wavefunction of a subsystem 6429:. Unsourced material may be challenged and 3852:fundamental conditional probability formula 3633:replaced with the conditional wavefunction 3200:The conditional wavefunction of a subsystem 15870: 15856: 15750: 15672:Quantum Physics Without Quantum Philosophy 14943:"New Support for Alternative Quantum View" 14756: 14741: 14294: 14244:European Journal for Philosophy of Science 14141: 14015:(pp. 33–42). Paris: Editions Albin Michel. 12940: 12312: 11791: 11606: 11334:Fankhauser Johannes, Dürr Patrick (2021). 11109: 10529:. Cambridge University Press. p. 26. 9206:in Princeton, many of the objections were 9202:, when Bohm's theory was presented at the 7422:{\displaystyle \mathbf {p} =m\mathbf {v} } 7096:, Everett's theory interprets this as our 7072: 6758:{\displaystyle \Delta x\Delta p\gtrsim h.} 3843:{\displaystyle \psi ^{\text{I}}(t,\cdot )} 3498:It follows immediately from the fact that 1933:{\displaystyle \psi (q,t)\in \mathbb {C} } 1836:1954, in which they introduced stochastic 994:of the universe, described by coordinates 911:which postulates that, in addition to the 887: 873: 18:Causal interpretation of quantum mechanics 15593: 15575: 15520:. Cambridge: Cambridge University Press. 15511: 15477: 15430: 15407: 15376: 15358: 15330:. Cambridge: Cambridge University Press. 15299: 15281: 15252: 15211: 15193: 15009: 14917: 14857: 14839: 14726: 14674: 14583: 14490: 14475: 14407: 14271: 14197: 14168: 13933: 13880: 13827: 13734: 13471:10.1093/acprof:oso/9780199560561.003.0019 13452: 13442: 13373:See section VI of Everett's dissertation 13247: 13195: 13066: 13003: 12961: 12878: 12829: 12767: 12664: 12619: 12601: 12556: 12431: 12417: 12382: 12368:Bohmian mechanics and quantum equilibrium 12339: 12289: 12233: 12180: 12127: 12074: 12021: 12003: 11985: 11983: 11947: 11894: 11805: 11735: 11682: 11629: 11581: 11563: 11467: 11414: 11369: 11351: 11222: 11092: 11066: 11009: 10956: 10911: 10893: 10832: 10779: 10714: 10661: 10495: 10473:, 1995, Volume 25, Number 2, pp. 329–347. 10400: 10343: 10245: 10227: 9536:{\displaystyle \omega _{c}=mc^{2}/\hbar } 9484:{\displaystyle \omega _{c}=mc^{2}/\hbar } 8769:{\displaystyle \mathbf {\dot {x}} (t)=/m} 8446:(with associated multiplication operator 6668:Learn how and when to remove this message 6449:Learn how and when to remove this message 6159: 6136:of motion, which are an extension of the 5692: 5598: 5583: 5568: 5559: 5545: 5375: 5305: 4594: 2801: 2632: 2583: 2569: 2180: 2171:For a spinless single particle moving in 2089: 2060: 2027: 1926: 1847: 15534: 14344: 13773: 12943:"On the Einstein Podolsky Rosen Paradox" 10643: 10641: 10639: 10637: 9078:had developed a hydrodynamic version of 8406:is. Then given the Hamiltonian operator 7100:responding to changes in the detector's 6901:reactive scattering can be found on the 6596: 6169: 4585:particle, spin space can be taken to be 1851: 15756:"Pilot Wave Theory and Quantum Realism" 15344: 15325: 14237: 13919: 13866: 13493: 13315: 13027: 12504: 12502: 11721: 11668: 11618:International Journal of Modern Physics 11615: 11549: 10942: 10879: 10549: 10522: 9100:has pointed out that, earlier in 1927, 7777:corresponds to the probability density 7148:, which was found to be unnecessary in 7081:and Bohm treated the wavefunction as a 5822: 14: 16657: 15660:, Routledge Chapman & Hall, 1993, 15401: 14966: 14791:de la Peña, Luis; Cetto, A.M. (1996). 14584:Wolchover, Natalie (11 October 2018). 14214:https://doi.org/10.1126/sciadv.aaw1634 13817: 13674:Bacciagaluppi, G., and Valentini, A., 13663:Causality and Chance in Modern Physics 13631:appeared after Bohm's death, in 1993; 12478: 12342:"Hidden Variables in Modern Cosmology" 12219: 12166: 11980: 10818: 10434:Causality and Chance in Modern Physics 10382: 9638:{\displaystyle mc^{2}\leftrightarrow } 9212:House Un-American Activities Committee 8889:{\displaystyle \rho (\mathbf {x} (t))} 8304:{\displaystyle {\frac {\nabla S}{m}}.} 5645:, so this spin space is 12-dimensional 4460:Bohm and Hiley: The Undivided Universe 15877: 15851: 15540:"Bohmian mechanics is a 'lost cause'" 14345:Hardesty, Larry (12 September 2014). 14295:Couder, Yves; Fort, Emmanuel (2006). 14013:Louis de Broglie—physicien et penseur 13686:See the brief summary by Towler, M., 13412: 13105: 12797:, newscientist.com, 19 February 2016. 12348:from the original on 11 December 2021 12271: 12269: 11936:Journal of Physics: Conference Series 11118: 10634: 10597: 10595: 10523:Holland, Peter R. (26 January 1995). 10496:Bohm, David; Hiley, Basil J. (1995). 9290:and preferred to call his theory the 8389:{\displaystyle {\frac {\nabla S}{m}}} 7770:{\displaystyle R^{2}(\mathbf {x} ,t)} 7493:. Thus, we have the guiding equation. 6704:and the momentum with an accuracy of 6284:making the parallel universes of the 1876:the interference pattern disappears. 916: 16665:Interpretations of quantum mechanics 15639:, Cambridge University Press, 2004, 15463: 15267: 15110: 15076: 15042: 14940: 14689: 14445: 14370: 13572: 13376:Theory of the Universal Wavefunction 13153: 12508: 12499: 10601: 10431: 10173: 9240: 9042: 7039:comments on these "empty" branches: 6646:adding citations to reliable sources 6613: 6427:adding citations to reliable sources 6394: 5088:{\displaystyle m_{k},e_{k},\mu _{k}} 15744:Stanford Encyclopedia of Philosophy 15260:Stanford Encyclopedia of Philosophy 14719:10.1146/annurev-fluid-010814-014506 14491:Wolchover, Natalie (24 June 2014). 14468:10.1146/annurev-fluid-010814-014506 14373:"The new wave of pilot-wave theory" 14228:, Springer, 1996, pp. 231−250. 13717: 13292:With a little help from my friends. 13112:The Journal of Physical Chemistry A 12963:10.1103/PhysicsPhysiqueFizika.1.195 12479:Musser, George (18 November 2013). 9302:theory and de Broglie–Bohm theory. 8776:in dependence on initial condition 6908:The Journal of Physical Chemistry A 6576: 5829:creation and annihilation operators 4454:A renewed interest in constructing 2166: 2047:is given followed by the setup for 1430:{\displaystyle \mathbf {\hat {P}} } 27:Interpretation of quantum mechanics 24: 15625: 13559:(1997), p. 133. James T. Cushing, 12977:Einstein; Podolsky; Rosen (1935). 12630:10.1023/B:JOSS.0000037234.80916.d0 12266: 11204:"Bohmian trajectories for photons" 10922:10.1023/B:FOPL.0000035670.31755.0a 10592: 10383:Passon, Oliver (1 November 2004). 10110:Fluid analogs in quantum mechanics 9262:, Bohmian mechanics considers the 8720: 8374: 8286: 8248: 8133: 8065: 8027: 8002: 7943: 7909: 7897: 7872: 7538: 7526: 7518: 7467: 7372: 7200:Einstein's light quanta hypothesis 6969: 6954: 6874: 6740: 6734: 6711: 6688: 6610:Heisenberg's uncertainty principle 5417: 5341: 4823: 4819: 4426:satisfies a Schrödinger equation. 2915: 2846: 2842: 2715: 2674: 2670: 2447: 2365:-th particle their velocities are 2267: 2080:, but configuration space becomes 1844:to quantum equilibrium (ρ → |ψ|). 1582: 1498: 1494: 1360: 1356: 1238: 1179: 419:Sum-over-histories (path integral) 105: 79: 35:Part of a series of articles about 25: 16681: 15717: 14989:10.1038/scientificamerican0594-58 14669:(4). IOP Publishing Ltd: 044004. 13820:Annuaire de l'Université de Sofia 13106:Wyatt, Robert (11 October 2007). 10344:Zeilinger, Anton (1 March 1999). 9595: 9530: 9478: 9032:, besides other features such as 8806:{\displaystyle \mathbf {x} (t=0)} 8536: 8355:, terminology introduced by Bohm. 7715: 7609:{\displaystyle \rho =|\psi |^{2}} 7250: 7215: 5446: 4814: 4564:in the 1950s and is also used by 4255:solves Schrödinger equation and, 2837: 2817:{\displaystyle \mathbb {R} ^{3N}} 2665: 2105:{\displaystyle \mathbb {R} ^{3N}} 1892: 1489: 1221: 1175: 56: 16639: 16638: 15736:Annual Review of Fluid Mechanics 15154:Bohm, David; B.J. Hiley (1993). 14934: 14819: 14784: 14750: 14735: 14699:Annual Review of Fluid Mechanics 14683: 14650: 14603: 14577: 14530: 14510: 14484: 14448:Annual Review of Fluid Mechanics 14439: 14364: 14338: 14288: 14231: 14218: 14206: 14185: 14157:Zeitschrift für Naturforschung A 14153:"Surrealistic Bohm Trajectories" 14118: 14087: 14031: 14018: 12732:(2010), Vol. 18, No. 4, 469–480. 10385:"How to teach quantum mechanics" 9563:{\displaystyle \leftrightarrow } 9315:William Simpson has suggested a 9088:Euler equations (fluid dynamics) 8949: 8870: 8784: 8730: 8699: 8696: 8171: 8149: 8075: 8012: 7953: 7927: 7882: 7819: 7791: 7754: 7697: 7669: 7646: 7437: 7415: 7404: 7365: 7326: 7318: 7287: 7254: 7243: 6788:Heisenberg uncertainty principle 6618: 6399: 5705:{\displaystyle \mathbb {C} ^{d}} 5461: 5452: 5388:{\displaystyle \mathbb {R} ^{3}} 5348: 5334: 5318:{\displaystyle \mathbb {R} ^{3}} 5278: 5143:{\displaystyle \mathbf {S} _{k}} 5130: 4999: 4990: 4959: 4785: 4764: 4643: 4607:{\displaystyle \mathbb {C} ^{2}} 4419:{\displaystyle \psi ^{\text{I}}} 4392:{\displaystyle \psi ^{\text{I}}} 4036:{\displaystyle \psi ^{\text{I}}} 4009:{\displaystyle \psi ^{\text{I}}} 3801:{\displaystyle Q^{\text{II}}(t)} 3653:{\displaystyle \psi ^{\text{I}}} 3379:{\displaystyle Q^{\text{II}}(t)} 2792:are now on configuration space, 2645:{\displaystyle \mathbb {R} ^{3}} 2596:{\displaystyle \mathbb {R} ^{3}} 2510: 2489: 2474: 2383: 2338:{\displaystyle \mathbf {Q} _{k}} 2325: 2287: 2217: 2193:{\displaystyle \mathbb {R} ^{3}} 2129: 2073:{\displaystyle \mathbb {R} ^{3}} 2040:{\displaystyle \mathbb {R} ^{3}} 1804:{\displaystyle |\psi (q,t)|^{2}} 1418: 1390: 1341: 1293: 1063:{\displaystyle \mathbf {Q} _{k}} 1050: 959: 14005: 13992: 13979: 13966: 13913: 13860: 13811: 13767: 13711: 13699: 13680: 13668: 13655: 13646: 13627:David Bohm and Basil J. Hiley, 13621: 13566: 13549: 13528: 13487: 13436: 13406: 13391:, Princeton Series in Physics, 13367: 13339: 13319:(1996). "Comment on Lockwood". 13309: 13281: 13183: 13167:. 10 March 2021. Archived from 13099: 13046: 13032:. Cambridge, Mass.: Blackwell. 13021: 13012: 12970: 12934: 12895: 12862: 12809: 12800: 12735: 12722: 12697: 12644: 12581: 12536: 12527: 12472: 12376: 12360: 12333: 12306: 12213: 12160: 12107: 12054: 11927: 11873: 11838: 11785: 11768: 11715: 11662: 11543: 11500: 11447: 11394: 11327: 11275: 11255: 11195: 11155: 11042: 10989: 10936: 10873: 10812: 10759: 10694: 10602:Bohm, David (15 January 1952). 10543: 10516: 10489: 10476: 8683:to Schrödinger's wave equation. 7224:{\displaystyle E=\hbar \omega } 7183:act back on the wave function. 6823:Einstein–Podolsky–Rosen paradox 6357:Measuring spin and polarization 5842: 5672:{\displaystyle (\cdot ,\cdot )} 3765:{\displaystyle Q^{\text{I}}(t)} 3734:conditional probability density 3725:{\displaystyle \psi (t,\cdot )} 3606:{\displaystyle Q^{\text{I}}(t)} 3390:of subsystem (I) is defined by 3343:{\displaystyle Q^{\text{I}}(t)} 2115: 16588:Relativistic quantum mechanics 15619:Bohmian mechanics on arxiv.org 15347:Foundations of Physics Letters 14518:"Quantum mechanics writ large" 13575:"On the impossible pilot wave" 13573:Bell, J. S. (1 October 1982). 12590:Journal of Statistical Physics 12513:. Cambridge University Press. 11966:10.1088/1742-6596/361/1/012008 11552:Foundations of Physics Letters 11164:Foundations of Physics Letters 10945:Foundations of Physics Letters 10882:Foundations of Physics Letters 10768:Journal of Statistical Physics 10650:Journal of Statistical Physics 10463: 10450: 10425: 10376: 10337: 10274: 10206: 10167: 10127: 9680: 9674: 9632: 9592: 9557: 9247:quantum equilibrium hypothesis 9036:or curved spatial geometries. 8967: 8962: 8959: 8953: 8945: 8938: 8883: 8880: 8874: 8866: 8800: 8788: 8755: 8752: 8749: 8740: 8734: 8726: 8717: 8711: 8705: 8630: 8618: 8598: 8592: 8586: 8574: 8562: 8556: 8541: 8506: 8500: 8497: 8494: 8488: 8482: 8459: 8181: 8167: 8159: 8145: 8089: 8085: 8071: 8062: 8022: 8008: 7963: 7949: 7937: 7923: 7892: 7878: 7834: 7829: 7815: 7808: 7801: 7787: 7764: 7750: 7707: 7693: 7679: 7665: 7656: 7642: 7596: 7587: 7560: 7544: 7339: 7314: 7297: 7283: 7186: 6966: 6549: 6540: 6479: 6470: 6333: 6324: 6152:, and also from the covariant 6134:Mathisson-Papapetrou equations 6081:{\displaystyle \gamma ^{\mu }} 6050:{\displaystyle {\bar {\psi }}} 6041: 5979: 5952: 5731: 5719: 5666: 5654: 5563: 5471: 5456: 5009: 4994: 4801: 4759: 4749: 4737: 4732: 4710: 4670: 4664: 4522: 4513: 4306: 4303: 4297: 4265: 4216: 4184: 4175: 4156: 4143: 4124: 4108: 4076: 3970: 3951: 3938: 3919: 3903: 3871: 3837: 3825: 3795: 3789: 3759: 3753: 3719: 3707: 3676: 3670: 3600: 3594: 3564: 3561: 3555: 3539: 3533: 3520: 3514: 3508: 3479: 3476: 3470: 3438: 3429: 3410: 3373: 3367: 3337: 3331: 3247: 3215: 3176: 3167: 3108: 3099: 3067: 3058: 3026: 3017: 2989: 2980: 2526: 2469: 2410: 2404: 2297: 2283: 2238: 2232: 2143:for particle positions, while 1979: 1973: 1919: 1907: 1791: 1786: 1774: 1767: 1743: 1737: 1728: 1705: 1644: 1632: 1626: 1620: 1611: 1599: 1522: 1510: 1463: 1451: 1272: 1260: 1215: 1203: 1169: 1163: 1106: 1100: 956:quantum equilibrium hypothesis 569:Relativistic quantum mechanics 101: 94: 75: 13: 1: 16566:Quantum statistical mechanics 16343:Quantum differential calculus 16265:Delayed-choice quantum eraser 16033:Symmetry in quantum mechanics 15222:10.1103/PhysRevLett.93.090402 14868:10.1016/S0375-9601(00)00186-9 14324:10.1103/PhysRevLett.97.154101 13349:; Lazarovici, Justin (2022). 12278:Classical and Quantum Gravity 12032:10.1103/PhysRevLett.93.090402 11241:10.1016/s0375-9601(01)00677-6 10160: 10067:{\displaystyle \lambda _{dB}} 10036:{\displaystyle \lambda _{dB}} 9925:{\displaystyle \lambda _{dB}} 6601:A specialized version of the 4483:Duffin–Kemmer–Petiau equation 4438: 4433: 3307:{\displaystyle q^{\text{II}}} 2200:, the particle's velocity is 609:Quantum statistical mechanics 15703:The quantum theory of motion 15684:Detlef Dürr, Stefan Teufel: 14049:, which takes reference to: 13536:The quantum theory of motion 13509:: 1043, 1143. Archived from 12485:blogs.scientificamerican.com 12405:10.1016/0375-9601(91)90330-b 10100:Local hidden-variable theory 10005:{\displaystyle \lambda _{F}} 9953:{\displaystyle \lambda _{c}} 9894:{\displaystyle \lambda _{F}} 9330:Hydrodynamic quantum analogs 9324:Hydrodynamic quantum analogs 9204:Institute for Advanced Study 8655:The Quantum Theory of Motion 6391:Collapse of the wavefunction 6211:probability density function 6117:{\displaystyle e_{\mu }^{i}} 5285:{\displaystyle \mathbf {A} } 4365:{\displaystyle q^{\text{I}}} 3280:{\displaystyle q^{\text{I}}} 3137:second law of thermodynamics 2136:{\displaystyle \mathbf {Q} } 1397:{\displaystyle \mathbf {j} } 923:. The theory is named after 7: 16368:Quantum stochastic calculus 16358:Quantum measurement problem 16280:Mach–Zehnder interferometer 15512:Silverman, Mark P. (1993). 15496:10.1088/0305-4470/38/26/013 15419:European Physical Journal D 15310:10.1088/0305-4470/37/40/015 15253:Goldstein, Sheldon (2001). 14628:10.1103/PhysRevE.102.013104 14039:Nonlocality in microsystems 12920:10.1088/0953-4075/37/18/N02 12848:10.1088/0953-4075/35/22/316 12683:10.1088/0305-4470/37/44/L02 12344:. Philosophy of Cosmology. 12146:10.1088/1751-8113/40/12/s08 11486:10.1088/0305-4470/37/49/011 11433:10.1088/0305-4470/35/47/311 11371:10.1016/j.shpsa.2020.12.002 10389:European Journal of Physics 10088: 9432:{\displaystyle \omega _{F}} 8402:emerges from that analysis. 6838:counterfactual definiteness 6814:, which in turn led to the 6562:{\displaystyle |\psi |^{2}} 6492:{\displaystyle |\psi |^{2}} 6346:{\displaystyle |\psi |^{2}} 6238:{\displaystyle \rho =R^{2}} 5811:, with the electromagnetic 4535:{\displaystyle |\psi |^{2}} 4063:can be written in the form 3189:{\displaystyle |\psi |^{2}} 3121:{\displaystyle |\psi |^{2}} 3080:{\displaystyle |\psi |^{2}} 3039:{\displaystyle |\psi |^{2}} 3002:{\displaystyle |\psi |^{2}} 2316:For many particles labeled 965: 579:Quantum information science 10: 16686: 15725:"Pilot-Wave Hydrodynamics" 15449:10.1140/epjd/e2007-00191-8 15326:Holland, Peter R. (1993). 15028:10.1103/PhysRevD.69.065014 14959: 14757:de Broglie, Louis (1987). 14692:"Pilot-Wave Hydrodynamics" 14555:10.1103/PhysRevE.92.013006 14256:10.1007/s13194-020-00342-5 13976:, Macmillan, 1971, p. 192. 13393:Princeton University Press 12450:10.1088/2058-7058/22/11/36 12340:Valentini, Antony (2013). 12252:10.1007/s10701-023-00695-w 12199:10.1007/s10701-022-00641-2 12093:10.1088/0305-4470/38/4/R01 10411:10.1088/0143-0807/25/6/008 10362:10.1103/RevModPhys.71.S288 9842:{\displaystyle \omega =ck} 9686:{\displaystyle \omega (k)} 9327: 9292:Ontological Interpretation 8993: 8468:{\displaystyle {\hat {f}}} 7992:Hamilton–Jacobi equation: 7090:many-worlds interpretation 6994:Moyal quantum trajectories 6869:superluminal communication 6842:many-worlds interpretation 6307: 6163: 5805:Stochastic electrodynamics 5800:Stochastic electrodynamics 4545: 2964: 1469:{\displaystyle \psi (q,t)} 16634: 16596: 16548: 16428:Quantum complexity theory 16406:Quantum cellular automata 16381: 16313: 16247: 16160: 16109: 16096:Path integral formulation 16063: 15928: 15885: 15387:10.1007/s10702-005-1128-1 15132:10.1080/09515089008573004 14941:Falk, Dan (16 May 2016). 14803:10.1007/978-94-015-8723-5 14371:Bush, John W. M. (2015). 14000:Scientific Correspondence 13974:The Born–Einstein Letters 13952:10.1007/s10701-004-1940-7 13899:10.1007/s10701-004-1940-7 13846:10.13140/RG.2.1.3663.8245 13753:10.1007/s10701-010-9480-9 13266:10.1007/s10701-004-2009-3 13227:; Wallace, David (2005). 13085:10.1088/1464-4266/4/4/344 12703:David Bohm, Basil Hiley: 11913:10.1016/j.aop.2011.11.010 11857:10.1007/978-3-319-07893-9 11824:10.1007/s10701-016-0043-6 11754:10.1142/S0219749911007344 11701:10.1142/s021974990900516x 11648:10.1142/s0217751x10047889 11592:10.1007/s10702-005-1128-1 11529:10.1007/s10701-010-9456-9 10975:10.1007/s10702-005-3957-3 10502:. Routledge. p. 24. 10350:Reviews of Modern Physics 10080:Surrealistic trajectories 9256:Hamilton-Jacobi equations 9092:stochastic interpretation 6888:Quantum trajectory method 6792:Copenhagen interpretation 6288:observable in principle. 5261:{\displaystyle s_{k}=1/2} 5174:–th particle's spin space 2961:Relation to the Born rule 1991:{\displaystyle q(t)\in Q} 1887: 1408:or probability flux, and 974:There is a configuration 960:mathematical formulations 16495:Quantum machine learning 16475:Quantum key distribution 16465:Quantum image processing 16455:Quantum error correction 16305:Wheeler's delayed choice 15797:25 November 2014 at the 15120:Philosophical Psychology 14690:Bush, John W.M. (2015). 14676:10.1088/1402-4896/acb408 14523:15 December 2017 at the 13635:by Sheldon Goldstein in 13420:(Report). Archived from 12300:10.1088/1361-6382/ad1fcb 11028:10.1103/physreva.60.2729 10120: 10105:Superfluid vacuum theory 9977:{\displaystyle \lambda } 9866:{\displaystyle \lambda } 7625:Hamilton–Jacobi equation 7198:can be derived by using 6810:to prove his now-famous 6720:{\displaystyle \Delta p} 6697:{\displaystyle \Delta x} 6523:Operators as observables 6372:Stern–Gerlach experiment 6286:chaotic inflation theory 6004:where the wave function 5634:{\displaystyle \otimes } 3388:conditional wavefunction 1112:{\displaystyle \phi (x)} 614:Quantum machine learning 367:Wheeler's delayed-choice 16411:Quantum finite automata 15817:, Cambridge University. 15781:"Bohmian-Mechanics.net" 15182:Physical Review Letters 14766:Annales de la Fondation 14238:Simpson, W.M.R (2021). 13494:Holland, Peter (2001). 13161:"Bittner Group Webpage" 12950:Physics Physique Fizika 12760:10.1126/science.1501466 11992:Physical Review Letters 11782:: Conf. Ser. 67 012035. 11780:2007 Journal of Physics 11305:10.1126/science.1202218 11176:10.1023/A:1012321402475 10307:10.1126/science.1202218 9198:According to physicist 9133:hidden-variables theory 7233:de Broglie's hypothesis 7073:Occam's-razor criticism 6298:quantum non-equilibrium 6180:quantum non-equilibrium 6166:Quantum non-equilibrium 5095:— the mass, charge and 4572: 1842:quantum non-equilibrium 1811:at some moment of time 324:Leggett–Garg inequality 16515:Quantum neural network 15837:The David Bohm Society 15604:10.1098/rspa.2004.1394 15099:10.1103/PhysRev.85.180 15065:10.1103/PhysRev.85.166 14910:10.1126/sciadv.1501466 14112:, pp.291–300, therein 13987:Physics and Philosophy 13922:Foundations of Physics 13869:Foundations of Physics 13723:Foundations of Physics 13579:Foundations of Physics 13236:Foundations of Physics 13005:10.1103/PhysRev.47.777 12509:Bell, John S. (1987). 12222:Foundations of Physics 12169:Foundations of Physics 11794:Foundations of Physics 11509:Foundations of Physics 11121:Foundations of Physics 11085:10.1098/rspa.2013.0699 10821:Physics and Philosophy 10733:10.1098/rspa.2011.0598 10620:10.1103/PhysRev.85.166 10484:Foundations of Physics 10471:Foundations of Physics 10456:D. Bohm and B. Hiley: 10256:10.1098/rspa.2004.1394 10196:10.1103/PhysRev.85.166 10068: 10037: 10006: 9978: 9954: 9926: 9895: 9867: 9843: 9813: 9744: 9687: 9639: 9605: 9564: 9537: 9485: 9433: 9307:philosopher of science 9286:sounded too much like 9193: 9123:de Broglie–Bohm Theory 9094:of quantum mechanics. 9080:Schrödinger's equation 9022: 8981: 8920: 8919:{\displaystyle d^{3}x} 8890: 8847: 8827: 8807: 8770: 8677: 8637: 8605: 8469: 8440: 8420: 8390: 8345: 8325: 8305: 8270: 8199: 7981: 7848: 7771: 7727: 7610: 7567: 7487: 7423: 7390: 7348: 7262: 7225: 7196:Schrödinger's equation 7142: 7134: 7070: 7061: 7050: 7029:universal wavefunction 7024: 6979: 6850:performed a series of 6759: 6721: 6698: 6603:double slit experiment 6563: 6493: 6347: 6271: 6239: 6203: 6182: 6160:Exploiting nonlocality 6150:energy-momentum tensor 6118: 6082: 6051: 6018: 5998: 5898: 5790: 5757: 5706: 5673: 5635: 5615: 5524: 5498: 5478: 5389: 5356: 5319: 5286: 5262: 5221: 5197: 5168: 5144: 5113: 5089: 5031: 4944: 4870: 4608: 4536: 4420: 4393: 4366: 4339: 4319: 4249: 4226: 4057: 4037: 4010: 3980: 3844: 3802: 3766: 3726: 3683: 3660:. Also, the fact that 3654: 3627: 3607: 3571: 3489: 3380: 3344: 3308: 3281: 3254: 3190: 3122: 3081: 3040: 3003: 2948: 2884: 2818: 2786: 2766: 2743: 2646: 2617: 2597: 2570:Schrödinger's equation 2559: 2536: 2359: 2339: 2307: 2194: 2157: 2137: 2106: 2074: 2041: 1992: 1957: 1934: 1868:is an illustration of 1866:double-slit experiment 1861: 1848:Double-slit experiment 1825: 1805: 1750: 1654: 1551: 1478:Schrödinger's equation 1470: 1431: 1398: 1376: 1113: 1084: 1064: 1035: 1015: 988: 901:de Broglie–Bohm theory 115: 16540:Quantum teleportation 16053:Wave–particle duality 15809:10 April 2016 at the 15742:"Bohmian Mechanics" ( 15730:18 March 2021 at the 15158:. London: Routledge. 14170:10.1515/zna-1992-1201 14135:10 March 2014 at the 13705:von Neumann, J. 1932 13693:22 March 2016 at the 13381:Bryce Seligman DeWitt 13333:10.1093/bjps/47.2.222 12487:. Scientific American 10851:10.17877/DE290R-14213 10151:causal interpretation 10147:Bohm's interpretation 10069: 10038: 10007: 9979: 9955: 9927: 9896: 9868: 9844: 9814: 9745: 9688: 9640: 9606: 9565: 9538: 9486: 9434: 9355:Hydrodynamic walkers 9282:. Later he felt that 9280:Causal Interpretation 9188: 9013: 9007:, author of the 1964 8982: 8921: 8891: 8848: 8846:{\displaystyle \psi } 8828: 8808: 8771: 8678: 8676:{\displaystyle \psi } 8638: 8636:{\displaystyle (v,w)} 8606: 8470: 8441: 8421: 8391: 8346: 8326: 8306: 8271: 8200: 7982: 7859:Continuity equation: 7849: 7772: 7728: 7611: 7568: 7488: 7424: 7391: 7349: 7263: 7226: 7135: 7133:{\displaystyle \psi } 7118: 7065: 7057: 7041: 7017:actual configuration 7014: 6980: 6918:University of Houston 6852:Bell test experiments 6816:Bell test experiments 6760: 6722: 6699: 6597:Different predictions 6587:uncertainty principle 6564: 6494: 6348: 6272: 6270:{\displaystyle \psi } 6240: 6204: 6173: 6156:equation of motion. 6119: 6083: 6057:is the corresponding 6052: 6019: 6017:{\displaystyle \psi } 5999: 5899: 5897:{\displaystyle u^{i}} 5839:does not take place. 5791: 5737: 5707: 5674: 5636: 5616: 5525: 5523:{\displaystyle \psi } 5499: 5479: 5390: 5357: 5320: 5287: 5263: 5222: 5198: 5196:{\displaystyle s_{k}} 5169: 5145: 5114: 5090: 5032: 4924: 4850: 4609: 4537: 4421: 4394: 4367: 4340: 4320: 4250: 4248:{\displaystyle \phi } 4227: 4058: 4056:{\displaystyle \psi } 4038: 4011: 3981: 3845: 3803: 3767: 3727: 3684: 3655: 3628: 3626:{\displaystyle \psi } 3608: 3572: 3490: 3381: 3345: 3309: 3282: 3255: 3191: 3123: 3082: 3041: 3004: 2949: 2864: 2819: 2787: 2767: 2765:{\displaystyle \psi } 2744: 2647: 2618: 2598: 2560: 2537: 2360: 2340: 2308: 2195: 2158: 2156:{\displaystyle \psi } 2138: 2121:We use variations of 2107: 2075: 2042: 1993: 1958: 1935: 1870:wave–particle duality 1855: 1826: 1806: 1751: 1655: 1531: 1471: 1432: 1399: 1377: 1114: 1085: 1065: 1036: 1016: 1014:{\displaystyle q^{k}} 989: 309:Elitzur–Vaidman 299:Davisson–Germer 116: 16571:Quantum field theory 16500:Quantum metamaterial 16445:Quantum cryptography 16175:Consistent histories 15776:– via YouTube. 15754:(30 November 2016). 15111:Bohm, David (1990). 13298:MIT Press/Bradford, 13028:Maudlin, T. (1994). 12356:– via YouTube. 11268:26 June 2011 at the 10752:3 March 2016 at the 10550:Holland, P. (2001). 10432:Bohm, David (1957). 10048: 10017: 9989: 9968: 9937: 9906: 9878: 9857: 9824: 9755: 9698: 9668: 9616: 9576: 9554: 9496: 9444: 9416: 9375:vacuum fluctuations 9030:quantum field theory 8934: 8900: 8860: 8837: 8817: 8780: 8690: 8667: 8615: 8479: 8450: 8430: 8410: 8368: 8335: 8315: 8280: 8213: 7996: 7863: 7781: 7737: 7636: 7577: 7509: 7433: 7400: 7358: 7277: 7239: 7206: 7124: 7114:many-worlds approach 7046:global wave function 6998:quantum trajectories 6937: 6903:Wyatt group web-site 6731: 6708: 6685: 6642:improve this section 6583:usual interpretation 6536: 6466: 6423:improve this section 6320: 6261: 6251:Schrödinger equation 6216: 6193: 6096: 6065: 6032: 6008: 5912: 5881: 5849:Riemannian manifolds 5823:Quantum field theory 5716: 5687: 5651: 5625: 5534: 5514: 5488: 5400: 5370: 5330: 5300: 5274: 5231: 5211: 5180: 5158: 5125: 5103: 5046: 4628: 4589: 4509: 4403: 4376: 4349: 4329: 4259: 4239: 4070: 4047: 4020: 3993: 3865: 3812: 3776: 3740: 3701: 3682:{\displaystyle Q(t)} 3664: 3637: 3617: 3581: 3502: 3397: 3354: 3318: 3291: 3264: 3209: 3163: 3095: 3054: 3013: 2976: 2831: 2796: 2776: 2756: 2659: 2627: 2607: 2578: 2549: 2372: 2349: 2320: 2207: 2175: 2147: 2125: 2084: 2055: 2022: 1967: 1947: 1901: 1815: 1763: 1664: 1483: 1445: 1412: 1386: 1123: 1094: 1074: 1045: 1025: 998: 978: 921:Schrödinger equation 574:Quantum field theory 486:Consistent histories 123:Schrödinger equation 50: 16670:Quantum measurement 16556:Quantum fluctuation 16525:Quantum programming 16485:Quantum logic gates 16470:Quantum information 16450:Quantum electronics 15910:Classical mechanics 15586:2005RSPSA.461..253V 15488:2005JPhA...38.6037S 15441:2007EPJD...44..319S 15369:2005FoPhL..18..549N 15292:2004JPhA...37.9549H 15255:"Bohmian Mechanics" 15204:2004PhRvL..93i0402D 15175:"Bohmian Mechanics" 15091:1952PhRv...85..180B 15057:1952PhRv...85..166B 15020:2004PhRvD..69f5014B 14981:1994SciAm.270e..58A 14969:Scientific American 14850:2000PhLA..268..224H 14711:2015AnRFM..47..269B 14460:2015AnRFM..47..269B 14427:on 25 November 2016 14392:2015PhT....68h..47B 14316:2006PhRvL..97o4101C 14124:Olival Freire jr.: 14065:1979NCimB..52...15P 13985:Werner Heisenberg, 13944:2005FoPh...35..177H 13891:2005FoPh...35..177H 13838:2012AUSFP..SE..112T 13789:1927ZPhy...40..322M 13745:2010FoPh...40.1333B 13729:(9–10): 1333–1340. 13591:1982FoPh...12..989B 13516:on 10 November 2011 13463:2008arXiv0811.0810V 13258:2005FoPh...35..517B 13124:2007JPCA..11110171. 13118:(41): 10171–10185. 13077:2002JOptB...4S.482A 13055:Journal of Optics B 12996:1935PhRv...47..777E 12941:Bell J. S. (1964). 12889:2001quant.ph..8038S 12840:2002JPhB...35.4751B 12793:Anil Ananthaswamy: 12675:2004JPhA...37L.547H 12612:2004JSP...116..959D 12567:1996quant.ph..1013D 12442:2009PhyW...22k..32V 12397:1991PhLA..158....1V 12244:2023FoPh...53...54F 12191:2022FoPh...52..116F 12138:2007JPhA...40.2997D 12085:2005JPhA...38R...1D 12014:2004PhRvL..93i0402D 11958:2012JPhCS.361a2008G 11905:2012AnPhy.327..421G 11816:2017FoPh...47..174S 11746:2010arXiv1002.3226N 11724:Int. J. Quantum Inf 11693:2008arXiv0811.1905N 11640:2010IJMPA..25.1477N 11574:2005FoPhL..18..549N 11521:2010FoPh...40..658H 11478:2004JPhA...3711935H 11462:(49): 11935–11943. 11425:2002JPhA...3510117D 11409:(47): 10117–10127. 11362:2021SHPSA..85...16F 11297:2011Sci...332.1170K 11291:(6034): 1170–1173. 11233:2001PhLA..290..205G 11188:, therein: section 11133:1996FoPh...26.1441G 11077:2013RSPSA.47030699D 11020:1999PhRvA..60.2729D 10967:2005FoPhL..18..123N 10904:2004FoPhL..17..363N 10843:2006quant.ph.11032P 10790:1992JSP....67..843D 10725:2012RSPSA.468..990T 10672:1992JSP....67..843D 10580:on 10 November 2011 10571:2001NCimB.116.1143H 10299:2011Sci...332.1170K 10293:(6034): 1170–1173. 10238:2005RSPSA.461..253V 10188:1952PhRv...85..166B 10115:Probability current 9785: 9715: 9349: 9264:continuity equation 9176:Copenhagen approach 9153:measurement problem 9061:quantum decoherence 8362:Newtonian mechanics 8276:and velocity field 7623:from above and the 7621:continuity equation 7504:continuity equation 6777:classical mechanics 6255:continuity equation 6247:quantum equilibrium 6113: 5945: 5861:boundary conditions 5772: 5205:spin quantum number 4914: 4562:Costa de Beauregard 3691:probability density 2928: 2004:classical mechanics 1942:configuration space 1721: 1595: 1406:probability current 948:measurement problem 362:Stern–Gerlach 159:Classical mechanics 16609:in popular culture 16391:Quantum algorithms 16239:Von Neumann–Wigner 16219:Objective collapse 15915:Old quantum theory 15688:, Springer, 2009, 15674:, Springer, 2012, 14073:10.1007/BF02743566 14053:Il Nuovo Cimento B 13797:10.1007/BF01400372 13599:10.1007/BF01889272 13534:Peter R. Holland: 12575:10.1007/BF00276801 12327:10.1007/BF00670477 11141:10.1007/BF02272366 11061:(2162): 20130699. 10798:10.1007/BF01049004 10680:10.1007/BF01049004 10133:Also known as the 10095:Madelung equations 10064: 10033: 10002: 9974: 9950: 9922: 9891: 9863: 9839: 9809: 9771: 9740: 9701: 9683: 9635: 9601: 9560: 9533: 9481: 9429: 9410:Trigger frequency 9347: 9299:Jean-Pierre Vigier 9084:Madelung equations 9026:special relativity 8977: 8916: 8886: 8843: 8823: 8803: 8766: 8673: 8633: 8601: 8465: 8436: 8416: 8386: 8341: 8321: 8301: 8266: 8195: 7977: 7844: 7767: 7723: 7606: 7563: 7483: 7419: 7386: 7344: 7258: 7221: 7150:special relativity 7146:luminiferous ether 7130: 6975: 6781:initial conditions 6755: 6717: 6694: 6559: 6489: 6343: 6267: 6235: 6199: 6183: 6154:Heisenberg picture 6114: 6099: 6078: 6047: 6014: 5994: 5931: 5894: 5833:stochastic process 5786: 5758: 5702: 5669: 5631: 5611: 5520: 5494: 5474: 5385: 5352: 5315: 5282: 5258: 5217: 5193: 5164: 5150:— the appropriate 5140: 5109: 5085: 5027: 5025: 4900: 4604: 4532: 4491:Jean-Pierre Vigier 4445:special relativity 4416: 4389: 4362: 4335: 4315: 4245: 4222: 4053: 4033: 4006: 3976: 3840: 3798: 3762: 3722: 3679: 3650: 3623: 3603: 3567: 3485: 3376: 3340: 3304: 3277: 3250: 3186: 3118: 3089:initial conditions 3077: 3036: 2999: 2944: 2914: 2814: 2782: 2762: 2739: 2642: 2613: 2593: 2555: 2532: 2355: 2335: 2303: 2190: 2153: 2133: 2102: 2070: 2037: 1988: 1953: 1930: 1862: 1860:of single photons. 1838:fluid fluctuations 1821: 1801: 1746: 1698: 1650: 1581: 1466: 1427: 1394: 1372: 1109: 1080: 1060: 1031: 1011: 984: 550:Von Neumann–Wigner 530:Objective-collapse 329:Mach–Zehnder 319:Leggett inequality 314:Franck–Hertz 164:Old quantum theory 111: 16652: 16651: 16626:Quantum mysticism 16604:Schrödinger's cat 16535:Quantum simulator 16505:Quantum metrology 16433:Quantum computing 16396:Quantum amplifier 16373:Quantum spacetime 16338:Quantum cosmology 16328:Quantum chemistry 16028:Scattering theory 15976:Zero-point energy 15971:Degenerate levels 15879:Quantum mechanics 15694:978-3-540-89343-1 15680:978-3-642-30690-7 15570:(2053): 253–272. 15560:Valentini, Antony 15527:978-0-521-44631-0 15472:(26): 6037–6049. 15337:978-0-521-48543-2 15276:(40): 9549–9556. 15165:978-0-415-12185-9 14998:Physical Review D 14828:Physics Letters A 14812:978-90-481-4646-8 14616:Physical Review E 14588:. Quanta Magazine 14543:Physical Review E 14516:John W. M. Bush: 14400:10.1063/PT.3.2882 14163:(12): 1175–1186. 14110:978-90-481-9421-6 14094:Olival Freire Jr. 13480:978-0-19-956056-1 13360:978-3-030-40067-5 13132:10.1021/jp079540+ 13039:978-0-631-18609-0 12914:(18): 3781–3783. 12520:978-0-521-33495-2 12385:Physics Letters A 12122:(12): 2997–3031. 11883:Annals of Physics 11866:978-3-319-07893-9 11730:(2011): 367–377. 11211:Physics Letters A 11127:(11): 1441–1455. 10998:Physical Review A 10565:(10): 1143–1172. 10536:978-0-521-48543-2 10509:978-0-415-12185-9 10443:978-0-8122-1002-6 10222:(2053): 253–272. 10143:Bohmian mechanics 10137:pilot wave theory 10077: 10076: 9656:harmony of phases 9260:quantum potential 9241:Bohmian mechanics 9234:Sheldon Goldstein 9180:Werner Heisenberg 9149:Schrödinger's cat 9111:quantum potential 9052:pilot wave theory 9043:Pilot-wave theory 9005:John Stewart Bell 8826:{\displaystyle S} 8702: 8590: 8559: 8539: 8462: 8439:{\displaystyle f} 8419:{\displaystyle H} 8384: 8353:quantum potential 8344:{\displaystyle R} 8324:{\displaystyle V} 8296: 8264: 8242: 8185: 8127: 8060: 8034: 7970: 7904: 7533: 7477: 7452: 6808:John Stewart Bell 6678: 6677: 6670: 6459: 6458: 6451: 6202:{\displaystyle R} 6142:conservation laws 6138:geodesic equation 6044: 5989: 5982: 5955: 5904:of an elementary 5809:quantum mechanics 5504:–th particle (in 5497:{\displaystyle k} 5449: 5220:{\displaystyle k} 5167:{\displaystyle k} 5112:{\displaystyle k} 4984: 4898: 4830: 4753: 4695: 4662: 4456:Lorentz-invariant 4413: 4386: 4359: 4338:{\displaystyle t} 4294: 4281: 4213: 4200: 4172: 4153: 4140: 4121: 4105: 4092: 4030: 4003: 3967: 3948: 3935: 3916: 3900: 3887: 3822: 3786: 3750: 3732:implies that the 3647: 3591: 3552: 3530: 3467: 3454: 3426: 3407: 3364: 3328: 3301: 3274: 3244: 3231: 2912: 2853: 2785:{\displaystyle V} 2712: 2681: 2616:{\displaystyle V} 2558:{\displaystyle N} 2463: 2431: 2402: 2358:{\displaystyle k} 2277: 2252: 2230: 1963:and trajectories 1956:{\displaystyle Q} 1858:weak measurements 1824:{\displaystyle t} 1740: 1708: 1696: 1579: 1505: 1439:momentum operator 1424: 1363: 1347: 1319: 1254: 1161: 1083:{\displaystyle N} 1034:{\displaystyle Q} 987:{\displaystyle q} 909:quantum mechanics 897: 896: 604:Scattering theory 584:Quantum computing 357:Schrödinger's cat 289:Bell's inequality 97: 72: 41:Quantum mechanics 16:(Redirected from 16677: 16642: 16641: 16353:Quantum geometry 16348:Quantum dynamics 16205:Superdeterminism 16137:Rarita–Schwinger 16086:Matrix mechanics 15941:Bra–ket notation 15872: 15865: 15858: 15849: 15848: 15777: 15769: 15734:Bush, J. W. M., 15699:Peter R. Holland 15615: 15597: 15579: 15577:quant-ph/0403034 15555: 15553: 15551: 15542:. Archived from 15536:Streater, Ray F. 15531: 15519: 15507: 15481: 15479:quant-ph/0412050 15460: 15434: 15432:quant-ph/0310096 15413: 15411: 15409:quant-ph/0412119 15398: 15380: 15362: 15360:quant-ph/0406173 15341: 15321: 15303: 15285: 15283:quant-ph/0406054 15264: 15249: 15215: 15197: 15195:quant-ph/0303156 15179: 15169: 15150: 15148: 15146: 15140: 15134:. Archived from 15117: 15102: 15068: 15039: 15013: 14992: 14953: 14952: 14938: 14932: 14931: 14921: 14898:Science Advances 14889: 14880: 14879: 14861: 14843: 14834:(4–6): 224–227. 14823: 14817: 14816: 14788: 14782: 14781: 14763: 14754: 14748: 14747: 14744:Gauthier-Villars 14739: 14733: 14732: 14730: 14696: 14687: 14681: 14680: 14678: 14654: 14648: 14647: 14607: 14601: 14600: 14595: 14593: 14581: 14575: 14574: 14534: 14528: 14514: 14508: 14507: 14505: 14503: 14488: 14482: 14481: 14479: 14443: 14437: 14436: 14434: 14432: 14426: 14420:. Archived from 14411: 14377: 14368: 14362: 14361: 14359: 14357: 14342: 14336: 14335: 14301: 14292: 14286: 14285: 14275: 14235: 14229: 14222: 14216: 14210: 14204: 14203: 14201: 14199:quant-ph/0010020 14189: 14183: 14182: 14172: 14148: 14139: 14122: 14116: 14091: 14085: 14084: 14035: 14029: 14022: 14016: 14009: 14003: 13996: 13990: 13983: 13977: 13970: 13964: 13963: 13937: 13935:quant-ph/0401017 13917: 13911: 13910: 13884: 13882:quant-ph/0401017 13864: 13858: 13857: 13831: 13815: 13809: 13808: 13783:(3–4): 322–326. 13771: 13765: 13764: 13738: 13715: 13709: 13703: 13697: 13684: 13678: 13672: 13666: 13659: 13653: 13650: 13644: 13625: 13619: 13618: 13570: 13564: 13553: 13547: 13532: 13526: 13525: 13523: 13521: 13515: 13500: 13491: 13485: 13484: 13456: 13440: 13434: 13433: 13431: 13429: 13414:Callender, Craig 13410: 13404: 13371: 13365: 13364: 13343: 13337: 13336: 13313: 13307: 13285: 13279: 13277: 13251: 13249:quant-ph/0403094 13233: 13225:Brown, Harvey R. 13221: 13202: 13201: 13199: 13187: 13181: 13180: 13178: 13176: 13171:on 5 August 2021 13157: 13151: 13150: 13148: 13146: 13103: 13097: 13096: 13070: 13068:quant-ph/0112005 13050: 13044: 13043: 13025: 13019: 13016: 13010: 13009: 13007: 12974: 12968: 12967: 12965: 12947: 12938: 12932: 12931: 12899: 12893: 12892: 12882: 12880:quant-ph/0108038 12866: 12860: 12859: 12833: 12831:quant-ph/0206196 12813: 12807: 12804: 12798: 12791: 12782: 12781: 12771: 12739: 12733: 12726: 12720: 12701: 12695: 12694: 12668: 12666:quant-ph/0401008 12648: 12642: 12641: 12623: 12605: 12603:quant-ph/0308038 12585: 12579: 12578: 12560: 12558:quant-ph/9601013 12551:(2–3): 379–397. 12540: 12534: 12531: 12525: 12524: 12506: 12497: 12496: 12494: 12492: 12476: 12470: 12469: 12435: 12415: 12409: 12408: 12380: 12374: 12364: 12358: 12357: 12355: 12353: 12337: 12331: 12330: 12310: 12304: 12303: 12293: 12273: 12264: 12263: 12237: 12217: 12211: 12210: 12184: 12164: 12158: 12157: 12131: 12129:quant-ph/0506173 12111: 12105: 12104: 12078: 12076:quant-ph/0407116 12058: 12052: 12051: 12025: 12007: 12005:quant-ph/0303156 11987: 11978: 11977: 11951: 11931: 11925: 11924: 11898: 11877: 11871: 11870: 11842: 11836: 11835: 11809: 11789: 11783: 11774:Hrvoje Nikolić: 11772: 11766: 11765: 11739: 11719: 11713: 11712: 11686: 11666: 11660: 11659: 11633: 11624:(7): 1477–1505. 11613: 11604: 11603: 11585: 11567: 11565:quant-ph/0406173 11547: 11541: 11540: 11504: 11498: 11497: 11471: 11469:quant-ph/0407089 11451: 11445: 11444: 11418: 11416:quant-ph/0202104 11398: 11392: 11391: 11373: 11355: 11331: 11325: 11324: 11279: 11273: 11259: 11253: 11252: 11226: 11224:quant-ph/0102071 11217:(5–6): 205–213. 11208: 11199: 11193: 11187: 11159: 11153: 11152: 11116: 11107: 11106: 11096: 11070: 11046: 11040: 11039: 11013: 11011:quant-ph/9801070 11004:(4): 2729–2736. 10993: 10987: 10986: 10960: 10958:quant-ph/0302152 10940: 10934: 10933: 10915: 10897: 10895:quant-ph/0208185 10877: 10871: 10870: 10836: 10834:quant-ph/0611032 10816: 10810: 10809: 10783: 10781:quant-ph/0308039 10774:(5–6): 843–907. 10763: 10757: 10744: 10718: 10698: 10692: 10691: 10665: 10663:quant-ph/0308039 10656:(5–6): 843–907. 10645: 10632: 10631: 10599: 10590: 10589: 10587: 10585: 10579: 10573:. Archived from 10556: 10547: 10541: 10540: 10520: 10514: 10513: 10493: 10487: 10480: 10474: 10467: 10461: 10454: 10448: 10447: 10429: 10423: 10422: 10404: 10402:quant-ph/0404128 10380: 10374: 10373: 10356:(2): S288–S297. 10341: 10335: 10334: 10278: 10272: 10267: 10249: 10231: 10229:quant-ph/0403034 10210: 10204: 10203: 10171: 10154: 10131: 10073: 10071: 10070: 10065: 10063: 10062: 10042: 10040: 10039: 10034: 10032: 10031: 10011: 10009: 10008: 10003: 10001: 10000: 9983: 9981: 9980: 9975: 9959: 9957: 9956: 9951: 9949: 9948: 9931: 9929: 9928: 9923: 9921: 9920: 9900: 9898: 9897: 9892: 9890: 9889: 9872: 9870: 9869: 9864: 9848: 9846: 9845: 9840: 9818: 9816: 9815: 9810: 9808: 9807: 9798: 9797: 9784: 9779: 9767: 9766: 9749: 9747: 9746: 9741: 9736: 9731: 9730: 9714: 9709: 9692: 9690: 9689: 9684: 9644: 9642: 9641: 9636: 9631: 9630: 9610: 9608: 9607: 9602: 9591: 9590: 9569: 9567: 9566: 9561: 9542: 9540: 9539: 9534: 9529: 9524: 9523: 9508: 9507: 9490: 9488: 9487: 9482: 9477: 9472: 9471: 9456: 9455: 9438: 9436: 9435: 9430: 9428: 9427: 9350: 9346: 9098:Peter R. Holland 9086:, being quantum 9065:John von Neumann 9048:Louis de Broglie 8986: 8984: 8983: 8978: 8976: 8975: 8970: 8952: 8941: 8925: 8923: 8922: 8917: 8912: 8911: 8895: 8893: 8892: 8887: 8873: 8856:The probability 8852: 8850: 8849: 8844: 8832: 8830: 8829: 8824: 8812: 8810: 8809: 8804: 8787: 8775: 8773: 8772: 8767: 8762: 8733: 8704: 8703: 8695: 8682: 8680: 8679: 8674: 8642: 8640: 8639: 8634: 8610: 8608: 8607: 8602: 8591: 8589: 8572: 8568: 8561: 8560: 8552: 8540: 8532: 8519: 8474: 8472: 8471: 8466: 8464: 8463: 8455: 8445: 8443: 8442: 8437: 8425: 8423: 8422: 8417: 8395: 8393: 8392: 8387: 8385: 8380: 8372: 8350: 8348: 8347: 8342: 8330: 8328: 8327: 8322: 8310: 8308: 8307: 8302: 8297: 8292: 8284: 8275: 8273: 8272: 8267: 8265: 8260: 8256: 8255: 8245: 8243: 8241: 8233: 8232: 8223: 8204: 8202: 8201: 8196: 8191: 8187: 8186: 8184: 8174: 8162: 8152: 8141: 8140: 8130: 8128: 8126: 8118: 8117: 8108: 8097: 8096: 8078: 8061: 8059: 8048: 8035: 8033: 8025: 8015: 8000: 7986: 7984: 7983: 7978: 7976: 7972: 7971: 7966: 7956: 7941: 7930: 7905: 7903: 7895: 7885: 7870: 7853: 7851: 7850: 7845: 7843: 7842: 7837: 7822: 7811: 7794: 7776: 7774: 7773: 7768: 7757: 7749: 7748: 7732: 7730: 7729: 7724: 7719: 7718: 7714: 7700: 7672: 7649: 7615: 7613: 7612: 7607: 7605: 7604: 7599: 7590: 7573:for the density 7572: 7570: 7569: 7564: 7559: 7558: 7534: 7532: 7524: 7516: 7492: 7490: 7489: 7484: 7482: 7478: 7473: 7465: 7453: 7445: 7440: 7428: 7426: 7425: 7420: 7418: 7407: 7396:. Assuming that 7395: 7393: 7392: 7387: 7382: 7368: 7353: 7351: 7350: 7345: 7343: 7342: 7329: 7321: 7290: 7267: 7265: 7264: 7259: 7257: 7246: 7230: 7228: 7227: 7222: 7177:Peter R. Holland 7165:tacit assumption 7139: 7137: 7136: 7131: 7079:Hugh Everett III 6984: 6982: 6981: 6976: 6962: 6961: 6952: 6951: 6916:'s group at the 6764: 6762: 6761: 6756: 6726: 6724: 6723: 6718: 6703: 6701: 6700: 6695: 6673: 6666: 6662: 6659: 6653: 6622: 6614: 6577:Hidden variables 6568: 6566: 6565: 6560: 6558: 6557: 6552: 6543: 6516:phenomenological 6498: 6496: 6495: 6490: 6488: 6487: 6482: 6473: 6454: 6447: 6443: 6440: 6434: 6403: 6395: 6352: 6350: 6349: 6344: 6342: 6341: 6336: 6327: 6281:Antony Valentini 6276: 6274: 6273: 6268: 6244: 6242: 6241: 6236: 6234: 6233: 6208: 6206: 6205: 6200: 6176:Antony Valentini 6174:Diagram made by 6123: 6121: 6120: 6115: 6112: 6107: 6087: 6085: 6084: 6079: 6077: 6076: 6056: 6054: 6053: 6048: 6046: 6045: 6037: 6023: 6021: 6020: 6015: 6003: 6001: 6000: 5995: 5990: 5988: 5984: 5983: 5975: 5971: 5967: 5966: 5957: 5956: 5948: 5944: 5939: 5929: 5924: 5923: 5903: 5901: 5900: 5895: 5893: 5892: 5813:zero-point field 5795: 5793: 5792: 5787: 5782: 5781: 5771: 5766: 5756: 5751: 5711: 5709: 5708: 5703: 5701: 5700: 5695: 5678: 5676: 5675: 5670: 5640: 5638: 5637: 5632: 5620: 5618: 5617: 5612: 5607: 5606: 5601: 5592: 5591: 5586: 5577: 5576: 5571: 5562: 5554: 5553: 5548: 5529: 5527: 5526: 5521: 5503: 5501: 5500: 5495: 5483: 5481: 5480: 5475: 5470: 5469: 5464: 5455: 5450: 5445: 5444: 5443: 5430: 5425: 5424: 5412: 5411: 5394: 5392: 5391: 5386: 5384: 5383: 5378: 5361: 5359: 5358: 5353: 5351: 5337: 5324: 5322: 5321: 5316: 5314: 5313: 5308: 5294:vector potential 5291: 5289: 5288: 5283: 5281: 5267: 5265: 5264: 5259: 5254: 5243: 5242: 5226: 5224: 5223: 5218: 5202: 5200: 5199: 5194: 5192: 5191: 5173: 5171: 5170: 5165: 5149: 5147: 5146: 5141: 5139: 5138: 5133: 5118: 5116: 5115: 5110: 5094: 5092: 5091: 5086: 5084: 5083: 5071: 5070: 5058: 5057: 5036: 5034: 5033: 5028: 5026: 5016: 5012: 5008: 5007: 5002: 4993: 4985: 4983: 4982: 4981: 4968: 4967: 4962: 4956: 4954: 4953: 4943: 4938: 4913: 4908: 4899: 4897: 4896: 4895: 4882: 4881: 4872: 4869: 4864: 4831: 4829: 4818: 4794: 4793: 4788: 4773: 4772: 4767: 4758: 4754: 4752: 4735: 4728: 4727: 4708: 4696: 4694: 4693: 4681: 4663: 4661: 4653: 4652: 4651: 4646: 4636: 4613: 4611: 4610: 4605: 4603: 4602: 4597: 4541: 4539: 4538: 4533: 4531: 4530: 4525: 4516: 4496:weak-measurement 4425: 4423: 4422: 4417: 4415: 4414: 4411: 4398: 4396: 4395: 4390: 4388: 4387: 4384: 4371: 4369: 4368: 4363: 4361: 4360: 4357: 4344: 4342: 4341: 4336: 4324: 4322: 4321: 4316: 4296: 4295: 4292: 4283: 4282: 4279: 4254: 4252: 4251: 4246: 4231: 4229: 4228: 4223: 4215: 4214: 4211: 4202: 4201: 4198: 4174: 4173: 4170: 4155: 4154: 4151: 4142: 4141: 4138: 4123: 4122: 4119: 4107: 4106: 4103: 4094: 4093: 4090: 4062: 4060: 4059: 4054: 4042: 4040: 4039: 4034: 4032: 4031: 4028: 4015: 4013: 4012: 4007: 4005: 4004: 4001: 3985: 3983: 3982: 3977: 3969: 3968: 3965: 3950: 3949: 3946: 3937: 3936: 3933: 3918: 3917: 3914: 3902: 3901: 3898: 3889: 3888: 3885: 3849: 3847: 3846: 3841: 3824: 3823: 3820: 3807: 3805: 3804: 3799: 3788: 3787: 3784: 3771: 3769: 3768: 3763: 3752: 3751: 3748: 3731: 3729: 3728: 3723: 3688: 3686: 3685: 3680: 3659: 3657: 3656: 3651: 3649: 3648: 3645: 3632: 3630: 3629: 3624: 3612: 3610: 3609: 3604: 3593: 3592: 3589: 3576: 3574: 3573: 3568: 3554: 3553: 3550: 3532: 3531: 3528: 3494: 3492: 3491: 3486: 3469: 3468: 3465: 3456: 3455: 3452: 3428: 3427: 3424: 3409: 3408: 3405: 3385: 3383: 3382: 3377: 3366: 3365: 3362: 3349: 3347: 3346: 3341: 3330: 3329: 3326: 3313: 3311: 3310: 3305: 3303: 3302: 3299: 3286: 3284: 3283: 3278: 3276: 3275: 3272: 3259: 3257: 3256: 3251: 3246: 3245: 3242: 3233: 3232: 3229: 3195: 3193: 3192: 3187: 3185: 3184: 3179: 3170: 3127: 3125: 3124: 3119: 3117: 3116: 3111: 3102: 3086: 3084: 3083: 3078: 3076: 3075: 3070: 3061: 3045: 3043: 3042: 3037: 3035: 3034: 3029: 3020: 3008: 3006: 3005: 3000: 2998: 2997: 2992: 2983: 2953: 2951: 2950: 2945: 2927: 2922: 2913: 2911: 2910: 2909: 2896: 2895: 2886: 2883: 2878: 2854: 2852: 2841: 2823: 2821: 2820: 2815: 2813: 2812: 2804: 2791: 2789: 2788: 2783: 2771: 2769: 2768: 2763: 2748: 2746: 2745: 2740: 2723: 2722: 2713: 2711: 2703: 2702: 2693: 2682: 2680: 2669: 2651: 2649: 2648: 2643: 2641: 2640: 2635: 2622: 2620: 2619: 2614: 2602: 2600: 2599: 2594: 2592: 2591: 2586: 2564: 2562: 2561: 2556: 2541: 2539: 2538: 2533: 2519: 2518: 2513: 2498: 2497: 2492: 2483: 2482: 2477: 2468: 2464: 2459: 2455: 2454: 2444: 2432: 2430: 2429: 2417: 2403: 2401: 2393: 2392: 2391: 2386: 2376: 2364: 2362: 2361: 2356: 2344: 2342: 2341: 2336: 2334: 2333: 2328: 2312: 2310: 2309: 2304: 2290: 2282: 2278: 2273: 2265: 2253: 2245: 2231: 2229: 2221: 2220: 2211: 2199: 2197: 2196: 2191: 2189: 2188: 2183: 2167:Guiding equation 2162: 2160: 2159: 2154: 2142: 2140: 2139: 2134: 2132: 2111: 2109: 2108: 2103: 2101: 2100: 2092: 2079: 2077: 2076: 2071: 2069: 2068: 2063: 2046: 2044: 2043: 2038: 2036: 2035: 2030: 1997: 1995: 1994: 1989: 1962: 1960: 1959: 1954: 1939: 1937: 1936: 1931: 1929: 1830: 1828: 1827: 1822: 1810: 1808: 1807: 1802: 1800: 1799: 1794: 1770: 1755: 1753: 1752: 1747: 1742: 1741: 1733: 1720: 1715: 1710: 1709: 1701: 1697: 1695: 1694: 1693: 1677: 1659: 1657: 1656: 1651: 1594: 1589: 1580: 1578: 1577: 1576: 1563: 1562: 1553: 1550: 1545: 1506: 1504: 1493: 1475: 1473: 1472: 1467: 1436: 1434: 1433: 1428: 1426: 1425: 1417: 1403: 1401: 1400: 1395: 1393: 1381: 1379: 1378: 1373: 1368: 1364: 1359: 1355: 1354: 1349: 1348: 1340: 1335: 1320: 1318: 1314: 1313: 1303: 1302: 1301: 1296: 1290: 1289: 1279: 1259: 1255: 1250: 1246: 1245: 1235: 1187: 1186: 1162: 1160: 1152: 1151: 1150: 1137: 1135: 1134: 1118: 1116: 1115: 1110: 1089: 1087: 1086: 1081: 1069: 1067: 1066: 1061: 1059: 1058: 1053: 1040: 1038: 1037: 1032: 1020: 1018: 1017: 1012: 1010: 1009: 993: 991: 990: 985: 927:(1892–1987) and 925:Louis de Broglie 917:guiding equation 889: 882: 875: 516:Superdeterminism 169:Bra–ket notation 120: 118: 117: 112: 104: 99: 98: 90: 78: 73: 71: 60: 32: 31: 21: 16685: 16684: 16680: 16679: 16678: 16676: 16675: 16674: 16655: 16654: 16653: 16648: 16630: 16616:Wigner's friend 16592: 16583:Quantum gravity 16544: 16530:Quantum sensing 16510:Quantum network 16490:Quantum machine 16460:Quantum imaging 16423:Quantum circuit 16418:Quantum channel 16377: 16323:Quantum biology 16309: 16285:Elitzur–Vaidman 16260:Davisson–Germer 16243: 16195:Hidden-variable 16185:de Broglie–Bohm 16162:Interpretations 16156: 16105: 16059: 15946:Complementarity 15924: 15881: 15876: 15846: 15811:Wayback Machine 15799:Wayback Machine 15772: 15732:Wayback Machine 15720: 15715: 15628: 15626:Further reading 15623: 15549: 15547: 15546:on 13 June 2006 15528: 15378:10.1.1.252.6803 15338: 15301:10.1.1.252.5757 15177: 15166: 15144: 15142: 15141:on 4 March 2016 15138: 15115: 15079:Physical Review 15045:Physical Review 14962: 14957: 14956: 14948:Quanta Magazine 14939: 14935: 14890: 14883: 14859:10.1.1.339.2104 14824: 14820: 14813: 14789: 14785: 14761: 14755: 14751: 14740: 14736: 14694: 14688: 14684: 14663:Physica Scripta 14655: 14651: 14608: 14604: 14591: 14589: 14582: 14578: 14535: 14531: 14525:Wayback Machine 14515: 14511: 14501: 14499: 14497:Quanta Magazine 14489: 14485: 14444: 14440: 14430: 14428: 14424: 14375: 14369: 14365: 14355: 14353: 14343: 14339: 14304:Phys. Rev. Lett 14299: 14293: 14289: 14236: 14232: 14223: 14219: 14211: 14207: 14190: 14186: 14149: 14142: 14137:Wayback Machine 14123: 14119: 14092: 14088: 14036: 14032: 14028:(1997), p. 133. 14024:F. David Peat, 14023: 14019: 14010: 14006: 13997: 13993: 13989:(1958), p. 133. 13984: 13980: 13971: 13967: 13918: 13914: 13865: 13861: 13816: 13812: 13772: 13768: 13716: 13712: 13704: 13700: 13695:Wayback Machine 13685: 13681: 13673: 13669: 13660: 13656: 13651: 13647: 13626: 13622: 13585:(10): 989–999. 13571: 13567: 13555:F. David Peat, 13554: 13550: 13533: 13529: 13519: 13517: 13513: 13503:Nuovo Cimento B 13498: 13492: 13488: 13481: 13441: 13437: 13427: 13425: 13424:on 12 June 2010 13411: 13407: 13385:R. Neill Graham 13379:, pp. 3–140 of 13372: 13368: 13361: 13344: 13340: 13314: 13310: 13286: 13282: 13231: 13222: 13205: 13188: 13184: 13174: 13172: 13159: 13158: 13154: 13144: 13142: 13104: 13100: 13051: 13047: 13040: 13026: 13022: 13018:Bell, page 115. 13017: 13013: 12990:(10): 777–780. 12975: 12971: 12945: 12939: 12935: 12900: 12896: 12867: 12863: 12814: 12810: 12805: 12801: 12792: 12785: 12754:(2): e1501466. 12740: 12736: 12730:European Review 12727: 12723: 12702: 12698: 12649: 12645: 12621:10.1.1.252.1653 12586: 12582: 12541: 12537: 12532: 12528: 12521: 12507: 12500: 12490: 12488: 12477: 12473: 12416: 12412: 12381: 12377: 12365: 12361: 12351: 12349: 12338: 12334: 12311: 12307: 12274: 12267: 12218: 12214: 12165: 12161: 12112: 12108: 12059: 12055: 11988: 11981: 11932: 11928: 11878: 11874: 11867: 11843: 11839: 11790: 11786: 11773: 11769: 11720: 11716: 11667: 11663: 11614: 11607: 11583:10.1.1.252.6803 11548: 11544: 11505: 11501: 11452: 11448: 11399: 11395: 11332: 11328: 11280: 11276: 11270:Wayback Machine 11260: 11256: 11206: 11200: 11196: 11160: 11156: 11117: 11110: 11047: 11043: 10994: 10990: 10941: 10937: 10878: 10874: 10817: 10813: 10764: 10760: 10754:Wayback Machine 10699: 10695: 10646: 10635: 10608:Physical Review 10600: 10593: 10583: 10581: 10577: 10559:Nuovo Cimento B 10554: 10548: 10544: 10537: 10521: 10517: 10510: 10494: 10490: 10481: 10477: 10468: 10464: 10455: 10451: 10444: 10430: 10426: 10381: 10377: 10342: 10338: 10279: 10275: 10216:Proc. R. Soc. A 10211: 10207: 10176:Physical Review 10172: 10168: 10163: 10158: 10157: 10132: 10128: 10123: 10091: 10082: 10055: 10051: 10049: 10046: 10045: 10024: 10020: 10018: 10015: 10014: 9996: 9992: 9990: 9987: 9986: 9969: 9966: 9965: 9944: 9940: 9938: 9935: 9934: 9913: 9909: 9907: 9904: 9903: 9885: 9881: 9879: 9876: 9875: 9858: 9855: 9854: 9825: 9822: 9821: 9803: 9799: 9793: 9789: 9780: 9775: 9762: 9758: 9756: 9753: 9752: 9732: 9726: 9722: 9710: 9705: 9699: 9696: 9695: 9669: 9666: 9665: 9626: 9622: 9617: 9614: 9613: 9586: 9582: 9577: 9574: 9573: 9555: 9552: 9551: 9525: 9519: 9515: 9503: 9499: 9497: 9494: 9493: 9473: 9467: 9463: 9451: 9447: 9445: 9442: 9441: 9423: 9419: 9417: 9414: 9413: 9405:zitterbewegung 9361:SED pilot wave 9332: 9326: 9276: 9243: 9172:Albert Einstein 9119:Brownian motion 9045: 9011:wrote in 1982: 8996: 8971: 8966: 8965: 8948: 8937: 8935: 8932: 8931: 8907: 8903: 8901: 8898: 8897: 8869: 8861: 8858: 8857: 8838: 8835: 8834: 8818: 8815: 8814: 8783: 8781: 8778: 8777: 8758: 8729: 8694: 8693: 8691: 8688: 8687: 8668: 8665: 8664: 8616: 8613: 8612: 8573: 8551: 8550: 8531: 8524: 8520: 8518: 8480: 8477: 8476: 8454: 8453: 8451: 8448: 8447: 8431: 8428: 8427: 8411: 8408: 8407: 8373: 8371: 8369: 8366: 8365: 8336: 8333: 8332: 8316: 8313: 8312: 8285: 8283: 8281: 8278: 8277: 8251: 8247: 8246: 8244: 8234: 8228: 8224: 8222: 8214: 8211: 8210: 8170: 8163: 8148: 8136: 8132: 8131: 8129: 8119: 8113: 8109: 8107: 8092: 8088: 8074: 8052: 8047: 8046: 8042: 8026: 8011: 8001: 7999: 7997: 7994: 7993: 7952: 7942: 7940: 7926: 7919: 7915: 7896: 7881: 7871: 7869: 7864: 7861: 7860: 7838: 7833: 7832: 7818: 7807: 7790: 7782: 7779: 7778: 7753: 7744: 7740: 7738: 7735: 7734: 7710: 7696: 7686: 7682: 7668: 7645: 7637: 7634: 7633: 7632:Decomposition: 7600: 7595: 7594: 7586: 7578: 7575: 7574: 7554: 7550: 7525: 7517: 7515: 7510: 7507: 7506: 7466: 7464: 7460: 7444: 7436: 7434: 7431: 7430: 7414: 7403: 7401: 7398: 7397: 7378: 7364: 7359: 7356: 7355: 7325: 7317: 7310: 7306: 7286: 7278: 7275: 7274: 7253: 7242: 7240: 7237: 7236: 7207: 7204: 7203: 7189: 7125: 7122: 7121: 7083:physically real 7075: 7006: 6996:as well as the 6957: 6953: 6944: 6940: 6938: 6935: 6934: 6924: 6914:Eric R. Bittner 6900: 6894:Robert E. Wyatt 6890: 6877: 6875:Classical limit 6800: 6732: 6729: 6728: 6709: 6706: 6705: 6686: 6683: 6682: 6674: 6663: 6657: 6654: 6639: 6623: 6612: 6599: 6579: 6553: 6548: 6547: 6539: 6537: 6534: 6533: 6525: 6483: 6478: 6477: 6469: 6467: 6464: 6463: 6455: 6444: 6438: 6435: 6420: 6404: 6393: 6384: 6359: 6337: 6332: 6331: 6323: 6321: 6318: 6317: 6310: 6262: 6259: 6258: 6229: 6225: 6217: 6214: 6213: 6194: 6191: 6190: 6168: 6162: 6108: 6103: 6097: 6094: 6093: 6072: 6068: 6066: 6063: 6062: 6036: 6035: 6033: 6030: 6029: 6009: 6006: 6005: 5974: 5973: 5972: 5962: 5958: 5947: 5946: 5940: 5935: 5930: 5928: 5919: 5915: 5913: 5910: 5909: 5888: 5884: 5882: 5879: 5878: 5845: 5825: 5802: 5777: 5773: 5767: 5762: 5752: 5741: 5717: 5714: 5713: 5696: 5691: 5690: 5688: 5685: 5684: 5652: 5649: 5648: 5626: 5623: 5622: 5602: 5597: 5596: 5587: 5582: 5581: 5572: 5567: 5566: 5558: 5549: 5544: 5543: 5535: 5532: 5531: 5515: 5512: 5511: 5489: 5486: 5485: 5465: 5460: 5459: 5451: 5439: 5435: 5431: 5429: 5420: 5416: 5407: 5403: 5401: 5398: 5397: 5379: 5374: 5373: 5371: 5368: 5367: 5347: 5333: 5331: 5328: 5327: 5309: 5304: 5303: 5301: 5298: 5297: 5277: 5275: 5272: 5271: 5250: 5238: 5234: 5232: 5229: 5228: 5212: 5209: 5208: 5187: 5183: 5181: 5178: 5177: 5159: 5156: 5155: 5134: 5129: 5128: 5126: 5123: 5122: 5104: 5101: 5100: 5097:magnetic moment 5079: 5075: 5066: 5062: 5053: 5049: 5047: 5044: 5043: 5024: 5023: 5003: 4998: 4997: 4989: 4977: 4973: 4969: 4963: 4958: 4957: 4955: 4949: 4945: 4939: 4928: 4909: 4904: 4891: 4887: 4883: 4877: 4873: 4871: 4865: 4854: 4846: 4842: 4835: 4822: 4817: 4808: 4807: 4789: 4784: 4783: 4768: 4763: 4762: 4736: 4723: 4719: 4709: 4707: 4703: 4689: 4685: 4680: 4673: 4654: 4647: 4642: 4641: 4637: 4635: 4631: 4629: 4626: 4625: 4620:Pauli spin term 4598: 4593: 4592: 4590: 4587: 4586: 4577:To incorporate 4575: 4550: 4526: 4521: 4520: 4512: 4510: 4507: 4506: 4441: 4436: 4410: 4406: 4404: 4401: 4400: 4383: 4379: 4377: 4374: 4373: 4356: 4352: 4350: 4347: 4346: 4330: 4327: 4326: 4291: 4287: 4278: 4274: 4260: 4257: 4256: 4240: 4237: 4236: 4210: 4206: 4197: 4193: 4169: 4165: 4150: 4146: 4137: 4133: 4118: 4114: 4102: 4098: 4089: 4085: 4071: 4068: 4067: 4048: 4045: 4044: 4027: 4023: 4021: 4018: 4017: 4000: 3996: 3994: 3991: 3990: 3964: 3960: 3945: 3941: 3932: 3928: 3913: 3909: 3897: 3893: 3884: 3880: 3866: 3863: 3862: 3819: 3815: 3813: 3810: 3809: 3783: 3779: 3777: 3774: 3773: 3747: 3743: 3741: 3738: 3737: 3702: 3699: 3698: 3689:is random with 3665: 3662: 3661: 3644: 3640: 3638: 3635: 3634: 3618: 3615: 3614: 3588: 3584: 3582: 3579: 3578: 3549: 3545: 3527: 3523: 3503: 3500: 3499: 3464: 3460: 3451: 3447: 3423: 3419: 3404: 3400: 3398: 3395: 3394: 3361: 3357: 3355: 3352: 3351: 3325: 3321: 3319: 3316: 3315: 3298: 3294: 3292: 3289: 3288: 3271: 3267: 3265: 3262: 3261: 3241: 3237: 3228: 3224: 3210: 3207: 3206: 3202: 3180: 3175: 3174: 3166: 3164: 3161: 3160: 3112: 3107: 3106: 3098: 3096: 3093: 3092: 3071: 3066: 3065: 3057: 3055: 3052: 3051: 3030: 3025: 3024: 3016: 3014: 3011: 3010: 2993: 2988: 2987: 2979: 2977: 2974: 2973: 2969: 2963: 2923: 2918: 2905: 2901: 2897: 2891: 2887: 2885: 2879: 2868: 2845: 2840: 2832: 2829: 2828: 2805: 2800: 2799: 2797: 2794: 2793: 2777: 2774: 2773: 2757: 2754: 2753: 2718: 2714: 2704: 2698: 2694: 2692: 2673: 2668: 2660: 2657: 2656: 2636: 2631: 2630: 2628: 2625: 2624: 2608: 2605: 2604: 2587: 2582: 2581: 2579: 2576: 2575: 2572: 2550: 2547: 2546: 2514: 2509: 2508: 2493: 2488: 2487: 2478: 2473: 2472: 2450: 2446: 2445: 2443: 2439: 2425: 2421: 2416: 2394: 2387: 2382: 2381: 2377: 2375: 2373: 2370: 2369: 2350: 2347: 2346: 2329: 2324: 2323: 2321: 2318: 2317: 2286: 2266: 2264: 2260: 2244: 2222: 2216: 2212: 2210: 2208: 2205: 2204: 2184: 2179: 2178: 2176: 2173: 2172: 2169: 2148: 2145: 2144: 2128: 2126: 2123: 2122: 2093: 2088: 2087: 2085: 2082: 2081: 2064: 2059: 2058: 2056: 2053: 2052: 2031: 2026: 2025: 2023: 2020: 2019: 1968: 1965: 1964: 1948: 1945: 1944: 1925: 1902: 1899: 1898: 1895: 1890: 1850: 1816: 1813: 1812: 1795: 1790: 1789: 1766: 1764: 1761: 1760: 1732: 1731: 1716: 1711: 1700: 1699: 1689: 1685: 1681: 1676: 1665: 1662: 1661: 1590: 1585: 1572: 1568: 1564: 1558: 1554: 1552: 1546: 1535: 1497: 1492: 1484: 1481: 1480: 1446: 1443: 1442: 1416: 1415: 1413: 1410: 1409: 1389: 1387: 1384: 1383: 1350: 1339: 1338: 1337: 1336: 1334: 1330: 1309: 1305: 1304: 1297: 1292: 1291: 1285: 1281: 1280: 1278: 1241: 1237: 1236: 1234: 1230: 1182: 1178: 1153: 1146: 1142: 1138: 1136: 1130: 1126: 1124: 1121: 1120: 1095: 1092: 1091: 1075: 1072: 1071: 1054: 1049: 1048: 1046: 1043: 1042: 1026: 1023: 1022: 1005: 1001: 999: 996: 995: 979: 976: 975: 968: 962:of the theory. 938:and explicitly 893: 864: 863: 862: 627: 619: 618: 564: 563:Advanced topics 556: 555: 554: 506:Hidden-variable 496:de Broglie–Bohm 475: 473:Interpretations 465: 464: 463: 433: 425: 424: 423: 381: 373: 372: 371: 338: 294:CHSH inequality 283: 275: 274: 273: 202:Complementarity 196: 188: 187: 186: 154: 125: 100: 89: 88: 74: 64: 59: 51: 48: 47: 28: 23: 22: 15: 12: 11: 5: 16683: 16673: 16672: 16667: 16650: 16649: 16647: 16646: 16635: 16632: 16631: 16629: 16628: 16623: 16618: 16613: 16612: 16611: 16600: 16598: 16594: 16593: 16591: 16590: 16585: 16580: 16579: 16578: 16568: 16563: 16561:Casimir effect 16558: 16552: 16550: 16546: 16545: 16543: 16542: 16537: 16532: 16527: 16522: 16520:Quantum optics 16517: 16512: 16507: 16502: 16497: 16492: 16487: 16482: 16477: 16472: 16467: 16462: 16457: 16452: 16447: 16442: 16441: 16440: 16430: 16425: 16420: 16415: 16414: 16413: 16403: 16398: 16393: 16387: 16385: 16379: 16378: 16376: 16375: 16370: 16365: 16360: 16355: 16350: 16345: 16340: 16335: 16330: 16325: 16319: 16317: 16311: 16310: 16308: 16307: 16302: 16297: 16295:Quantum eraser 16292: 16287: 16282: 16277: 16272: 16267: 16262: 16257: 16251: 16249: 16245: 16244: 16242: 16241: 16236: 16231: 16226: 16221: 16216: 16211: 16210: 16209: 16208: 16207: 16192: 16187: 16182: 16177: 16172: 16166: 16164: 16158: 16157: 16155: 16154: 16149: 16144: 16139: 16134: 16129: 16124: 16119: 16113: 16111: 16107: 16106: 16104: 16103: 16098: 16093: 16088: 16083: 16078: 16073: 16067: 16065: 16061: 16060: 16058: 16057: 16056: 16055: 16050: 16040: 16035: 16030: 16025: 16020: 16015: 16010: 16005: 16000: 15995: 15990: 15985: 15980: 15979: 15978: 15973: 15968: 15963: 15953: 15951:Density matrix 15948: 15943: 15938: 15932: 15930: 15926: 15925: 15923: 15922: 15917: 15912: 15907: 15906: 15905: 15895: 15889: 15887: 15883: 15882: 15875: 15874: 15867: 15860: 15852: 15845: 15844: 15839: 15834: 15829: 15824: 15818: 15801: 15789: 15784: 15778: 15770: 15761:PBS Space Time 15748: 15739: 15721: 15719: 15718:External links 15716: 15714: 15713: 15696: 15682: 15668: 15647: 15629: 15627: 15624: 15622: 15621: 15616: 15595:10.1.1.252.849 15556: 15532: 15526: 15509: 15461: 15425:(2): 319–326. 15414: 15399: 15353:(6): 549–561. 15342: 15336: 15323: 15265: 15250: 15170: 15164: 15151: 15126:(2): 271–286. 15108: 15085:(2): 180–193. 15074: 15051:(2): 166–179. 15040: 15011:hep-th/0304105 14993: 14963: 14961: 14958: 14955: 14954: 14933: 14881: 14818: 14811: 14783: 14772:(4): 399–421. 14749: 14734: 14705:(1): 269–292. 14682: 14649: 14602: 14576: 14529: 14509: 14483: 14454:(1): 269–292. 14438: 14363: 14337: 14310:(15): 154101. 14287: 14230: 14217: 14205: 14184: 14140: 14117: 14086: 14030: 14017: 14004: 13991: 13978: 13965: 13928:(2): 177–196. 13912: 13875:(2): 177–196. 13859: 13810: 13766: 13710: 13698: 13679: 13667: 13654: 13645: 13620: 13565: 13548: 13527: 13486: 13479: 13435: 13405: 13366: 13359: 13338: 13327:(2): 222–228. 13317:Deutsch, David 13308: 13288:Daniel Dennett 13280: 13242:(4): 517–540. 13203: 13182: 13165:k2.chem.uh.edu 13152: 13098: 13061:(4): 482–488. 13045: 13038: 13020: 13011: 12969: 12933: 12894: 12861: 12808: 12799: 12783: 12734: 12721: 12696: 12643: 12580: 12535: 12526: 12519: 12498: 12471: 12410: 12375: 12359: 12332: 12321:(4): 221–230. 12305: 12265: 12212: 12159: 12106: 12053: 11979: 11926: 11889:(2): 421–437. 11872: 11865: 11851:. p. 95. 11837: 11800:(2): 174–207. 11784: 11767: 11714: 11677:(3): 595–602. 11661: 11605: 11558:(6): 549–561. 11542: 11515:(6): 658–678. 11499: 11446: 11393: 11326: 11274: 11254: 11194: 11190:3. Conclusions 11170:(4): 395–400. 11154: 11108: 11041: 10988: 10951:(2): 123–138. 10935: 10913:10.1.1.253.838 10888:(4): 363–380. 10872: 10811: 10758: 10693: 10633: 10614:(2): 166–179. 10591: 10542: 10535: 10515: 10508: 10488: 10475: 10462: 10449: 10442: 10424: 10395:(6): 765–769. 10375: 10336: 10273: 10247:10.1.1.252.849 10205: 10182:(2): 166–179. 10165: 10164: 10162: 10159: 10156: 10155: 10125: 10124: 10122: 10119: 10118: 10117: 10112: 10107: 10102: 10097: 10090: 10087: 10081: 10078: 10075: 10074: 10061: 10058: 10054: 10043: 10030: 10027: 10023: 10012: 9999: 9995: 9984: 9973: 9961: 9960: 9947: 9943: 9932: 9919: 9916: 9912: 9901: 9888: 9884: 9873: 9862: 9850: 9849: 9838: 9835: 9832: 9829: 9819: 9806: 9802: 9796: 9792: 9788: 9783: 9778: 9774: 9770: 9765: 9761: 9750: 9739: 9735: 9729: 9725: 9721: 9718: 9713: 9708: 9704: 9693: 9682: 9679: 9676: 9673: 9661: 9660: 9657: 9654: 9651: 9647: 9646: 9634: 9629: 9625: 9621: 9611: 9600: 9597: 9594: 9589: 9585: 9581: 9571: 9559: 9548: 9544: 9543: 9532: 9528: 9522: 9518: 9514: 9511: 9506: 9502: 9491: 9480: 9476: 9470: 9466: 9462: 9459: 9454: 9450: 9439: 9426: 9422: 9411: 9407: 9406: 9403: 9401:zitterbewegung 9398: 9395: 9391: 9390: 9387: 9384: 9381: 9377: 9376: 9373: 9372:internal clock 9370: 9369:bath vibration 9367: 9363: 9362: 9359: 9356: 9353: 9328:Main article: 9325: 9322: 9275: 9272: 9242: 9239: 9184:Wolfgang Pauli 9157:Bell's theorem 9076:Erwin Madelung 9056:Wolfgang Pauli 9050:presented his 9044: 9041: 9009:Bell's theorem 8995: 8992: 8991: 8990: 8989: 8988: 8974: 8969: 8964: 8961: 8958: 8955: 8951: 8947: 8944: 8940: 8915: 8910: 8906: 8885: 8882: 8879: 8876: 8872: 8868: 8865: 8854: 8842: 8822: 8802: 8799: 8796: 8793: 8790: 8786: 8765: 8761: 8757: 8754: 8751: 8748: 8745: 8742: 8739: 8736: 8732: 8728: 8725: 8722: 8719: 8716: 8713: 8710: 8707: 8701: 8698: 8684: 8672: 8661: 8650: 8649: 8645: 8644: 8632: 8629: 8626: 8623: 8620: 8600: 8597: 8594: 8588: 8585: 8582: 8579: 8576: 8571: 8567: 8564: 8558: 8555: 8549: 8546: 8543: 8538: 8535: 8530: 8527: 8523: 8517: 8514: 8511: 8508: 8505: 8502: 8499: 8496: 8493: 8490: 8487: 8484: 8461: 8458: 8435: 8415: 8403: 8398: 8397: 8383: 8379: 8376: 8357: 8356: 8340: 8320: 8311:The potential 8300: 8295: 8291: 8288: 8263: 8259: 8254: 8250: 8240: 8237: 8231: 8227: 8221: 8218: 8206: 8205: 8194: 8190: 8183: 8180: 8177: 8173: 8169: 8166: 8161: 8158: 8155: 8151: 8147: 8144: 8139: 8135: 8125: 8122: 8116: 8112: 8106: 8103: 8100: 8095: 8091: 8087: 8084: 8081: 8077: 8073: 8070: 8067: 8064: 8058: 8055: 8051: 8045: 8041: 8038: 8032: 8029: 8024: 8021: 8018: 8014: 8010: 8007: 8004: 7989: 7988: 7975: 7969: 7965: 7962: 7959: 7955: 7951: 7948: 7945: 7939: 7936: 7933: 7929: 7925: 7922: 7918: 7914: 7911: 7908: 7902: 7899: 7894: 7891: 7888: 7884: 7880: 7877: 7874: 7868: 7856: 7855: 7841: 7836: 7831: 7828: 7825: 7821: 7817: 7814: 7810: 7806: 7803: 7800: 7797: 7793: 7789: 7786: 7766: 7763: 7760: 7756: 7752: 7747: 7743: 7722: 7717: 7713: 7709: 7706: 7703: 7699: 7695: 7692: 7689: 7685: 7681: 7678: 7675: 7671: 7667: 7664: 7661: 7658: 7655: 7652: 7648: 7644: 7641: 7629: 7628: 7617: 7603: 7598: 7593: 7589: 7585: 7582: 7562: 7557: 7553: 7549: 7546: 7543: 7540: 7537: 7531: 7528: 7523: 7520: 7514: 7498: 7497: 7494: 7481: 7476: 7472: 7469: 7463: 7459: 7456: 7451: 7448: 7443: 7439: 7417: 7413: 7410: 7406: 7385: 7381: 7377: 7374: 7371: 7367: 7363: 7354:. Notice that 7341: 7338: 7335: 7332: 7328: 7324: 7320: 7316: 7313: 7309: 7305: 7302: 7299: 7296: 7293: 7289: 7285: 7282: 7270: 7269: 7256: 7252: 7249: 7245: 7220: 7217: 7214: 7211: 7188: 7185: 7129: 7094:Geiger counter 7074: 7071: 7005: 7002: 6974: 6971: 6968: 6965: 6960: 6956: 6950: 6947: 6943: 6922: 6898: 6889: 6886: 6876: 6873: 6834:Bell's theorem 6828:Decades later 6806:: it inspired 6799: 6796: 6754: 6751: 6748: 6745: 6742: 6739: 6736: 6716: 6713: 6693: 6690: 6676: 6675: 6658:September 2024 6626: 6624: 6617: 6611: 6608: 6598: 6595: 6578: 6575: 6556: 6551: 6546: 6542: 6524: 6521: 6503:for details). 6486: 6481: 6476: 6472: 6457: 6456: 6439:September 2024 6407: 6405: 6398: 6392: 6389: 6383: 6380: 6358: 6355: 6340: 6335: 6330: 6326: 6309: 6306: 6266: 6232: 6228: 6224: 6221: 6198: 6164:Main article: 6161: 6158: 6111: 6106: 6102: 6090:Dirac matrices 6075: 6071: 6043: 6040: 6013: 5993: 5987: 5981: 5978: 5970: 5965: 5961: 5954: 5951: 5943: 5938: 5934: 5927: 5922: 5918: 5891: 5887: 5844: 5841: 5824: 5821: 5801: 5798: 5797: 5796: 5785: 5780: 5776: 5770: 5765: 5761: 5755: 5750: 5747: 5744: 5740: 5736: 5733: 5730: 5727: 5724: 5721: 5699: 5694: 5683:in spin space 5668: 5665: 5662: 5659: 5656: 5646: 5643:tensor product 5630: 5610: 5605: 5600: 5595: 5590: 5585: 5580: 5575: 5570: 5565: 5561: 5557: 5552: 5547: 5542: 5539: 5519: 5509: 5493: 5473: 5468: 5463: 5458: 5454: 5448: 5442: 5438: 5434: 5428: 5423: 5419: 5415: 5410: 5406: 5395: 5382: 5377: 5364:magnetic field 5350: 5346: 5343: 5340: 5336: 5325: 5312: 5307: 5280: 5269: 5257: 5253: 5249: 5246: 5241: 5237: 5227:–th particle ( 5216: 5190: 5186: 5175: 5163: 5154:acting in the 5137: 5132: 5120: 5108: 5082: 5078: 5074: 5069: 5065: 5061: 5056: 5052: 5022: 5019: 5015: 5011: 5006: 5001: 4996: 4992: 4988: 4980: 4976: 4972: 4966: 4961: 4952: 4948: 4942: 4937: 4934: 4931: 4927: 4923: 4920: 4917: 4912: 4907: 4903: 4894: 4890: 4886: 4880: 4876: 4868: 4863: 4860: 4857: 4853: 4849: 4845: 4841: 4838: 4836: 4834: 4828: 4825: 4821: 4816: 4813: 4810: 4809: 4806: 4803: 4800: 4797: 4792: 4787: 4782: 4779: 4776: 4771: 4766: 4761: 4757: 4751: 4748: 4745: 4742: 4739: 4734: 4731: 4726: 4722: 4718: 4715: 4712: 4706: 4702: 4699: 4692: 4688: 4684: 4679: 4676: 4674: 4672: 4669: 4666: 4660: 4657: 4650: 4645: 4640: 4634: 4633: 4616:inner products 4601: 4596: 4574: 4571: 4558:Yakir Aharonov 4556:. It draws on 4529: 4524: 4519: 4515: 4449:Dirac equation 4440: 4437: 4435: 4432: 4409: 4382: 4355: 4334: 4314: 4311: 4308: 4305: 4302: 4299: 4290: 4286: 4277: 4273: 4270: 4267: 4264: 4244: 4233: 4232: 4221: 4218: 4209: 4205: 4196: 4192: 4189: 4186: 4183: 4180: 4177: 4168: 4164: 4161: 4158: 4149: 4145: 4136: 4132: 4129: 4126: 4117: 4113: 4110: 4101: 4097: 4088: 4084: 4081: 4078: 4075: 4052: 4026: 3999: 3987: 3986: 3975: 3972: 3963: 3959: 3956: 3953: 3944: 3940: 3931: 3927: 3924: 3921: 3912: 3908: 3905: 3896: 3892: 3883: 3879: 3876: 3873: 3870: 3839: 3836: 3833: 3830: 3827: 3818: 3797: 3794: 3791: 3782: 3761: 3758: 3755: 3746: 3721: 3718: 3715: 3712: 3709: 3706: 3695:square modulus 3678: 3675: 3672: 3669: 3643: 3622: 3602: 3599: 3596: 3587: 3566: 3563: 3560: 3557: 3548: 3544: 3541: 3538: 3535: 3526: 3522: 3519: 3516: 3513: 3510: 3507: 3496: 3495: 3484: 3481: 3478: 3475: 3472: 3463: 3459: 3450: 3446: 3443: 3440: 3437: 3434: 3431: 3422: 3418: 3415: 3412: 3403: 3375: 3372: 3369: 3360: 3339: 3336: 3333: 3324: 3297: 3270: 3249: 3240: 3236: 3227: 3223: 3220: 3217: 3214: 3201: 3198: 3183: 3178: 3173: 3169: 3115: 3110: 3105: 3101: 3074: 3069: 3064: 3060: 3033: 3028: 3023: 3019: 2996: 2991: 2986: 2982: 2965:Main article: 2962: 2959: 2955: 2954: 2943: 2940: 2937: 2934: 2931: 2926: 2921: 2917: 2908: 2904: 2900: 2894: 2890: 2882: 2877: 2874: 2871: 2867: 2863: 2860: 2857: 2851: 2848: 2844: 2839: 2836: 2811: 2808: 2803: 2781: 2761: 2750: 2749: 2738: 2735: 2732: 2729: 2726: 2721: 2717: 2710: 2707: 2701: 2697: 2691: 2688: 2685: 2679: 2676: 2672: 2667: 2664: 2639: 2634: 2612: 2590: 2585: 2571: 2568: 2554: 2543: 2542: 2531: 2528: 2525: 2522: 2517: 2512: 2507: 2504: 2501: 2496: 2491: 2486: 2481: 2476: 2471: 2467: 2462: 2458: 2453: 2449: 2442: 2438: 2435: 2428: 2424: 2420: 2415: 2412: 2409: 2406: 2400: 2397: 2390: 2385: 2380: 2354: 2332: 2327: 2314: 2313: 2302: 2299: 2296: 2293: 2289: 2285: 2281: 2276: 2272: 2269: 2263: 2259: 2256: 2251: 2248: 2243: 2240: 2237: 2234: 2228: 2225: 2219: 2215: 2187: 2182: 2168: 2165: 2152: 2131: 2099: 2096: 2091: 2067: 2062: 2034: 2029: 1987: 1984: 1981: 1978: 1975: 1972: 1952: 1928: 1924: 1921: 1918: 1915: 1912: 1909: 1906: 1894: 1893:The pilot wave 1891: 1889: 1886: 1849: 1846: 1833: 1832: 1820: 1798: 1793: 1788: 1785: 1782: 1779: 1776: 1773: 1769: 1757: 1745: 1739: 1736: 1730: 1727: 1724: 1719: 1714: 1707: 1704: 1692: 1688: 1684: 1680: 1675: 1672: 1669: 1649: 1646: 1643: 1640: 1637: 1634: 1631: 1628: 1625: 1622: 1619: 1616: 1613: 1610: 1607: 1604: 1601: 1598: 1593: 1588: 1584: 1575: 1571: 1567: 1561: 1557: 1549: 1544: 1541: 1538: 1534: 1530: 1527: 1524: 1521: 1518: 1515: 1512: 1509: 1503: 1500: 1496: 1491: 1488: 1465: 1462: 1459: 1456: 1453: 1450: 1423: 1420: 1392: 1371: 1367: 1362: 1358: 1353: 1346: 1343: 1333: 1329: 1326: 1323: 1317: 1312: 1308: 1300: 1295: 1288: 1284: 1277: 1274: 1271: 1268: 1265: 1262: 1258: 1253: 1249: 1244: 1240: 1233: 1229: 1226: 1223: 1220: 1217: 1214: 1211: 1208: 1205: 1202: 1199: 1196: 1193: 1190: 1185: 1181: 1177: 1174: 1171: 1168: 1165: 1159: 1156: 1149: 1145: 1141: 1133: 1129: 1108: 1105: 1102: 1099: 1079: 1057: 1052: 1030: 1008: 1004: 983: 967: 964: 934:The theory is 905:interpretation 895: 894: 892: 891: 884: 877: 869: 866: 865: 861: 860: 855: 850: 845: 840: 835: 830: 825: 820: 815: 810: 805: 800: 795: 790: 785: 780: 775: 770: 765: 760: 755: 750: 745: 740: 735: 730: 725: 720: 715: 710: 705: 700: 695: 690: 685: 680: 675: 670: 665: 660: 655: 650: 645: 640: 635: 629: 628: 625: 624: 621: 620: 617: 616: 611: 606: 601: 599:Density matrix 596: 591: 586: 581: 576: 571: 565: 562: 561: 558: 557: 553: 552: 547: 542: 537: 532: 527: 522: 521: 520: 519: 518: 503: 498: 493: 488: 483: 477: 476: 471: 470: 467: 466: 462: 461: 456: 451: 446: 441: 435: 434: 431: 430: 427: 426: 422: 421: 416: 411: 406: 401: 396: 390: 389: 388: 382: 379: 378: 375: 374: 370: 369: 364: 359: 353: 352: 351: 350: 349: 347:Delayed-choice 342:Quantum eraser 337: 336: 331: 326: 321: 316: 311: 306: 301: 296: 291: 285: 284: 281: 280: 277: 276: 272: 271: 270: 269: 259: 254: 249: 244: 239: 234: 232:Quantum number 229: 224: 219: 214: 209: 204: 198: 197: 194: 193: 190: 189: 185: 184: 179: 173: 172: 171: 166: 161: 155: 152: 151: 148: 147: 146: 145: 140: 135: 127: 126: 121: 110: 107: 103: 96: 93: 87: 84: 81: 77: 70: 67: 63: 58: 55: 44: 43: 37: 36: 26: 9: 6: 4: 3: 2: 16682: 16671: 16668: 16666: 16663: 16662: 16660: 16645: 16637: 16636: 16633: 16627: 16624: 16622: 16619: 16617: 16614: 16610: 16607: 16606: 16605: 16602: 16601: 16599: 16595: 16589: 16586: 16584: 16581: 16577: 16574: 16573: 16572: 16569: 16567: 16564: 16562: 16559: 16557: 16554: 16553: 16551: 16547: 16541: 16538: 16536: 16533: 16531: 16528: 16526: 16523: 16521: 16518: 16516: 16513: 16511: 16508: 16506: 16503: 16501: 16498: 16496: 16493: 16491: 16488: 16486: 16483: 16481: 16480:Quantum logic 16478: 16476: 16473: 16471: 16468: 16466: 16463: 16461: 16458: 16456: 16453: 16451: 16448: 16446: 16443: 16439: 16436: 16435: 16434: 16431: 16429: 16426: 16424: 16421: 16419: 16416: 16412: 16409: 16408: 16407: 16404: 16402: 16399: 16397: 16394: 16392: 16389: 16388: 16386: 16384: 16380: 16374: 16371: 16369: 16366: 16364: 16361: 16359: 16356: 16354: 16351: 16349: 16346: 16344: 16341: 16339: 16336: 16334: 16333:Quantum chaos 16331: 16329: 16326: 16324: 16321: 16320: 16318: 16316: 16312: 16306: 16303: 16301: 16300:Stern–Gerlach 16298: 16296: 16293: 16291: 16288: 16286: 16283: 16281: 16278: 16276: 16273: 16271: 16268: 16266: 16263: 16261: 16258: 16256: 16253: 16252: 16250: 16246: 16240: 16237: 16235: 16234:Transactional 16232: 16230: 16227: 16225: 16224:Quantum logic 16222: 16220: 16217: 16215: 16212: 16206: 16203: 16202: 16201: 16198: 16197: 16196: 16193: 16191: 16188: 16186: 16183: 16181: 16178: 16176: 16173: 16171: 16168: 16167: 16165: 16163: 16159: 16153: 16150: 16148: 16145: 16143: 16140: 16138: 16135: 16133: 16130: 16128: 16125: 16123: 16120: 16118: 16115: 16114: 16112: 16108: 16102: 16099: 16097: 16094: 16092: 16089: 16087: 16084: 16082: 16079: 16077: 16074: 16072: 16069: 16068: 16066: 16062: 16054: 16051: 16049: 16046: 16045: 16044: 16043:Wave function 16041: 16039: 16036: 16034: 16031: 16029: 16026: 16024: 16021: 16019: 16018:Superposition 16016: 16014: 16013:Quantum state 16011: 16009: 16006: 16004: 16001: 15999: 15996: 15994: 15991: 15989: 15986: 15984: 15981: 15977: 15974: 15972: 15969: 15967: 15966:Excited state 15964: 15962: 15959: 15958: 15957: 15954: 15952: 15949: 15947: 15944: 15942: 15939: 15937: 15934: 15933: 15931: 15927: 15921: 15918: 15916: 15913: 15911: 15908: 15904: 15901: 15900: 15899: 15896: 15894: 15891: 15890: 15888: 15884: 15880: 15873: 15868: 15866: 15861: 15859: 15854: 15853: 15850: 15843: 15840: 15838: 15835: 15833: 15830: 15828: 15825: 15822: 15819: 15816: 15812: 15808: 15805: 15802: 15800: 15796: 15793: 15790: 15788: 15785: 15782: 15779: 15775: 15771: 15767: 15763: 15762: 15757: 15753: 15749: 15747: 15745: 15740: 15737: 15733: 15729: 15726: 15723: 15722: 15712: 15711:0-521-48543-6 15708: 15704: 15700: 15697: 15695: 15691: 15687: 15683: 15681: 15677: 15673: 15669: 15667: 15666:0-415-06588-7 15663: 15659: 15655: 15651: 15648: 15646: 15645:0-521-81862-1 15642: 15638: 15634: 15631: 15630: 15620: 15617: 15613: 15609: 15605: 15601: 15596: 15591: 15587: 15583: 15578: 15573: 15569: 15565: 15561: 15557: 15545: 15541: 15537: 15533: 15529: 15523: 15518: 15517: 15510: 15505: 15501: 15497: 15493: 15489: 15485: 15480: 15475: 15471: 15467: 15462: 15458: 15454: 15450: 15446: 15442: 15438: 15433: 15428: 15424: 15420: 15415: 15410: 15405: 15400: 15396: 15392: 15388: 15384: 15379: 15374: 15370: 15366: 15361: 15356: 15352: 15348: 15343: 15339: 15333: 15329: 15324: 15319: 15315: 15311: 15307: 15302: 15297: 15293: 15289: 15284: 15279: 15275: 15271: 15266: 15262: 15261: 15256: 15251: 15247: 15243: 15239: 15235: 15231: 15227: 15223: 15219: 15214: 15213:10.1.1.8.8444 15209: 15205: 15201: 15196: 15191: 15188:(9): 090402. 15187: 15183: 15176: 15171: 15167: 15161: 15157: 15152: 15137: 15133: 15129: 15125: 15121: 15114: 15109: 15106: 15100: 15096: 15092: 15088: 15084: 15080: 15075: 15072: 15066: 15062: 15058: 15054: 15050: 15046: 15041: 15037: 15033: 15029: 15025: 15021: 15017: 15012: 15007: 15004:(6): 065014. 15003: 14999: 14994: 14990: 14986: 14982: 14978: 14974: 14970: 14965: 14964: 14950: 14949: 14944: 14937: 14929: 14925: 14920: 14915: 14911: 14907: 14903: 14899: 14895: 14888: 14886: 14877: 14873: 14869: 14865: 14860: 14855: 14851: 14847: 14842: 14841:gr-qc/9906084 14837: 14833: 14829: 14822: 14814: 14808: 14804: 14800: 14796: 14795: 14787: 14779: 14775: 14771: 14767: 14760: 14753: 14745: 14738: 14729: 14724: 14720: 14716: 14712: 14708: 14704: 14700: 14693: 14686: 14677: 14672: 14668: 14664: 14660: 14653: 14645: 14641: 14637: 14633: 14629: 14625: 14622:(1): 013104. 14621: 14617: 14613: 14606: 14599: 14587: 14580: 14572: 14568: 14564: 14560: 14556: 14552: 14549:(1): 013006. 14548: 14544: 14540: 14533: 14526: 14522: 14519: 14513: 14498: 14494: 14487: 14478: 14473: 14469: 14465: 14461: 14457: 14453: 14449: 14442: 14423: 14419: 14415: 14410: 14409:1721.1/110524 14405: 14401: 14397: 14393: 14389: 14385: 14381: 14380:Physics Today 14374: 14367: 14352: 14348: 14341: 14333: 14329: 14325: 14321: 14317: 14313: 14309: 14305: 14298: 14291: 14283: 14279: 14274: 14269: 14265: 14261: 14257: 14253: 14249: 14245: 14241: 14234: 14227: 14221: 14215: 14209: 14200: 14195: 14188: 14180: 14176: 14171: 14166: 14162: 14158: 14154: 14147: 14145: 14138: 14134: 14131: 14127: 14121: 14115: 14111: 14107: 14103: 14099: 14095: 14090: 14082: 14078: 14074: 14070: 14066: 14062: 14058: 14054: 14048: 14044: 14040: 14034: 14027: 14021: 14014: 14008: 14001: 13995: 13988: 13982: 13975: 13969: 13961: 13957: 13953: 13949: 13945: 13941: 13936: 13931: 13927: 13923: 13916: 13908: 13904: 13900: 13896: 13892: 13888: 13883: 13878: 13874: 13870: 13863: 13855: 13851: 13847: 13843: 13839: 13835: 13830: 13825: 13821: 13814: 13806: 13802: 13798: 13794: 13790: 13786: 13782: 13779: 13778: 13770: 13762: 13758: 13754: 13750: 13746: 13742: 13737: 13732: 13728: 13724: 13720: 13714: 13708: 13702: 13696: 13692: 13689: 13683: 13677: 13671: 13665:(1957). p. x. 13664: 13658: 13649: 13642: 13638: 13637:Physics Today 13634: 13630: 13624: 13616: 13612: 13608: 13604: 13600: 13596: 13592: 13588: 13584: 13580: 13576: 13569: 13562: 13558: 13552: 13545: 13544:0-521-48543-6 13541: 13537: 13531: 13512: 13508: 13504: 13497: 13490: 13482: 13476: 13472: 13468: 13464: 13460: 13455: 13450: 13446: 13439: 13423: 13419: 13415: 13409: 13402: 13401:0-691-08131-X 13398: 13394: 13390: 13386: 13382: 13378: 13377: 13370: 13362: 13356: 13352: 13348: 13342: 13334: 13330: 13326: 13322: 13318: 13312: 13305: 13304:0-262-68117-X 13301: 13297: 13293: 13289: 13284: 13275: 13271: 13267: 13263: 13259: 13255: 13250: 13245: 13241: 13237: 13230: 13226: 13220: 13218: 13216: 13214: 13212: 13210: 13208: 13198: 13193: 13186: 13170: 13166: 13162: 13156: 13141: 13137: 13133: 13129: 13125: 13121: 13117: 13113: 13109: 13102: 13094: 13090: 13086: 13082: 13078: 13074: 13069: 13064: 13060: 13056: 13049: 13041: 13035: 13031: 13024: 13015: 13006: 13001: 12997: 12993: 12989: 12986: 12985: 12980: 12973: 12964: 12959: 12955: 12951: 12944: 12937: 12929: 12925: 12921: 12917: 12913: 12909: 12905: 12898: 12890: 12886: 12881: 12876: 12872: 12865: 12857: 12853: 12849: 12845: 12841: 12837: 12832: 12827: 12823: 12819: 12812: 12803: 12796: 12790: 12788: 12779: 12775: 12770: 12765: 12761: 12757: 12753: 12749: 12745: 12738: 12731: 12725: 12718: 12714: 12713:0-203-98038-7 12710: 12706: 12700: 12692: 12688: 12684: 12680: 12676: 12672: 12667: 12662: 12658: 12654: 12647: 12639: 12635: 12631: 12627: 12622: 12617: 12613: 12609: 12604: 12599: 12595: 12591: 12584: 12576: 12572: 12568: 12564: 12559: 12554: 12550: 12546: 12539: 12530: 12522: 12516: 12512: 12505: 12503: 12486: 12482: 12475: 12467: 12463: 12459: 12455: 12451: 12447: 12443: 12439: 12434: 12429: 12426:(11): 32–37. 12425: 12421: 12420:Physics World 12414: 12406: 12402: 12398: 12394: 12390: 12386: 12379: 12373: 12369: 12363: 12347: 12343: 12336: 12328: 12324: 12320: 12316: 12309: 12301: 12297: 12292: 12287: 12284:(6): 065011. 12283: 12279: 12272: 12270: 12261: 12257: 12253: 12249: 12245: 12241: 12236: 12231: 12227: 12223: 12216: 12208: 12204: 12200: 12196: 12192: 12188: 12183: 12178: 12174: 12170: 12163: 12155: 12151: 12147: 12143: 12139: 12135: 12130: 12125: 12121: 12117: 12110: 12102: 12098: 12094: 12090: 12086: 12082: 12077: 12072: 12068: 12064: 12057: 12049: 12045: 12041: 12037: 12033: 12029: 12024: 12023:10.1.1.8.8444 12019: 12015: 12011: 12006: 12001: 11998:(9): 090402. 11997: 11993: 11986: 11984: 11975: 11971: 11967: 11963: 11959: 11955: 11950: 11945: 11942:(1): 012008. 11941: 11937: 11930: 11922: 11918: 11914: 11910: 11906: 11902: 11897: 11892: 11888: 11884: 11876: 11868: 11862: 11858: 11854: 11850: 11849: 11841: 11833: 11829: 11825: 11821: 11817: 11813: 11808: 11803: 11799: 11795: 11788: 11781: 11777: 11771: 11763: 11759: 11755: 11751: 11747: 11743: 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10351: 10347: 10340: 10332: 10328: 10324: 10320: 10316: 10312: 10308: 10304: 10300: 10296: 10292: 10288: 10284: 10277: 10270: 10265: 10261: 10257: 10253: 10248: 10243: 10239: 10235: 10230: 10225: 10221: 10217: 10209: 10202: 10197: 10193: 10189: 10185: 10181: 10177: 10170: 10166: 10152: 10148: 10144: 10140: 10139: 10138: 10130: 10126: 10116: 10113: 10111: 10108: 10106: 10103: 10101: 10098: 10096: 10093: 10092: 10086: 10059: 10056: 10052: 10044: 10028: 10025: 10021: 10013: 9997: 9993: 9985: 9971: 9963: 9962: 9945: 9941: 9933: 9917: 9914: 9910: 9902: 9886: 9882: 9874: 9860: 9852: 9851: 9836: 9833: 9830: 9827: 9820: 9804: 9800: 9794: 9790: 9786: 9781: 9776: 9772: 9768: 9763: 9759: 9751: 9737: 9733: 9727: 9723: 9719: 9716: 9711: 9706: 9702: 9694: 9677: 9671: 9663: 9662: 9658: 9655: 9652: 9649: 9648: 9627: 9623: 9619: 9612: 9598: 9587: 9583: 9579: 9572: 9549: 9546: 9545: 9526: 9520: 9516: 9512: 9509: 9504: 9500: 9492: 9474: 9468: 9464: 9460: 9457: 9452: 9448: 9440: 9424: 9420: 9412: 9409: 9408: 9404: 9402: 9399: 9396: 9393: 9392: 9388: 9386:monochromatic 9385: 9383:monochromatic 9382: 9379: 9378: 9374: 9371: 9368: 9365: 9364: 9360: 9357: 9354: 9352: 9351: 9345: 9342: 9340: 9339: 9331: 9321: 9318: 9313: 9311: 9308: 9303: 9300: 9295: 9293: 9289: 9288:deterministic 9285: 9281: 9271: 9267: 9265: 9261: 9257: 9253: 9248: 9238: 9235: 9230: 9228: 9223: 9220: 9215: 9213: 9209: 9205: 9201: 9196: 9192: 9187: 9185: 9181: 9177: 9173: 9168: 9166: 9162: 9158: 9154: 9150: 9146: 9142: 9138: 9134: 9129: 9126: 9124: 9120: 9114: 9112: 9108: 9103: 9099: 9095: 9093: 9089: 9085: 9081: 9077: 9072: 9070: 9066: 9062: 9057: 9053: 9049: 9040: 9037: 9035: 9031: 9027: 9021: 9017: 9012: 9010: 9006: 9002: 8972: 8956: 8942: 8929: 8913: 8908: 8904: 8877: 8863: 8855: 8840: 8833:the phase of 8820: 8797: 8794: 8791: 8763: 8759: 8746: 8743: 8737: 8723: 8714: 8708: 8685: 8670: 8662: 8659: 8658: 8656: 8652: 8651: 8647: 8646: 8627: 8624: 8621: 8595: 8583: 8580: 8577: 8569: 8565: 8553: 8547: 8544: 8533: 8528: 8525: 8521: 8515: 8512: 8509: 8503: 8491: 8485: 8456: 8433: 8413: 8404: 8400: 8399: 8381: 8377: 8363: 8359: 8358: 8354: 8338: 8318: 8298: 8293: 8289: 8261: 8257: 8252: 8238: 8235: 8229: 8225: 8219: 8216: 8208: 8207: 8192: 8188: 8178: 8175: 8164: 8156: 8153: 8142: 8137: 8123: 8120: 8114: 8110: 8104: 8101: 8098: 8093: 8082: 8079: 8068: 8056: 8053: 8049: 8043: 8039: 8036: 8030: 8019: 8016: 8005: 7991: 7990: 7973: 7967: 7960: 7957: 7946: 7934: 7931: 7920: 7916: 7912: 7906: 7900: 7889: 7886: 7875: 7866: 7858: 7857: 7839: 7826: 7823: 7812: 7804: 7798: 7795: 7784: 7761: 7758: 7745: 7741: 7720: 7711: 7704: 7701: 7690: 7687: 7683: 7676: 7673: 7662: 7659: 7653: 7650: 7639: 7631: 7630: 7626: 7622: 7618: 7601: 7591: 7583: 7580: 7555: 7551: 7547: 7541: 7535: 7529: 7521: 7512: 7505: 7500: 7499: 7495: 7479: 7474: 7470: 7461: 7457: 7454: 7449: 7446: 7441: 7411: 7408: 7383: 7379: 7375: 7369: 7361: 7336: 7333: 7330: 7322: 7311: 7307: 7303: 7300: 7294: 7291: 7280: 7272: 7271: 7247: 7234: 7218: 7212: 7209: 7201: 7197: 7194: 7193: 7192: 7184: 7182: 7178: 7175:According to 7173: 7171: 7166: 7162: 7159:According to 7157: 7155: 7154:Occam's razor 7151: 7147: 7141: 7127: 7117: 7115: 7111: 7107: 7103: 7099: 7095: 7091: 7087: 7084: 7080: 7069: 7064: 7060: 7056: 7054: 7053:David Deutsch 7049: 7047: 7040: 7038: 7037:H. Dieter Zeh 7034: 7030: 7023: 7020: 7013: 7011: 7001: 6999: 6995: 6990: 6987: 6972: 6963: 6958: 6948: 6945: 6941: 6930: 6928: 6919: 6915: 6911: 6909: 6904: 6895: 6885: 6883: 6872: 6870: 6866: 6860: 6856: 6853: 6849: 6845: 6843: 6839: 6835: 6831: 6826: 6824: 6819: 6817: 6813: 6809: 6805: 6795: 6793: 6789: 6784: 6782: 6778: 6773: 6771: 6765: 6752: 6749: 6746: 6743: 6737: 6714: 6691: 6672: 6669: 6661: 6651: 6647: 6643: 6637: 6636: 6632: 6627:This section 6625: 6621: 6616: 6615: 6607: 6604: 6594: 6590: 6588: 6584: 6574: 6570: 6554: 6544: 6529: 6520: 6517: 6512: 6508: 6504: 6502: 6484: 6474: 6453: 6450: 6442: 6432: 6428: 6424: 6418: 6417: 6413: 6408:This section 6406: 6402: 6397: 6396: 6388: 6379: 6375: 6373: 6368: 6364: 6354: 6338: 6328: 6314: 6305: 6303: 6299: 6295: 6289: 6287: 6282: 6278: 6264: 6256: 6252: 6248: 6230: 6226: 6222: 6219: 6212: 6196: 6187: 6181: 6177: 6172: 6167: 6157: 6155: 6151: 6147: 6143: 6139: 6135: 6131: 6127: 6109: 6104: 6100: 6091: 6073: 6069: 6060: 6038: 6027: 6011: 5991: 5985: 5976: 5968: 5963: 5959: 5949: 5941: 5936: 5932: 5925: 5920: 5916: 5907: 5889: 5885: 5877: 5876:four-velocity 5872: 5869: 5868:vector bundle 5864: 5862: 5858: 5854: 5850: 5840: 5836: 5834: 5830: 5820: 5818: 5814: 5810: 5806: 5783: 5778: 5774: 5768: 5763: 5759: 5753: 5748: 5745: 5742: 5738: 5734: 5728: 5725: 5722: 5697: 5682: 5681:inner product 5663: 5660: 5657: 5647: 5644: 5628: 5608: 5603: 5593: 5588: 5578: 5573: 5555: 5550: 5540: 5537: 5517: 5510: 5507: 5491: 5466: 5440: 5436: 5432: 5426: 5421: 5413: 5408: 5404: 5396: 5380: 5365: 5344: 5338: 5326: 5310: 5295: 5270: 5268:for electron) 5255: 5251: 5247: 5244: 5239: 5235: 5214: 5206: 5188: 5184: 5176: 5161: 5153: 5152:spin operator 5135: 5121: 5106: 5098: 5080: 5076: 5072: 5067: 5063: 5059: 5054: 5050: 5042: 5041: 5040: 5037: 5020: 5017: 5013: 5004: 4986: 4978: 4974: 4970: 4964: 4950: 4946: 4940: 4935: 4932: 4929: 4925: 4921: 4918: 4915: 4910: 4905: 4901: 4892: 4888: 4884: 4878: 4874: 4866: 4861: 4858: 4855: 4851: 4847: 4843: 4839: 4837: 4832: 4826: 4811: 4804: 4798: 4795: 4790: 4780: 4777: 4774: 4769: 4755: 4746: 4743: 4740: 4729: 4724: 4720: 4716: 4713: 4704: 4700: 4697: 4690: 4686: 4682: 4677: 4675: 4667: 4658: 4655: 4648: 4638: 4623: 4621: 4617: 4599: 4584: 4580: 4570: 4567: 4563: 4559: 4555: 4549: 4544: 4527: 4517: 4502: 4499: 4497: 4492: 4488: 4484: 4480: 4475: 4471: 4468: 4463: 4461: 4457: 4452: 4450: 4446: 4431: 4427: 4407: 4380: 4353: 4332: 4312: 4309: 4300: 4288: 4284: 4275: 4271: 4268: 4262: 4242: 4219: 4207: 4203: 4194: 4190: 4187: 4181: 4178: 4166: 4162: 4159: 4147: 4134: 4130: 4127: 4115: 4111: 4099: 4095: 4086: 4082: 4079: 4073: 4066: 4065: 4064: 4050: 4024: 3997: 3973: 3961: 3957: 3954: 3942: 3929: 3925: 3922: 3910: 3906: 3894: 3890: 3881: 3877: 3874: 3868: 3861: 3860: 3859: 3855: 3853: 3834: 3831: 3828: 3816: 3792: 3780: 3756: 3744: 3735: 3716: 3713: 3710: 3704: 3696: 3693:given by the 3692: 3673: 3667: 3641: 3620: 3597: 3585: 3558: 3546: 3542: 3536: 3524: 3517: 3511: 3505: 3482: 3473: 3461: 3457: 3448: 3444: 3441: 3435: 3432: 3420: 3416: 3413: 3401: 3393: 3392: 3391: 3389: 3370: 3358: 3334: 3322: 3295: 3268: 3238: 3234: 3225: 3221: 3218: 3212: 3197: 3181: 3171: 3157: 3152: 3150: 3146: 3141: 3138: 3134: 3129: 3113: 3103: 3090: 3072: 3062: 3047: 3031: 3021: 2994: 2984: 2968: 2958: 2941: 2938: 2935: 2932: 2929: 2924: 2919: 2906: 2902: 2898: 2892: 2888: 2880: 2875: 2872: 2869: 2865: 2861: 2858: 2855: 2849: 2834: 2827: 2826: 2825: 2809: 2806: 2779: 2759: 2736: 2733: 2730: 2727: 2724: 2719: 2708: 2705: 2699: 2695: 2689: 2686: 2683: 2677: 2662: 2655: 2654: 2653: 2637: 2610: 2588: 2567: 2552: 2529: 2523: 2520: 2515: 2505: 2502: 2499: 2494: 2484: 2479: 2465: 2460: 2456: 2451: 2440: 2436: 2433: 2426: 2422: 2418: 2413: 2407: 2398: 2395: 2388: 2378: 2368: 2367: 2366: 2352: 2330: 2300: 2294: 2291: 2279: 2274: 2270: 2261: 2257: 2254: 2249: 2246: 2241: 2235: 2226: 2223: 2213: 2203: 2202: 2201: 2185: 2164: 2150: 2119: 2117: 2113: 2097: 2094: 2065: 2050: 2032: 2016: 2014: 2008: 2005: 2000: 1985: 1982: 1976: 1970: 1950: 1943: 1922: 1916: 1913: 1910: 1904: 1885: 1881: 1877: 1873: 1871: 1867: 1859: 1854: 1845: 1843: 1839: 1818: 1796: 1783: 1780: 1777: 1771: 1758: 1734: 1725: 1722: 1717: 1712: 1702: 1690: 1686: 1682: 1678: 1673: 1670: 1667: 1647: 1641: 1638: 1635: 1629: 1623: 1617: 1614: 1608: 1605: 1602: 1596: 1591: 1586: 1573: 1569: 1565: 1559: 1555: 1547: 1542: 1539: 1536: 1532: 1528: 1525: 1519: 1516: 1513: 1507: 1501: 1486: 1479: 1460: 1457: 1454: 1448: 1440: 1407: 1369: 1365: 1351: 1331: 1327: 1324: 1321: 1315: 1310: 1306: 1298: 1286: 1282: 1275: 1269: 1266: 1263: 1256: 1251: 1247: 1242: 1231: 1227: 1224: 1218: 1212: 1209: 1206: 1200: 1197: 1194: 1191: 1188: 1183: 1172: 1166: 1157: 1154: 1147: 1143: 1139: 1131: 1127: 1103: 1097: 1077: 1055: 1028: 1006: 1002: 981: 973: 972: 971: 963: 961: 957: 953: 949: 943: 941: 937: 936:deterministic 932: 931:(1917–1992). 930: 926: 922: 918: 914: 910: 906: 902: 890: 885: 883: 878: 876: 871: 870: 868: 867: 859: 856: 854: 851: 849: 846: 844: 841: 839: 836: 834: 831: 829: 826: 824: 821: 819: 816: 814: 811: 809: 806: 804: 801: 799: 796: 794: 791: 789: 786: 784: 781: 779: 776: 774: 771: 769: 766: 764: 761: 759: 756: 754: 751: 749: 746: 744: 741: 739: 736: 734: 731: 729: 726: 724: 721: 719: 716: 714: 711: 709: 706: 704: 701: 699: 696: 694: 691: 689: 686: 684: 681: 679: 676: 674: 671: 669: 666: 664: 661: 659: 656: 654: 651: 649: 646: 644: 641: 639: 636: 634: 631: 630: 623: 622: 615: 612: 610: 607: 605: 602: 600: 597: 595: 592: 590: 589:Quantum chaos 587: 585: 582: 580: 577: 575: 572: 570: 567: 566: 560: 559: 551: 548: 546: 545:Transactional 543: 541: 538: 536: 535:Quantum logic 533: 531: 528: 526: 523: 517: 514: 513: 512: 509: 508: 507: 504: 502: 499: 497: 494: 492: 489: 487: 484: 482: 479: 478: 474: 469: 468: 460: 457: 455: 452: 450: 447: 445: 442: 440: 437: 436: 429: 428: 420: 417: 415: 412: 410: 407: 405: 402: 400: 397: 395: 392: 391: 387: 384: 383: 377: 376: 368: 365: 363: 360: 358: 355: 354: 348: 345: 344: 343: 340: 339: 335: 332: 330: 327: 325: 322: 320: 317: 315: 312: 310: 307: 305: 302: 300: 297: 295: 292: 290: 287: 286: 279: 278: 268: 265: 264: 263: 262:Wave function 260: 258: 255: 253: 250: 248: 245: 243: 242:Superposition 240: 238: 235: 233: 230: 228: 225: 223: 220: 218: 215: 213: 210: 208: 205: 203: 200: 199: 192: 191: 183: 180: 178: 175: 174: 170: 167: 165: 162: 160: 157: 156: 150: 149: 144: 141: 139: 136: 134: 131: 130: 129: 128: 124: 91: 85: 68: 65: 61: 53: 46: 45: 42: 39: 38: 34: 33: 30: 19: 16363:Quantum mind 16275:Franck–Hertz 16184: 16117:Klein–Gordon 16071:Formulations 16064:Formulations 15993:Interference 15983:Entanglement 15961:Ground state 15956:Energy level 15929:Fundamentals 15893:Introduction 15759: 15752:O'Dowd, Matt 15743: 15735: 15702: 15685: 15671: 15657: 15636: 15633:John S. Bell 15567: 15563: 15548:. Retrieved 15544:the original 15515: 15469: 15465: 15422: 15418: 15350: 15346: 15327: 15273: 15269: 15258: 15185: 15181: 15155: 15143:. Retrieved 15136:the original 15123: 15119: 15082: 15078: 15048: 15044: 15001: 14997: 14975:(5): 58–67. 14972: 14968: 14946: 14936: 14919:10072/100637 14901: 14897: 14831: 14827: 14821: 14797:. Springer. 14793: 14786: 14769: 14765: 14752: 14743: 14737: 14728:1721.1/89790 14702: 14698: 14685: 14666: 14662: 14652: 14619: 14615: 14605: 14597: 14590:. Retrieved 14579: 14546: 14542: 14532: 14512: 14500:. Retrieved 14496: 14486: 14477:1721.1/89790 14451: 14447: 14441: 14429:. Retrieved 14422:the original 14383: 14379: 14366: 14354:. Retrieved 14351:news.mit.edu 14350: 14340: 14307: 14303: 14290: 14247: 14243: 14233: 14225: 14220: 14208: 14187: 14160: 14156: 14125: 14120: 14101: 14097: 14089: 14056: 14052: 14042: 14038: 14033: 14025: 14020: 14012: 14007: 13999: 13994: 13986: 13981: 13973: 13968: 13925: 13921: 13915: 13872: 13868: 13862: 13819: 13813: 13780: 13775: 13769: 13726: 13722: 13719:Bub, Jeffrey 13713: 13706: 13701: 13682: 13670: 13662: 13657: 13648: 13640: 13636: 13628: 13623: 13582: 13578: 13568: 13560: 13556: 13551: 13535: 13530: 13518:. Retrieved 13511:the original 13506: 13502: 13489: 13444: 13438: 13426:. Retrieved 13422:the original 13408: 13388: 13375: 13369: 13353:. Springer. 13350: 13347:Dürr, Detlef 13341: 13324: 13320: 13311: 13295: 13291: 13283: 13239: 13235: 13185: 13173:. Retrieved 13169:the original 13164: 13155: 13143:. 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