Abstract
This chapter takes us into the domains of quantum nonlocality. The journey starts with a brief introduction to Bohm’s causal theory of quantum mechanics, which serves to further discuss the nonlocality contained in the Schrödinger descriptionQuantum regime!and Schrödinger description. A critical discussion of this causal and deterministic approach illustrates the virtues and limitations associated with the BohmianCausality interpretationInterpretation of quantum mechanics. The tools developed in previous chapters are then applied to a more detailed analysis of quantum nonlocality, both for single-particle and bipartite systems. Some important results are derived, which throw light on the relationship between the quantum potentialQuantum potential and linearity, fluctuations, and nonlocalityQuantum potential!and nonlocality. The bipartite system is further analyzed and the connections between nonlocality, entanglementEntanglement and noncommutativity are disclosed for general continuous variables.
Why is the pilot-wave picture ignored in the text books? Should it not be taught, not as the only way, but as an antidote to the prevailing complacency? To show that vagueness, subjectivity, and indeterminism Indeterminism, are not forced on us by experimental facts, but by deliberate theoretical choice?
Bell (1987, page 160)
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
It is much less known that almost simultaneously and independently, a similar result was published by Solomon (1933).
- 2.
Take for example the spin projections along three different directions: \( \hat{A}=\hat{S}_{x},\) \(\hat{B}=\hat{S}_{y}\) and \(\hat{C}=(\hat{A}+\hat{B})/ \sqrt{2}=(\hat{S}_{x}+\hat{S}_{y})/\sqrt{2}\). If the system possesses spin \( 1/2\), the eigenvalues of each of these operators are the same and equal to \( \pm 1;\) clearly the eigenvalues of \(\hat{C}\) are not the linear combination \( (\pm 1\pm 1)/\sqrt{2}.\)
- 3.
Since Hermann’s argument is little known, and is just the same discovered by Bell 30 years later, it seems of interest to transcribe it here: “Suppose we have an ensemble of physical systems, with \( \mathfrak {R}\) and \(\mathfrak {S}\) physical quantities that can be measured on this ensemble; the expectation value of \(\mathfrak {R}\) (Expt(\(\mathfrak {R}\))) is the average value of all measurementMeasurement outcomes that will be obtained when measuring \(\mathfrak {R}\) on all systems of the ensemble, and is also the value that is expected to be obtained when measuring \(\mathfrak {R}\) on an arbitrary element of this ensemble. Von Neumann requires that for this expectation value-function Expt(\(\mathfrak {R}\)), defined using an ensemble of physical systems and producing a number for every physical quantity, Expt(\(\mathfrak {R}+\mathfrak {S}\)) = Expt(\(\mathfrak {R}) + \text {Expt}(\mathfrak {S}\)). In words: The expectation value of a sum of physical quantities is equal to the sum of the expectation values of both quantities. With this assumption the proof of von Neumann either succeeds or fails.” “For classical physics this requirement is trivial and also for those quantum mechanical observables that \([\)commute\(]\)... Not trivial however is the relation for quantum mechanical quantities for which indeterminacy relations hold. In fact the sum of two such quantities is not even defined: Because a sharp measurement of one of them excludes sharp measurementMeasurement of the other one and thus because both quantities cannot have sharp values at the same time, the commonly used definition of the sum of two quantities breaks down.”
- 4.
Feyerabend noticed that the postulates used in von Neumann’s derivation did not exclude dispersive hidden variables. Now if the hidden variablesHidden variables added to qm had an irreducible dispersion, the quantum variables themselves should continue to be dispersive and things remained essentially the same, except that the theorem needed some reformulation. Mugur-Schächter, on her part, argued that the demonstration was not as general as assumed, since it presupposes that the distribution of the hidden variables (once more, distributed variables) has properties similar to those of the quantum distribution.
- 5.
Without pretending to undermine de Broglie’s credit for his seminal contribution, in most of this book we shall refer to Bohm’s theory, for short, as is customary in present-day literature. An alternative form of qm, similar to Bohm’s, had been proposed many years earlier by Madelung (1927).
- 6.
The well-known book by Bell containing the collection of his articles on the foundations of qm (Bell 1987)was very influential in the revival of Bohm’s theoryBell!and Bohm’s theory. Bell appreciated the objective, deterministic and causal aspects of the pilot-wave theory. It was the search for an answer to the question: Is it that any hidden-variables theory is by necessity nonlocal? what prompted Bell’s work leading to his now famous inequalities.
- 7.
This is achieved by simply inverting the reasoning in the derivations. Two crucial postulates are needed: one is of course the guidance equation; the second demand serves to introduce the quantum potential \(V_{Q}\) into Eq. (8.9) on the basis of an appropriately contrived argument. The simplest procedure is to consider the quantum potentialQuantum potential as an empirical—and thus phenomenological—expression, and to proceed from there on. There exist all sorts of interpretations and ‘derivations’ of the quantum potential, as commented in footnote 4.14.
- 8.
The kinetic origin of the quantum potentialQuantum potential is discussed in de la Peña et al. (2011).De la Peña, L. To the varied proposals to derive the quantum potential cited in footnote 4.14, one should add those of Dürr et al. (1992)Van@Ván, P. Goldstein, S., and Ván and Fülöp (2003)Fülöp, T., as well as.the thermodynamic approach of Grössing (2008, 2009)Grössing, G.. A somewhat bolder one is that of Floyd (2002)Floyd, E. R., who proposes a trajectory description based on a peculiar quantum potential containing derivatives of third order. An interesting point of this theory is that it contains extra parameters that allow for a distribution of the velocity \(\varvec{v},\) resulting in a more realisticDescription!realist description. Salesi (1996) and Recami and Salesi (1998) propose that the quantum potential can be derived by considering the energy associated with the internal zitterbewegung (considered as the antecedent of the spin). A similar proposal is made by Esposito (1999)Esposito, S., who associates the quantum potential with the (internal) kinetic energy due to a generalized spin; see also Yang (2006)Yang, C. D.. For these authors, the notions of spin, zitterbewegungZitterbewegung and quantum potential are intimately related. Garbaczewski (1992)Garbaczewski, P. offers a nice derivation of the quantum potential as due to the fluctuationsSpin!and fluctuations of the momentumFluctuations!momentum. In Carroll, R. Carroll (2007, 2010), additional arguments are introduced about the origin of the quantum potentialQuantum potential!origin, related to Fisher information.
- 9.
The field \(\psi \) differs in essence from those known to classical physics. In contrast to the gravitational or the electromagnetic field, for example, it does not have a generating source. Moreover, it affects the particle (by guiding it) but is not affected by it. This lack of reciprocity in the field-particle influence led de Broglie (1956)De Broglie, L. (and afterwards Bohm himself) to regard the pilot-wave theory as just a step towards a necessarily more developed theory. [See item 15 in Bell (1987)].
- 10.
This assertion requires some qualification. It is not too difficult to find (both in orthodox textbooks and in research papers, and of course also in popular works), arguments that bear implicitly or explicitly on the notion of trajectory. For example, in discussions on van der WaalsBoyer!and van der Waals forces or molecular forces a drawing is sometimes made of atoms with well-localized orbiting (point) electrons, and the Hamiltonian is written accordingly. True, at some moment an average is taken, but nevertheless the discussion refers, or at least seems to refer, to orbiting point particles. Another example is an atom or a particle in a Stern-Gerlach experiment, which in every analysis is considered to follow a definite trajectory.
- 11.
It is this statistical treatment what engenders ‘indistinguishability’, and this occurrs regardless of whether the system is classical or quantum. This, for instance, explains the use of the notion of indistinguishabilityIndistinguishability to solve the Gibbs paradoxGibbs paradox in classical statistical physics (see e.g. Mandl 1988).
- 12.
Also Nelson’s theoryNelson theory!and nonlocality Nelson theory and more generally the stochastic description of qm have been successfully used to investigate quantum trajectories, as shown by the examples in Chap. 2.
- 13.
Further examples can be seen in Holland (1993), Lopreore and Wyatt (1999, 2000)Wyatt, R. E.—who have generated what they call the ‘quantum trajectory method’—; Suñé and Oriols (2000)Suñé, J., Matzkin and Nurock (2008),Nurock, V. Matzkin, A. Sanz et al. (2002)Sanz, A. S. Miret-Artés, S., Philippidis et al. (1982), and Kumar Chattaraj (2010).
- 14.
- 15.
Here it is in place to recall the argument against the Copenhagen interpretationInterpretation raised by Einstein (1953), considering the stationary states of an infinite one-dimensional square well potential. The spatial part of the wave function can be written in the form \(\varphi =N\sin kx.\) From Eq. (8.11) it follows that \(v=0,\) hence there is no flow velocity. However, by writing the wave function in the form \(\psi =(N/2i)(e^{ikx}-e^{-ikx}),\) it can be interpreted as referring to two similar subensembles of particles, traveling to the right and to the left, with velocities \(\pm \hbar k/m.\) Thus, it is the net (mean) velocity that is null. Einstein used this example to argue that the statistical reading is the single one that can be made in the limit of high energies. Since passing to this limit does not change the nature of the problem, Einstein concluded that one should consider the wave function as describing an ensemble, not an individual particle.
- 16.
- 17.
The possibility that the result of a measurement depends on both the system under observation and the measuring apparatus is also present at the classical level. A common example of the class of nondisturbing classical procedures is a photocell detector that checks the presence or absence of somebody before closing the door of an elevator. An example of the second class could be a ‘tail or head’ detector for tossed coins which operates by inserting a card to stop and receive the coins. Of course this second mechanism can be replaced with more elaborate optical procedures that do not disturb the observed coins. This is a matter of the measurer’s skills and of the existing technical possibilities.
- 18.
A first peculiarity of \(V_{Q}\) is that it is independent of the field’s strength, or rather of the intensity (\(\sim \!\rho \)) of the wave\(.\) This follows from the fact that \(V_{Q}(\rho )=V_{Q}(A\rho )\) for any constant \(A,\) and indicates that the effects due to the particles do not depend on the number of particles present; but on their distribution. That there are forces within the classical realm, particularly in the hydrodynamical analogy, with similar peculiarities does not suffice to surmount the problem, since in the hydrodynamical case there is a medium that supports and transmits the presure and the stresses. By contrast, in the quantum single-particle problem we are not dealing with a collective system; the ‘collection’ may be a conceptual ensemble, devoid of physical existence.
- 19.
Bell inequalities is a collective name referring to a number of inequalities (such as the chsh-type inequalities) that involve correlations between variables of the constituents of a composite system and are violated by qm, reputedly due to the nonlocal properties of the quantum description.
- 20.
The transmitted particles are among those that gain enough energy to travel not through the barrier, but over it (see e.g. Lopreore and Wyatt 1999).
- 21.
This feature is not exclusive of the present example, but rather the general rule: \(V_{Q}\) does not decay as \(x\rightarrow \pm \infty ,\) i.e., at far distances from the particles.
- 22.
The effective interaction potential introduced via \(V_{Q}(\varvec{x}_{1}, \varvec{x}_{2})\), which remains ‘hidden’ in the depths of the Schrödinger equation, formally transforms the original noninteracting system into an interacting one [see Eq. (8.52)]. By contrast, the possible nonfactorizability of \(\exp (iS)\) does not manifest itself as a formal interaction potential in Eq. (8.52). The nonlocal effect of this kind of entanglement is manifested when a description in terms of forces is made, as we have seen.
- 23.
We use here the term covariance to refer to a two-point momentum \(\langle \hat{F}_{1}\hat{G}_{2}\rangle \), even if the product \(\langle \hat{F}_{1}\rangle \langle \hat{G}_{2}\rangle \) differs from zero. In the literature the term ‘correlation’Correlation is frequently used for \(\langle \hat{F}_{1} \hat{G}_{2}\rangle \), so we use it here when convenient.
- 24.
From this perspective, the conclusions reached regarding the dispersive and nonlocal features of the quantum potential (a quantity that depends on \(u\) only) become evident.
- 25.
When the operator \(\hat{A}\) does not correspond to a single particle, this statement ceases to be true. For example, for \(\hat{A}=A_{1}( \hat{x}_{1})A_{2}(\hat{x}_{2}),\) the entanglement is revealed in the covariance Covariance \(\left\langle A_{1}A_{2}\right\rangle \) even though none of the variables is momentum-dependent. In fact, the point here is to show that the present approach allows to reach conclusions about entanglement by focusing on single-particle variables, rather than on correlations between variables of the two subsystems, as is customarily done [see discussion following Eq. (8.96)].
- 26.
The \(p\)-local mean value of a dynamic variable \(g\) is defined, in analogy with Eq. (4.50), as its partial average over the configuration space, using the distribution \(Q\),
$$\begin{aligned} \left\langle g\right\rangle (p_{1},p_{2})=\left\langle g\right\rangle _{p}= \frac{1}{\rho _{p}}\int gQ(x_{1},x_{2},p_{1},p_{2})dx_{1}dx_{2}. \end{aligned}$$ - 27.
The fact that the kind of variables that may exhibit nonlocality is representation-dependent does not mean that the very existence of nonlocality is representation-dependent. Indeed, for any entangled state \( \left| \psi \right\rangle \) there will always be some variable exhibiting nonlocal features; which one depends on the representation used to project \(\left| \psi \right\rangle \).
- 28.
Equation (8.87) is just Eq. (8.92) with \(\hat{A}_{i}=f(\hat{x} _{i})\), yet Eq. (8.86) differs from the structure of (8.92). To see this consider in particular Eq. (8.84) with \(n=1;\) then \( P_{i}(\pi _{i},\partial _{i}^{k}\pi _{i})=m_{i}v_{i}(x_{i})\ne \left\langle x_{i}\right| \hat{p}_{i}\left| x_{i}\right\rangle \), so indeed (8.86) is not Eq. (8.92) for \(\hat{A}=\hat{g}_{i}\) and \(\alpha =x_{i}\).
- 29.
Notice that the use of a fixed representation for both elements of the composite system, i.e. \(\{ \left| \alpha \right\rangle _{1}\} ,\) \(\{ \left| \beta \right\rangle _{2}\} \), is a matter of necessity when discussing entanglement. The same applies when considering measurements on a system. In fact, given an \((\alpha ,\beta )\)-representation, the distribution function \(\rho \left( \alpha ,\beta \right) \) is defined as the joint probabilityProbability!joint density that determines the probability of obtaining the values \(\alpha \) and \(\beta \) when performing the projective measurements corresponding to the proyectors \(\Pi _{A_{1}}^{\alpha }=\left| \alpha \right\rangle \left\langle \alpha \right| \) \(\in \mathcal {H}_{1}\) and \(\Pi _{B_{2}}^{\beta }=\left| \beta \right\rangle \left\langle \beta \right| \) \(\in \mathcal {H}_{2},\) respectively. Thus the representation used is linked with the variables that are measured in a certain experiment.
- 30.
This conclusion is in line with the results obtained in Sect. 7.2.5. Specifically, the discussion following Eq. (7.63) tells us that for entanglement to become manifest through a correlation, both dynamical variables involved (i.e., \(F,G\), the equivalent of \(A_{1},B_{2}\) in the present case) must have nondiagonal elements in a given representation (the energy representation, in that case).
- 31.
Note that, as already remarked in connection with von Neumann’s theorem [see Eq. (8.2)], the equality
$$\begin{aligned} \langle \hat{C}\rangle =\langle \hat{a}\hat{b}\rangle +\langle \hat{a}\hat{B} \rangle +\langle \hat{A}\hat{b}\rangle -\langle \hat{A}\hat{B}\rangle \end{aligned}$$does not hold in general if the operators in the terms of the sum do not commute. This important restriction needs to be borne in mind when attempting to apply (8.94) [or (8.96)] to draw conclusions aboutAccardi L. Fargue, D. correlations. See e.g. Accardi (1984).
- 32.
With an eye put on note (31), making sure that the average is taken over the same distribution in each term.
- 33.
The extra correlations that lead to the violation of a Bell inequality exist also in the case of photons, due to the correlations between the excitations of the field and the corresponding modes of the zpf; see e.g. Casado, A. Casado et al. (1998).
References
Accardi, L.: The probabilistic roots of the quantum mechanical paradoxes. In: Diner, S., Fargue, D., Lochak, G., Selleri, F. (eds.) The Wave-Particle Dualism. Springer, Dordrecht (1984)
Aharonov, Y., Vaidman, L.: Bohmian Mechanics and Quantum Theory: An Appraisal, p. 141. Kluwer, Dordrecht (1996)
Bacciagaluppi, G., Valentini, A.: Quantum Theory at the Crossroads. Reconsidering the 1927 Solvay Conference. Cambridge University Press, Cambridge (2009)
Bell, J.S.: On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38, 447 (1966). Reprinted in Bell 1987
Bell, J.S.: Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press, Cambridge (1987)
Belinfante, F.J.: A Survey of Hidden-Variables Theories. Pergamon, Oxford (1973)
Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden” variables. I. Phys. Rev. 85, 166 (1952a)
Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden” variables. II. Phys. Rev. 85, 180 (1952b)
Bohm, D.: Comments on an article of Takabayasi concerning the formulation of quantum mechanics with classical pictures. Progr. Theor. Phys. 9, 273 (1953)
Bohm, D., Hiley, D.J.: The Undivided Universe. An Ontological Interpretation of Quantum Theory. Routledge, London (1995)
Bohm, D., Hiley, B.J.: Statistical mechanics and the ontological interpretation. Found. Phys. 26, 823 (1996)
Bohm, D., Vigier, J.-P.: Model of the causal interpretation of quantum theory in terms of a fluid with irregular fluctuations. Phys. Rev. 96, 208 (1954)
Brown, H., Harré, R. (eds.): Philosophical Foundations of Quantum Field Theories. Clarendon Press, Oxford (1988)
Brun, T.A.: A simple model of quantum trajectories. Am. J. Phys. 70, 719 (2002) quant-ph/0108132v1
Bub, J.: Von Neumann’s ‘no hidden variables’ proof: a re-appraisal. Found. Phys. 40, 1333 (2010)
Carroll, R.: Some remarks on Ricci flow and the quantum potential. (2007). arXiv:math-ph/0703065v2
Carroll, R.: On the Emergence Theme of Physics. World Scientific, Singapore (2010)
Casado, A., Marshall, T.W., Santos, E.: Type II parametric downconversion in the Wigner-function formalism: entanglement and Bell’s inequalities. J. Opt. Soc. Am. B 15, 1572 (1998)
Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A.: Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 23, 880 (1969)
*Cole, D.C., Zou, Y.: Quantum mechanical ground state of hydrogen obtained from classical electrodynamics. Phys. Lett. A 317, 14 (2003)
*Cole, D.C., Zou, Y.: Analysis of orbital decay time for the classical hydrogen atom interacting with circularly polarized electromagnetic radiation. Phys. Rev. E 69, 016601 (2004a)
*Cole, D.C., Zou, Y.: Simulation study of aspects of the classical hydrogen atom interacting with electromagnetic radiation: circular orbits. J. Sci. Comput. 20, 43 (2004b)
*Cole, D.C., Zou, Y.: Simulation study of aspects of the classical hydrogen atom interacting with electromagnetic radiation: elliptical orbits. J. Sci. Comput. 20, 379 (2004c)
*Cole, D.C., Zou, Y.: Perturbation analysis and simulation study of the effects of phase on the classical hydrogen atom interacting with circularly polarized electromagnetic radiation. J. Sci. Comput. 21, 145 (2004d)
*Cole, D.C., Zou, Y.: Subharmonic resonance behavior for the classical hydrogen atomic system. J. Sci. Comput. 39, 1 (2009)
Cushing, J.T.: Causal quantum theory: why a nonstarter? In: Selleri, F. (ed.) The Wave-Particle Duality. Plenum, New York (1992)
de Broglie, L.: Sur la possibilité de relier les phénomènes d’interférences et de diffraction à la théorie des quanta de lumière. C. R. Acad. Sci. Paris 183, 447 (1926a)
de Broglie, L.: Les principes de la nouvelle Mécanique ondulatoire, série VI. J. Phys. 7(11), 321–333 (1926b)
de Broglie, L.: La structure atomique de la matière et du rayonnement et la Mécanique ondulatoire. C. R. Acad. Sci. Paris 184, 273 (1927a)
de Broglie, L.: Sur le rôle des ondes continues en Mécanique ondulatoire. C. R. Acad. Sci. Paris 185, 380 (1927b)
de Broglie, L.: La Mécanique ondulatoire et la structure atomique de la matière et du rayonnement. J. Phys. Rad. 8, 225 (1927c)
de Broglie, L.: Électrons et Photons: Rapports et discussions du Cinquième Conseil de Physique. Gauthier-Villars, Paris (1928)
de Broglie, L.: Une tentative d’interprétation causale et non linéare de la Mécanique Ondulatoire: La Théorie de la Double Solution. Gauthier-Villars, Paris (1956)
de Broglie, L.: Recherches sur la Théorie des Quanta. Réédition du texte de 1924. Masson, Paris (1963)
*de la Peña, L., Cetto, A.M., Valdés-Hernández, A., França, H.M.: Genesis of quantum nonlocalities. Phys. Lett. A 375, 1720 (2011). arXiv:quant-phys:1110.4641v1
Dewdney, C., Hardy, L., Squires, E.J.: How late measurements of quantum trajectories can fool a detector. Phys. Lett. A 184, 6 (1993)
Dewdney, C., Hiley, B.J.: A Quantum potential description of one-dimensional time-dependent scattering from square barriers and square wells. Found. Phys. 12, 27 (1982)
Dürr, D., Fusseder, W., Goldstein, S., Zanghi, N.: Comment on “Sur-realistic Bohm trajectories”. Z. Naturforsch. 48a, 1261 (1993)
Dürr, D., Goldstein, S., Zanghi, N.: Quantum mechanics, randomness, and deterministic reality. Phys. Lett. A 172, 6 (1992)
Dürr, D., Teufel, S.: Bohmian Mechanics. The Physics and Mathematics of Quantum Theory. Springer, Dordrecht (2009)
Einstein, A.: Elementare Überlegungen zur Interpretation der Grundlagen der Quanten-Mechanik. In: Scientific Papers, Presented to Max Born. Oliver & Boyd, Edinburgh (1953)
Englert, B.G., Scully, M.O., Süssmann, G., Walther, H.: Surrealistic Bohm Trajectories. Z. Naturforsch. 47a, 1175 (1992)
Esposito, S.: Photon wave mechanics: a de Broglie-Bohm approach. Found. Phys. Lett. 12, 533 (1999)
Feyerabend, P.K.: Eine Bemerkung zum Neumannschen Beweis. Zeit. Phys. 145, 421 (1956)
Floyd, E.R.: Classical limit of the trajectory representation of quantum mechanics, loss of information and residual indeterminacy, Int. J. Mod. Phys. A 15, 1363 (2000). arXiv:quant-ph/9907092v3
Floyd, E.R.: The philosophy of the trajectory representation of quantum mechanics. In: Amoroso, R.L., et al. (eds.) Gravitation and Cosmology: From the Hubble Radius to the Planck Scale, Fundamental Theories of Physics, vol. 126, Part IV, p. 401. Kluwer Academic, Netherlands (2002)
Garbaczewski, P.: Derivation of the quantum potential from realistic Brownian particle motions. Phys. Lett. A 162, 129 (1992)
Gleason, A.M.: Measures on the closed subspaces of a Hilbert space. J. Math. Mech. 6, 885 (1957)
Goldstein, H., Poole, Ch., Safko, J.: Classical Mechanics, 3rd edn. Pearson, New Jersey (2002)
Griffiths, R.B.: Consistent interpretation of quantum mechanics using quantum trajectories. Phys. Rev. Lett. 70, 2201 (1993)
Grössing, G.: A classical explanation of quantization. Phys. Lett. A 372, 4556 (2008)
Grössing, G.: On the thermodynamic origin of the quantum potential. Physica A 388, 811 (2009)
Gudder, S.: On hidden-variable theories. J. Math. Phys. 11, 431 (1970)
Gull, S., Lasenby, A., Doran, Ch.: Electron paths, tunnelling and diffraction in the spacetime algebra. Found. Phys. 23, 1329 (1993)
Heisenberg, W.: The development of the interpretation of the quantum theory. In: Pauli, W. (ed.) Niels Bohr and the Development of Physics. Pergamon Press, London (1955)
Hermann, G.: Die naturphilosophischen Grundlagen der Quantenmechanik, section 7. Abhandlungen der Fries’schen Schule 6 (1935) (English translation: Seevinck, M.: www.phys.uu.nl/igg/seevinck/trans.pdf)
Hernández-Zapata, S., Hernández-Zapata, E.: Classical and non-relativistic limits of a Lorentz-Invariant Bohmian model for a system of spinless particles. Found. Phys. 40, 532 (2010)
Holland, P.R.: The Quantum Theory of Motion. An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics. Cambridge University Press, Cambridge (1993)
Huang, W.Ch.-W., Batelaan, H.: Dynamics underlying the gaussian distribution of the classical harmonic oscillator in zero-point radiation. (2012). arXiv:quant-ph:1206.5323v1
Jammer, M.: The Philosophy of Quantum Mechanics. Wiley, NY (1974)
Kochen, S., Specker, E.P.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59 (1967)
Kumar Chattaraj, P.: Quantum Trajectories. CRC Press, NY (2010)
Landau, L.J.: On the violation of Bell’s inequality in quantum theory. Phys. Lett. A 120, 54 (1987)
Lopreore, C.L., Wyatt, R.E.: Quantum wave packet dynamics with trajectories. Phys. Rev. Lett. 82, 5190 (1999)
Lopreore, C.L., Wyatt, R.E.: Lopreore and Wyatt reply. Phys. Rev. Lett. 85, 895 (2000)
Madelung, E.: Quantentheorie in hydrodynamischer form. Zeit. Phys. 43, 354 (1927)
Mandl, F.: Statistical Physics. Wiley, NY (1988)
Matzkin, A., Nurock, V.: Classical and Bohmian trajectories in semiclassical systems: mismatch in dynamics, mismatch in reality? Stud. Hist. Philos. Sci. B 39, 17 (2008)
Mittelstaedt, P.: Objectification. In: Greenberger, D., Hentschel, K., Weinert, F. (eds.) Compendium of Quantum Physics. Springer, Berlin (2009)
Mugur-Schächter, M.: Étude du Caractère Complet de la Théorie Quantique. Gauthier-Villars, Paris (1964)
Nikolić, H.: Bohmian mechanics in relativistic quantum mechanics, quantum field theory and string theory. J. Phys. Conf. Ser. 67, 012035 (2007)
Omnès, R.: The Interpretation of Quantum Mechanics. Princeton University Press, Princeton (1994)
Oriols, X., Mompart, J.: Applied Bohmian Mechanics. From Nanoscale Systems to Cosmology. Pan Stanford Publishing Pte. Ltd., Singapore (2012)
Passon, O.: Why Isn’t Every Physicist a Bohmian? (2005). arXiv:quant-ph/0412119v2
Pauli, W., George, A. (ed.) Louis de Broglie Physicien et Penseur, vol. 33. Albin Michel, Paris (1952)
Peres, A.: Generalized Kochen-Specker theorem. Found. Phys. 41, 807 (1996)
Philippidis, C., Dewdney, C., Hiley, B.J.: Quantum interference and the quantum potential. Nuovo Cimento 52B, 15 (1979)
Philippidis, C., Dewdney, C., Kaye, R.D.: The Aharonov-Bohm effect and the quantum potential. Nuovo Cimento 71B, 75 (1982)
Recami, E., Salesi, G.: Kinematics and hydrodynamics of spinning particles. Phys. Rev. A 57, 98 (1998)
Revzen, M., Lokajíček, M., Mann, A.: Bell’s inequality and operators’ noncommutativity. Quantum Semiclass. Opt. 9, 501 (1997)
Salesi, G.: Spin and Madelung fluid. Mod. Phys. Lett. A 11, 1815 (1996). arXiv:0906.4147v1
Santos, E.: On the possibility of local hidden-variable theories. Phys. Lett. A 53, 432 (1975)
Sanz, A.S., Borondo, F., Miret-Artés, S.: Particle diffraction studied using quantum trajectories. J. Phys.: Condens. Matter 14, 6109 (2002)
Solomon, J.: Sur l’indéterminisme de la Mécanique quantique. J. Phys. 4, 34 (1933)
Suñé, J., Oriols, X.: Comment on “quantum wave packet dynamics with trajectories”. Phys. Rev. Lett. 85, 894 (2000)
Thiounn, M.: Doctorate thesis, Cahiers de Physique 174, 53 (1965); for further works on the theory of the double solution see C. R. Acad. Sci. Paris 262B, 657 (1966), Portugaliae, Physica 4, 85 (1966)
Towler, M.D.: De Broglie-Bohm Pilot Wave Theory and the Foundations of Quantum Mechanics. A Graduate Lecture Course. Cambridge University, Cambridge (2009). www.tcm.phy.cam.ac.uk/mdt26/pilot_waves.html
Tsirelson, B.S.: Quantum generalizations of Bell’s inequality. Lett. Math. Phys. 4, 93 (1980)
Ván, P., Fülöp, T.: Stability of stationary solutions of the Schrödinger-Langevin equation (2003). arXiv:quant-ph/0304190v4
van Kampen, N.G.: Ten theorems about quantum mechanical measurements. Physica A 153, 97 (1988)
Vasudevan, R., Parthasarathy, K.V., Ramanathan, R.: Quantum Mechanics. A Stochastic Approach. Alpha Science, Oxford (2008)
von Neumann, J.: Mathematische Grundlagen der Quantenmechanik. Springer, Berlin (1932). English Trans: Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton (1932/1955)
Wang, X.-S.: Derivation of the Schrödinger equation based on a fluidic continuum model of vacuum and a sink model of particles. (2006). arXiv:0610224
Wick, D.: The Infamous Boundary. Seven Decades of Heresy in Quantum Physics. Copernicus-Springer, New York (1995)
Yang, C.D.: Modeling quantum harmonic oscillator in complex domain. Chaos Solitons Fractals 30, 342 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
de la Peña, L., Cetto, A.M., Valdés Hernández, A. (2015). Causality, Nonlocality, and Entanglement in Quantum Mechanics. In: The Emerging Quantum. Springer, Cham. https://doi.org/10.1007/978-3-319-07893-9_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-07893-9_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-07892-2
Online ISBN: 978-3-319-07893-9
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)