Abstract
The ontology of Bohmian mechanics includes both the universal wave function (living in 3N-dimensional configuration space) and particles (living in ordinary 3-dimensional physical space). Proposals for understanding the physical significance of the wave function in this theory have included the idea of regarding it as a physically-real field in its 3N-dimensional space, as well as the idea of regarding it as a law of nature. Here we introduce and explore a third possibility in which the configuration space wave function is simply eliminated—replaced by a set of single-particle pilot-wave fields living in ordinary physical space. Such a re-formulation of the Bohmian pilot-wave theory can exactly reproduce the statistical predictions of ordinary quantum theory. But this comes at the rather high ontological price of introducing an infinite network of interacting potential fields (living in 3-dimensional space) which influence the particles’ motion through the pilot-wave fields. We thus introduce an alternative approach which aims at achieving empirical adequacy (like that enjoyed by GRW type theories) with a more modest ontological complexity, and provide some preliminary evidence for optimism regarding the (once popular but prematurely-abandoned) program of trying to replace the (philosophically puzzling) configuration space wave function with a (totally unproblematic) set of fields in ordinary physical space.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
See also chapter 8 of Heisenberg (1958)
In response to Einstein’s remark that Bohm’s way “seems too cheap to me,” for example, Max Born wrote that he thought “this theory was quite in line with [Einstein’s] own ideas, to interpret the quantum mechanical formulae in a simple, deterministic way...” (Born 1971, pp. 192–193)
It is not entirely clear, but Peter Holland may endorse this kind of view: “a complete and accurate account of the motions of particles moving in accordance with the laws of quantum mechanics must be directly connected with multidimensional waves dynamically evolving in configuration space.” (Holland 1995, p. 321). Antony Valentini has also made comments suggesting that the main innovation of quantum mechanics is the need to accept the wave function as a new kind of causal agent which physically affects particles despite its living in a high-dimensional space.
We emphasize that the adjective “non-local” has two different (but related) meanings. First, we use “non-local” in the phrase “non-local beable” to denote an object (posited as physically real in some candidate theory) to stress that the object does not assign values to regions in 3-dimentional space (or 3+1 spacetime). And second, we use the word “non-local” to describe the special, faster-than-light, type of causal influence that Bell’s theorem shows must exist Bell (1994). In particular, we stress that despite proving the existence of non-locality (in the second sense) Bell’s theorem does not show that empirically viable theories must include beables that are non-local (in the first sense). Indeed, one of the key points of our paper is a demonstration that the non-locality required by Bell’s theorem can actually be embedded in a theory of exclusively local beables.
For example, Dürr et al. (1996) write: “We propose that the wave function belongs to an altogether different category of existence than that of substantive physical entities [–] that the wave function is a component of physical law rather than of the reality described by the law.”
In this paper we will use bold letters to indicate a point in the configuration space, while a non-bold variable indicates a point in the physical space. Capital letters denote the actual positions of particles, while lowercase letters denote generic positions. In principle, positions in physical space have three coordinates; however, for simplicity only, we will often consider a 1D physical space. The symbol \(\nabla _i\) accounts for the gradient in a 3D physical space or \(\nabla _i= \partial / \partial x_i\) in a 1D physical space.
The appropriate generalization for \(N>2\) particles moving in 3 spatial dimensions is trivial; dealing with systems of particles with spin in a fully general way probably requires working instead in terms of the Bohmian conditional density matrices defined by Dürr et al. (2005), a possibility we set aside for future work.
Note that, as a result of Eqs. (9) and (10), the conditional wave functions are not normalized in the usual way. It would be easy enough to adjust the definition to yield normalized conditional wave functions; but since any overall multiplicative factor cancels out anyway in Eq. (11), we set aside this needless complication.
We assume here that the spacing, \(| \lambda (a_n - a_{n+1})|\), between adjacent possible pointer positions is small compared to the width, \(w\), of \(\beta \).
On one hand, the potential \(V[x,X_2(t),t]\) in (28) produces correlations between the two particles \(X_1(t)\) and \(X_2(t)\). The dependence of \(V[x,X_2(t),t]\) on \(x\) and \(X_2(t)\) imposes a restriction on the speed of such interaction. For example, the retarded electromagnetic potentials ensures that there is no superluminal electromagnetic influence between particles due to \(V[x,X_2(t),t]\). On the other hand, such restriction on the speed of the interaction between particles is not present in the new potentials \(A_1(x,X_2(t),t)\) and \(B_1(x,X_2(t),t)\) in (28). Thus, the particle \(X_1(t)\) have an instantaneous (non-local) interaction with \(X_2(t)\) due to the potentials \(A_1(x,X_2(t),t)\) and \(B_1(x,X_2(t),t)\). Of course, insofar as the usual Bohmian particle trajectories, and hence the usual statistical predictions of quantum mechanics, are reproduced, we know that this non-locality will not support superluminal communication.
See the website http://europe.uab.es/bitlles.
In fact, \(\varPsi \) is determined by the one-particle wave function and the associated potentials for any one particle. The complete ontic state—comprising the one-particle wave function and associated potentials for all particles, plus the particle positions themselves—thus contains a tremendous amount of redundancy. This is yet another strong piece of evidence suggesting that empirical viability should be able to be achieved, even with a greatly reduced ontic complexity.
References
Alarcón, A., Yaro, S., Cartoixà, X., & Oriols, X. (2013). Computation of many-particle quantum trajectories with exchange interaction: Application to the simulation of nanoelectronic devices. Journal of Physics: Condensed Matter, 25, 325601.
Albareda, G., Suñé, J., & Oriols, X. (2009). Many-particle hamiltonian for open systems with full coulomb interaction: Application to classical and quantum time-dependent simulations of nanoscale electron devices. Physical Review B, 79, 075315.
Albareda, G., Lopez, H., Cartoixà, X., Suñé, J., & Oriols, X. (2010). Time-dependent boundary conditions with lead-sample Coulomb correlations: Application to classical and quantum nanoscale electron device simulators. Physical Review B, 82, 085301.
Albareda, G., Marian, D., Benali, A., Yaro, S., Zanghì, N., & Oriols, X. (2013). Time-resolved electron transport with quantum trajectories. Journal of Computational Electronics, 12(3), 405–419.
Albert, D. Z. (1996). Elementary quantum metaphysics. In J. T. Cushing, A. Fine, & S. Goldstein (Eds.), Bohmian mechanics and quantum theory: an appraisal (pp. 277–284). Dordrecht: Kluwer.
Allori, V., Goldstein, S., Tumulka, R., & Zanghì, N. (2013). Predictions and Primitive Ontology in Quantum Foundations: A Study of Examples. British Journal for the Philosophy of Science. doi:10.1093/bjps/axs048.
Bacciagaluppi, G., & Valentini, A. (2009). Quantum theory at the crossroads. Cambridge: Cambridge University Press.
Bassi, A., Lochan, K., Satin, S., Singh, T. P., & Ulbricht, H. (2013). Models of wave-function collapse, underlying theories, and experimental tests. Review of Modern Physics, 85, 471–527.
Bell, J. S. (1994). Speakable and unspeakable in quantum mechanics (2nd ed.). Cambridge: Cambridge University Press.
Bohm, D. (1987). Causality & chance in modern physics. Philadelphia: University of Pennsylvania Press.
Born, I. (1971). The Born-Einstein Letters. New York: Walker and Company.
Cohen, I. B. (1999). A guide to Newton’s Principia. In Isaac Newton. The Principia (pp. 60–64 & 277–280), (I. B. Cohen & A. Whitman, Trans.). Berkely, CA: University of California Press.
Darrigol, O. (2000). Electrodynamics from Ampère to Einstein. Oxford: Oxford University Press.
Dürr, D., Goldstein, S., & Zanghì, N. (1992). Quantum equilibrium and the origin of absolute uncertainty. Journal of Statistical Physics, 67, 843–907.
Dürr, D., Goldstein, S., & Zanghì, N. (1996). Bohmian mechanics and the meaning of the wave function. In R. S. Cohen, M. Horne, & J. Stachel (Eds.), Experimental metaphysics—quantum mechanical studies in honor of Abner Shimony (Vol. I, pp. 25–38). Dordrecht: Kluwer Academic Publishers.
Dürr, D., Goldstein, S., & Zanghì, N. (2004). Quantum equilibrium and the role of operators as observables in quantum theory. Journal of Statistical Physics, 116, 959–1055.
Dürr, D., Goldstein, S., Tumulka, R., & Zanghì, N. (2005). On the role of density matrices in Bohmian mechanics. Foundations of Physics, 35, 449–467.
Ghirardi, G. C., Rimini, A., & Weber, T. (1986). Unified dynamics for microscopic and macroscopic systems. Physical Review D, 34, 470–491.
Heisenberg, W. (1955). The development of the interpretation of the quantum theory. In W. Pauli, L. Rosenfeld, & V. Weisskopf (Eds.), Niels Bohr and the development of physics (pp. 12–29). New York: McGraw-Hill.
Heisenberg, W. (1958). Physics and philosophy: the revolution in modern science. New York: Harper.
Hesse, M. B. (2005). Forces and fields: The concept of action at a distance in the history of physics. Mineola, NY: Dover Publications.
Holland, P. (1995). The quantum theory of motion. Cambridge: Cambridge University Press.
Howard, D. (1990). ‘Nicht sein kann was nicht sein darf’, or the prehistory of EPR, 1909–1935: Einstein’s early worries about the quantum mechanics of composite systems. In A. I. Miller (Ed.), Sixty-two years of uncertainty (pp. 61–111). New York: Plenum Press.
Hunt, B. (1994). The Maxwellians. Ithaca, NY: Cornell University Press.
Janiak, A. (2009). Newton’s philosophy. In E.N. Zalta (Ed.), The stanford encyclopedia of philosophy. Retrieved from http://plato.stanford.edu/entries/newton-philosophy. Accessed 28 April 2014.
Maudlin, T. (2007). Completeness, supervenience and ontology. Journal of Physics A, 40, 3151–3171.
McMullin, E. (1989). The explanation of distant action: historical notes. In J. T. Cushing & E. McMullin (Eds.), Philosophical consequences of quantum theory (pp. 272–302). Notre Dame, IN: University of Notre Dame Press.
Mermin, N. D. (2009). What’s bad about this habit. Physics today, 62(5), 8–9.
Norsen, T. (2010). The theory of (exclusively) local beables. Foundations of Physics, 40, 1858–1884.
Oriols, X. (2007). Quantum trajectory approach to time dependent transport in mesoscopic systems with electron–electron interactions. Physical Review Letters, 98, 066803.
Oriols, X., & Mompart, J. (2012). Applied Bohmian mechanics: From nanoscale systems to cosmology. Singapore: Pan Stanford Publishing.
Pusey, M. F., Barrett, J., & Rudolph, T. (2012). On the reality of the quantum state. Nature Physics, 8, 475–478.
Traversa, F. L., Buccafurri, E., Alarcón, A., Albareda, G., Clerc, R., Calmon, F., et al. (2011). Time-dependent many-particle simulation for resonant tunneling diodes: Interpretation of an analytical small-signal equivalent circuit. IEEE Transanction on Electronic Devices, 58, 2104–2112.
von Neumann, J. (1955). Mathematical foundations of quantum mechanics (R. T. Beyer, Trans.). Princeton: Princeton University Press. (Original title: Mathematische Grundlagen der Quantenmechanik, Berlin 1932).
Wessels, L. (1979). Schrödinger’s route to wave mechanics. Studies in History and Philosophy of Science Part A, 10(4), 311–340.
Acknowledgments
We gratefully acknowledge Nino Zanghì and Albert Solé for reading a preliminary version of this paper and for very useful discussions. D.M. is supported in part by INFN and acknowledges support of COST action (MP1006) through STSM. X.O. acknowledge support from the “Ministerio de Ciencia e Innovación” through the Spanish Project TEC2012-31330 and by the Grant agreement no: 604391 of the Flagship initiative “Graphene-Based Revolutions in ICT and Beyond”.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Norsen, T., Marian, D. & Oriols, X. Can the wave function in configuration space be replaced by single-particle wave functions in physical space?. Synthese 192, 3125–3151 (2015). https://doi.org/10.1007/s11229-014-0577-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-014-0577-0