Abstract
If the classical structure of space-time is assumed to define an a priori scenario for the formulation of quantum theory (QT), the coordinate representation of the solutions \(\psi (\vec x,t)(\psi (\vec x_1 , \ldots ,\vec x_N ,t))\) of the Schroedinger equation of a quantum system containing one (N) massive scalar particle has a preferred status. Let us consider all of the solutions admitting a multipolar expansion of the probability density function \(\rho (\vec x,t) = \left| {(\psi (\vec x,t)} \right|^2\) (and more generally of the Wigner function) around a space-time trajectory \(\vec x_c (t)\) to be properly selected. For every normalized solution \(\left( {\smallint d^3 x\rho (\vec x,t) = 1} \right)\) there is a privileged trajectory implying the vanishing of the dipole moment of the multipolar expansion: it is given by the expectation value of the position operator \(\left\langle {\psi (t)\left| {\hat \vec x} \right|\psi (t)} \right\rangle = \vec x_c (t)\). Then, the special subset of solutions \(\psi (\vec x,t)\) which satisfy Ehrenfest’s Theorem (named thereby Ehrenfest monopole wave functions (EMWF)), have the important property that this privileged classical trajectory \(\vec x_c (t)\) is determined by a closed Newtonian equation of motion where the effective force is the Newtonian force plus non-Newtonian terms (of order ħ 2 or higher) depending on the higher multipoles of the probability distribution ρ. Note that the superposition of two EMWFs is not an EMWF, a result to be strongly hoped for, given the possible unwanted implications concerning classical spatial perception. These results can be extended to N-particle systems in such a way that, when N classical trajectories with all the dipole moments vanishing and satisfying Ehrenfest theorem are associated with the normalized wave functions of the N-body system, we get a natural transition from the 3N-dimensional configuration space to the space-time. Moreover, these results can be extended to relativistic quantum mechanics. Consequently, in suitable states of N quantum particle which are EMWF, we get the “emergence” of corresponding “classical particles” following Newton-like trajectories in space-time. Note that all this holds true in the standard framework of quantum mechanics, i.e. assuming, in particular, the validity of Born’s rule and the individual system interpretation of the wave function (no ensemble interpretation). These results are valid without any approximation (like ħ → 0, big quantum numbers, etc.). Moreover, we do not commit ourselves to any specific ontological interpretation of quantum theory (such as, e.g., the Bohmian one). We will argue that, in substantial agreement with Bohr’s viewpoint, the macroscopic description of the preparation, certain intermediate steps and the detection of the final outcome of experiments involving massive particles are dominated by these classical “effective” trajectories. This approach can be applied to the point of view of de-coherence in the case of a diagonal reduced density matrix ρ red (an improper mixture) depending on the position variables of a massive particle and of a pointer. When both the particle and the pointer wave functions appearing in ρ red are EMWF, the expectation value of the particle and pointer position variables becomes a statistical average on a classical ensemble. In these cases an improper quantum mixture becomes a classical statistical one, thus providing a particular answer to an open problem of de-coherence about the emergence of classicality.
Similar content being viewed by others
References
G. Bacciagaluppi, Measurement and Classical Regime in Quantum Mechanics in The Oxford Handbook of Philosophy of Physics, edited by R. Batterman (Oxford University Press, Oxford, 2013).
M. Schlosshauer, K. Camilleri, What classicality? Decoherence and Bohr’s Classical Concepts, arXiv:1009.4072.
M. Schlosshauer, K.Camilleri, The quantum-to-classical transition: Bohr’s doctrine of classical concepts, emergent classicality, and decoherence, arXiv:0804.1609.
M. Schlosshauer, Rev. Mod. Phys. 76, 1267 (2004).
M. Schlosshauer, Decoherence and the Quantum-to-Classical Transition (Springer, Berlin, 2007).
K. Camilleri, Stud. Hist. Phil. Mod. Phys. 40, 290 (2009).
N.P. Landsman, Between Classical and Quantum, in Handbook of the Philosophy of Science, Vol. 2: Philosophy of Physics, edited by J. Butterfield, J. Earman (Elsevier, Amsterdam, 2006) p. 417--553, arXiv:quant-ph/0506082.
O. Hay, A. Peres, Phys. Rev. A 58, 116 (1998) arXiv:quant-ph/9712044.
P. Ehrenfest, Z. Phys. 45, 455 (1927).
R.J. Cotter, Time-of-Flight Mass Spectrometry (American Chemical Society, 1994) ISBN 0-8412-3474-4.
Time-of-Flight Techniques for the Investigation of Kinetic Energy Distribution of Ions and Neutrals Desorbed by Core Excitations, http://www.osti.gov./bridge/servlets/pur1/764241-IyAfiC/webviewable/764241.pdf.
D. Bohm, B.J. Hiley, The Undivided Universe (Routledge, London, 1993).
P.R. Holland, The Quantum Theory of Motion (Cambridge University Press, Cambridge, 1993).
D. Dürr, S.Teufel, Bohmian Mechanics (Springer, Berlin, 2009).
G. Grübl, S. Kreidl, Bohmian Mechanics group, University of Innsbruck, http://bohm-mechanics.uibk.ac.at.
M. Towler, De Broglie-Bohm Pilot-Wave Theory and the Foundations of Quantum Mechanics, graduate lecture course, University of Cambridge (2009) http://www.tcm.phy.cam.ac.uk/mdt26/pilotwaves.html.
E. Deotto, G.C. Ghirardi, Found. Phys. 28, 1 (1998).
G.C. Ghirardi, A. Rimini, T. Weber, Phys. Rev. D 34, 470 (1986).
G.C. Ghirardi, An Attempt at a Macrorealistic Quantum World View, in Proceedings of the Symposium The Interpretations of Quantum Theory: Where Do we Stand ?, edited by L. Accardi (Istituto della Enciclopedia Italiana, Academy for Advanced Studies USA at Columbia University, Fordham U.P., New York, 1992).
G.C. Ghirardi, Collapse Theories, in Stanford Encyclopedia of Philosophy (2002).
G. Bacciagaluppi, The Role of Decoherence in Quantum Mechanics, in Stanford Encyclopedia of Philosophy, edited by E.N. Zalta (2003) (Fall 2008 edition, http://plato.stanford.edu/archives/fall2008entries/qm-decoherence/).
H. Janssen, Reconstructing Reality. Enviroment-Induced Decoherence, the Measurement Problem and the Emergence of Definitness in Quantum Mechanics. A Critical Assessment, Master’s thesis, theoretical physics, Redbout University, Nijmegen (2008) (philsci-archive.pitt.edu/4224/1/scriptie.pdf).
S.L. Adler, Stud. Hist. Phil. Mod. Phys. 34, 135 (2003).
M. Pauri, Found. Phys. 41, 1677 (2011).
D. Alba, H.W. Crater, L. Lusanna, J. Math. Phys. 52, 062301 (2011) arXiv:0907.1816.
L. Lusanna, Int. J. Mod. Phys. A 12, 645 (1997).
D. Alba, H.W. Crater, L. Lusanna, Can. J. Phys. 88, 379 (2010).
D. Alba, H.W. Crater, L. Lusanna, Can. J. Phys. 88, 425 (2010).
G. Friesecke, M. Koppen, J. Math. Phys. 50, 082102 (2009).
G. Friesecke, B. Schmidt, A Sharp Version of Ehrenfest’s Theorem for General Self-Adjoint Operators, arXiv:1003.3372.
R.W. Robinett, Phys. Rep. 392, 1 (2004) arXiv:quant-ph/0401031.
M. Dickson, Non-Relativistic Quantum Mechanics, in Philosophy of Physics, edited by J. Butterfield, J. Earman (Elsevier, Amsterdam, 2007) Part A, p. 275.
W.G. Dixon, Acta Phys. Pol. Proc. Suppl. B1, 27 (2008).
D. Alba, L. Lusanna, M. Pauri, New Directions in Non-Relativistic and Relativistic Rotational and Multipole Kinematics for N-Body and Continuous Systems, in Atomic and Molecular Clusters: New Research, edited by Y.L. Ping (Nova Science, New York, 2006) arXiv:hep-th/0505005.
D. Alba, L. Lusanna, M. Pauri, J. Math. Phys. 43, 1677 (2002) arXiv:hep-th/0102087.
D. Alba, L. Lusanna, M. Pauri, J. Math. Phys. 46, 062505 (2004) arXiv:hep-th/0402181.
S. Kocsis, B. Braverman, S. Ravets, M.J. Stevens, R.P. Mirin, L.K. Shalm, A.M. Steinberg, Science 332, 1170 (2011).
H.W. Crater, L. Lusanna, On Relativistic Entanglement and Localization of Particles and on their Comparison with the Non-Relativistic Theory, arXiv:1306.6524.
A.V. Rau, J.A. Dunningham, K. Burnett, Science 301, 1081 (2003).
A.V. Rau, J.A. Dunningham, K. Burnett, J. Mod. Opt. 51, 2323 (2004).
P.A. Knott, J. Sindt, J.A. Dunningham, J. Phys. B 46, 095501 (2013).
P. Busch, T. Heinonen, P. Lahti, Phys. Rep. 452, 155 (2007).
P. Busch, P.J. Lahti, Phys. Rev. D 29, 1634 (1984).
A. Peruzzo, P. Shadbolt, N. Brunner, S. Popescu, J.L. O’Brien, Science 338, 634 (2012).
P. Storey, C. Cohen-Tannoudji, J. Phys. II France 4, 1999 (1994).
M. Kasevich, S. Chu, Phys. Rev. Lett. 67, 181 (1991).
H. Müller, A. Peters, S. Chu, Nature 463, 926 (2010).
H. Müller, A. Peters, S. Chu, Comment on: Does an Atom Interferometer Test the Gravitational Redshift at the Compton Frequency?, arXiv:1112.6039.
H. Müller, A. Peters, S. Chu, Testing the Gravitational Redshift with Atomic Gravimeters?, arXiv:1106.3412.
H. Müller, A. Peters, S. Chu, Class. Quantum Grav. 28, 145017 (2011) arXiv:1012.1194.
H. Müller, A. Peters, S. Chu, Nature 467, E1 (2010) arXiv:1009.0602.
H. Yabuki, Int. J. Mod. Phys. 25, 159 (1986).
E. Gindensperger, C. Meier, J.A. Beswick, J. Chem. Phys. 113, 9369 (2000).
E. Gindensperger, C. Meier, J.A. Beswick, J. Chem. Phys. 116, 8 (2002).
B. Poirer, Chem. Phys. 370, 4 (2010).
D.A. Deckert, D.Dürr, P.Pickl, J. Phys. Chem. A 111, 10325 (2007).
N. Takemoto, A. Becker, J. Chem. Phys. 134, 074309 (2011).
K.H. Hughes, G. Parlant (Editors), CCP6 Workshop on Quantum Trajectories, July 12-14, 2010, Bangor University UK, www.ccp6.ac.uk/booklets/CCP6-2011_Quantum_Trajectories.pdf.
C.L. Lopreore, R.E. Wyatt, Phys. Rev. Lett. 82, 5190 (1999).
R.E. Wyatt, E.R. Bittner, J. Chem. Phys. 113, 8898 (2000).
R.E. Wyatt, Quantum Dynamics with Trajectories: Introduction to Quantum Hydrodynamics (Springer, Berlin, 2005).
A.R. Vargas, Open Quantum Systems and Quantum Information Theory, Dissertation thesis at the Ulm University, http://vts.uni-ulm.de/docs/2011/7581/vts_7581_10841.pdf.
A.E. Allahverdyan, R. Balian, Theo M. Nieuwenhuizen, Understanding Quantum Measurement from the Solution of Dynamical Models, arXiv:1107.2138.
G. Gualdi, C.P. Koch, Approximating Open Quantum System Dynamics in a Controlled and Efficient Way: A Microscopic Approach to Decoherence (2011) arXiv:1111.4959.
J. Prior, A.W. Chin, S.F. Huelga, M.B. Plenio, Phys. Rev. Lett. 105, 050404 (2010) arXiv:1003.5503.
P. Ghose, A.S. Majumdar, S. Ghua, J. Sau, Phys. Lett. A 290, 205 (2001).
A.S. Sanz, M. Davidovic, M. Bozic, S. Miret-Artes, Ann. Phys. (NY) 325, 763 (2010) arXiv:0907.2667.
W.P. Schleich, Quantum Optics in Phase Space (Wiley, Berlin, 2001).
A. Polkovnikov, Ann. Phys. 325, 1790 (2010).
L. Lusanna, From Clock Synchronization to Dark Matter as a Relativistic Inertial Effect, Lecture at the Black Objects in Supergravity School BOSS2011, Frascati, 9-13 May 2011, Springer Proc. Phys., Vol. 144 (Spinger, Berlin, 2013) pp. 267--343, arXiv:1205.2481.
M. Pauri, Invariant localization and mass-spin relations in the Hamiltonian formulation of classical relativistic dynamics, Parma University preprint IFPR-T-019 (1971) unpublished.
M. Pauri, Canonical (possibly Lagrangian) Realizations of the Poincaré Group with increasing Mass-spin Trajectories, Invited Talk at the International Colloquium, Group Theoretical Methods in Physics, Cocoyoc, Mexico, 1980, in Lecture Notes in Physics, edited by K.B. Wolf, Vol. 135 (Springer-Verlag, Berlin, 1980).
D. Alba, H.W. Crater, L. Lusanna, J. Phys. A 40, 9585 (2007) arXiv:gr-qc/0610200.
C. Lammerzahl, J. Math. Phys. 34, 3918 (1993).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lusanna, L., Pauri, M. On the transition from the quantum to the classical regime for massive scalar particles: A spatiotemporal approach. Eur. Phys. J. Plus 129, 178 (2014). https://doi.org/10.1140/epjp/i2014-14178-y
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/i2014-14178-y