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A Wave Collapse in the Causal Interpretation of Quantum Theory

A central consequence of the causal interpretation (CI) is that trajectories do not cross [2, 3]. But what happens to the motion of quantum particles if the wavefunction becomes singular and indefinite? Here, an example is presented for the Gross–Pitaevskii equation with a harmonic potential and a time-dependent nonlinear potential. At certain time steps, where , the squared wavefunction becomes infinitely large and in an infinitesimal region and a singularity occurs.
In the causal interpretation, every particle has a definite position and momentum at all times. The trajectories within the wavepacket are not directly measurable but are guided by a quantum potential. Both quantities could be derived from the wavefunction.
In general, crossing trajectories are possible only for multivalued phase functions or for multidimensional fields, in which the orbits in the phase space are mapped into the lower-dimensional configuration space [2].
In the case of an indefinite wavefunction, the gradient of the phase function becomes multivalued and the trajectories can cross. The particle, which starts to the left of the peak of the wavefunction, would be found on the right side of the peak after the singularity. For these very special cases the motion of trajectories does not describe a deterministic process, because the wavefunction is irregular. Thus it must be kept in mind that this is an artificial model of a dynamic system, because the equation of the trajectories is derived from a wavefunction that is not completely definite in -space. But, roughly speaking, one could attempt to interpret the term in the Gross–Pitaevskii equation as a potential term for a measuring device that detects the position of a particle at certain times , with , which introduces a certain type of wave collapse.
The graphic on the left shows the density (blue), the velocity (gray), and the quantum potential (red). On the right, you can see the density and the trajectories in -space.

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Solitons are approximately stable waves that behave as quasi-particles; only for a few cases the equations of motion in the Bohm approach are fully integrable. A notable case is the solitary wave solution of an effective one-dimensional (1D) Bose–Einstein condensate treated by the causal interpretation [1]. The effective Gross–Pitaevskii equation with a harmonic potential, applied here, is:
, where is the complex conjugate, , and , …. The solitary solution can be written:
with .
In the eikonal representation of the complex-valued wave , the gradient of the phase is the particle velocity. This is a linear function with respect to , from which the equation of motion is derived by solving: , where the analytic solution is given by , with the integration constant . The initial positions are distributed linearly around the point with . It may appear that the presence of the term in the amplitude and in the velocity term of the soliton will lead to singularities at finite times (). The velocity field is multivalued and infinite at these times. With the potential , the Gross–Pitaevskii equation is the nonlinear analog of the Schrödinger equation.
I am grateful to Detlef Dürr and his collaborator Dustin Lazarovici for helpful suggestions.
References
[1] S. Rajendran, P. Muruganandam, and M. Lakshmanan, "Bright and Dark Solitons in a Quasi 1D Bose–Einstein Condensates Modelled by 1D Gross–Pitaevskii Equation with Time-Dependent Parameters," Physica D, 239, 2010 pp. 366–386 or arXiv:0812.5073[nlin.PS].
[2] P. Holland, The Quantum Theory of Motion, Cambridge: Cambridge University Press, 1993.
[3] B-G. Englert, M. O. Scully, G. Sussman, and H. Walther, "Surrealistic Bohm Trajectories," Zeitschrift für Naturforschung, 47a, 1992 pp. 1175–1186.
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