Wavefunction collapse via a nonlocal relativistic variational principle

Abstract

We propose, as an alternative theory of quantum mechanics, a relativistically covariant variational principle (VP) capable of describing both wavefunction collapse and, as an appropriate limiting case, evolution of the wavefunction according to the standard quantum mechanical (SQM) wave equation. This results in a nonlinear, nonlocal, time-symmetric hidden-variable theory; the hidden variable is the phase of the wavefunction, which affects the dynamics via zitterbewegung. The VP is \delta (A_1 + \epsilon A_2) = 0, in which A_1 and A_2 are positive definite integrals (over all spacetime) of functions of the wavefunction \psi. A_1 is quadratic in deviations of the wavefunction from compliance with the SQM wave equation. A_2 is a measure of the uncertainty of the wavefunction, driving collapse by penalizing certain kinds of superpositions. We also show that A_1 limits the rate of collapse, and that it enforces the Born rule, with suitable assumptions and approximations. Since the VP optimizes a function {\psi} of both space and time, the theory is not "causal" in the usual sense. Because it is not clear how Nature solves the optimization problem (e.g., whether a global or a local minimum is sought), we cannot yet say whether it is deterministic.