A pilot-wave theory for interacting quantum fields without preferred foliation
In plain words
Bohmian mechanics (pilot-wave theory) gives particles definite paths guided by the wave function and reproduces quantum predictions, but its nonlocal guidance seems to need a preferred notion of 'now' across space, which relativity lacks. Whether a fully relativistic version exists for interacting fields is open.
Precise statement
Bohmian mechanics for entangled particles and its field-theory extensions (Bell-type quantum field theories with stochastic particle creation) use a preferred spacelike foliation; Durr, Goldstein, Norsen, Struyve and Zanghi (Proc. R. Soc. A 2014) showed that for fixed particle number the foliation can be generated covariantly from the wave function. Determine whether a hidden-variable theory for an interacting relativistic QFT (e.g. QED) exists whose only extra spacetime structure is fixed covariantly by the wave function, with well-defined beables through particle creation and annihilation, reproducing all quantum predictions; or prove that such a theory needs structure not determined by the dynamics.
What would settle it
An explicit covariant construction for an interacting QFT with a proof of empirical equivalence, or a no-go theorem under stated assumptions.
Status in the literature
A covariant foliation law for fixed particle number was given in 2014 (doi:10.1098/rspa.2013.0699); no accepted construction for interacting QFT was found in the sources checked.