Hi,Demystifier, a quick question if you have time.
The guidance equation is a function of instantaneous positions of all other particles, which themselves have positions which are unknown/unknowable. OK, that seems reasonable to me (as being a source of apparent indeterminism).
Now, the issue with that we are saying the "instantaneous position" of other particles, which gives rise to non-local influences. Why couldn't the equation be a function of their positions, but NOT their instantaneous positions? Perhaps the influence is time-delayed so that c is respected. Wouldn't that allow sufficient room for the equations to have a similar effect (i.e. apparent indeterminism but actually is deterministic)? Or perhaps it is not a past configuration that controls, but a future one. The question is whether the guidance equation demands instantaneous positions only for the influence, or could there be other solutions as well?
(Hopefully my question makes sense.)
Your question make sense. It is a question Dirac ask himself in 1932, answered in a paper on relativistic quantum mechanics. It is the first time in the history a physicist introduce as many time variable as space coordinates variables, i.e. one couple (r,t) for each particles. Tomonaga and Schwinger then generalized this idea (a little bit enhanced by Dirak-Fock-Podolsky in their multi-time formulation of quantum mechanics) to fields. This means they associate infinitely continuous time variable to infinitely continuous coordinates positions of the field in space. The result was a Nobel and the birth of quantum field theory.
In the case of Bohm, this have been tried, and the authors claim they have a relativistic extension of bohmian mechanics. Google : multi-time Dewdney Horton to see the details. In short, they stop to describe the path of particles and start describing their world lines. Then it is relativistic if the state of the particles is'nt entangled. If it is entengled, they describe the world line of the system taken as a whole, if I remember well.