From a quantum mechanical point of view I'm not convinced that this is valid way to think about the problem, and if I may be so bold, this question is verging back on the topics in the Einstein-Bohr debates.

The main issue I have is with the concept of the 'updating' of the wave function, which by posing the question gives us a concept that there is some information or state that exists and is meaningful between measurements, even if it is is not accessible. So in other words, I think that you are implicitly asking about a form of hidden variable treatment.

What we have in quantum mechanics is only the results of measurements, from which we have to infer properties. From this point of view, the only 'recalculation' that would be required of the particle is at the point of measurement, although even this explanation seems unsound and too anthropomorphic for my liking.

Returning however to the analogy, we can see the difference between a system with repeated measurements and long periods without measurements (ideal projective, or weak measurements) in the change in trajectories from quantum (exhibiting superposition states for example) in the limit of few measurements with large spacing, to classical (a discrete trajectory exhibiting a definite path) in the limit of repeated measurements on timescales commensurate with the evolution, to Zeno-type freezing, in the limit of repeated strong measurements on timescales short compared to the characteristic evolution timescales.

So I suppose to summarise my point, if a particle were able to perform the kind of 'recalculation' that you mention, then I would believe that this must imply a hidden variable description, of the kind that is ruled out by Bell-type inequalities.

Finally I mentioned Einstein-Bohr debates. Your question reminds me of Heisenberg's insight when trying to justify the probabilistic interpretations of quantum mechanics with the observation of particle tracks in cloud chambers. This was seen as a huge problem for the early quantum treatments, as we were being led to believe that particles behaved in entirely bizarre (from the classical viewpoint) ways, yet here were subatomic particles following largely classical trajectories. Heisenberg's great insight was that the repeated measurements from the gas prevented the quantum effects, and hence by repeatedly localising the particle, the classical trajectory results. This neatly explains the observations of tracks in cloud chambers, whilst preserving the (observed) quantum effects of superposition.

I do not know Andrew, wave function updating does not require hidden variables, I think by the way that the issue of hidden variables was effectively laid to rest (Gleason's theorem and all that).

The wave function determines the probability of presence somewhere, which in turn determines the most likely trajectory, nothing else is needed (apart from context-dependent integration boundaries)

The alternative explanations to the bullet somehow "knowing" where to go do not look very convincing to me, especially with the calculation of statistical deviations from the true path conforming exactly to observations on electron trajectories (with e.g. the size of deviations being exactly calculable in terms of expected statistical frequency and matching observations very closely), but of course i'm open to different interpretations, which is why I posted this.

I'm not sure whether this a question in physics or philosophy. On the physics side these sort of questions are best tackled by looking at them in four dimensions. Since time is not (completely) separated from space, this point of view is in some sense more respectful to the symmetries of spacetime...

Moreover, within this view, it is clear why this question doesn't make any sense... The trajectory of the bullet (in 4D) is just an extremum of the action - the 'best' choice in the space of trajectories - no 'recalculation' is needed. Another advantage of this framework is that there is no need for a "collapse of the wavefunction", a concept that has no relativistic counterpart. The only sacrifice one is required to make is to accept the existence of many world lines...

Tomer, sorry I do not understand your answer.

What you call the 'best' choice of trajectories - the question is precisely how this 'best' is determined, and when (considering that no memory of the past exists other than embedded in boundary conditions within certain applicable equations ) The experiments I am referring to seem to say that the 'best' trajectory is recalculated or reset anew at each instant of time, consistent with the wave function determining the most probable locations at every instant of time - my original question exactly

Not sure what your phrase 'extremum of the action' exactly means , mathematically speaking - what equation does this phrase refer to ? If it means a peak of likelihood of a certain trajectory from within a ensemble of possible trajectories, then the problem is recast in a different frame : how is the individual trajectory or trajectory component chosen and when ? How is the trajectory kept to in time (it could 'decohere' in several ways)

By 'many world lines' I take it you refer to the Everett interpretation?, but the same question applies: how is then an individual world line chosen and hewed to (maintained)? The Everett interpretation still being, incidentally, quite controversial.

Let me put it in different (but equivalent) words: Physics is a framework of describing natural phenomena using mathematics. You fire a bullet, a physicist will tell you where it will hit. The methods he will use to compute that need not be directly connected with the real world - in contrast to what Einstein believed. More explicitly, nature doesn't compute equations as things happen. All a physicist can do is cook up some description of what is happening, and check that it is consistent.

Perhaps you meant to ask a different question, regarding the quantization of spacetime itself?