Does a particle "exist" between emission and detection? In considering a simple Young's double slit setup:
The classical idea that a particle must exist with defined position and momentum between the source and detection plate leads to philosophical angst, and Quantum Mechanics is most often interpreted (and in fact seems to make the most sense) if particles are not, in fact, "lumps of potato" between source and detection, but instead can only be said to truly exist at the point of detection. 
In other words, the question of particle history, i.e. "What was particle 'b' DOING before I detected it?" is nonsensical in QM, and a ton of effort is spent getting us to stop thinking in this way.
My question is this: 
Feynman's path integral approach and the improvement of Weak Measurements both seem to suggest that, while we may not be able to detect exactly which path a specific particle has taken to high degree of accuracy, there is in fact a particle that is taking a path in the first place.  This seems contra to my understanding of QM, which explicitly works best if the very idea of particles taking defined paths of any kind is jettisoned as quickly as possible.  
Further reinforcing the idea that particles are indeed, in some way, "lumps of potato" even when no attempt is made at weak measurement along their flight, is the matter of transit time:  
Photons (or electrons, et al) generated at our experiment's particle source will not show up randomly at my detection plate placed 5 meters away; instead they will arrive in a predictable and timely manner such that I may start my watch when the photon is emitted and predict when it will hit my detector, as the photon will obediently arrive in the time dictated by c/5m.   The very fact that the photon arrives in a repeatable and predictable period of time suggests that it exists during that interval period of time in some way, shape or form.  
That the path integral approach works, that transit time is obeyed and predictable, and that weak measurements may be used to build up the "paths most traveled" all offer what seem to be at least circumstantial evidence that something does exist in the classical sense during the period of time between emission and detection, and I'm hoping the community can set me on the right path (pun intended) in reconciling these contradictions.
 A: A few points first


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*Feynman path integral never says that a particle takes a definite path. Instead it says that the inner product between the initial and the final states is a sum over all paths in the position space. It never implies that one path was taken.

*Similarly weak measurements too don't violate the postulates of the quantum theory in any manner. It is a measurement scheme opposed to projective measurement, working completely by the established laws of the existing quantum theory and is dependent on the coupling of the system to the apparatus and doesn't contradict anything, although surely as is defined in the original paper, it might give results which are different from those as measured by projective measurements. The question whether weak values are actually physical is not solved yet.
Coming to the actual question,
So you know that a photon coming from a distance of 5m will reach you in 5/c seconds from the special theory of relativity. The speed of light is constant and that is the law of nature. But that doesn't imply that the photon reaches you in 5/c seconds! Two ways to understand this


*

*If you measure the exact time at which the photon comes to hit the detector for a large number of readings on a detector, then the photons will have an uncertainity in time interval which is estimated by


$$ \Delta T \Delta E \approx \hbar$$
The physical implications of this relation is that for a small enough time interval, the energy of a system can be violated and thus there will be a spread in the detection time interval.


*The difference between the Feynman path integral and classical physics is that there is a sum over all paths taken in the Feynman path integral while classical mechanics follows the path of least action in the physical world. If you believe in Feynman's formulation, then the photon can take an infinite number of amount of paths to travel the 5m as stated. In that case an exact detection of time at which the photon reaches the screen will tell you which path the photon takes as it travels always at c. So for quantum mechanics to be followed, there will be a time spread in the detection of photon.


IMO the question whether a particle exists in between detection and origin is meaningless. Also IMO weak values in the context of weak measurements are not physical. And also the arguments you put up don't really give any evidence whether a particle exists in the meantime. Although philosophically satisfying to have a definite answer, the question doesn't really achieve anything.
