Yeah, that is true. But this is true for a path of any length, for example A and B could be meters away from each other. This is why I was saying that particles don't just do little jumps on the order of the planck length."
Lets not forget the distinction between the nature of motion of the photon with no rest mass, and the nature of the motion of a particle with mass.

A photon travels at the speed of light, so it covers a fixed distance per time unit. If it moves in discrete increments (alternate paths notwithstanding) it moves either exactly one Plank length at a time or by some exact proportion to it.

A massive particle on the other hand, although composed of bound energy, does not move at the speed of light. What I have researched so far leads me to believe that the Plank unit governing the motion of a massive particle seems much more likely to be the Plank energy.

deBroglie said that all motion requires frequency. It struck me as profound.

I think spacetime decides how particle should travel. The thing is we do not know about the structure of spacetime other than it is three dimensional and continuous. By looking at how particle travels should reveal more about spacetime. I think base on special relativity spacetime would contract at high speed and distance changes. (http://en.wikipedia.org/wiki/Length_contraction)

I think spacetime decides how particle should travel.
Absolutely. But spacetime also needs information from the particle.

In case #1 the particle is moving at some velocity and we say that spacetime moves the particle.

In case #2 accelerate the particle to twice it’s initial velocity. Now spacetime moves the particle at twice the original velocity.

So how does spacetime ultimately decide at what velocity to move the particle? How does it decide which velocity of the two cases to use, or of an infinity of other possible cases? Spacetime uses information (kinetic energy etc) of the particle itself.

Are you asking about the destination or the trajectory? The trajectory is determined because that particular EOM minimize the action using Hamilton's principle. I think the final position however is confined by the trajectory on the phase space

Absolutely. But spacetime also needs information from the particle.

In case #1 the particle is moving at some velocity and we say that spacetime moves the particle.

In case #2 accelerate the particle to twice it’s initial velocity. Now spacetime moves the particle at twice the original velocity.

So how does spacetime ultimately decide at what velocity to move the particle? How does it decide which velocity of the two cases to use, or of an infinity of other possible cases? Spacetime uses information (kinetic energy etc) of the particle itself.
We'll need to see the form energy takes apart from the particle itself. This is a famous equation by Einstein (http://en.wikipedia.org/wiki/Mass–energy_equivalence). Let's say the particle is at rest, and then some energy is transfer to it, how does it decide where its next position should be.

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Actually energy should have a direction heh, else the model would not work.

sophiecentaur