Nagarjuna, a buddhist philsopher around 200 AD, in India, wrote on the impossibility of motion in the context of justifying Buddhist ontology (lack of svabhava); this is the content of the second section of his Madhymakarika (Verses on the Middle Way); Westerhoff, in his translation writes:
We might want to want to note that matters don't improve if we assume that time consists of discrete indivisible atoms; if we regard t as an atomic moment between the last moment of rest t0 and the first moment of motion t1, we are again faced with the problem of where to locate t in the exhausative division of the temporal duration into past, present & future. Since the beginning of motion cannot be in the past or future, or best bet is the present motion. But then since t is atomic, it cannot be the moment of present motion, since nothing moves during t: there can be no changes during an atomic moment of time.
He adds that this argument is implicit in Candrakirtis commentary.
Our best theory of motion in the small is QM; and it seems to me that it is at least consonant with this. If a particle is precisely located, so we know where it is and what time it is there - we have no knowledge of how it moves: momentum is indefinite. The usual explanation is that the particle can be anywhere, and this is justified, at least in Heisenbergs writings as being the effect of a 'large' disturbance.
It appears to me though, that one can perhaps take this as face value; momentum, in some places (ie by way of Noether) is called the generator of translations; for there to be no change in an atomic unit of time, a temporal atom, it seems to me we cannot have momentum; that is momentum must be 'un-valued'.
One might think, for a particle to be at rest, it simply needs to have zero momentum; but this won't do; as Newtons laws suggest this is synonymous with uniform translation - one merely needs to change to a different inertial frame (stationary frame, in Einsteins usage, and to my mind a more descriptive term).
Thus, if we take Candrakirtis commentary at face value, and thus allow for atomic temporal units, as well as atomic particles - can an atomic particle carry momentum (atomic or not) in an atomic unit of time?