10:30 PM
Last night dream has a thought experiment where Einstein is walking across a bridge trying to outran a light pulse. The bridge is also subjected to pretty strong pulses of gravity thus it flexes and wanes like a wave (to be detailed later in a diagram)
(Of course he never outran it)
But what's interesting about this dream is a regular spacing of spacetime slices is taken on this curved spacetime (as the bridge has a pulse of gravity travelling through it just behind where Einstein is, so representing the whole region as a spacetime manifold it will have some curvature)
Under these regular spacing, the worldline of the light pulse appeared as a sequence of dots regularly spaced along the geodesics of the manifold. And from that I am starting to suspect why momentum has to be tied with wavenumber
So classically in Newtonian spacetime, velocity measures how frequently something repeats in space when all 1 second snapshots are overlaid on top of each other
Here the particle in all snapshots is at some well defined position (x,y,z)
Now, using De Broglie's suggestion, a particle position instead get smeared out into probability amplitudes to form a wavefunction. Thus looking at the diagram of all time slices overlaid on top of each other, it now looks like a continuous wave where the wavelengths of each component (as determined by the Fourier transform) are exactly how far the peaks and trough are spaced out
Likewise taking space slices instead of time slices, and run the argument similarly, the position of the particle and the peaks/troughs of the wave repeats every e.g. 1m. Thus this is the frequency
We knew that infinitesimal displacement and infinitesimal time displacement is a generator of momentum and energy, and both of these quantities are conserved if translation and time translation are conserved respectively as told by Noether's theorem.
Now moving to SR and GR, the velocity and momentum 4 vector directs the direction and magnitude on some object in spacetime, thus momentum and energy is united to form the energy-momentum
and thus, with these concepts in mind, and h supposed to be the smallest possible scale, the reason why: $$E=hf, p=\frac{h}{\lambda}$$ becomes apparent
Anything else is too long thus I am going to do it in another room
@vzn Never thought I can get something this concrete out of dreams. I wonder if it actually holds in the mathematical framework...
So I guess, in a timeless language, "what is motion?" Motion is how the same object is replicated in space when we take away time. Thus if momentum is discrete, we would be seeing an object teleporting from one point to the next as time passes
