Well, yeah. The acceleration must be quantized because energy is quantized, which implies that the force is quantized as well. Quantizing force has profound effects on the aggregate behaviour. If you program, you can see this by comparing an analytical graph of x(t) to a quantized simulation; the simulation, especially in limit cases such as F <<>-2 range will be different when it feels the impulse than when it 'should' feel the average impulse, and with F -> 0 the difference will actually be significant, and each small difference pushes the next difference yet further from the continuous F=ma approximation.
Not only that, but you have to figure out how the quantized force knows when to land the impulses. If this wasn't gravity, then it would be easy; the force carriers are quantized and it applies forces upon collision. However, we can't determine whether the force-carrier description or the bent-space description works better for gravity, and even if is force carriers we haven't found the graviton and can't be sure of its properties.
I'd guess that like many things, quantized gravitational accelerations would happen probabilistically, but this pushes the actual behaviour yet further from the continuous F=ma, again because each acceleration affects when the next acceleration will be felt; late impulses will allow the object to fly farther from the analytical-continuous position
Moreover, all these possibilities must be normalized for conservation of energy. As the object is feeling these random forces, they must somehow work out to be proportional to the change in potential energy. Alternatively, the potential energy calculation needs to be altered in light of the new behaviour.
So, at the very least, as compared to continuous F=ma, objects drift off true, then feel different forces, and then the forces are somehow corrected for conservation of energy, which then feeds back by drifting it off true... Instead of x(t) = d + vt + at2/2, it's a complex partial differential equation, and a(t) is basically going to be Dirac deltas. All of this is a direct consequence of quantized energy.
This is such a basic analysis I'm surprised I haven't seen it anywhere before. I guess that means I'm grateful to Tech Review for prodding me into doing something I already should have done.
Subscribe to: Post Comments (Atom)
Post a Comment