Source of Quantum Indeterminacy
As far as I know there is no formal definition of quantum indeterminacy and we do not know if there is any difference between certain random quantum events, i.e photons passing through a beam splitter or radioactive decay. We also do not know the source of all quantum indeterminacy.
The question I proposed to my professor was suppose we have a finite phase space of the initial conditions of a single particle. This particle independent of our ignorance has a single unique lagrangian mapping its worldline but because of our ignorance it has many possible lagrangians proportional to the volume of the phase space. If the phase space approaches zero then certain lagrangians become less and less probable to the point where we have a single lagrangian. According to QM this is not correct, QM is really random independent of ignorance, but why? What is the source of this weirdness?
Well to start we cannot have a single particle system even in theory because of the vacuum which brings me to a another question. Can we even compute probability regions of the positions of the out-states in QED? Even if we know the precise initial conditions of the electron and positron, i.e the position, momentum, and lagrangian. Once they annihilate they produce a virtual particle a mathematical approximation in perturbation theory consisting of 4-momentum and propagators respectively "gluing" the in-state and out-state. If we assume energy is conserved at all times regardless of the ignorance involved with the energy-time uncertainty, we must then assume information is conserved. The initial conditions must exist somewhere in the excitation of fields where our perturbative approximation is insufficient to retrieve such information.