PeterDonis said:
If you want to do relativistic quantum mechanics, you don't use Schrodinger's Equation; you use quantum field theory.
The Schrödinger equation works fine in the relativistic field theory too. There are, of course, the field-theoretic problems with handling an infinite number of degrees of freedom, but this is not different from non-relativistic field theory. Despite being relativistic, the momentum of, say, a scalar field follows the same formula ##\pi(x) = \dot{\phi}(x)## and the energy is quadratic in this momentum. The spatial partial derivatives are functions on the configuration space, which is the space of functions ##\phi(x)## (but not of t).
One can get rid of these problems with an infinite number of degrees of freedom using a regularization with a finite lattice on a big cube with periodic boundary conditions, then one can apply standard the Schrödinger picture with the Schrödinger equation for ##\vec{q} = (\phi(n_i,n_j,n_k)),\vec{p} = (\pi(n_i,n_j,n_k)), H=\frac12 |\vec{p}|^2 + V(\vec{q})## where the potential V contains also dependencies on the finite differences approximating the spatial partial derivatives.
MacCrimmon said:
Is there a reciprocal theory to understand POVM using superposition?
The best way to understand POVMs is to see how they describe approximate measurements of position and momentum together. To measure ##\hat{p}## and ##\hat{q}## at the same time approximately, prepare another test particle in the ground state of some harmonic oscillator and measure ##\hat{p}+\hat{p}_{test}## and ##\hat{q}-\hat{q}_{test}##. They commute, so that you can measure them at the same time, in the same measurement. But the result is some two-particle state, which does not define any pure one-particle wave function. So you cannot use it for preparation of some one-particle state which would give the same measurement result again. (The mathematics of this representation on the p-q plane is very beautiful, the wave functions are not only continuous but, modulo a simple weight factor ##e^{-\frac12 z\bar{z}}##, holomorph (complex analytic) in ##z=p+iq##. The analog of the basis are known as coherent states.)
The situation with time is similar. You have no operator for time measurement, the POVM does not give you as the result a state where the same measurement immediately after this would give the same result. There are no clocks giving accurate time, all clocks with some nonzero probability go sometimes even backward in Schrödinger time.
MacCrimmon said:
Well... Could it be (and I am asking not speculating) that entangled particles seem to be influencing each other nonlocally because they are still sharing an absolute time reference. This would in no way be a contradiction or violation of GR or SR. And may not be limited to quantum particles.?
This is essentially how realistic interpretations explain the violations of the Bell inequalities. They use a hidden preferred frame. The key is that from the Schrödinger equation follows a continuity equation for the probability in configuration space ##\rho(q)=|\psi{q}|^2##:
$$ \partial_t \rho + \nabla (\rho \vec{v}(q)) = 0. $$
with the Bohmian velocity v. That means, one can define a Bohmian trajectory by ##\dot{q}(t) = \vec{v}(q)##. The velocity ##\vec{v}(\vec{q})## depends on the whole configuration ##\vec{q}## at that moment of absolute time. So, absolute time is indeed essential to explanation of the non-local quantum effects.
This is obviously not in contradiction with SR, but simply the Lorentz ether interpretation of SR which has a preferred frame. It is in some weak conflict with GR, given that GR has solutions which don't allow to define a global time-like time coordinate, like wormholes or Goedel's universe with causal loops. So, this requires a minor modification of GR which excludes such things. Such a generalization has been proposed in Schmelzer, I. (2012). A Generalization of the Lorentz Ether to Gravity with General-Relativistic Limit. Advances in Applied Clifford Algebras 22(1), 203-242, resp. arxiv:gr-qc/0205035. It is, roughly, a Lorentz ether interpretation of the field-theoretic version of GR on a flat background ##\mathbb{R}^4## in harmonic coordinates.