Time evolution operator and Klein-Gordon equation The basis of classical QM is the postulate of a time evolution operator
$$|\alpha,t_0;t\rangle=U(t,t')|\alpha,t_0;t'\rangle$$
Is it correct to interpret this postulate as

All future states are determined by the presence, the past is irrelevant.

And infer that the first try to achieve a relativistic QM with the Klein-Gordon equation was doomed to failure because it violates this principle as the second order differential equation requires the first time derivative to determine future states?
I'm asking because I mostly read a non-positive semi definite probability density has been the issue with the Klein Gordan equation, not this violation. Maybe I'm wrong and this isn't even a violation?
 A: When we talk about the future being entirely determined by the present state of a system (the Markov property—terminology borrowed from classical stochastic processes), we have to be careful about specifying what we mean by the current "state" of a system. For the purposes of the Klein-Gordon equation (which was actually the very first attempt at a wave equation in quantum mechanics; Schrödinger considered it and rejected it before developing the Schrödinger equation as its nonrelativistic limit), the state of a system is specified by both the value of the field at all points in space, $\phi(\vec{x},0)$, and its time derivative, $\dot{\phi}(\vec{x},t)$.  (More generally we can substitute the values on some fixed spacelike hypersurface $\Sigma$.)  With this initial data given, the Cauchy problem—finding the subsequent time evolution at all points in space—is well defined.
The practical problems with the Klein-Gordon equation have nothing to do with the need to specify both $\phi$ and $\dot{\phi}$ as initial conditions.  The difficulties with the Klein-Gordon equation come from the fact that it does not have a sensible interpretation as a single-particle wave function; there is no way to have $|\phi(\vec{x},t)|^{2}$ represent the probability density for finding a single particle at the position $\vec{x}$ at time $t$, for a theory with nontrivial interactions.  Frankly, solutions of the Dirac equation do not have a coherent interpretation as a single-particle wave function either, although (unlike for the Klein-Gordon equation), sometimes the single-particle Dirac equation is a reasonable solution.  The ultimate reason that this kind of interpretation fails for any relativistic wave equation is that in a relativistic theory, it must be possible to create and destroy particle-antiparticle pairs, and so only a second quantize theory that allows for such creation and annihilation can give a complete picture of the interactions.
