Dear reader,
In Dec. 2, 2008 @ 08:28 GMT I wrote several posts presenting very briefly the main points of my essay: a mathematical framework for foundational discussions (the World Theory), the Strong Causality Principle, the Smooth Quantum Mechanics and delayed initial conditions, minds flowing in the Frozen River, and a Free-Will Hypothesis.
In this essay, I intended to put together several ideas, because I think that, together, they propose an interesting view about the nature of time. The drawback was that I was compelled to present too briefly my ideas, as well as the connection between them, since I wanted to respect the 5000 words rule.
Because I don't know how will be the final theory, I tried to be as general as I could, when discussing the problems related to time, space, causality, and physical law. It is difficult to maintain generality when discuss alternative incompatible viewpoints, therefore I cooked up a mathematical structure named World, which is consistent, but general enough. In this structure, spacetime is a topological manifold, and the physical law is a sheaf (think of it as a collection of possible local solutions) over this spacetime. The matter field is a global section of the law sheaf. To determine the matter field, we need something like initial/boundary conditions. I abstract (in the cited article about World Theory) the conditions imposed both by the physical laws and the observations in such a manner that it become clear that the initial instant to which we apply an initial condition is not absolute.
On the other hand, trying to find an alternative to the discontinuity of the wavefunction collapse in Quantum Mechanics, I constructed a mechanism that uses this "relativity of the initial time" used in an initial condition. My explanation recover the unitary evolution at a higher level, while maintaining the appearance of the collapse, but replaces the discontinuous jump with a set of delayed initial conditions. This mechanism becomes clear when we account for the entanglement of the observed system with the preparation device.
As a consequence of the Smooth Quantum Mechanics, the matter field is determined gradually by each measurement, but we are not able to access the entire set of delayed initial conditions. We collect them in a "registry", which, being incomplete, can be employed only for probabilistic predictions. Therefore, even though the wavefunction evolves deterministically, it appears to us as collapsing, and as having a probabilistic behavior.
Returning to the mathematical structure named world, in the Smooth Quantum Mechanics we see that the matter field may be never determined, it is only a solution of the evolution equation and an incomplete registry of delayed initial conditions.
On the other hand, it seems that the observer, by making a choice of the observable, can impose a delayed initial condition, determining incompletely determined parts of the initial conditions in the past. The global solution don't preexist, only the set of possible solutions, which is gradually reduced by the observations, and the observer's choices of the observables.
This suggests the possibility that the free-will manifests in this deterministic world by the freedom to choose yet-to-be-determined initial conditions. This mechanism, if it exists, works also within the standard Quantum Mechanics, only that in the smooth version is more dramatic. In order to test this free-will hypothesis, I propose an experiment, which relies strongly on our progress in science and technology. Therefore, I suggest waiting until we can perform this experiment, or find another proof, before drawing a final conclusion about the free-will.
All the best,
Cristi Stoica