When we integrate over all possible paths we find an interference pattern. Some paths are fit to survive the environment challenge posed by the double slit and others self desstruct, even on a photon by photon basis... Now to extend this analogy to extinction. Extinction is as if we suddenly changed the environment say by closing one of the slits, or more subtly by just detecting which slit a photon goes through. (by collapsing one set of waves at the slit) In that case the entire interference pattern is lost. The analogy breaks down in that every path through the other slit now survives. I guess the better analogy would be to close both slits. Then no path survives. So I do not think it is correct to apply the idea of natural selection to such catastrophic events that entire species and 80-90% of all life are destroyed.
In the case of the relationship of path integrals with natural selection, I think it is helpful to pick-up the double slit screen and throw it in the dumpster (or putting it in the equipment room where you can use it another day).
I think you can extend the path integral formulation to natural selection, but you have to consider the interpretation (or meaning) of the path integral formulation and adjust the scale to the phenomena you are trying to describe. So, for instance, if you want to describe the whole universe, the path integral formulation applies on a massive scale where each 'particle' represents a universe (in which case you are talking many universes), or it can also apply to the evolution of the universe where each 'particle' is represented by a possibility (or probability) and the universe is the summation of those possibilities (or probabilities). Traditionally, most physicists have interpreted the path integral formulation in terms of the latter (with a few saying that the other possibilities/probabilities actually existing in their own right).
Now, with regard to natural selection, I'm thinking in terms of the latter interpretation in combination of the scale. So, if you apply this to natural selection, what we are saying in effect is that there are many possibilities/probabilities that evolution can go (extinction, proliferation, bifurcation, etc). All of these paths face the whole of life, they face individual species, individual orders of life, etc. What the path integral does is provide a mathematical means of showing which path is more probable, which happens to be the path where the route is the most direct. In other words, natural selection as it occurs is the shortest path that nature should follow. If there were a shorter path, nature would surely follow that other path.
Now, where is God in the path integral. Well, the path integral is well-defined not just in terms of the possible paths and 'distances' of each path, it is also defined by the source and destination of all the collective paths. So, in case of the whole universe, let's suppose that God is the beginning of the path, and God is also the end of the path (as is said in Christianity where in the beginning is God, and at the end God is "all in all"). Okay, now we have a path integral based on all the possibilities from getting from the Source (God) to the Destination (God). There are many paths, in all likelihood, but all the paths that exist must make their way to the Destination (God).
So, we being the curious types, focus on just one of those paths, and lo and behold we find our little universe. The beginning is God who causes the big bang (naturally, of course) and the universe moves about naturally as if there is no intervention from the Creator until it arrives at its Omega Point (of some sort). It looks like it all happened by chance and by all accounts - naturally (or path integrally), but this universe wouldn't exist unless it also happened to be one which would make the path from the Source to the Destination.
So, does that mean that God is not involved during the routing of our path (i.e., the evolution of the universe)? No. Rather, the universe is responding to the Path Integral in the Sky (God: the Logos), and the universe is making its way based on the criteria of that path integral. Of course, it looks natural, and that is because it IS natural. Natural in that it is moving in a path integral manner.
If God has some specific objective that he wants for our universe to meet (e.g., the introduction of good religion and ethics to humanity), all that God has to do is will to be and it is so. That is, God can create any path integral demand to the universe that he wants and when he wants. So, if you look at the history of earth, for example, if God wanted humans to evolve from apes, then he simply eliminates all the paths from the main Source and Destination path integral that do not have humanity evolving from apes. He eliminates them by simply adding criteria to the Destination. So, for example:
Former PI) The Source is God and the Destination is God. Any path that can accomplish this is a valid path.
Replacement PI) The Source is God and the Destination is God which can only be obtained by meeting these x,y,z objectives.
By changing the path destination based on certain criteria, the former paths that could have reached that former destination, no longer can because they do not meet x,y,z objectives. The source is specific to a set of criteria, and only the paths that meet those criteria are elgible to reach the source. The paths that no longer reach the new destination are not possibilities, and therefore they never existed in the first place (hmm... "it will be as though they never been..."). Well, I digress...
In any case, God can have full control of the universe with these criteria added, and this is how a perfect God can be involved in the creation of a world that is dominated by natural selection and evil. The natural selection and evil is not a criteria that has been eliminated (at least not yet), and therefore they form part of the shortest path that this universe will and did take in its history. If the criteria was added to eliminate natural selection, then surely we wouldn't have evolved through the use of natural selection but some other means. Perhaps no paths exist without natural selection, and therefore God choose not to eliminate it at this point. In any case, it hasn't been eliminated as an option, and since it forms part of the shortest path, it is fully operational in the world (along with the evils that come with it). Perhaps God first judged the mind of consciousness and found that natural selection was needed because of the first conscious mindset (Adam?). These are questions we'll just have to wait (hopefully as long as possible) before we have them answered.