Sunday, February 07, 2010

Fuzzy after dinner thoughts one quiet Sunday afternoon.

Brave Knight Sir Roger Penrose doesn’t know which way his Hilbert space vector is going to point next

I am hoping to kick this habit of evolution vs. anti-evilution debate posting, an activity that I have engaged in for nearly two years now: I honestly don’t think I can make much further useful progress on the question. So, I plan to make only two more posts on the debate in order to wind it up. My mind is now turning (back) to other issues.

With the authoritative and excellent help of Sir Roger Penrose’s books I have been pondering the vexed question of the evolution (“evolution” as in “change”, not as in “evilution”) of the quantum mechanical state vector. The following is a resume of my current thinking so far, thinking which I have to admit is rather at the impressionistic sketch stage. (There is no guarantee as to the correctness of the following; after all I’m not a “Sir”. Also, today was one of those days when, as H. G. Wells put it in "The Time Machine", I was experiencing "that luxurious after-dinner atmosphere, when thought runs gracefully free of the trammels of precision")

The orientation of the QM state vector in Hilbert space changes in a smooth and deterministic way until a measuring system from without hits that vector and it then appears to undergo a random discontinuous jump according to probabilities calculated with the “projection hypothesis”. Ostensibly, Hilbert space is Mathematically Isotropic. Moreover, there seems to be a complementary symmetry between position and momentum (or time and energy) and this symmetry is especially clear with relativistic renditions of quantum mechanics. So given these symmetries, on the face of it there seems be no reason why we shouldn’t observe those strange mixtures of state that involve macroscopic objects occupying two positions at once, much as do photons in Young’s slits experiment; for given the isotropy of Hilbert space obviously spatially mixed macroscopic states are no less preferred than spatially unmixed states.

It has to be admitted that decoherence theory may provide an excellent solution to this problem. Entanglement with “hot” macroscopic measuring tools will appear to collapse the wavefunction. Furthermore, thermodynamics and entanglement will ensure that for macroscopic objects some balance between a spread of momenta and distances is maintained, and this perhaps is precisely what we see in the seemingly unambiguous states of the macroscopic world. Decoherence theory still remains a very good candidate explaining the apparent random jumping of the state vector and the preference for certain kinds of macro-state. The big bonuses of this theory are that it preserves frame independence, other symmetries and above all that holy grail of reductionist science: An “in principle” determinism – the closed system. However, it has to be said that at the level of the outermost frame there are no outside “hot” measuring contexts which will help maintain the state vector in some balance between momentum and position.

For reasons I have given here, I am inclined to favour the notion of real discontinuous random jumps in the state vector. Moreover, that the Schrödinger equation is not Lorentz invariant may hint that there is an actual asymmetry between space and momentum. In fact my own attempt at explaining gravity depends on a non-linear quantum equation that makes use of postulated underlying asymmetries in frame that are only partially compensated for by Einstein relativity. Gravity is something that operates in x-space and not in momentum space; hence there is an asymmetry between space and momentum. Thus gravity is implicated as a factor explaining why we only observe spatially unambiguous macro states. Penrose’s ideas about why the state vector jumps randomly may be right: It may jump to ensure that we don’t get quantum superpositions which imply spatially ambiguous macroscopic distributions of matter entailing gravitational energy divergences violating the allowed uncertainty in energy. Such a view entails the end of the reductionist's closed system.

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