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This is where first of two major difficulties arose:

To determine what will happen in a specific experiment involving a specific particle and experimental setting – an observation must be made. This means that, in absence of an observing and measuring human, flanked by all necessary measurement instrumentation – outcome of wavefunction cannot be settled. It just continues to evolve in time, describing a dizzyingly growing repertoire of options. Only a measurement (=the involvement of a human or, at least, a measuring device which can be read by a human) reduces wavefunction to a single solution, collapses it.

A wavefunction is a function. Its REAL result (the selection in reality of one of its values) is determined by a human, equipped with an apparatus. Is it recursive (TM computable and compatible)? In a way, it is. Its values can be effectively and mechanically computed. The value selected by measurement (thus terminating propagation of function and its evolution in time by zeroing its other terms, bar one selected) is one of values which can be determined by an effective-mechanical method. So, how should we treat measurement? No interpretation of quantum mechanics gives us a satisfactory answer. It seems that a probabilistic automaton which will deal with semi recursive functions will tackle wavefunction without any discernible difficulties – but a new element must be introduced to account for measurement and resulting collapse. Perhaps a "boundary" or a "catastrophic" automaton will do trick.

The view that quantum process is computable seems to be further supported by mathematical techniques which were developed to deal with application of Schrodinger equation to a multi-electron system (atoms more complex than hydrogen and helium). The Hartree-Fok method assumes that electrons move independent of each other and of nucleus. They are allowed to interact only through average electrical field (which is charge of nucleus and charge distribution of other electrons). Each electron has its own wavefunction (known as: "orbital") – which is a rendition of Pauli Exclusion Principle.

The problem starts with fact that electric field is unknown. It depends on charge distribution of electrons which, in turn, can be learnt from wavefunctions. But solutions of wavefunctions require a proper knowledge of field itself!

Thus, SE is solved by successive approximations. First, a field is guessed, wavefunctions are calculated, charge distribution is derived and fed into same equation in an ITERATIVE process to yield a better approximation of field. This process is repeated until final charge and electrical field distribution agree with input to SE.

Recursion and iteration are close cousins. The Hartree-Fok method demonstrates recursive nature of functions involved. We can say SE is a partial differential equation which is solvable (asymptotically) by iterations which can be run on a computer. Whatever computers can do – TMs can do. Therefore, Hartree-Fok method is effective and mechanical. There is no reason, in principle, why a Quantum Turing Machine could not be constructed to solve SEs or resulting wavefunctions. Its special nature will set it apart from a classical TM: it will be a probabilistic automaton with catastrophic behaviour or very strong boundary conditions (akin, perhaps, to mathematics of phase transitions).

Classical TMs (CTMs, Turing called them Logical Computing Machines) are macroscopic, Quantum TMs (QTMs) will be microscopic. Perhaps, while CTMs will deal exclusively with recursive functions (effective or mechanical methods of calculation) – QTMs could deal with half-effective, semi-recursive, probabilistic, catastrophic and other methods of calculations (other types of functions).

The third level is Universe itself, where all functions have their values. From point of view of Universe (the equivalent of an infinite TM), all functions are recursive, for all of them there are effective-mechanical methods of solution. The Universe is domain or set of all values of all functions and its very existence guarantees that there are effective and mechanical methods to solve them all. No decision problem can exist on this scale (or all decision problems are positively solved). The Universe is made up only of proven, provable propositions and of theorems. This is a reminder of our finiteness and to say otherwise would, surely, be intellectual vanity.

Sam Vaknin is the author of Malignant Self Love - Narcissism Revisited and After the Rain - How the West Lost the East. He is a columnist for Central Europe Review, United Press International (UPI) and eBookWeb and the editor of mental health and Central East Europe categories in The Open Directory, Suite101 and searcheurope.com.

Visit Sam's Web site at http://samvak.tripod.com