Is the measurement problem an example of essential incompleteness?
Two strong remarks have been made about the Quantum Mechanics:
1) it is internally consistent
2) it is complete
One of the mathematicians who were at Princeton when Einstein was there was Kurt Gödel. He was able to show that in any number theory where the axioms were considered complete that there would be theorems that would be true but not derived from the set of axioms. The set of axioms will always be essentially incomplete. To help demonstrate the proof, he used an old unsolved problem since Euclid regarding the inability to derive a formula for prime factorization and inverted it.
So the question that arises when putting the Quantum Mechanics in a formal representation that seeks to ensure that it is internally consistent and complete is: Does essential incompleteness have a place here as well?
If the answer is yes, this would mean there are true theorems that can not be derived from the fundamental axioms. This means there will be observations from experiments that are not solvable from the fundamental axioms of Quantum Mechanics.
One potential candidate could be “the measurement problem”. This is a problem that has not yet been solved intrinsically from the postulates of Quantum Mechanics and may represent an example of something true but not subject to proof from the basic axioms.
The measurement problem deals with the question of how a purely random process tends to give a classical result far more often than it should. Einstein, quite late in his career, brought the question up to Max Born:
“In a letter dated 1954, Albert Einstein wrote to Max Born “Let
phi1 and phi 2 be solutions of the same Schrödinger equation.. . ..When the system is a macrosystem and when phi1 and phi2 are
‘narrow’ with respect to the macrocoordinates, then in by far the
greater number of cases this is no longer true for phi= phi1 + phi 2. Narrowness with respect to macrocoordinates is not only independent of the principles of quantum mechanics, but, moreover, incompatible with them.” (The translation from Born (1969) quoted here is due to Joos (1986).)”
Source is a footnote from:
AND THE QUANTUM ORIGINS OF THE CLASSICAL
Wojciech Hubert Zurek
PDF (60 pages): http://arxiv.org/PS_cache/quant-ph/p.../0105127v3.pdf