|
| related topics |
| {time, systems, information} |
| {particle, mechanics, theory} |
| {algorithm, log, probability} |
| {let, theorem, proof} |
| {energy, gaussian, time} |
| {theory, mechanics, state} |
| {vol, operators, histories} |
| {measurement, state, measurements} |
| {wave, scattering, interference} |
| {bell, inequality, local} |
|
From Heisenberg to Goedel via Chaitin
C. S. Calude, M. A. Stay
abstract: In 1927 Heisenberg discovered that the ``more precisely the position is
determined, the less precisely the momentum is known in this instant, and vice
versa''. Four years later G\"odel showed that a finitely specified, consistent
formal system which is large enough to include arithmetic is incomplete. As
both results express some kind of impossibility it is natural to ask whether
there is any relation between them, and, indeed, this question has been
repeatedly asked for a long time. The main interest seems to have been in
possible implications of incompleteness to physics. In this note we will take
interest in the {\it converse} implication and will offer a positive answer to
the question: Does uncertainty imply incompleteness? We will show that
algorithmic randomness is equivalent to a ``formal uncertainty principle''
which implies Chaitin's information-theoretic incompleteness. We also show that
the derived uncertainty relation, for many computers, is physical. In fact, the
formal uncertainty principle applies to {\it all} systems governed by the wave
equation, not just quantum waves. This fact supports the conjecture that
uncertainty implies randomness not only in mathematics, but also in physics.
- oai_identifier:
- oai:arXiv.org:quant-ph/0402197
- categories:
- quant-ph
- comments:
- Small changes
- doi:
- 10.1007/s10773-005-7081-8
- arxiv_id:
- quant-ph/0402197
- journal_ref:
- Int. J. Th. Phys. 44(7). 2005. pp. 1053--1065
- report_no:
- CDMTCS RR 235: February 2004
- created:
- 2004-02-25
- updated:
- 2006-07-11
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