230 research outputs found
From Heisenberg to Goedel via Chaitin
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.Comment: Small change
A Complete Theory of Everything (will be subjective)
Increasingly encompassing models have been suggested for our world. Theories
range from generally accepted to increasingly speculative to apparently bogus.
The progression of theories from ego- to geo- to helio-centric models to
universe and multiverse theories and beyond was accompanied by a dramatic
increase in the sizes of the postulated worlds, with humans being expelled from
their center to ever more remote and random locations. Rather than leading to a
true theory of everything, this trend faces a turning point after which the
predictive power of such theories decreases (actually to zero). Incorporating
the location and other capacities of the observer into such theories avoids
this problem and allows to distinguish meaningful from predictively meaningless
theories. This also leads to a truly complete theory of everything consisting
of a (conventional objective) theory of everything plus a (novel subjective)
observer process. The observer localization is neither based on the
controversial anthropic principle, nor has it anything to do with the
quantum-mechanical observation process. The suggested principle is extended to
more practical (partial, approximate, probabilistic, parametric) world models
(rather than theories of everything). Finally, I provide a justification of
Ockham's razor, and criticize the anthropic principle, the doomsday argument,
the no free lunch theorem, and the falsifiability dogma.Comment: 26 LaTeX page
Thermodynamic picture of the glassy state gained from exactly solvable models
A picture for thermodynamics of the glassy state was introduced recently by
us (Phys. Rev. Lett. {\bf 79} (1997) 1317; {\bf 80} (1998) 5580). It starts by
assuming that one extra parameter, the effective temperature, is needed to
describe the glassy state. This approach connects responses of macroscopic
observables to a field change with their temporal fluctuations, and with the
fluctuation-dissipation relation, in a generalized, non-equilibrium way.
Similar universal relations do not hold between energy fluctuations and the
specific heat.
In the present paper the underlying arguments are discussed in greater
length. The main part of the paper involves details of the exact dynamical
solution of two simple models introduced recently: uncoupled harmonic
oscillators subject to parallel Monte Carlo dynamics, and independent spherical
spins in a random field with such dynamics. At low temperature the relaxation
time of both models diverges as an Arrhenius law, which causes glassy behavior
in typical situations. In the glassy regime we are able to verify the above
mentioned relations for the thermodynamics of the glassy state.
In the course of the analysis it is argued that stretched exponential
behavior is not a fundamental property of the glassy state, though it may be
useful for fitting in a limited parameter regime.Comment: revised version, 38 pages, 9 figure
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