On 4/12/2017 2:34 pm, Russell Standish wrote:
On Mon, Dec 04, 2017 at 02:11:11PM +1100, Bruce Kellett wrote:
On 3/12/2017 9:03 am, Russell Standish wrote:
The point being that the uncertainty in the coin's initial position is
itself due to the amplification of quantum uncertainty by classical
chaos.
That may happen in some cases, but just looking at the numbers says that
normal thermal motions will far outweigh the effect of any residual quantum
uncertainty. In most cases where the Lyanpunov exponents lead to classical
chaos, there is more than enough classical thermal uncertainty in the
initial conditions so that any residual quantum uncertainty is irrelevant.
But surely, classical thermal uncertainty is just due to amplification
of quantum uncertainty by means of molecular chaos.
But molecular chaos, /a là/ Eherenfest, Maxwell, Boltzmann /et al./ is
essentially a classical phenomenon, due to the random motions of atoms
or molecules in the kinetic theory. Although these are, in some sense,
quantum objects, the momenta involved at normal temperatures are such
the uncertainty principle considerations are irrelevant. So there is no
quantum uncertainty involved in standard molecular chaos, or in the
random thermal motion of molecules in liquids or gasses.
If the uncertainty in initial conditions is reduced by measurement to
something like exp(-λt)w, where w is the coin's thickness, λ the
system's maximal lyapunov exponent and t the time of flight, then the
coin can be treated deterministically, with the outcome of the toss
known once initial conditions specified to that level of accuracy.
But in the general case, the initial conditions are not so precisely
known. With MWI, an observer is in a superposition of many different
(albeit decohered) quantum universes, and no God can point to one of
them and say that is the real world. So the outcome of the coin toss
can be traced back to the effect of quantum fluctuations during the
setup of the experiment.
That is the contention, but it is fanciful. Quantum uncertainties only lead
to distinct, non-interacting, worlds if the initial quantum effect is
amplified in such a way that decoherence can lead to the (effective)
diagonalization of the density matrix. That does not happen for just any
quantum interaction. So even if the coin tosser is split into disjoint
worlds by someone doing a quantum optics experiment in the next room, that
is completely irrelevant. One cannot ascribe the uncertainty in the outcome
of the coin toss to the quantum experiment next door -- the uncertainty in
the toss outcome is solely due to the lack of sufficiently detailed
knowledge of the initial conditions. And that uncertainty is purely
classical in origin. The fact that there might be many different coin
tossers in different worlds does not affect the random influences on the
coin toss in a particular world. And we are concerned only with the
particular world which we happen to inhabit -- the others are disjoint and,
by definition, irrelevant.
If quantum uncertainties were to add up in the way you suggest, it would
seem that thermal randomness, being much larger, would also add up in such a
way that nothing would ever be predictable. But we know that the world is,
by and large, classical and predictable, so quantum uncertainties must tend
to cancel out in the way that thermal uncertainties do. Brownian motion is
an essentially classical phenomenon, and it is important only for
microscopic objects. Brownian effects cancel out in the large.
There is additional (observer inherent) coarse-graining, which means
we can make useful predictions on average of the behaviour of
macroscopic thermodynamic variables.
So the molecular chaos becomes tamed in classical thermodynamics.
But it tends to fail for variables that aren't coarse grained (eg the
state of a tossed coin after it lands), as these states are finely
distributed throughout phase space.
Yes, if we do not have a 'soft catch' situation, the final position of
the coin is sensitively dependent on the rotational speed, angular
momentum, and angle of impact with the hard surface. But it would be
hard to put this down to purely quantum effects -- we are talking about
macroscopic objects and classical phase space here.
Bruce
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