On 3/12/2017 9:03 am, Russell Standish wrote:
On Fri, Dec 01, 2017 at 11:49:06AM +1100, Bruce Kellett wrote:
On 1/12/2017 8:57 am, Bruce Kellett wrote:
The coin is classical, consisting of something of the order of 10^22
atoms. Indeterminacy in position as given by the Heisenberg Uncertainty
Principle, is undetectably small.
I think it is worth while to put some (approximate) numbers around this. The
reduced Planck constant, h-bar, is approximately 10^{-27} g.cm^2/s. The
Uncertainty Principle is
delta(x)*delta(p) >= h-bar/2.
For a coin weighing approximately 10 g and moving at 1 cm/s, the momentum is
mv = 10 g.cm/s. Taking the momentum uncertainty to be of this order, the
uncertainty in position, delta(x) is of the order of 10^{-28} cm. A typical
atom has a diameter of about 10^{-8} cm, so the uncertainty in position is
approximately 20 orders of magnitude less than the atomic diameter. That is
why quantum uncertainties are irrelevant for macroscopic objects.
Uncertainties do not add up coherently for macroscopic objects --
macroscopic objects act as a unit, and the HUP is irrelevant, even for small
coins.
Bruce
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.
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.
In terms of Brent's comment about the magician tossing and catching a
coin in a deterministic (to the magician) way, I believe that involves
a slight of hand. The way I remember doing this as a kid is to flick
the coin enough to make it wobble, and appear spinning to someone
else, but not to flip over completely. Maybe there is a way of doing
it with a flipping coin where the number of flips can be
predetermined. We can say that these are techniques that reduce the
intrinsic Lypanunov exponent. Apparently it is much harder to achieve
this result if the coin lands on a hard surface, rather than the
tosser's hand, which is why that is usually insisted upon.
I think it is most likely that the magician knows the way the coin is
lying initially, and simply counts the number of spins, catching the
coin after the appropriate even or odd number. As you suggest, this
trick would not work if you allow the coin to land on a hard surface,
because one wouldn't have sufficient control the angle of impact or the
subsequent bouncing. But once again, the trick is unlikely to rely on
controlling the initial conditions of the toss all that finely -- one
simply has to control the landing by a soft catch. Lyapunov exponents
are unlikely to have anything to do with it.
Bruce
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to everything-list+unsubscr...@googlegroups.com.
To post to this group, send email to everything-list@googlegroups.com.
Visit this group at https://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.
