From: firth@sei.cmu.edu (Robert Firth)
Subject: Night Thoughts on Bell's Inequality
Organization: Software Engineering Institute
Date: Fri, 9 Oct 1992 14:27:43 GMT

Night Thoughts on Bell's Inequality
-----------------------------------

There are many situations in the natural world where we are unable to
predict the outcome of an experiment, and there have been ever since the
first Sumerian tossed the first die.  The classical theory of probability
allows us, in most controlled situations, to predict not only the set of
possible outcomes, but also the approximate frequency, over the long term,
with which each can be expected to occur.  I believe Kolmogorov has put
this on a firm foundation, and shall move on.

For example, a roulette wheel can be thought of as a device for
generating a "random number" between 0 and 36, and indeed it had
better do just that, if the house is to thrive.  We explain this
by saying the wheel is "true" - it has been carefully machined so
that the differences between slots are as small as possible.  The
probabilities are the same for all observers, and so objective.
Indeed, if one observer in the casino were able to predict the
fall of the ball, our instinct might be to assume that he was
a privileged observer, but one who knew that the game was rigged.

In classical mechanics, the system is deterministic.  The number that
comes up is an inevitable consequence of the prior state of the
system, and a perfect observer, knowing the exact position and
momentum of all relevant particles, could indeed predict the outcome.
Well, that was Lagrange's view, and it's rather gone out of fashion;
but it's still a valid model for probabilistic events: the outcome
is determined, but by visible attributes of the system that are too
small to measure or that cannot be measured with sufficient precision.

Another device for generating a "random number" is the function
Random() found in your computer.  This generates a number between,
say, 0 and 65535, and if it's a good generator, no observer could
in reasonable time acquire enough information to predict the result
of the next invocation.

But, if we take the lid off the computer, we see that the system
is still deterministic: it works by means of an algorithm, more
or less complicated, that operates on a "seed", a static value
hidden inside the function.  In other words, a "hidden variable":
state of the system forever invisible to an external observer.

On to quantum mechanics.  If we fire a photon at some kind of state
splitter, a polarising filter or a half-silvered mirror, here again
we are unable to predict the outcome of any single trial, but we
have equations that allow us to predict the frequency of the outcomes
over many trials.  So the photon is another "random number" generator,
yielding the result 0 or 1, left or right, with frequency asymptotic to 1/2.

What models do we have, that purport to explain this phenomenon?

One model, clearly, is the classical one: the behaviour of the photon
is determined, by attributes that we could measure in principle but
that currently elude our intelligence and our instruments.  This, I
think, is the model Einstein badly wanted to reestablish.  It might
be called a "visible variable" theory.

A second model is the computer model: the behaviour is determined
by hidden state that is in principle unobservable, since we can't
pry the lid off the photon and look inside.  This is the "hidden
variable" theory.

One obvious remark, is that there must be quite a lot of hidden state.
After all, the photon doesn't know in advance what experiment it will
be subjected to, and so must have enough state to determine the outcome
of any possible experiment, including, say, one with a zillion beam
splitters.  Using the computer analogy, it's as if the photon had an
internal look-up table, and reasons thus: "Hmm, what's this?  Ah, a
horizontal polarising filter.  That's experiment #4567, so let's see...
wait a nanosecond... oh-ho, I'm supposed to be absorbed.  Bummer, dude."

Does a photon contain an infinite amount of hidden state?  Well, why
not - after all, a classical particle contains an infinite amount of
visible state.  The positions and momenta are real values that may
be measured with arbitrary precision: in effect, six floating-point
values in registers of infinite width.  Drop a classical particle
into an infinite triangular pinball machine, and you can read off the
mantissa of one of those registers, left to right, one bit at a time.

No, I don't find that argument convincing, but I'm setting you up.

Qunatum mechanics, in some of its interpretations, offers us a third
model for the phenomenon, and it is simply this:  there is no state,
hidden or visible; the macroscopic nondeterminism is a consequence
of a microscopic nondeterminism, and the latter is irreducible.  It's
fuzzy dice all the way down.

In imagination, then, the photon carries with it a fuzzy die, and when
it hits the filter, it rolls the die. "Hmm, a horizontal polarising
filter.  Let's see - I roll a 3, and that means - oh-ho, it means I
get absorbed.  Any chance of a saving roll against dexterity?"  "No
way, sucker" - replies the Dungeon Master.  And that's all, folks;
there is absolutely no explanation for the dice.  Alas, poor Albert!

Though this is a digression, I should in fairness point out yet a fourth
picture, the one drawn by Cramer's "Transactional Interpretation".  The
photon is a superposition of an "offer wave", propagating forwards in
time, and a "confirmation wave", propagating backwards in time.  Hence,
the photon simply follows the track of the advanced wave, like a thread
through a labyrinth.  So, when it hits a half-silvered mirror, the
situation is not symmetrical - there is an advanced wave coming from
the left, propagated backwards from the scintillation event in the left
detector, and there is none from the right.  So it knows to jump left.

In the mechanics of Lagrange, the behaviour of a system is completely
determined if we are given either the initial or the final boundary
conditions.  In quantum mechanics, the behaviour is similarly determined
if we are given *both* the initial and the final boundary conditions.
Somebody seems to rediscover that, in different formalism, every decade
or so, and it's repeatedly forgotten.  Which suggests that what people
who fret over fuzzy dice really want is not determinism, but causality,
and asymmetric causality at that.  Read the first sentence of this
paragraph again.  How many textbooks include "or the final" in their
statement of the case, as they should?  Don't ask me to explain this -
I have never lived in that world, and have no idea what makes it so
comfortable.

So much for one photon.  What about two correlated photons?  What we
observe is clear: the photons may jump either left or right, but they'll
both jump the same way.  How?

For the fuzzy dice model, this is a catastrophe.  We seem to be asserting
that nobody, not the experimenter, not the photon itself, not even God,
knows which way Castor will jump.  But each photon, miraculously, seems
to know how the *other* will jump.  We have two strictly correlated
"random" events, as if they had thrown one die for the pair of them.
However you slice it, this is preposterous - it is collusion across
a spacelike interval, which is politically correct doublespeak for
"instantaneous action at a distance".

The hidden variable model, by contrast, is much cleaner.  Of course
each photon knows which way the other will jump - they were in contact,
and plotted the whole thing together.  Each photon is carrying state
away from the point of contact, enough state to ensure they remain
correlated, like synchronized swimmers heading for opposite ends of
the pool.

Alas, no; and at last we come to Bell's inequality.  How clever must
these photons be, how much state must they contain, to reproduce the
perfect correlation we observe?  Let us make the most generous assumption:
each photon may contain an algorithm of arbitrary complexity working on
an arbitrary amount of state.  The photon is a universal Turing machine.

Nevertheless, one can set up an experimental arrangement where no
algorithm, no prior state, is enough to maintain the corelations.  The
best algorithm can maintain only partial correlation, and any worse
one will maintain even less correlation - hence the inequality.  But
the correlations are indeed maintained.

This refutes not just current hidden variable theories, but all possible
hidden variable theories that work on the assumption of correlation
maintained by prior collusion.  That is, all "local" hidden variable
theories.  Since the theories cannot be patched, their adherents are
busy looking for loopholes in the experiments.  Good luck, folks.

The principle of locality was enunciated thus by Einstein:

	"But on one point we should, in my opinion, absolutely
	 hold fast: the real factual situation of system S1 is
	 independent of what is done with system S2, which is
	 spatially separated from the former."

And it is this principle that we must abandon, in *any* theory that
is able to explain what we observe.

So, any hidden variable theory that saves the phenomena must be a
nonlocal hidden variable theory, in which the two photons share state,
and a state change produced by Castor's interaction is visible to
Pollux, at the other end of the labyrinth.  In other words, correlation
between hidden variables is maintained across spacelike intervals.

And that, I think, explodes the whole thing.  We are not asked to
choose between a simple preposterous explanation ("fuzzy dice") and 
a complex reasonable explanation ("hidden variables").  We are asked
to choose between a simple preposterous explanation and a complex
explanation equally preposterous - and hence, surely, an explanation
needlessly complex.

Hidden variable theories predict nothing new, explain nothing new,
and cannot even save the common-sense notion of locality that seems
to be the issue generating all the emotion.  I submit, then, that
hidden variables should be relegated to the same scientific Valhalla
as epicycles and phlogiston: they fought well, and are now history.

Robert Firth