From: naturman@uctvax.uct.ac.za
Newsgroups: sci.physics
Subject: GROSS MISCONCEPTIONS ABOUT PROBABILITY, DETERMINISM AND LOCALITY
Date: 12 Oct 92 13:31:38 +0200
Organization: University of Cape Town

From the responses to my postings on QM which I have received, it is clear to 
me that many people are grossly ignorant of basic mathematics. Here are some 
examples of the gross misconceptions regarding probability which I have 
encountered:

Misconception (1): "Probability is independant of knowledge." To see that this 
is false consider the case of a man who throws a coin over a wall. For him, the
probability that the coin landed heads up is 0.5 and the probability that it 
landed tails up is 0.5. For a man on the other side of the wall who can see the
coin, there are two cases: If it landed heads prob(heads) = 1 and prob(tails) 
= 0; if it landed tails then prob(tails) = 1 and prob(heads) = 0.

Misconception (2): "If the probability of something occuring is 0 then it can't
occur, and if the probability of something occurring is 1 then it must occur."
This nonsense is related to the belief in the Value Rule in interpretations
of QM. The Value Rule says that "if the probabilty of an observable Q having
value q in state Psi is zero, then Q cannot have value q in state Psi". The
Value Rule is usually only considered in realist interpretations of QM. It can
be reworded for non-realist interpretations as follows: "if the probability of
measuring Q in state Psi and obtaining value q, is zero, then we can never
obtain the value q when measuring Q in state Psi". Most people recognize that
the Value Rule fails for observables Q whose corresponding operator Q^ has 
an uncountable spectrum of eigenvalues. But there are some ignoramuses out 
there that seem to think that it does hold in such a case. To see that it
doesn't consider the 1-D case and let Q be the position of a particle along an
axis. Let Psi, for example, be given by a Gaussian wavefunction centered at the
origin. Then no matter what value q we consider, the probability of Q having
value q (on measurement, if you are a non-realist) is zero. Yet when you
measure you will still get some value, and in particular that value had
probability zero. (Tough, if you don't like it.) To see that the Value Rule
also fails for a countable spectrum, let Psi be as before but this time let Q
be the signum of the position of the particle along the axis. (The signum
function, sgn, works as follows: sgn(r) = -1 if r<0, sgn(r) = 0 if r=0, sgn(r)
= 1 if r>0.) Then prob(Q = -1) = prob(Q = 1) = 0.5 and prob(Q = 0) = 0.
However, it is certainly possible to obtain a position of zero for the particle
along the axis and this then gives Q = 0 as well. (Double tough if you don't
like it.)

Other unfounded assumptions related to misconception (2) are:
(a) The Eigenvector Rule: "If Psi is an eigenstate of Q with eigenvalue q, 
then Q must have value q in Psi."
(b) The Spectrum Rule: "The only values which Q can have are the eigenvalues 
of the operator Q^ corresponding to Q."

In special cases these can be shown to hold, but in general, they are 
unnecessary extraneous assumptions which should be shaved off with Occam's
razor. (In fact, if we take a realist approach, an example can be constructed
in which (a) fails.) We should make the least amount of assumptions which are
needed to fit experimental results

Misconception (3): "In a realist interpretation, the mean value of an
observable should equal the expectation value of the observable." Let's call
this the Expectation Rule. This rule is the culprit in the Bell Inequality
controversy. The Bell Inequality shows that if we accept this rule locality
must be violated. Lack of understanding about the calculation of expectation
values, has resulted in people performing experiments which they thought were
tests of the Bell Inequality against predictions of pure Quantum mechanics. In
fact all experiments carried out have only tested expectation values. One
cannot test full mean values in realist interpretations.

Misconception (4): "Realism logically implies determinism while non-realism 
implies non-determinism." This is false. There are realist theories that are 
non-determistic. Moreover, in some of them we can derive the Bell inequality, 
and so the Bell Inequality has nothing to do with determinism. However, 
non-deterministic realist theories have problems agreeing with pure QM. 
Experiments still need to be done to confirm whether pure QM or these theories 
are correct. There are also non-realist theories that are deterministic.

Misconception (5): "Only realist interpretations suffer from non-locality." 
Certainly false, as the Einstein-Podolsky-Rosen argument shows. (If you don't
know what it is then look it up.)

Misconception (6): "All interpretations suffer from non-locality." Obviously, 
a different class of people to those who believe (5). Well this looked like it 
was true until I pointed out that the Expectation Rule need not be true. (That 
the Expec. Rule need not be true is obvious to any pure mathematician who 
understands the difference between expectation values and mean values, but I 
seem to be the first person to realize that it is the culprit in the Bell 
Inequality fiasco. Some other people posting to sci.physics did come close to 
this realization, though.)

Now about locality again. There is no physical evidence of spooky interactions
that would be able to cause violation of locality. This strongly suggests that
interpretations that require locality violation are wrong. The only 
interpretations which avoid non-locality are ones which are realist, keep 
functionality (= functional correspondence = Functional Composition Principle, 
whatever you want to call it) and deny the Expectation Rule. As for the Value
Rule, it is false and so we don't have a choice about it. (Notice that keeping
functionality while recognizing the falsity of the Value Rule, avoids the
Kochen-Specker paradox and avoids ontological and environmental non-locality.)

What about (non-)determism. Well we can have theories which satisfy the above 
and which are non-deterministic. But these theories are not consistent with 
pure QM. We need experimental evidence to decide whether they are acceptable.
A deterministic theory avoids any problems that we know of.

Also note that while we can't have free-will in a deterministic world,
non-determinism ain't so hot either. A non-deterministic world differs from a
deterministic one in that some things happen which are not determined by prior
events but just happen. If you think that you can get free-will out of that
then you are a weineramus. (Sorry Benjamin I couldn't resist.)
-- 
_______________________________________________________________________________
|                                                                             |
| Fluffy, the Quantum Kitty                                                   |
| c/o Colin Naturman             "`All right,' said the Cat; and this time it |
| naturman@uctvax.uct.ac.za        vanished quite slowly, beginning with the  |
|                                  end of the tail and ending with the grin,  |
|                                  which remained some time after the rest of |
|                                  it had gone."                              |
|                                                                             |
|                          Lewis Carroll,  Alice's Adventures in Wonderland   |
|_____________________________________________________________________________|

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