Many posts on this forum are accessed hundreds or even thousands
of times. But only a half-dozen or so posters account for maybe 90%
of the posts, and these half-dozen "regulars" vehemently espouse views
which I imagine that 99% of practicing physicists would emphatically reject.
Of course, that doesn't imply that the regulars are wrong
(though for other reasons I do think that they are wrong).
But I wonder who might be the hundreds who regularly read
their posts. Are they unconvinced (either of the established views or
of the views of the regulars) and seeking enlightenment? Do they consider
the forum analogous to a "Flat Earth Society" which they read for their
amusement? I have no idea, but in case it is the former, I thought it
might be interesting to try a different way of explaining why what the
regulars deprecatingly call "Bellians" think as they do.
I will illustrate with a parable. An experimenter named Alice
has an instrument with a switch with two positions labeled "A" and "A`".
She believes that it measures one of two physical quantities, with the same
labels. (In physics texts, each of the two quantities is usually the spin of
a particle along a specified axis.) Each measurement yields one
of the two values +1 or -1. Because the switch has two mutually exclusive
positions, she cannot measure both A and A' simultaneously.
Similarly, another experimenter named Bob in a laboratory far
from Alice has an identical instrument which can measure one (and only one)
of two quantities B and B', again obtaining result +1 or -1 for each
measurement.
Alice and Bob can collaborate by prearranging that each will make
one of his (or hers) measurements at a specified time, record the results,
and later compare them. If, say, Alice measures A' and Bob measures B,
denote the joint measurement as A' B. The four possible results of A' B
are (+1, +1), (+1, -1), (-1, +1), (-1, -1), where the first number in
each parenthesis refers to Alice's result and the second to Bob's.
By doing this numerous times, they can estimate
the probability of each possible result (a, b). Thus they can obtain,
to an arbitrary accuracy, a probability distribution which we'll denote
pA' B = pA' B( . , . ) which assigns a probability pA' B(a,b) to each of
the four possibilities (a,b). The TeX facility produces rather ugly
results, so I find it clearer to avoid it when possible, but when
using TeX, I'll write pA' B as
Similarly, Alice and Bob can approximate to arbitrary accuracy three
other possible probability distributions denoted similarly as
pAB, pAB', and pA' B' . These four distributions will be called
"marginal" probability distributions, for reasons which will be made
clear later. They can conveniently be written as matrices, for example
they might find that
which means that p(1,1) = 3/8, p(1,-1) = 1/8, p(-1,1) = 1/8, p(-1,-1) = 3/8,
and that
To skip ahead a bit, these are the (hypothetical) "marginals"
predicted by quantum mechanics for a particular entangled state similar
to the so-called "singlet" state which will be specified later.
The others are
and
Alice and Bob find this method of generating the marginals rather
tedious, and they wonder if with more refined instruments they might be able
to measure all four of the quantities A, A', B, B' simultaneously.
Then super-salesman Gideon (aka "God") appears, claiming to be able to sell
them just such an instrument, which they can barely afford.
He also tries to sell them an expensive course in his self-proclaimed
"Gideon University" to teach them how to use this fantastic instrument.
Then Alice and Bob meet mathematician Mark, who tells them that
he used to work for Gideon and that his instrument and university course
are nothing but a scam. Alice and Bob want to believe Gideon, and so
are skeptical of Mark.
Mark explains that he has a mathematical proof that no such instrument
can exist. It starts as follows.
Suppose there were such an instrument. Then by using it one could
approximate to arbitrary accuracy a probability distribution
p(a, a', b, b')
on four-tuples of variables (a, a', b, b'), where each variable can take the
value +1 or -1, where it is hoped that the notation will seem self-explanatory.
For example,
p(1, -1, -1, 1) = 1/4
represents the probability 1/4 that Gideon's instrument indicates
A = 1, A' = -1, B = -1, B' = 1 .
If Gideon's instrument is measuring what he claims, then from
his measurements we could deduce the four marginals as follows.
,
where the sum ranges over all four possible values for (a' , b' ),
and similarly for the other three marginals.
This explains why the marginals might deserve that name.
ASSUMING THAT GIDEON's CLAIMS ARE CORRECT,
they are slight generalizations of the standard notion of
"marginal distribution" in probability theory. But if we aren't sure
that p actually exists (i.e., that Gideon's instrument works as he claims),
then they really shouldn't be called "marginals", though that name is
still convenient so I won't change the terminology.
[Incidentally, another regular poster has a similar serious omission
in a submitted paper on another topic---he assumes without proof that
a mathematical object which his theory requires must exist. It's
easy to say "let p = p( . , . , . , . ) be a probability distribution
whose marginals are the given pAB( . , . ), pAB', etc.", but just
referring to it in this way doesn't prove that such a p exists!]
That's how Mark's claimed proof starts. The rest is just a usual proof
of some version of Bell's Theorem. From the marginals, we can calculate
the so-called "correlations" like
` C_{AB} := p_{AB}(1,1) + p_{AB}(-1,-1) - p_{AB}(1,-1) - p_{AB}(-1,1)
(These are also often denoted by notation like
or
which you may have seen in books or posts of the regulars.)
Note that the CAB, etc can be obtained
from the corresponding pAB, etc, but not conversely. Thus a theory
that only yields the CAB, etc. (like Dr. Christian's, so far as I know), cannot be said to be
complete unless it is augmented by explanation as to how to obtain the
corresponding pAB, etc. Quantum mechanics
does give the pAB, etc.
For example, I calculated the ones given above using the rules of quantum mechanics.
To elaborate a little more, Bell's Theorem states that given
the marginals pAB, etc., a
necessary condition that p exists
(hence that Gideon's instrument can do what he claims) is that the
CAB, etc., satisfy some inequality
(there are several, called Bell inequalities).
One such inequality is
and you can easily verify that the marginals given explicitly above
don't satisfy it. Hence Gideon is full of baloney.
I've seen some regulars make puzzling statements on the order of
"a mathematical inequality can NEVER be violated, and therefore quantum
mechanics cannot violate Bell's inequality". I can't imagine that
they mean the words which they have written. I am sure that they are
smarter than that and must have meant something else.
Mathematical inequalities generally make some assumptions which
are necessary for their validity. In our case, the assumption is that
Gideon's instrument actually does what he claims, or more mathematically,
that p can exist. If the inequality is violated, that just means that
the hypotheses for the inequality are violated---in our case that
Mark is right that Gideon's instrument is impossible.
In case the marginals are those predicted by quantum mechanics,
the violation of a Bell inequality just means that quantum mechanics
cannot be reproduced by a theory which assumes that A, A', B, and B'
can all be measured at the same time. This is what almost all physicists
mean by a (local) *realistic* theory. (I wouldn't be surprised if all those
who don't mean that are regulars of this forum.) A (local) "realistic"
theory assumes that what instruments measure is "really" there, whether
measured or not. Even if Alice can't measure A and A' at the same time
because of her antiquated equipment, a "realistic" theory assumes that
someone like Gideon can.
You are probably wondering where the marginals quoted above came from.
The example given above is essentially that of Asher Peres' generally
excellent book Quantum Theory: Concepts and Methods", pp. 162 ff.
Peres assumes the particular quantum state consisting of two entangled
qubits given by
In typical and hopefully self-explanatory physics notation, this state
might be denoted
where the subscript A refers to Alice's part of the qubit and B to Bob's.
(Neither are Peres' notation, which is not as explicit
as it should be.)
Peres only calculates the CAB, etc., not the marginals pAB.
I calculated those. I hope I got them right, but if not it doesn't matter
because the correlations derived from them are the same as Peres'.
To close, I suggest that the hundreds of lurking readers can probably
resolve any disputes to their satisfaction by asking the following questions
of the regulars:
1) Do they agree that the marginals quoted above are predicted by quantum
mechanics?
2) Do they agree that Gideon's p yielding those marginals is logically
impossible?
3) Do they agree that quantum mechanics is not (locally) "realistic"
in the sense of the above definition, i.e., that p cannot exist.
Of course, if the regulars have some different definition of "locally
realistic" the above discussion may not apply to that definition.
You can't argue with a
definition ! But the definition used by
the "Bellians" is the one which I just gave.
Under that definition, it seems to me that the regulars should admit that the much-ridiculed
"Bellian" viewpoint is reasonable and consistent and tone down their rhetoric.
If I knew what definition of "realistic" theory the regulars are using,
perhaps I could also admit that their viewpoint is reasonable and consistent.
Should the difference be merely one of definition, no argument is possible,
and insulting rhetoric is inappropriate.