[N.B. this is from a usenet posting to sci.philosophy.meta from May,
1996. Dan M. is another usenet poster. The exposition of Bell's
theorem is paraphrased from Tim Maudlin's _Quantum Nonlocality and
Relativity_, ch. 1 (q.v.) --Michael Huemer]
Dan M. and I have been arguing, with little headway, about this.
I say that the theory of relativity is false, and so is the
orthodox (Copenhagen) interpretation of quantum mechanics
(hereafter, QM). The following will be a simple explanation that
anyone can understand of why.
First, I will explain the EPR experiment (originally conceived by
Einstein to refute QM). Then I will explain why the results of the
experiment refute quantum mechanics. Finally (this is the hard
part), I will explain why they also refute the theory of
relativity.
I. THE EPR EXPERIMENT
(If you're already familiar with this, skip to section II.)
This was originally a thought experiment, but since was performed
as an actual experiment. You have a light-filter called a
polarizer. What it does is, when you shoot photons at it, each
photon either passes through it, or gets absorbed (with a certain
probability of each). It also matters how the polarizer is
oriented: for a photon in a given state, its probability of
passing through a polarizer varies as you rotate the polarizer.
It's possible to have a pair of photons that are in a state such
that, quantum mechanics predicts, if they both encounter a
polarizer oriented the same way, *either* they will both pass, *or*
they will both get absorbed. There is a 100% probability that the
two photons will do the same thing. However, whether they will
both pass or both get absorbed is also, according to QM, random --
there is a 50% probability that they both pass, and a 50%
probability that they both get absorbed. This probability is
irreducible -- that is, it is not due in any way to our ignorance
of relevant causal factors; the way the world works is inherently
random.
Let us say the photons 'agree' if they both pass or both get
absorbed, and they 'disagree' if one passes and the other doesn't.
Then in this case, the photons are guaranteed (100% probability) to
agree.
One more thing: If you take one of the polarizers and rotate it by
30 degrees (relative to the other), then the probability of
'agreement' goes down to 3/4. If, instead, you rotate one of the
polarizers by 60 degrees, then the probability of agreement will be
1/4. Finally, if you rotate one of the polarizers by 90 degrees,
then the probability of agreement is 0 (i.e. they always disagree).
These statistics have been experimentally verified.
Now, Einstein's thought experiment went like this: You shoot the
two photons off in opposite directions. When they get a good
distance apart, you simultaneously let them both encounter
polarizers with the same orientation. At the time they encounter
the polarizers, the photons are far enough apart that no signal,
travelling at or below the speed of light, could travel from the
location of one of the photons to the location of the other before
the latter had been measured (i.e., in time to 'tell' the other
photon what to do). This kind of separation is called 'space-like
separation' -- two events are separated in such a way that a light
signal could not connect them. This is significant because, in the
theory of relativity, it is impossible for any event to have any
influence on a space-like separated event (i.e. no influence can be
propagated faster than the speed of light).
With that set up, let's turn to how Einstein (correctly, IMO) used
this thought experiment to refute QM.
II. EPR REFUTES THE COPENHAGEN INTERPRETATION OF QUANTUM MECHANICS
The argument is really pretty simple. Many scientists have a
prejudice against simple arguments, but if you can overcome that,
imagine the following:
Suppose you and a friend decide to play a little game. The object
of the game is to mimic the behavior of photons (hey, you're
smarter than a couple of photons). It goes like this: You and the
friend start out in a room together. While you are in the room,
you are free to communicate all you like, and to devise any
strategy you like for playing the game. After that, you both leave
the room through opposite doors. You walk about a mile, and then
each of you encounters a questioner. The questioner says to each
of you one of three things: "30?", "60?", or "90?" (analog to the
orientation of polarizers to the vertical). You are required to
answer either "Pass" or "Absorbed". However, you are not
permitted, after you have left the room, to communicate with your
partner any more. Furthermore, you do not know what question you
are going to be asked until you are actually asked, and you do not
know what question your friend is asked. And he cannot communicate
to you either what question he was asked or what he answered.
This game will be played over and over, about a thousand times.
Now, your task (in collaboration with your friend) will be to
devise a strategy -- any strategy at all, subject to the
constraints just indicated -- for choosing how to answer the
questions that will reproduce the statistics the photons with the
polarizers reproduce. That is:
1. When both of you are asked the same question, your answers
must agree 100% of the time.
2. When you are asked questions that differ by 30 (i.e., one
of you is asked "30?" and the other is asked "60?", or one of
you is asked "60?" and the other is asked "90?"), your answers
agree 3/4 of the time.
3. When you are asked questions that differ by 60 (i.e., one
of you is asked "30?" and the other is asked "90?"), then your
answers agree only 1/4 of the time.
All of this is an exact analogy to the EPR experiment, only more
picturesque.
One more thing: each of you has a coin in his pocket which you may
flip if you want to. If you like, you can also have a many-sided
die to roll, to help you generate random numbers, if you think this
will help you.
Now, the orthodox interpretation of quantum mechanics describes the
'strategy' that the photons play. So why not start there? Can you
win this game by taking the strategy that (QM says) the photons
take?
Clearly no. The photons' strategy, allegedly, is the following:
While in the room, you both agree that you will answer all
questions in the same way; however, you do not decide which way
that will be. That is, you both agree that if asked "30?" you will
agree in your answers, but you don't know yet whether you will both
say "pass" or both say "absorbed," and that won't be decided until
the time comes. Then you both leave the room. When you meet your
questioner, he asks you "30?" let's say. You flip a coin to
determine how to answer. End of story.
Obviously, this strategy does not work -- it does not satisfy
constraint (1) above. That is, if you play this strategy, then
when both of you are asked "30?", your answers will agree only 50%
of the time, not 100% of the time as desired. For when you are
asked this question, you know that IF your partner was just asked
"30?" and HE answered "pass," then you *have to* answer "pass" as
well. Likewise, if he was just asked "30?" and he answered
"absorbed," then you have to answer "absorbed." If you have no
idea what answer he gave, then you have no idea how to answer. The
only way you could be guaranteed of giving the same answer as your
partner, would be if you knew what answer he gave (or would have
given if he was asked this question). According to the current
strategy, however, you don't know this -- for (a) you and he did
not decide upon this while in the room; and (b) he has not
communicated with you in any way since then, so there's no way you
could know what answer he decided to give.
Given the constraints of the game, the only possible strategy for
satisfying condition (1) is that, in each iteration of the game,
you decide while in the room how you're both going to answer each
question. You cannot decide later and hope to have your answers
correspond 100% of the time without communication.
This refutes the orthodox interpretation of quantum mechanics --
Einstein was right. The standard interpretation simply cannot
explain a basic fact -- it cannot explain the 100% correlations in
this experiment.
For those of you who are more into QM: When the photons leave the
room, they are both in a 'superposition' -- neither of them has a
determinate polarization. Let's say photon A is the first to be
measured. When photon A reaches its polarizer, it passes, let's
say. Its state therefore collapses; it is no longer in a
superposition, but in a definite state. Question: what is the
state of photon B at this time, just prior to photon B's being
measured? If photon B is in a superposition still, then there is
only a 50% chance that it will pass; hence, only a 50% chance of
agreement -- which is contrary to the 100% prediction. But if
photon B is not in a superposition still, then the measurement at
the location of A has instantaneously changed its state -- hence,
action at a distance.
III. EPR REFUTES RELATIVITY
Now, this is the startling thing -- John Bell proved that there is
no strategy that you can pick that will satisfy conditions 1, 2,
and 3, subject to my constraints -- that is, there is *no way* that
you and your friend can play this game and win, if you are not
allowed to communicate after you leave the room.
Okay, here's the mathematical part. As we just established, in
order to have any hope of winning, you and your friend must decide,
while in the room, how you're going to answer each question.
Now, even if you weren't convinced by the above argument, consider
this: Let's suppose for the moment that I was wrong about that,
and it is possible to win by flipping a coin after you leave the
room. Then you might as well just flip the coin while you're IN
the room, and tell your partner what result you got. Having more
information cannot possibly DEGRADE his performance, since in the
worst case, he can just ignore it. So even if there is a strategy
that could work by deciding how to answer *after* you leave the
room, there must be a corresponding strategy *at least as good* in
which you decide while *in* the room.
That being established, let's consider the possible ways you and
your friend could decide to answer the questions, on any given
iteration of the game ("P" stands for "Pass" and "A" for
"Absorbed"):
1. PPP 5. AAA (a)
2. PPA 6. AAP (b)
3. PAP 7. APA (c)
4. APP 8. PAA (d)
Strategy 1 means you both will answer "Pass" if asked "30?", "Pass"
if asked "60?" and "Pass" if asked "90?" Strategy 2 is you answer
"pass" is asked "30?", "pass" if asked "60?", and "absorbed" if asked
"90?" Etc. (We here ignore any possible strategies in which you
don't both answer the same way to all questions.) These are the only
logical possibilities.
Of course, you will have to use different strategies on different
iterations of the game (you cannot, for example, always choose #1).
Now let a = the proportion of times you pick either strategy 1 or
5 (i.e., lump #1 and #5 together).
Let b = the proportion of times you and your friend use either 2 or
6.
Let c = the proportion with which you use 3 or 7.
And let d = the proportion with which you use 4 or 8.
Now, since you must always use 1,2,3,4,5,6,7, or 8, we know that
a + b + c + d = 1.
Also, when one of you is asked "30?" and the other is asked "60?",
your answers have to have a 3/4 probability of being the same.
They will be the same only under strategies 1,2,5, and 6.
Therefore,
a + b = 3/4.
Next, when one of you is asked "60?" and the other is asked "90?",
your answers will agree iff you have picked strategy 1,4,5, or 8.
Since you are supposed to agree 3/4 of the time if these are the
questions, that means you have to pick 1,4,5, or 8 3/4 of the time.
Ergo,
a + d = 3/4.
Finally, when one of you is asked "30?" and the other is asked
"90?" your answers must agree 1/4 of the time. You can only
achieve this objective by choosing 1,5,3, or 7 1/4 of the time.
Hence,
a + c = 1/4.
This gives you 4 equations in 4 unknowns -- enough to solve and
determine what your strategy should be. Solve these equations
(left as an exercise for the reader). Surprisingly, (if I've done
my calculations right) you will obtain the following values:
a = 3/8
b = 3/8
c = -1/8
d = 3/8.
But this is impossible. It is not possible to choose strategy 3 or
7 *negative one eighth of the time*!
Therefore, you cannot succeed. That is to say, you and your friend
cannot reproduce the statistics that quantum mechanics predicts,
subject to the constraints we delineated (namely, that you can't
communicate after you leave the room, and you can't know beforehand
which questions you're going to be asked).
What does this prove? It proves that you *have to communicate*.
Of course, if you are allowed to communicate with each other after
you leave the room -- say you're given a pair of walkie talkies --
then there is absolutely no problem. Then you can just have
whichever of you encounters the questioner first call up the other
one, tell him what question you were asked and how you answered,
and the other can then roll his dice to get the appropriate
probability of agreement and answer accordingly.
The above is Bell's Theorem. Bell's Theorem is widely
misunderstood. I think Dan misunderstands it. It is sometimes
thought to be a proof of something different (viz. that the EPR
experiment can't be explained with 'hidden variables'). What it
proves is that to produce the correlations predicted by QM in the
EPR experiment, you have to communicate after you leave the room;
hence, that the photons must 'communicate' (i.e. influence each
other) faster than the speed of light.
This isn't speculation, and it's not a 'philosophical' or
'metaphysical' point that I've personally invented. This is a
proven result that anyone can understand, and, what's more, the
statistical predictions used in the above (the 1/4, 3/4, etc.) have
been experimentally verified by Alain Aspect. So there is no
escape from this conclusion: There is faster-than-light causal
influence.
This refutes the theory of relativity. The theory of relativity
entails that the photons could not 'communicate' after they were
outside each other's light cones (at space-like separation),
because (a) nothing can travel faster than the speed of light; (b)
more importantly, according to relativity, there is never any
objective fact about which event happened first, when the two
events are space-like separated (the temporal order is relative to
a reference-frame). Therefore, the theory of relativity is false.
Finally, for you QM enthusiasts, here's one more, even simpler
argument that QM is an inherently non-local theory ('non-local'
means the theory requires instantaneous action at a distance).
Here is a formulation of what locality requires for a probabilistic
theory: Let A and B be space-like separated events. Let K be a
complete description of all the events preceding A (i.e. in A's
backward light cone). Then the objective probability of A given K
should be identical to the probability of A given K and B:
P(A|K) = P(A|K,B)
In other words, no event outside A's backward light cone should
alter the probability of A occurring. In QM, this condition is
obviously violated, where A and B are the measurement-results for
the two photons. Hence, QM is, straightforwardly, a non-local
theory.
_ _ _
Dan, if you're reading this, what is your response to these
arguments? So far, I am not aware of any. I do not believe there
is any response. To recap, there have been three main arguments,
all based on the EPR experiment:
1. The Copenhagen interpretation of QM could not explain the EPR
correlations. (section II)
2. Bell's Theorem proves that the correlations require action at a
distance (non-locality). (section III)
3. It's obvious anyway that QM requires this, inasmuch as the
theory requires that events outside A's absolute past alter the
probability of A. 2 & 3 make QM inconsistent with relativity.
Hence, I conclude that relativity is false (from 2,3), and the
Copenhagen interpretation is false (from 1).
Geez, I hope this wasn't too long . . .