Another FTL question (sigh) what about Bell?
Jesse M. Mundis
Bells "proof" that the underlying "deep reality" is in fact superluminal
and non-local?
As I understand the argument, he assumes space is local, and subluminal,
then points to the EPR Paradox and with a bit of algebra succedes in
doing a proof by contradiction of the non-locality of space. The argument
I read was very convincing. I would appreciate any intelligent comments
on this perplexing little idea.
Jesse Mundis
je...@pawl.rpi.edu
John Baez
I am disturbed by your brief summary of it... first, `space
is local' doesn't mean anything to me, and second, if Bell
had assumed `space is local' and proved `the non-locality
of space', we would be in deep trouble, since there would
be a contradiction here!!
Bell's theorem points out something strange about quantum
mechanics (as compared to classical mechanics, anyway), but
no contradiction is involved. Essentially, it amounts to
this. Suppose one has a system consisting of 2 parts, A
and B. In classical mechanics one describes the state of
system A by a point in a `phase space' X(A), and one
describes the state of system B by a point in a space X(B).
States of the whole system are described by pairs of points,
one in X(A) and X(B). To be crude, to say what's going on in
the whole system you just need to say what's going on in
both parts. (Note: to the connosieur, the `states' above
are known as `pure states', and one says `pure states in
the whole system are always products of pure states of
the 2 subsystems'.)
In quantum mechanics, one describes states by unit vectors
in a Hilbert space, so instead one would have a Hilbert
space H(A), a Hilbert space H(B), and states of the
whole system would be unit vectors in a Hilbert space
H(A) x H(B) , the `tensor product' of H(A) and H(B) .
(Again, an aside to the connosieur: there's a special
symbol for tensor product, a circle with an x on it, but I
can't type it here! Also, (pure) states are really unit
vectors modulo a phase factor, but that won't matter here.)
Now, the kicker is that there are unit vectors in
H(A) x H(B) which are not `tensor products' of a unit
vector in H(A) and a unit vector in H(B). These
correspond to states of the whole system which CANNOT be
described by separately saying everything that's going
on in A and everything that's going on in B. The
correlated state of 2 electrons (or photons in the
original EPR `paradox') is one such state. It has
`correlation information' which is not given by saying
what each of the photons is doing separately. This is
sometimes called the `failure of local realism'.
Bell's accomplishment was to show that this implies
that there are some inequalities satisfied by classical
systems that need not be satisfied by quantum systems.
The Aspect experiment (I believe it's called) verified
that indeed the real world breaks these inequalities,
hence is quantum.
It's important to note that none of this implies any
superluminal communication. Quacks love to claim that
somehow it might.
Someone who loves math might enjoy my brief note,
`Bell's inequality for C*-algebras', Lett. Math. Phys.
13 (1987) pp. 135 - 136. Most people would find the
Scientific American article on the subject, or the
Physics today article, much more enlightening.
The world is strange but apparently not self-contradictory.
Steven Daryl McCullough
> I agree that Bell's argument is perfectly convincing, but
> I am disturbed by your brief summary of it... first, `space
> is local' doesn't mean anything to me, and second, if Bell
> had assumed `space is local' and proved `the non-locality
> of space', we would be in deep trouble, since there would
> be a contradiction here!!
>
John, I have no reason to doubt your version of the physics of
Bell's theorem, but I have to disagree that coming up with a
contradiction in a theoretical analysis means we are in "deep
trouble". If Bell assumed that "space is local", or in your
words, that "local realism holds", and then proceeded to derive
a contradiction, that simply shows that he made a false assumption.
Therefore local realism fails. That seems to me to be a perfectly
valid form of reasoning. Maybe that means I've been hanging out
with logicians for too long. 8^)
> [...stuff deleted...]
> It's important to note that none of this implies any
> superluminal communication. Quacks love to claim that
> somehow it might.
I don't think it is fair to call all such people quacks. In the last
year or so there was a paper in an honest-to-God physics journal, either
Physical Review, or Physical Review Letters, I think, in which the authors
claim to derive the possibility of faster-than-light communication using
an EPR-like experiment involving parity-violating weak decays. I was not
completely convinced by their treatment, because they used an ad-hoc mixture
of nonrelativistic quantum mechanics and particle physics, but the authors
were certainly *not* quacks (at least not in the sense of people who have
only sham knowledge of their subject matter).
Anyway, the EPR experiment seems to contradict special relativity, at
least if one uses a naive interpretation of quantum mechanics (and, in
my humble opinion, *every* interpretation so far concocted is a naive
interpretation). According to Von-Neumann's theory of quantum measurement,
a measurement results in a "sudden collapse" of the wave function into
an eigenstate of the measured quantity. The problem for special relativity
is that the notion of a "sudden collapse" is not a covariant notion: what
is a simultaneous collapse in one frame will not be a simultaneous collapse
in another.
> [...stuff deleted...]
> The world is strange but apparently not self-contradictory.
There is an easy proof that the world is not self-contradictory:
Assume that it is self-contradictory. Contradiction! Therefore our
assumption must be false. Therefore the world is not self-contradictory.
Isn't logic great? 8^)
Daryl McCullough
(The man who proved the hookup theorem)
John Baez
>> if Bell
>> had assumed `space is local' and proved `the non-locality
>> of space', we would be in deep trouble, since there would
>> be a contradiction here!!
>>
>words, that "local realism holds", and then proceeded to derive
>a contradiction, that simply shows that he made a false assumption.
>Therefore local realism fails. That seems to me to be a perfectly
>valid form of reasoning.
Point well taken. You're right, he showed that local
realism fails. I was trying to say that Bell's result
was not a `paradox' (an as-yet-unresolved seeming
contradiction), just a curious fact. But I didn't say
it right.
>> It's important to note that none of this implies any
>> superluminal communication. Quacks love to claim that
>> somehow it might.
>
>I don't think it is fair to call all such people quacks. In the last
>year or so there was a paper in an honest-to-God physics journal, either
>Physical Review, or Physical Review Letters, I think, in which the authors
>claim to derive the possibility of faster-than-light communication using
>an EPR-like experiment involving parity-violating weak decays.
While it is true that quacks love to claim that Bell's inequality
implies superluminal communication, I didn't mean to insinuate
that anyone studying the possibility of superluminal communication
and Bell's inequality is a quack. (Maybe I've been hanging around
logicians too long.)
I've heard of but haven't read that paper. It's interesting
to hear that
> they used an ad-hoc mixture
>of nonrelativistic quantum mechanics and particle physics,
because that would certainly explain how they got such a
strange result. If one uses straight quantum field theory, which has
special relativity built into it, one couldn't get superluminal
communication.
>Anyway, the EPR experiment seems to contradict special relativity
Since you hang around with logicians I'll give you the benefit
of a literal interpretation. Yes, the EPR experiment
may `seem' to contradict special relativity - I'll only
be upset if you say it DOES.
> at least if one uses a naive interpretation of quantum mechanics
>(and, in my humble opinion, *every* interpretation so far concocted
> is a naive interpretation).
Here I agree; the EPR `paradox' forces one into a slightly less
naive interpretation of quantum mechanics.
To reassure the populace, let me say again: despite suddenly
`collapsing' ghost-like wave-functions which appear in some
interpretations of quantum mechanics, no-one has managed to
send signals faster than light, or show some mathematical
contradiction in axioms for special relativity & quantum mechanics.
If those guys above do it using parity-violating weak decays
I'll quit my job and go into the time-machine business (since
after all superluminal communication is equivalent to communication
from future to past upon a change of Lorentz frames).
This is not to say that all is well in the marriage between
quantum theory and special relativity. Quantum field theory
has been notoriously difficult to make rigorous. The difficulties,
however, seem to have little to do with Bell's inequality,
rather with the assumption that arbitrarily short distances
make sense (the continuum nature of spacetime).
>There is an easy proof that the world is not self-contradictory:
>Assume that it is self-contradictory. Contradiction! Therefore our
>assumption must be false. Therefore the world is not self-contradictory.
>Isn't logic great? 8^)
Logic is so... *logical*.
It's worth noting... to digress from physics.. that any
proof of consistency of ANY mathematical system is worthless
if one doesn't trust the consistency of the methods of
proof... an observation much simpler than Goedel's theorem
that tends to be ignored.
Okay, Daryl, what's the hookup theorem? I read sci.logic if
that's where it belongs... why's there no sci.illogic, come to
think of it?
Steven Daryl McCullough
> which has special relativity built into it, one couldn't get superluminal
> communication.
>
I don't buy that argument. Sure quantum field theory predicts
that fields evolve according to rules which are completely consistent
with special relativity. The part that seems to violate SR (yes, I mean
*seems* and not *does*; I'm not that confident in my own arguments) is
in measurements: does detecting a particle "collapse" the field values
everywhere, simultaneously, or not? If so, then quantum field theory
has some of the same trouble with SR as nonrelativistic quantum mechanics
does.
> To reassure the populace, let me say again: despite suddenly
> `collapsing' ghost-like wave-functions which appear in some
> interpretations of quantum mechanics, no-one has managed to
> send signals faster than light, or show some mathematical
> contradiction in axioms for special relativity & quantum mechanics.
When you say that there are no contradictions, do you mean in the
evolution equations for the fields (or wave functions)? I will sort
of agree with that, although to be picky, one might consider the occurrence
of infinities in quantum field theories as a sign of inconsistency; I don't
think that such slight inconsistencies are so serious, since renormalization
seems to fix things up okay. However, the *complete* story for quantum
field theory must include the interpretation of the fields or wave functions,
which must include a theory of what happens when a measurement is made.
The interpretation with ghostly wave functions (I wish this posting had
been made in time for Halloween 8^) which collapse upon observation,
as weird as it is, is, as far as I know (1) in agreement with every
experiment, and (2) inconsistent with SR (in the sense that the collapse
cannot be made in a Lorentz-invariant way). Most physicists get along fine
without considering the meaning of QM, so they don't run into any problems
with SR.
> Okay, Daryl, what's the hookup theorem?
Actually, it's of no particular interest to physics or logic. I was just
grasping for a tag line.
Daryl McCullough
(The man who proved the hook-up theorem)
John Baez
>evolution equations for the fields (or wave functions)? I will sort
>of agree with that, although to be picky, one might consider the occurrence
>of infinities in quantum field theories as a sign of inconsistency; I don't
>think that such slight inconsistencies are so serious, since renormalization
>seems to fix things up okay.
This `picky' point is something I consider much more worrisome
than superluminal communication, and it's this I was referring to
I said that there were problems with merging SR and QM.
It seems to me that the more one studies quantum field theory,
the more one realizes the problems associated with
renormalization are very deep. Renormalization cannot be said
to be in a satisfactory state. But renormalization the way some
physicists (I'm a mathematician) do it is so nonrigorous, one
need not be surprised that they come up with infinities which
then need to be swept under the rug. (They feel free to do
things like subtract infinity from infinity and get a finite
answer!) I meant to say that while there are various approaches
to rigorous QFT, none have PROVEN any contradictions among the
basic desiderata for interacting quantum fields, so the hope
is still that QFT can be done in some sensible way. One glimmer
of hope is that interacting quantum fields have been constructed
in 2-dimensional spacetime. People interested in this sort of
thing can glance at `Introduction to Axiomatic Quantum Field
Theory' by Bogoliubov, Logunov and Todorov, or `Quantum Physics:
A Functional Integral Point of View' by Glimm & Jaffe, or
`Introduction to Algebraic and Constructive Field Theory', by
Segal, Zhou and myself --- when the thing finally gets published!
However, the *complete* story for quantum
>field theory must include the interpretation of the fields or wave functions,
>which must include a theory of what happens when a measurement is made.
>The interpretation with ghostly wave functions (I wish this posting had
>been made in time for Halloween 8^) which collapse upon observation,
>as weird as it is, is, as far as I know (1) in agreement with every
>experiment, and (2) inconsistent with SR (in the sense that the collapse
>cannot be made in a Lorentz-invariant way). Most physicists get along fine
>without considering the meaning of QM, so they don't run into any problems
>with SR.
I don't buy the ghostly wave function collapse story. Basically,
I think its in the same league as the `angels pushing around
the planets' theory. In other words, it makes no falsifiable
predictions. You seem to agree, since you say that it is both
in agreement with every experiment and in disagreement with SR.
An honest theory that was in disagreement with SR would make
predictions incompatible with SR, and when tested either it or
SR would come tumbling down. A theory of Lorentz-noninvariant
angels (or ghosts!) which are not observable, however...
I want to eat lunch, so I'm not going to get into MY naive
interpretation of quantum mechanics. Suffice it to say that
for a while I was into the many-worlds interpretation, but then
I became a bit more of a postivist and have essentially thrown
out all the rest of the worlds! The `one-world interpretation?'
You note that most physicists get along fine without worrying
about the meaning of QM. It's this sort of observation that
led me to my interpretation of QM ... I concluded that most
`interpretations' supply a lot of stuff that's not actually
needed to apply the theory. I agree though that a complete
story must say something about measurement.
I'm surprised that a huge crowd of disputatious net users
hasn't pounced on our fairly civil discussion yet. Regards.
Arshad Mahmood
>....
>is local' doesn't mean anything to me, and second, if Bell
>had assumed `space is local' and proved `the non-locality
>of space', we would be in deep trouble, since there would
>be a contradiction here!!
>
Proof by Contradiction.
A. Mahmood
Laboratory for the Foundations of Computer Science
Edinburgh University
Scotland
Steven Daryl McCullough
> to be in a satisfactory state. [...more stuff deleted...]
> One glimmer
> of hope is that interacting quantum fields have been constructed
> in 2-dimensional spacetime.
Also, does anyone happen to know whether superstring theory is finite?
That was supposed to be one of its advantages: that it did not suffer
from the divergences plaguing quantum gravity; but I'm not sure whether
the resulting theory is finite, or merely renormalizable.
> [...more editing...]
> I don't buy the ghostly wave function collapse story. Basically,
> I think its in the same league as the `angels pushing around
> the planets' theory. In other words, it makes no falsifiable
> predictions.
I don't see how you can say that. It is *pure* quantum mechanics,
with only the Schroedinger equation to describe the evolution of
the wave function which makes no predictions (since it is impossible
to measure the wave function). However, the combination of Schroedinger
to describe wave function evolution, plus the doctrine of wave function
collapse, to describe measurements, makes plenty of falsifiable
predictions; every prediction made by quantum mechanics is actually a
prediction made by this combination.
Even the very first prediction made by quantum mechanics, namely the
energy levels of the hydrogen atom, required something like the
wave-function collapse. Schroedinger's equation only allows one to
calculate the energy levels for energy eigenstates. If you consider
mixed states, the energy can be made to be absolutely anything you
please. To compare quantum mechanics with experiment, it *seems* as if
measuring energy forces the system into an energy eigenstate. This
observation gave rise to the doctrine of wave function collapse. So,
far from being unfalsifiable, wave function collapse was born in an
attempt to link quantum mechanics with experiment, and, as far as I
know, has been very successful.
To say that a theory is unfalsifiable is to say that there is no
experiment that could prove it wrong. That is why the "theory"
explaining the motions of the planets by invoking angels pushing
them around is not a good theory---*any* motion of the planets is
consistent with such a theory. In contrast, the doctrine of wave
function collapse makes very definite predictions which can be
tested. For example, if tests of Bell's theorem had gone
the other way, then that would have spelled doom for the doctrine of
wave function collapse. It would not have spelled doom for quantum
mechanics, which would still have survived as a statistical approximation
to an underlying deterministic physics. In contrast, it seems to me that
the Many Worlds Interpretation *is* unfalsifiable; how could you *disprove*
the existence of alternate worlds?
Perhaps you mean that there is no experiment that can distinguish between
the doctrine of wave function collapse and competing theories. Or perhaps
you mean that the doctrine is *unverifiable*: there is no way to prove
it true. But it is wrong to say it is unfalsifiable.
Daryl McCullough
(The Once and Future Physicist)
Lord Snooty @ The Giant Poisoned Electric Head
> Also, does anyone happen to know whether superstring theory is finite?
> That was supposed to be one of its advantages: that it did not suffer
> from the divergences plaguing quantum gravity; but I'm not sure whether
> the resulting theory is finite, or merely renormalizable.
According to Davies' et al. book "Superstrings - A Theory Of Everything?"
superstring theory is "purely local" and therefore "exciting". Maybe this
answers the question. It's a good read in any case.
--
...........................................................................
Andrew Palfreyman a wet bird never flies at night time sucks
and...@dtg.nsc.com there are always two sides to a broken window
John Baez
>That was supposed to be one of its advantages: that it did not suffer
>from the divergences plaguing quantum gravity; but I'm not sure whether
>the resulting theory is finite, or merely renormalizable.
I'm not following string theory closely, but it seems very
far from having been proved finite; when I last looked they
didn't know how to do the perturbation theory past the one-loop
level, which would seem to indicate that they didn't know if
it was renormalizable to all orders.
Thus I broadcast the following request: could someone who is
really INTO string theory explain roughly where it's at???
Has is been formulated in a manifestly covariant (coordinate-
free) manner? Has it been shown to be renormalizable? For that
matter, to what extent has a unique `it' been settled upon?
I recall open strings, closed strings, heterotic strings...
and I recall that people weren't sure which 6-dimensional
Calabi-Yau manifold (whatever that is exactly) was supposed
to represent the extra curled-up dimensions of spacetime.
>> I don't buy the ghostly wave function collapse story. Basically,
>> I think its in the same league as the `angels pushing around
>> the planets' theory. In other words, it makes no falsifiable
>> predictions.
>
>I don't see how you can say that. It is *pure* quantum mechanics,
>with only the Schroedinger equation to describe the evolution of
>the wave function which makes no predictions (since it is impossible
>to measure the wave function). However, the combination of Schroedinger
>to describe wave function evolution, plus the doctrine of wave function
>collapse, to describe measurements, makes plenty of falsifiable
>predictions; every prediction made by quantum mechanics is actually a
>prediction made by this combination.
>
>Even the very first prediction made by quantum mechanics, namely the
>energy levels of the hydrogen atom, required something like the
>wave-function collapse. Schroedinger's equation only allows one to
>calculate the energy levels for energy eigenstates. If you consider
>mixed states, the energy can be made to be absolutely anything you
>please. To compare quantum mechanics with experiment, it *seems* as if
>measuring energy forces the system into an energy eigenstate. This
>observation gave rise to the doctrine of wave function collapse. So,
>far from being unfalsifiable, wave function collapse was born in an
>attempt to link quantum mechanics with experiment, and, as far as I
>know, has been very successful.
Okay, it sounds like we actually agree for the most part,
but we'll probably need a number of rounds to get our
signals straightened out here. I slice the cake somewhat
differently than you. Let's consider Schroedinger's equation
for an electron, to be specific. For me, the basic testable
part of QM consists of Schroedinger's equation along with the
rules for feeding in input data to the equation and getting
predictive output. In a summary form these are: 1) one takes
all the information one knows about the electron and fits a
psi to it (for example, if one knows the energy and something
about the angular momentum of an electron in a hydrogen atom
one picks the appropriate eigenstate to be psi... if, as is
usual, one doesn't know enough to completely determine psi
one should use Jaynes' maximal entropy rule... but this is
typically considered under statistical mechanics); 2) to figure
out the expected value of an observable one does the good old
<psi | A | psi> routine (in particular if A is a projection
one get probabilities). This is all fairly uncontroversial
and is what physicists do every day; it's eminently falsifiable
and beautifully in agreement with experiments.
For me the evil `ghostly wave function collapse' sneaks in
when one tries to attribute a physical reality to psi of a
sort analogous to the electric field. Then we get the
problem that there is a privileged class of physical
processes called `measurements' which can all of a sudden
make psi equal to zero (POP!) far far way from where the
measurement happens. This is scary both because one will
eventually want things to fit in with SR, but also because the
plain old nonrelativistic Schroedinger's equation doesn't
predict this kind of behaviour for psi; according to it psi
changes rather smoothly.
My point, as usual rather sloppily expressed, was that regarding
psi as a physical field that changes due to the measurement
process in some manner called `collapse' is on the order of
angels pushing the planets: nothing comes of it that can't
be explained by basic setup outlined above. E.g. one can't
do any measurements to determine WHEN psi `collapses'. (I
suspect I can get into trouble on such a bold and vague
statement. Here's an example of the sort of thing I mean.
One releases a free electron in a huge box. According to
QM it's wave function spreads out and bounces around. Now at
one end of the box you look to see if the electron is there
at some randomly chosen time T. According to the collapse
story this changes the wave function everywhere at time T -
regardless of whether the electron is found to be there, or
found NOT to be there, hence somewhere else. (In the collapse
theory the latter will boost the value of psi away from
where the measurement occurs.) Is there a way that you can
do an experiment at the other end of the box to tell WHEN
this collapse occurs?)
Here's an analogy that has nothing to do with QM. When I'm
looking for my other blue-green sock, and all I know is that
it's in the house SOMEWHERE, I can describe my state of ignorance
by a `sock probability distribution' smeared out over the house.
(Perhaps with lower values in the refrigerator than in the
bedroom.) Now say I find the sock under the dresser drawer.
This `collapses' the sock function: instantaneously I know, with
superluminal speed, that the sock function equals zero in the
living room. Nonetheless I don't feel there's the slightest
contradiction between SR and sock mechanics. (Though sometimes
they seem to disappear, which would violate conservation of
mass.)
Sho Kuwamoto
>when one tries to attribute a physical reality to psi of a
>sort analogous to the electric field. Then we get the
>problem that there is a privileged class of physical
>processes called `measurements' which can all of a sudden
>make psi equal to zero (POP!) far far way from where the
Can you really treat this wave function as a true field? I'm not sure
exactly what I mean by this, but, for example, the E&M field has
energy which couples to gravity. Does a wave function act so simply?
I tend to think of a wave function as a complex function which has a
value at every point in space, but *not* as a physical field. Perhaps
I think this way only because you can't take a meter and measure it.
Or maybe there is some fundamental distinction.
-Sho
--
s...@physics.purdue.edu
Matt Crawford
) Can you really treat this wave function as a true field? I'm not sure
) exactly what I mean by this, but, for example, the E&M field has
) energy which couples to gravity. Does a wave function act so simply?
Since the wave function includes information representing the amplitude
for finding a particle in a certain place, the rest-energy and kinetic
energy of that particle also have an amplitude to be in a certain place.
A quantum theory of gravity, if we had one, would couple to this wave
function.
If you think about the particle in quantum terms but gravity in
classical terms, you have to wonder how finding the particle in your
detector *here* eliminates its gravitational effects *there*.
________________________________________________________
Matt Crawford ma...@oddjob.uchicago.edu