Bell's Theorem?
Lester Welch
accepted?
I believe the following simple example to be consistent with
Christian's disproof:
Let q= cos(a)sin(b) I +cos(a) cos(b) J + sin(a) K
where I,J,K are the basis of the quaternions. a,b are real and
arbitrary.
Then q^2 = -1 and q is isomorphic to i, the complex imaginary. Thus
in quantum mechanics, whereever we have i we can replace it with q -
with identical results and yet it has two hidden variables - a and b.
If two particles are entangled then a,b of one particle is related to
a,b of the second. If two particles are unentangled then a,b of one
particle is completely unrelated to a,b of the second.
Jim Black
Joy Christian's counterexample is invalid.
He considers an experiment where the measured result at each detector
is
either +1 or -1. But in his model of this experiment, he equates
these
*measured quantities* to bivectors. Not some intermediate
theoretical
construct, but the actual measured results.
(He also states that the bivectors have "values +/- 1," which seems
to
equate bivectors with scalars.)
He then computes the correlation of the measurements using a Clifford
product to get a correlation that matches experiment. But the
correlation
isn't something you directly measure; it's something you compute from
the
measured quantities, which are real numbers, using the ordinary real
number
product. What he has done, changing both the algebraic type of the
measured
quantities and the means by which the correlation is computed, amounts
to
altering the experiment.
--
Jim E. Black
student
> Is Joy Christian disproof of Bell's theorem (arXiv:quant-ph/0703179)
> accepted?
No, he makes a fundamental category error, and there is
certainly no 'disproof'! Bell's theorem is about sums and
products of measurement results taking values in {1,-1}.
Christian's eprint assumes instead that measurement
results are Grassmann numbers, which is fine, but
irrelevant. One could as well assume that they were
Hermitian operators, and derive a result entirely
analogous to his, and just as irrelevant to Bell's
theorem.
The only result of interest in the eprint is that some
2-spin system states can be modelled via Grassmann
numbers. See also the note by Diosi in this regard:
http://arxiv.org/abs/0708.0664
> I believe the following simple example to be consistent with
> Christian's disproof:
>
> Let q= cos(a)sin(b) I +cos(a) cos(b) J + sin(a) K
>
> where I,J,K are the basis of the quaternions. a,b are real and
> arbitrary.
>
> Then q^2 = -1 and q is isomorphic to i, the complex imaginary. Thus
> in quantum mechanics, whereever we have i we can replace it with q -
> with identical results and yet it has two hidden variables - a and b.
>
> If two particles are entangled then a,b of one particle is related to
> a,b of the second. If two particles are unentangled then a,b of one
> particle is completely unrelated to a,b of the second.
This example is, again, irrelen\vant to Bell's theorem. In
experiments,
the spin is up or down. Label these 1 and -1. Bell's theorem is
about the averages of such labellings.
Jack
> Is Joy Christian disproof of Bell's theorem (arXiv:quant-ph/0703179) accepted?
No, he is a crackpot. As for his specific error, I haven't studied it
in detail, but people think his hidden variables are nonlocal after
all.
Bell's theorem does not depend on the nature of the hidden variables,
as long as they are local. For a simple proof of the theorem see
http://onqm.blogspot.com/2009/07/simple-proof-of-bells-theorem.html
The only requirements for the theorem to hold are that the results of
one measurement don't depend on the direction chosen for the other,
and that there is only one outcome for each measurement, so that we
can talk about what the result of a measurement _would have been_.
harry
> On Sep 6, 2:59锟絘m, Lester Welch <lester.we...@gmail.com> wrote:
>
> > Is Joy Christian disproof of Bell's theorem (arXiv:quant-ph/0703179)
> > accepted?
>
> > I believe the following simple example to be consistent with
> > Christian's disproof:
>
> > Let q= cos(a)sin(b) I +cos(a) cos(b) J + sin(a) K
>
> > where I,J,K are the basis of the quaternions. a,b are real and
> > arbitrary.
>
> > in quantum mechanics, whereever we have i we can replace it with q -
> > with identical results and yet it has two hidden variables - a and b.
>
> > If two particles are entangled then a,b of one particle is related to
> > particle is completely unrelated to a,b of the second.
>
> Joy Christian's counterexample is invalid.
>
> He considers an experiment where the measured result at each detector
> is
> these *measured quantities* to bivectors. 锟絅ot some intermediate
> theoretical construct, but the actual measured results.
Yes, that's as much as I understood as well.
> (He also states that the bivectors have "values +/- 1," which seems
> to equate bivectors with scalars.)
Hmmm... giving him the benefit of the doubt (rather little doubt, as
he's an expert in the field), I guess him to mean that these are unit
vectors that have +1 or -1 as possible values. Then they correspond to
either "spin up" or "spin down, just as the typical measurement
results of QM.
> He then computes the correlation of the measurements using a Clifford
> product to get a correlation that matches experiment. 锟紹ut the
> correlation
> isn't something you directly measure; it's something you compute from
> the
> measured quantities, which are real numbers, using the ordinary real
> number product.
Please explain why you claim that instead of for example "spin up" and
spin down" (which correspond to vectors), real numbers are measured.
> What he has done, changing both the algebraic type of the
> measured
> quantities and the means by which the correlation is computed, amounts
> to altering the experiment.
It appears to me that you are mistaken; but if not, please elaborate!
Note: this was helpful fo me, for now I start to understand a little
of Joy Christian's first* paper on this. :-)
Thanks,
Harald
*For later clarifications see his overview on:
http://www.perimeterinstitute.ca/index.php?option=com_content&task=view&id=30&Itemid=72&pi=Joy_Christian
Lester Welch
Thanks. From that irrefutable source, Wikipedia [ :-) ]
"Later, Bell's theorem would prove (in the opinion of most physicists
and contrary to Einstein's assertion) that local hidden variables are
impossible."
So, regardless of whether Joy Christian is right or not, does Bell's
theorem address the question of hidden variables?
Ilja
> Please explain why you claim that instead of for example "spin up" and
> spin down" (which correspond to vectors), real numbers are measured.
"Spin up" and "spin down" correspond to two different labels, thus, a
quite simple set {u,d} of two elements.
To identify them with anything more complicated than a set of two
elements, like {-1,1}, seems a waste of allowed values. But, of
course,
you can feel free to identify them with any subset of any vector
space,
"up" as one element, and "down" with -"up".
But the more important point is that the description of the experiment
completely specifies what has to be done with the labels "u" and
"d" to obtain E(a,b). Namely, you have to look if the values
of A and B are equal (above "u" or above "d"), and in this case
memorize a +1 for this run of the experiment, or if they are unequal,
then you have to memorize a -1. And then you have to compute
the average over all runs.
This is the _definition_ of the E(a,b) used in Bell's inequality,
thus, Christian is not free to change it. But he changes it.
If you compute, instead of this E(a,b), something else, you are
not explaining the violation of the BI.
> It appears to me that you are mistaken; but if not, please elaborate!
I also think Christian is mistaken, and for the same reasons.
Ilja
> Thanks. �From that irrefutable source, Wikipedia [ �:-) ]
>
> "Later, Bell's theorem would prove (in the opinion of most physicists
> and contrary to Einstein's assertion) that local hidden variables are
> impossible."
>
> So, regardless of whether Joy Christian is right or not, does Bell's
> theorem address the question of hidden variables?
It does address the problem of local hidden variables.
The problem of hidden variables in general (possibly nonlocal,
that means with a hidden preferred frame in the relativistic domain)
was addressed by Bohm with an explicit example:
de Broglie-Bohm pilot wave theory.
Jim Black
> On Sep 9, 11:09�am, Jim Black <trams...@yahoo.com> wrote:
[regarding content at
http://www.perimeterinstitute.ca/index.php?option=com_content&task=view&id=30&Itemid=72&pi=Joy_Christian]
>> Joy Christian's counterexample is invalid.
>>
>> He considers an experiment where the measured result at each detector is
>> either +1 or -1. �But in his model of this experiment, he equates
>> these *measured quantities* to bivectors. �Not some intermediate
>> theoretical construct, but the actual measured results.
>
> Yes, that's as much as I understood as well.
>
>> (He also states that the bivectors have "values +/- 1," which seems
>> to equate bivectors with scalars.)
>
> Hmmm... giving him the benefit of the doubt (rather little doubt, as
> he's an expert in the field), I guess him to mean that these are unit
> vectors that have +1 or -1 as possible values. Then they correspond to
> either "spin up" or "spin down, just as the typical measurement
> results of QM.
To be clear, unit bivectors. Also known as two-forms or antisymmetric
tensors of rank two in three dimensions. They can be represented as
pseudovectors -- like vectors but they don't flip sign under a parity
transformation.
>> He then computes the correlation of the measurements using a Clifford
>> product to get a correlation that matches experiment. �But the correlation
>> isn't something you directly measure; it's something you compute from the
>> measured quantities, which are real numbers, using the ordinary real
>> number product.
>
> Please explain why you claim that instead of for example "spin up" and
> spin down" (which correspond to vectors), real numbers are measured.
You can express the measured spin values as bivectors if you want to.
What you cannot do is compute the expectation value of the Clifford
product of those bivectors and pretend it's the same as the
expectation value of real numbers which Bell's theorem is concerned
with.
So let us express the spin values as bivectors and ask: Does quantum
mechanics agree with Joy Christian's result?
In what follows, x, y, and z mean the unit vectors in the x, y, and z
directions. In this notation, the rules for the Clifford algebra Cl_
{3,0} can be summarized as:
x^2 = y^2 = z^2 = 1
xy = -yx, xz = -zx, yz = -zy
The usual associative and distributive axioms apply. Addition is
commutative, and it's perfectly okay to add any combination of
scalars, vectors, and all the possible products of vectors. We define
I = xyz, as in Joy Christian's paper.
Let detector A measure spin along the z-axis. We can represent "spin
up" as the bivector Iz = xy, and "spin down" as the bivector -xy. In
the real number representation, these measurements are represented as
+1 for spin up and -1 for spin down.
Let detector B measure spin along the axis
cos(theta) z + sin(theta) x.
Spin aligned with the axis (+1 in the numeric rep.) is then
represented by
I [cos(theta) z + sin(theta) x] = cos(theta) xy + sin(theta) yz,
and the opposite spin (-1) is
- cos(theta) xy - sin(theta) yz.
According to quantum mechanics, the probabilities of the various
measurement outcomes are:
++ : (1/2) sin^2(theta/2)
+- : (1/2) cos^2(theta/2)
-+ : (1/2) cos^2(theta/2)
-- : (1/2) sin^2(theta/2)
where "+-" indicates +1 at A and -1 at B.
The value of the Clifford product of the spins is:
++ : xy * [cos(theta) xy + sin(theta) yz]
= - cos(theta) + sin(theta) xz
+- : + cos(theta) - sin(theta) xz
-+ : + cos(theta) - sin(theta) xz
-- : - cos(theta) + sin(theta) xz
Therefore quantum mechanics predicts an expectation value of:
cos^2(theta) - sin(theta) cos(theta) xz
Whereas Joy Christian's model predicts:
-cos(theta)
--
Jim E. Black (domain in headers)
How to filter out stupid arguments in 40tude Dialog:
!markread,ignore From "Name" +"<email address>"
[X] Watch/Ignore works on subthreads
Jim Black
> On Sep 9, 11:09=A0am, Jim Black <trams...@yahoo.com> wrote:
[regarding content at
http://www.perimeterinstitute.ca/index.php?option=3Dcom_content&task=3Dvi=
ew&id=3D30&Itemid=3D72&pi=3DJoy_Christian]
>> Joy Christian's counterexample is invalid.
>>
>> He considers an experiment where the measured result at each detector =
is
>> either +1 or -1. =A0But in his model of this experiment, he equates
>> these *measured quantities* to bivectors. =A0Not some intermediate
>> theoretical construct, but the actual measured results.
>=20
> Yes, that's as much as I understood as well.
>=20
>> (He also states that the bivectors have "values +/- 1," which seems
>> to equate bivectors with scalars.)
>=20
> Hmmm... giving him the benefit of the doubt (rather little doubt, as
> he's an expert in the field), I guess him to mean that these are unit
> vectors that have +1 or -1 as possible values. Then they correspond to
> either "spin up" or "spin down, just as the typical measurement
> results of QM.
To be clear, unit bivectors. Also known as two-forms or antisymmetric
tensors of rank two in three dimensions. They can be represented as
pseudovectors -- like vectors but they don't flip sign under a parity
transformation.
>> He then computes the correlation of the measurements using a Clifford
>> product to get a correlation that matches experiment. =A0But the corre=
lation
>> isn't something you directly measure; it's something you compute from =
the
>> measured quantities, which are real numbers, using the ordinary real
>> number product.
>=20
> Please explain why you claim that instead of for example "spin up" and
> spin down" (which correspond to vectors), real numbers are measured.
You can express the measured spin values as bivectors if you want to. Wh=
at
you cannot do is compute the expectation value of the Clifford product of
those bivectors and pretend it's the same as the expectation value of rea=
l
numbers which Bell's theorem is concerned with.
So let us express the spin values as bivectors and ask: Does quantum
mechanics agree with Joy Christian's result?
In what follows, x, y, and z mean the unit vectors in the x, y, and z
directions. In this notation, the rules for the Clifford algebra Cl_{3,0=
}
can be summarized as:
x^2 =3D y^2 =3D z^2 =3D 1
xy =3D -yx, xz =3D -zx, yz =3D -zy
The usual associative and distributive axioms apply. Addition is
commutative, and it's perfectly okay to add any combination of scalars,
vectors, and all the possible products of vectors. We define I =3D xyz, =
as
in Joy Christian's paper.
Let detector A measure spin along the z-axis. We can represent "spin up"
as the bivector Iz =3D xy, and "spin down" as the bivector -xy. In the r=
eal
number representation, these measurements are represented as +1 for spin =
up
and -1 for spin down.
Let detector B measure spin along the axis
cos(theta) z + sin(theta) x.
Spin aligned with the axis (+1 in the numeric rep.) is then represented b=
y
I [cos(theta) z + sin(theta) x] =3D cos(theta) xy + sin(theta) yz,
and the opposite spin (-1) is
- cos(theta) xy - sin(theta) yz.
According to quantum mechanics, the probabilities of the various
measurement outcomes are:
++ : (1/2) sin^2(theta/2)
+- : (1/2) cos^2(theta/2)
-+ : (1/2) cos^2(theta/2)
-- : (1/2) sin^2(theta/2)
where "+-" indicates +1 at A and -1 at B.
The value of the Clifford product of the spins is:
++ : xy * [cos(theta) xy + sin(theta) yz] =3D - cos(theta) + sin(theta)=
xz
+- : + cos(theta) - sin(theta) xz
-+ : + cos(theta) - sin(theta) xz
-- : - cos(theta) + sin(theta) xz
Therefore quantum mechanics predicts an expectation value of:
cos^2(theta) - sin(theta) cos(theta) xz
Whereas Joy Christian's model predicts:
-cos(theta)
--=20
Jim E. Black (domain in headers)
How to filter out stupid arguments in 40tude Dialog:
!markread,ignore From "Name" +"<email address>"
[X] Watch/Ignore works on subthreads
--0-886472827-1252898200=:15412--
harry
> On Thu, 10 Sep 2009 16:38:01 +0000 (UTC), harry wrote:
> > On Sep 9, 11:09�am, Jim Black <trams...@yahoo.com> wrote:
>
Jim thanks very much for the clarification; it is reassuring to see
that you now don't suggest anymore that he confuses bivectors with
scalars, and went along with him in the use of bivectors. But
regretfully (or happily, depending on one's bias), also your new
criticism is wrong IMHO. You specified that "addition is commutative",
and apparently you put that statement to use in your reproduction of
Christian's model.
To the contrary, Joy Christian elaborated on the fact that the
addition of his observables is NOT commutative; this is a main
discussion point in his follow-up papers. As he puts it in
http://arxiv.org/pdf/0707.1333v2 :
"To be sure, the non-commutativity of observables plays a central role
in our model".
Therefore I think to have reason to suspect that you have not
correctly reproduced his model, despite my lack of understanding of
this topic. Indeed, it would be quite surprising if such an expert
would make the kind of error that you suggest without noticing it over
the course of two years while in consultation with leaders in the
field. Thus I simply assume his math to be correct; my doubt is about
the physical meaning of his "beables".
Regards,
Harald
Jim Black
> Jim thanks very much for the clarification; it is reassuring to see
> that you now don't suggest anymore that he confuses bivectors with
> scalars, and went along with him in the use of bivectors. But
> regretfully (or happily, depending on one's bias), also your new
> criticism is wrong IMHO. You specified that "addition is commutative",
> and apparently you put that statement to use in your reproduction of
> Christian's model.
> To the contrary, Joy Christian elaborated on the fact that the
> addition of his observables is NOT commutative; this is a main
g/pdf/0707.1333v2:
>
> "To be sure, the non-commutativity of observables plays a central role
> in our model".
>
> Therefore I think to have reason to suspect that you have not
> correctly reproduced his model, despite my lack of understanding of
> this topic. Indeed, it would be quite surprising if such an expert
> would make the kind of error that you suggest without noticing it over
> the course of two years while in consultation with leaders in the
> field. Thus I simply assume his math to be correct; my doubt is about
> the physical meaning of his "beables".
It's multiplication, not addition, that's non-commutative here.
Nevertheless, the example calculation probably does not correspond to
the algebra(s?) Christian has in mind. The math in Christian's paper
is quite unclear, and I'm not sure I want to muddle through it. But
the calculation was not the main point. The main point was this:
>> You can express the measured spin values as bivectors if you want to.
>> What you cannot do is compute the expectation value of the Clifford
>> product of those bivectors and pretend it's the same as the
>> expectation value of
[the product of]
>> real numbers which Bell's theorem is concerned
>> with.
This is what Christian has done, and it's a conceptual error, not a
mathematical one. I don't know whether Christian's math is
consistent, but with a critical conceptual mistake, it doesn't matter.
--
Jim E. Black
Lester Welch
From the abstract of Carl Brans' article:
"Bell's theorem does not eliminate fully causal hidden variables"
International Journal of Theoretical Physics, Volume 27, 219.
(February, 1988)
"Bell's theorem applies only to a hybrid universe in which hidden
variables determine only part of the outcomes of experiments. When
applied to a fully causal hidden variable theory, in which detector
settings as well as their interaction with particles during
observation are determined by the variables, Bell's analysis must be
modified. The result is that a fully causal hidden variable model can
be produced for which a properly chosen spread of hidden variables
gives precisely the same prediction as standard quantum theory."
Ilja
Superdeterminism, where the hidden variable settings define also
the free choices of the experimenters, makes the notion of causality
meaningless.
Assume you have an FTL phone. You can talk with somebody on the
Mars without any propagation delay. You think in this case Einstein
causality would be proven wrong? You err. With superdeterminism,
your observations that you can talk with the Mars station without time
delay, and the content of the correspondence can later be verified by
other information channels, means nothing.
Thus, with superdeterminism, Einstein causality becomes unfalsifiable
and therefore contentless. So, superdeterminism does not save
Einstein causality.
harry
"Jim Black" <tram...@yahoo.com> wrote in message
news:20ef93c6-5d47-4c62...@v23g2000pro.googlegroups.com...
> On Sep 14, 11:36 pm, harry <harald.vanlin...@epfl.ch> wrote:
>> Jim thanks very much for the clarification; it is reassuring to see
>> that you now don't suggest anymore that he confuses bivectors with
>> scalars, and went along with him in the use of bivectors. But
>> regretfully (or happily, depending on one's bias), also your new
>> criticism is wrong IMHO. You specified that "addition is commutative",
>> and apparently you put that statement to use in your reproduction of
>> Christian's model.
>> To the contrary, Joy Christian elaborated on the fact that the
>> addition of his observables is NOT commutative; this is a main
>> discussion point in his follow-up papers. As he puts it
>> inhttp://arxiv.or=
> g/pdf/0707.1333v2:
>>
>> "To be sure, the non-commutativity of observables plays a central role
>> in our model".
>>
>> Therefore I think to have reason to suspect that you have not
>> correctly reproduced his model, despite my lack of understanding of
>> this topic. Indeed, it would be quite surprising if such an expert
>> would make the kind of error that you suggest without noticing it over
>> the course of two years while in consultation with leaders in the
>> field. Thus I simply assume his math to be correct; my doubt is about
>> the physical meaning of his "beables".
>
> It's multiplication, not addition, that's non-commutative here.
Sorry for the glitch.
> Nevertheless, the example calculation probably does not correspond to
> the algebra(s?) Christian has in mind. The math in Christian's paper
> is quite unclear, and I'm not sure I want to muddle through it. But
> the calculation was not the main point. The main point was this:
>
>>> You can express the measured spin values as bivectors if you want to.
>>> What you cannot do is compute the expectation value of the Clifford
>>> product of those bivectors and pretend it's the same as the
>>> expectation value of
>
> [the product of]
>
>>> real numbers which Bell's theorem is concerned
>>> with.
>
> This is what Christian has done, and it's a conceptual error, not a
> mathematical one. I don't know whether Christian's math is
> consistent, but with a critical conceptual mistake, it doesn't matter.
Thanks for the clarification. I now verified with Bell's original paper that
Bell's theorem according to Bell is not really about real numbers (which is
mathematics), but about the foundation of physics question if deterministic
local realism can be compatible with quantum mechanics. For sure that's also
the question that Joy Christian discusses.
Regretfully I can't (yet) picture Joy Christian's "beables", thus I'm not
convinced either.
Best regards,
Harald
Benjamin Schulz
> On Sep 6, 5:59 am, Lester Welch <lester.we...@gmail.com> wrote:
>> Is Joy Christian disproof of Bell's theorem (arXiv:quant-ph/0703179) accepted?
>
> No, he is a crackpot. As for his specific error, I haven't studied it
> in detail, but people think his hidden variables are nonlocal after
> all.
>
> Bell's theorem does not depend on the nature of the hidden variables,
> as long as they are local. For a simple proof of the theorem see
>
> http://onqm.blogspot.com/2009/07/simple-proof-of-bells-theorem.html
This proof is flawed. You have there only one probability measure P.
Yet the expectation values of quantum mechanics generate an axis
dependent probability measure. That is, you have a whole family of
functions P_ab(A and B), where A and B are the axes and a and b are the
events.
>
> The only requirements for the theorem to hold are that the results of
> one measurement don't depend on the direction chosen for the other,
> and that there is only one outcome for each measurement, so that we
> can talk about what the result of a measurement _would have been_.
>
That is wrong. Due to the above structure (the family of probability
measures), Bell's theorem can only be proven if you assume that
1) the measurement result of A does not depend on the axis of B and vice
versa
2) the measurement result of A is statistically independent of the
measurement result at B and vice versa.
It can be shown, that condition 2 has nothing to do with signaling but
says that there exists an event C at the particle source which is
equivalent to the events A and B (that is, the event C determines A and B).
For detailed proofs of this, see:
E. Nelson, The Locality Problem in Stochastic Mechanics, in: New
Techniques and Ideas in Quantum Measurement Theory, edited by D.
Greenberger, Ann. N. Y. Acad. Sci. 480, 533 (1986)
W. G. Faris, Probability in Quantum Mechanics, appendix to: D. Wick, The
Infamous Boundary: Seven Decades of Controversy in Quantum Physics
(Birkhauser, Boston, 1995)
B. Schulz, Ann. Phys. (Berlin) 18, 231 (2009).
harry
"Lester Welch" <lester...@gmail.com> wrote in message
news:ecfadb80-7cc1-4d16...@r9g2000yqa.googlegroups.com...
> Is Joy Christian disproof of Bell's theorem (arXiv:quant-ph/0703179)
> accepted?
>
> I believe the following simple example to be consistent with
> Christian's disproof:
His disproof (if it truly is one) is generally not well understood.
However, recently he wrote a paper that is meant to be a bit clearer:
http://arxiv.org/abs/0904.4259
I'm mostly hindered by unfamiliarity with the usual notations but I think
that I now - at last - grasp what he is talking about. At the risk of
misrepresentation, I will try to sketch the essence of his paper in a few
simple sentences.
Bell illustrated the hidden variables as 3D unit vectors (points on a
spherical surface); the measurement results of the interplay of such spacial
vectors with a direction of measurement are "EPR elements of reality", which
are correlated with that of a distant "entangled" particle.
According to Christian, Bell's correlated EPR elements of reality should
also be regarded as 3D unit vectors. He argues that the "simultaneous
existence of elements of reality at a remote location for all components of
spin (the one that was actually predicted as well as all the others which
could have been predicted) is at the heart of the EPR argument".
However, the integral that Bell used does not describe correlations between
two such 3D vectors ("between points of a 2-sphere") but instead "describes
[correlations] between the points of the real line".
Christian finds that, when accounting for the right geometry ("topology") of
the correlations, entanglement is an illusion.
As I need some time to let it sink in, I don't know yet what to think of
this. Probability calculus can be very tricky!
Regards,
Harald
Benjamin Schulz
> That is, you have a whole family of
> functions P_ab(A and B), where A and B are the axes and a and b are the
> events.
I'm sorry. In the sentence above is a mistake. It should read
That is, you have a whole family of functions P_ab(A and B), where a and
a are the axes and A and B are the events.
Joy Christian
> "Lester Welch" <lester.we...@gmail.com> wrote in message
>
> news:ecfadb80-7cc1-4d16...@r9g2000yqa.googlegroups.com...
>
> > Is Joy Christian disproof of Bell's theorem (arXiv:quant-ph/0703179)
> > accepted?
>
Several people have asked me to respond to the comments made here
about my counterexample to Bell�s theorem,
so here is my response. In my view Bell�s theorem is based on a
serious topological error. The error lies in the very first
equation of Bell�s famous paper. He associates numbers +1 and -1 with
the end results of an EPR-type experiment, and
writes them as A ( a, L ) = +1 or -1. What could be wrong with such an
innocent assumption? Well, the problem is that A
and B are supposed to represent values of the EPR elements of reality
(or spin components). But EPR-Bohm elements
of reality have a very specific topological structure---they live on a
unit 2-sphere (i.e., on the surface of a unit ball). This
topological structure differs from the topological structure presumed
by Bell in the functions A ( a, L ) = +1 or -1, which live
on a unit 0-sphere, not 2-sphere. Thus Bell�s theorem simply does not
apply to the EPR argument, unless one modifies
his main assumption by writing his function as A ( a, L ) = +1 or -1
about a. After all, no one has ever observed a �click� in
an experiment other than about some experimental direction a. With
this simple change the function A now takes on
values in a topological 2-sphere, not the real line, thereby correctly
representing the EPR elements of reality. The values
of the spin components are still +1 or -1, but they now reside on the
surface of a unit ball. This, in essence, is the only
change I have made in any of my papers. But once this change is made,
no contradiction with quantum mechanics arises.
In fact I have been able to reproduced many complicated quantum
mechanical results by implementing this corrected
assumption. And I have done this in a manifestly local and realistic
manner. Hence the title �disproof of Bell�s theorem.�
Here is how locality is rigorously maintained. Although EPR
correlations are correlations between the points of a unit
2-sphere, the 2-sphere itself is not closed under multiplication of
its points, and hence it cannot respect Bell�s factorizability
(or locality) condition. But 3-sphere (whose equator is a 2-sphere) is
closed under multiplication, and hence the local EPR
elements of reality must be taken as equatorial points of a unit 3-
sphere. This is completely consistent with EPR�s criteria
of reality, locality, and completeness. The EPR-Bohm correlations are
then simply correlations between the equatorial points
of a unit 3-sphere. The most natural language to demonstrate this is
the language of Clifford or geometric algebra. The paper
http://arxiv.org/abs/quant-ph/0703179 is written using this language.
However, for those not well versed in Clifford algebra,
my arguments may also be formulated without using this language, as
seen in http://arxiv.org/abs/0904.4259 .
I appreciate the effort that Jim Black has made to understand my work.
I would like to demonstrate however why his argument
is invalid. To begin with, it is not clear which quantum state he has
in mind. If it is the singlet state, then quantum mechanics
does not predict what he claims. It is well known that quantum
mechanics predicts the expectation value of � cos(theta) for his
experimental setup. The corresponding calculation within my model is
done as follows: The observables, in his notation, are
mu.A = +/- xy and mu.B = +/- [ cos(theta) xy - sin(theta) yz ]
The product then is
(mu.A) (mu.B) = - cos(theta) -/+ xz sin(theta)
When the hidden variable mu (i.e., -/+ ) is summed over, we get � cos
(theta), just as in quantum mechanics .
If, on the other hand, the state considered by Jim is not the singlet
state, then the calculations are more involved and I can address
that case in more detail if necessary. But one cannot simply take one
quantum state and one set of observables within my model and
compare them with another state and another set of observables within
quantum mechanics. You have to compare apples with apples.
For those interested in delving deeper, in http://arxiv.org/abs/0904.4259
one can see nine different explicit examples with much more
complicated observables than the simple one seen above. In each of
these examples quantum mechanical predictions are exactly
reproduced. For example sixteen different predictions of a
rotationally non-invariant entangled state (the Hardy state) have been
explicitly worked out on the pages 14 to 17 of http://arxiv.org/abs/0904.4259
.
Joy Christian
Ah Ha
wrote:
> On 22 Sep, 23:14, "harry" <harald.NOTTHISvanlin...@epfl.ch> wrote:
>
> > "Lester Welch" <lester.we...@gmail.com> wrote in message
>
> >news:ecfadb80-7cc1-4d16...@r9g2000yqa.googlegroups.com...
>
> > > Is Joy Christian disproof of Bell's theorem (arXiv:quant-ph/0703179)
> > > accepted?
>
> Several people have asked me to respond to the comments made here
> so here is my response. In my view Bell's theorem is based on a
> serious topological error.
>
[[Mod. note -- 42 excessively-quoted lines snipped. -- jt]]
> I appreciate the effort that Jim Black has made to understand my work.
> I would like to demonstrate however why his argument
> is invalid. To begin with, it is not clear which quantum state he has
> in mind. If it is the singlet state, then quantum mechanics
> mechanics predicts the expectation value of M-BM- -- cos(theta) for his
> experimental setup. The corresponding calculation within my model is
> done as follows: The observables, in his notation, are
>
>
> The product then is
>
> M-BM- M-BM- (mu.A) (mu.B) = - cos(theta) M-BM- -/+ xz sin(theta)
>
> When the hidden variable mu (i.e., -/+ ) is summed over, we get -- cos
> (theta), just as in quantum mechanics .
>
> If, on the other hand, the state considered by Jim is not the singlet
> state, then the calculations are more involved and I can address
> quantum state and one set of observables within my model and
> compare them with another state and another set of observables within
> quantum mechanics. You have to compare apples with apples.
> one can see nine different explicit examples with much more
> complicated observables than the simple one seen above. In each of
> these examples quantum mechanical predictions are exactly
> reproduced. For example sixteen different predictions of a
> rotationally non-invariant entangled state (the Hardy state) have been
> .
>
> Joy Christian
Hello Joy,
I am so glad to see you have joined this discussion and enjoyed
reading your very clear response. I find your mathematics sound
and interesting, and physicists need to consider your proposed
application to physics carefully in an open discussion. SPR could
be just the right forum for this. I especially admire you for joining
the discussion after the derisive name calling. (Is someone
moderating these postings? It looks like graffiti scrawled in a back
alley. Is there no civility or decorum? I hope that you will delete
all occurrences immediately before further damage is done. )
Jenny Harrison
Professor of Mathematics
University of California, Berkeley
[[Mod. note -- Yes, there are moderators here. See the FAQ for details:
http://www.astro.multivax.de:8000/spr/spr.html
We try to strike a balance between the group decending into chaos and
the posting of nonsense, and choking off discussion. Are decisions are
of course subjective, and sometimes whichever (randomly-chosen) one of
us is processing a submission may err (in either direction). Caveat
reader!
-- jt]]
Joy Christian
>
> I am so glad to see you have joined this discussion and enjoyed
> reading your very clear response. I find your mathematics sound
> and interesting, and physicists need to consider your proposed
> application to physics carefully in an open discussion. SPR could
> be just the right forum for this. I especially admire you for joining
> the discussion after the derisive name calling. (Is someone
> moderating these postings? It looks like graffiti scrawled in a back
> alley. Is there no civility or decorum? I hope that you will delete
> all occurrences immediately before further damage is done. )
>
> Jenny Harrison
> Professor of Mathematics
> University of California, Berkeley
Hello Jenny,
I am glad you liked my response. I would be happy to address any further
questions. There is also a talk I gave recently. It may provide further
clarification of some of the issues raised above. It can be found among
other talks posted at the FQXi website: http://fqxi.org/conference/talks
.
Joy Christian
student
wrote:
> Several people have asked me to respond to the comments made here
> about my counterexample to Bell�s theorem,
> so here is my response. In my view Bell�s theorem is based on a
> serious topological error. The error lies in the very first
> equation of Bell�s famous paper. He associates numbers +1 and -1 with
> the end results of an EPR-type experiment, and
> writes them as A ( a, L ) = +1 or -1. What could be wrong with such an
> innocent assumption?
Nothing as such. An assumption is an assumption. Inequalities are
rigorously derived on the basis of this assumption.
> Well, the problem is that A
> and B are supposed to represent values of the EPR elements of reality
> (or spin components). But EPR-Bohm elements
> of reality have a very specific topological structure---they live on a
> unit 2-sphere (i.e., on the surface of a unit ball). This
> topological structure differs from the topological structure presumed
> by Bell in the functions A ( a, L ) = +1 or -1, which live
> on a unit 0-sphere, not 2-sphere. Thus Bell�s theorem simply does not
> apply to the EPR argument, unless one modifies
> his main assumption by writing his function as A ( a, L ) = +1 or -1
> about a. After all, no one has ever observed a �click� in
> an experiment other than about some experimental direction a.
One observes results, for a given direction a, which can be
labelled as +1 or -1. Bell's theorem is a rigorous statement
about such results, based effectively on the assumption that
there is a hidden variable L which predetermines them. If your
model does not have such a prediction (i.e., functions A(a,L)
and B(b,L), then it does not use this assumption. However,
this does not constitute a 'disproof' of Bell theorem, it simply
means your model is not a 'local realistic' model as defined
by Bell.
> With
> this simple change the function A now takes on
> values in a topological 2-sphere, not the real line, thereby correctly
> representing the EPR elements of reality. The values
> of the spin components are still +1 or -1, but they now reside on the
> surface of a unit ball. This, in essence, is the only
> change I have made in any of my papers. But once this change is made,
> no contradiction with quantum mechanics arises.
> In fact I have been able to reproduced many complicated quantum
> mechanical results by implementing this corrected
> assumption. And I have done this in a manifestly local and realistic
> manner. Hence the title �disproof of Bell�s theorem.�
As you are perfectly well aware, most experts in the field do not
by any means consider that the above constitutes a 'local realistic'
model in the sense of Bell. In fact, you are explicitly making a
different definition of 'local realistic' to Bell (and everyone else).
I could redefine 2 to be different from 1+1, but this does not
disprove that 2=1+1 in the standard defintion!
Let me use the same logic to 'generalise' your model to give a
'local realistic' model of all QM (not just spin):
I represent
every measurement by a Hermitian matrix A having the
measurement results as eigenvalues relative to some basis.
After all, no one has ever observed a measurement result
without choosing a basis. With this simple change the
function A now takes values on a suitable space of matrices,
thereby correctly representing the EPR elements of reality.
The values of the outcomes are still the eigenvalues of A, but
they now reside in a Hermitian matrix. This is, in essence,
the only change I have made to your paper (one also replaces
your strange measure theory, that you gloss over above,
by density operators). Once this change is made no
contradiction with quantum mechanics arises. Indeed,
surprise, surprise, it IS quantum mechanics!! And I have
done this in a manifestly local and realistic way. Hence
the title "This spoof of Bell's theorem".
As I mention in an earlier comment, there is something
of value in your papers, in that they yield a
Grassmannian model for two spin-1/2 systems which in
some sense is 'simpler' that the model based on spin
operators. However, claims that either model is
'realistic' is misleading, and inconsistent with the
rest of the literature on Bell's theorem. Maybe you
should think of a better term (eg, 'nonrealistic' would
be quite appropriate).
Jack
> Jack schrieb:
> > Bell's theorem does not depend on the nature of the hidden variables, as long as they are local. �For a simple proof of the theorem see
> >http://onqm.blogspot.com/2009/07/simple-proof-of-bells-theorem.html
>
> This proof is flawed. You have there only one probability measure P.
> Yet the expectation values of quantum mechanics generate an axis dependent probability measure.
I'm not sure what you mean by having 'only one probability measure P'
but I suppose you didn't understand my notation. I want it to be as
clear as possible so I want to know if that is the case.
When I write, for example, P(A+ & B-), it means the probability that
the result _would be_ + _if measured along direction A_ and would be -
if measured along direction B, both pertaining to Alice's particle.
(Counterfactual definiteness is obviously assumed.) So it is already
a function of the directions as well as the measured outcomes. If I
write P(A+ ; B-), that means the probability that the result would be
+ if measured along direction A by Alice for her particle, and would
be - if measured along direction B by Bob for his particle. I guess
in your notation it would be called P_AB(+ and -).
> > The only requirements for the theorem to hold are that the results of one measurement don't depend on the direction chosen for the other, and that there is only one outcome for each measurement, so that we can talk about what the result of a measurement _would have been_.
> That is wrong. Due to the above structure (the family of probability measures), Bell's theorem can only be proven if you assume that
>
> 1) the measurement result of A does not depend on the axis of B and vice versa
That is the same as the first condition I listed.
> 2) the measurement result of A is statistically independent of the measurement result at B and vice versa.
> It can be shown, that condition 2 has nothing to do with signaling but says that there exists an event C at the particle source which is equivalent to the events A and B (that is, the event C determines A and B).
That doesn't make any sense. If C determines events A and B, then A
and B don't have to be independent of each other. In fact, the EPR
correlation guarantees that they aren't independent; if both
measurement directions are the same, then the result of A has to be
the opposite of the result of B. That is correlation, not
independence.
In any case, as I said, there has to be only one actual outcome for
each measurement in order for the theorem to hold. That's why it
doesn't apply to the Many-Worlds Interpretation (MWI).
Jack
wrote:
> He associates numbers +1 and -1 with the end results of an EPR-type experiment, and writes them as A ( a, L ) = +1 or -1. What could be wrong with such an innocent assumption? Well, the problem is that A and B are supposed to represent values of the EPR elements of reality (or spin components). But EPR-Bohm elements of reality have a very specific topological structure---they live on a unit 2-sphere (i.e., on the surface of a unit ball).
The end results of the experiment are just binary values. You could
label them +/-, or cat/dog, or whatever. They are just macroscopic
results and are in no way describable the way you are saying. The
only way things could be more complicated is if there is more than one
actual outcome, or in other words, in a Many-Worlds Interpretation
(which can indeed be local).
Joy Christian
>
> Nothing as such. �An assumption is an assumption. �Inequalities are
> rigorously derived on the basis of this assumption.
>
Bell�s theorem has foundational significance only within the context
of the EPR argument. Bell�s assumption is incompatible with the EPR
criteria of locality, reality, and completeness. This becomes evident
when one considers these criteria collectively within the coherence of
the EPR argument. An inequality derived using a faulty assumption
cannot have any relevance for the question of local realism. Just as
von Neumann�s theorem could not rule out all hidden variable theories
because of its faulty assumption, Bell�s theorem cannot---and does
not---rule out a local-realistic theory. These issues are discussed in
more detail on the pages 3 to 7 of http://arxiv.org/abs/0904.4259
>
> One observes results, for a given direction a, which can be
> labelled as +1 or -1. �Bell's theorem is a rigorous statement
> about such results, based effectively on the assumption that
> there is a hidden variable L which predetermines them. �If your
> model does not have such a prediction (i.e., functions A(a,L)
> and B(b,L), then it does not use this assumption. However,
> this does not constitute a 'disproof' of Bell theorem, it simply
> means your model is not a 'local realistic' model as defined
> by Bell.
>
Given L and a, my model does predict a definite binary outcome --- +1
or -1.
The only difference is that my beables take their values on a unit 2-
sphere,
thereby correctly representing the EPR elements of reality---whereas
Bell�s
beables take their values on a subset of the real line---thereby
incorrectly
representing the EPR elements of reality. This is better explained in
the
paper: http://arxiv.org/abs/0904.4259 . I fail to see why the numbers
+1 and
-1 should be regarded as less real when they are taken to have the
topology
of a 2-sphere, and more real when they are taken to have the topology
of the
real line. After all, EPR elements of reality are not lined up as a
real line.
They are organized as points of a unit 2-sphere. My models are
completely
consistent with the EPR criteria of locality, reality, and
completeness. They
also rigorously respect Bell�s own criterion of locality, or
factorizability.
Hence I do not recognize my model in some of your assertions.
>
> your strange measure theory, that you gloss over above,
>
I have only used the measure theory that was used by von Neumann in
his pioneering book, and by Bell himself in the first two of his
papers. I do
not find anything strange about it. Further discussion of this theory
can be
found on the page 8 of http://arxiv.org/abs/0904.4259
Joy Christian
harry
> On Sep 24, 7:07�pm, Joy Christian <joy.christ...@wolfson.ox.ac.uk>
> wrote:
>
> > Several people have asked me to respond to the comments made here
> > so here is my response. In my view Bell�s theorem is based on a
> > serious topological error. The error lies in the very first
> > the end results of an EPR-type experiment, and
> > writes them as A ( a, L ) = +1 or -1. What could be wrong with such an
> > innocent assumption?
>
> Nothing as such. �An assumption is an assumption. �Inequalities are
> rigorously derived on the basis of this assumption.
Assumptions can be faulty; if a physics problem is wrongly translated
into math, then in principle the solution is wrong as well.
As I stressed earlier, Bell's theorem concerns the question if
deterministic local realism can be compatible with quantum mechanics.
Bell in his 1964 introduction:
"In this note that idea [of EPR that QM should be supplemented by
additional variables] will be formulated mathematically and shown to
be incompatible with the statistical predictions of quantum
mecanics."
and his generalization in that same note:
"for at least one quantum mechanical state [..], the statistical
predictions of quantum mechanics are incompatible with separable
predetermination."
[..]
Harald
Ilja
it seems to me that you have not addressed the main argument against
your approach at all. I will try to explain it in some more detail.
What you have to explain, with a local realistic model, are the
observed facts. These observed facts are probability distributions drho
(A,B|a,b) of the results of the measurements A,B, which are simply
binary numbers {-1,1}, in dependence of the orientations a,b of the
measurement devices.
In Bell's inequalities, one uses, instead of these probability
distributions, certain averages (expectation values)
E(a,b) = E(AB|a,b) = int AB drho(A,B|a,b)
where in this particular case the integral over all possible values of
A and B reduces to a simple sum.
These are only a special case of a more general formula for
expectation values of arbitrary functions f(A,B) on the measurement
results
E(f|a,b) = int f(A,B) drho(A,B|a,b) (1)
A realistic explanation of such an expectation value in Bell's sense
would be defined by a space of beables L, a probability measure drho
(l) on it which does not depend on a,b (which formalizes the observer-
independence of reality), and a rule which defines which measurement
result A,B is obtained for a given state of the beables l in L and
given orientations a,b of the measurement instruments.
E(f|a,b) = int f(A(l,a,b),B(l,a,b)) drho(l) (2)
which is local if A depends only on a and B only on b.
The problem with your model is that it does not fit into this scheme.
What we see is an "explanation" of the form
E(f|a,b) = int f'(A'(l,a),B'(l,b)) drho(l) (3)
While your A',B' have some connection A'=A(l,a)a, B'=B(l,b)b with
those functions we need, in the realistic explanation, and even the
function f'(A',B') = f(A,B) (a x b) allows to recover the function f
(A,B)=AB we need. But you don't recover it. That would be
E(f|a,b) = int <f'(A'(l,a),B'(l,b)),a x b> drho(l) (4)
and define a rewriting of (2) which hides its nature as being an
explanation of (1) for all imaginable functions f(A,B), but compute
something different.
The key phrase in 0904.4259 seems to be
"On the counterexample side, the question then is: how should one
correctly calculate the correlations between the EPR elements of
reality in general?"
The point is that this question does not stand. There is no "correct"
or "incorrect" choice of the function f(A,B), we can compute E(f|ab)
for every function f(A,B) we like. Bell's theorem uses a particular
choice f(A,B) = AB, but he could have chosen to prove some inequality
violated by quantum theory for f(A,B) = sin(AB) arctan(B+2A) as well.
The inequality would be probably a different one, but the same
experiments (which, essentially, measure some frequencies which
approximate drho(A,B|a,b)) would allow to test them too - but now with
another rule (1) how to compute the expectation values E(a,b) for the
new version of Bell's inequality.
And the realistic explanation has to explain why for the
_particular_given_choice_ of f(A,B)=AB we obtain a violation of the
particular inequality derived for these E(f|a,b). And you are not free
to change this function f, to decide that it is incorrect and has to
be replaced by something else, namely f'(A',B') = Aa x Bb, even if
this looks simpler and more natural than the original formula
rewritten in the particular representation (4).
The point can be made in other terms: A realistic model would have to
explain the observed probability distribution drho(A,B|a,b). So you
have to derive drho(A,B|a,b) from your model so that we can use (1) to
obtain the E(a,b) for f(A,B)=AB. I see no way to do this.
Ah Ha
>
> As I stressed earlier, Bell's theorem concerns the question if
> deterministic local realism can be compatible with quantum mechanics.
>
> Bell in his 1964 introduction:
>
> "In this note that idea [of EPR that QM should be supplemented by
> additional variables] will be formulated mathematically and shown to
> be incompatible with the statistical predictions of quantum
> mecanics."
As a mathematician looking in on this, I wonder if the problem comes
down to the definition of the term "variable". It appears to me from
this discussion, that Bell defines variables to be elements of a
field, in this case real numbers, and Christian broadens this
definition to include the Clifford, or perhaps exterior, algebra (over
the same field) where non-commutativity of rotations in 3-space is
naturally represented.
Physicists are quite used to using separable Hilbert spaces of
differential forms to model physics. However, there are advantages
to representing physical structures by reversing the variance and
using locally convex topological vector spaces of "differential
chains" and their operator algebras instead of differential forms and
their operator algebras. Physical realism is one. Differential
chains can be thought of as "distributions without the test
functions". Indeed, there exists a powerful new "operator calculus"
of differential chains that offers mathematical support to
Christian's idea. The theory has been many years in the making, and I
plan to submit a formal announcement to a mathematics journal
shortly. The mathematics of Christian's idea will be seen as sound,
once the entire calculus is revealed. It builds upon what one might
call "hidden variables", namely discrete sections of the exterior
tangent bundle over a manifold, and higher order versions of these
sections, such as dipoles and quadrupoles. This provides a new way,
beyond fiber bundles, to move from the exterior algebra, which is
defined on the tangent space of each point in a manifold, to
structures globally defined on the manifold where these experiments
are taking place.
Jenny Harrison
Jim Black
> I appreciate the effort that Jim Black has made to understand my work.
> I would like to demonstrate however why his argument
> is invalid. To begin with, it is not clear which quantum state he has
> in mind. If it is the singlet state,
It is.
> then quantum mechanics
> does not predict what he claims. It is well known that quantum
> mechanics predicts the expectation value of -- cos(theta) for his
> experimental setup.
But the expectation value of what? The point was that you can label
the observed states as +1 and -1, or +2 and -2, or with bivectors, or
whatever labels you choose, but when you do so, that changes the
expectation value of their product. For example, if the states are
labeled +/- 2, the expectation value of their product, according to
QM, is -4 cos(theta).
If you label them with unit bivectors, in a Clifford algebra, aligned
with the direction of measured spin, then QM predicts the funny
expectation value I derived. If you label them with bivectors in your
algebra, then I'm not sure what you get, but I'm hardly convinced it
will be -cos(theta).
> The corresponding calculation within my model is
> done as follows: The observables, in his notation, are
>
> mu.A = +/- xy and mu.B = +/- [ cos(theta) xy - sin(theta) yz ]
>
> The product then is
>
> (mu.A) (mu.B) = - cos(theta) -/+ xz sin(theta)
>
> When the hidden variable mu (i.e., -/+ ) is summed over, we get -- cos
> (theta), just as in quantum mechanics .
It isn't very well hidden if you can figure out its value from the
product of two observables, is it?
All of this fuss could have been avoided if you simply calculated the
expected number of counts in each combination of output channels
(i.e., ++, +-, -+, and --) in your original paper, and from there
calculated the correlation. Since these counts are measured in
experiments, your model must be able to predict them if it is
complete. Have you done this calculation anywhere?
--
Jim E. Black
Joy Christian
Your assertion suggests that you have not read my papers. You will
find the answer to your puzzle here: http://arxiv.org/abs/0904.4259
Joy Christian
Igor Khavkine
> On Sep 24, 2:07 am, Joy Christian <joy.christ...@wolfson.ox.ac.uk> wrote:
> > Several people have asked me to respond to the comments made here
> > about my counterexample to Bell's theorem,
> > so here is my response. In my view Bell's theorem is based on a
> > serious topological error.
> Hello Joy,
>
> I am so glad to see you have joined this discussion and enjoyed
> reading your very clear response. I find your mathematics sound
> and interesting, and physicists need to consider your proposed
> application to physics carefully in an open discussion.
Hello, Jenny. I'm glad to hear that you find Joy's mathematics sound.
Looking through some of his papers, I personally find them hard to
follow because of his use of non-standard terminology. Perhaps you
could
help us translate some of what he's written into standard mathematical
language.
For example, in arXiv:0904.4259v2 equation (148) illustrates a group
homomorphism between a Hilbert space (as an additive group) and S^3
(as
the group of unit quaternions, or equivalently SU(2)). He even makes
it
sound like an isomorphism, though he doesn't use that word. Now,
that's
my reading of the surrounding text. Perhaps my reading is wrong, but
as
I understand his statement, it cannot be correct. Simply put, a
Hilbert
space is commutative under addition, while SU(2) is not a commutative
group, therefore no isomorphism can exist between them. In my mind,
this kind of error casts serious doubt on the construction in Joy's
framework, which follows (148), of an arbitrarily entangle state in an
arbitrary Hilbert space. Could you or Joy clarify this point?
The above is just one example. I've noticed a number of places where
I've found Joy's terminology confusing.
Also, a couple of questions directly to Joy.
(1) Does your model for 2 spin-1/2 particles differ from the
corresponding quantum mechanical one? Specifically, you claim that
your
model can reproduce quantum mechanical correlations, but does your
model
have any features that are detectable by an experimentalist that are
different from the corresponding quantum model?
(2) If I am an experimentalist, I can construct an apparatus that will
measure the spin of a particle along a given axis and display the
words
"up" or "down" to show the outcome. I set up two of these measurement
devices and send entangled particles at them. After many repetitions,
I've collected the frequencies of the occurrence of each of four
possible outcomes: (up,up), (up,down), (down,up) and (down,down). In
principle, I could stop here without doing any more data analysis, or
I
could go on to compute a Bell-type correlation. In the parts of your
papers that I've looked at, I've only seen your compute expectations
and
correlations. How does your probability framework yield the four
probabilities for each possible experimental outcome?
Thanks.
Igor
Benjamin Schulz
> On Sep 21, 3:18 am, Benjamin Schulz <Benjamin_Sch...@gmx.de> wrote:
>
> > Jack schrieb:
> > > Bell's theorem does not depend on the nature of the hidden variables, as long as they are local. For a simple proof of the theorem see
> > >http://onqm.blogspot.com/2009/07/simple-proof-of-bells-theorem.html
>
> > This proof is flawed. You have there only one probability measure P.
> > Yet the expectation values of quantum mechanics generate an axis dependent probability measure.
> > That is, you have a whole family of functions P_ab(A and B), where a and b are the axes and A and B are the events.
>
> I'm not sure what you mean by having 'only one probability measure P'
> but I suppose you didn't understand my notation. I want it to be as
> clear as possible so I want to know if that is the case.
>
> When I write, for example, P(A+ & B-), it means the probability that
> the result _would be_ + _if measured along direction A_ and would be -
> if measured along direction B, both pertaining to Alice's particle.
>From your answer, I see that you have not taken an advanced graduate
course on probability theory. I think that I indeed do understand
your notation.
Let me explain your error more precisely:
Say, you have a function f, that takes some argument A. Furthermore,
Assume that A depends on adirection (this is an unit vector we call
x). That is you have something like
1) f(A(x))
Now, say you have an unit vector a and an entire family of functions
f. For every unit vector a, there exists a function f. This function f
now takes some argument A. That is you have something like
2) f_a(A)
It is clear that 1) and 2) are completely different mathematical
objects. Hence, you can prove something for 1) that you cannot prove
for 2).
Now when it comes to Bell's theorem, you want to prove an inequality
that is violated by quantum mechanics after all. That is, you have to
be careful, that the probability functions and the events for whichyou
derive your inequality, have the same mathematical structure that is
given by quantum mechanics.
Now the quantum mechanical correlation coefficient defines for each
chosen axis q an own probabability measure P_a for events A. That is,
you have something like P_a(A).
However, in your proof you give an inequality for an object like P(A
(a)). Hence, you show an inequality that is located in an entirely
different probabiltiy space than is given by quantum mechanics. Your
inequality may look similar to the inequality that is violated by
quantum mechanics. Yet your inequality lies in a completely different
probability space, and therefore has nothing to do with bell's
inequality or quantum mechanics after all.
Now let's define four events:
Aup:=spin_1up AND (spin2down OR spin2up)
Adown:=spin_1down AND (spin2down OR spin2up)
Bup:= (spin1down OR spin1up) AND spin_2up
Bdown:= (spin1down OR spin1up) AND spin_2down
Now say, you measure spin 1in direction a and spin 2 in direction b.
Now let a fixed and b2 be another direction at spin 2. Then, the no-
signalling conditions for b are
3) P_ab(Aup)=P_ab2(Aup)
4) P_ab(Adown)=P_ab2(Adown)
Smilarly, let b fixed and a2 another direction at station 1. Then you
get other no-signalling conditions_
5) P_ab(Bup)=P_a2b(Aup)
6) P_ab(Bdown)=P_a2b(Bdown)
However, the conditions 3)-6) are not sufficient to derive Bell's
inequalities. Bell's inequalities can only be shown, if another
condition is applied, namley that of statistical independence of A and
B.
7) P_ab(Aup and Bdown)=P_ab(Aup)*P_Ab(Bdown)
8) P_ab(Adown and Bup)=P_ab(Adown)*P_Ab(Bup)
Unfortunately, 7) and 8) have nothing to do with signalling. However,
7) and 8) have something to do with determinism.
> That doesn't make any sense. If C determines events A and B, then A
> and B don't have to be independent of each other.
Unfortunately, in probability theory, such things make sense and
indeed happen often. We say, an event C is equivalent to an event D,
if
P(C)=P(C AND D)=P(D)
Hence, everytime when C happens, D happens and vice-versa.
Now the conditions 7) and 8) (which are needed to derive Bell's
inequality) would just imply that the events A and B are equivalent to
an event C in the common past cone of A and B.
If there is no such preparation event C in the past cone of A and B
that is equivalent to A and B, the conditions 7) and 8) can be
violated. And in fact this is the case in quantum mechanics.
However, it is clear, that the violation of 7) and 8) has nothing to
do with signalling. Indeed one can easily construct hidden variable
theories, that respect the no-signalling conditions 3)-6) but violate
7) and 8) and therefore Bell's inequalities.
All this is rigorously reviewed in
B. Schulz, Ann. Phys. 2009, vol. 18, no. 4, pp. 231--270
http://www3.interscience.wiley.com/journal/122268387/abstract
However, the man who discovered all that was Princeton mathematician
Edward Nelson in
The Locality Problem in Stochastic Mechanics, Annals of the New York
Academy of Sciences, vol. 480, issue 1 New Technique, pp. 533-538
http://www3.interscience.wiley.com/journal/119493682/abstract
With best wishes,
Benjamin.
Ilja
> As a mathematician looking in on this, I wonder if the problem comes
> down to the definition of the term "variable". � It appears to me from
> this discussion, that Bell defines variables to be elements of a
> field, in this case real numbers, and Christian broadens this
> definition to include the Clifford, or perhaps exterior, algebra (over
> the same field) where non-commutativity of rotations in 3-space is
> naturally represented.
No, it has nothing to do with it.
In Bell's approach, the hidden variables (beables) may be anything,
you can use an arbitrary mathematical space Lambda to specify
the states of reality lambda in Lambda. If there is some product
structure
in Lambda or not, if it is commutative or not, is not specified.
But a realistic model of something of course needs some connection
to the something it is supposed to explain. In this case, what
one has to explain are the observable measurement results
(detector clicked or not - two possible values at the two
places A,B in {-1,1}) in dependence
of the parameters the experimenters can fix by their free
decision (the orientation of the measurement devices a,b).
One is not free to replace the observations to be explained
by something else.
And if the experimenters decide to compute, from their
data (probability distributions p(A,B|a,b))
expectation values of the product
E(AB|a,b) = sum AB p(A,B|a,b),
and these expectation values have strange properties,
we cannot consider some other expectation values of
other functions
E(f|a,b) = sum f(A,B) p(A,B|a,b)
and name this a realistic model.
Jack
> On 25 Sep., 16:40, Jack <jackmal...@yahoo.com> wrote:
> > When I write, for example, P(A+ & B-), it means the probability that the result _would be_ + _if measured along direction A_ and would be - if measured along direction B, both pertaining to Alice's particle.
> I think that I indeed do understand your notation.
As your response below shows, you did not. Hopefully you will this
time.
> Say, you have a function f, that takes some argument A. Furthermore, Assume that A depends on a direction (this is an unit vector we call x). That is you have something like
>
> 1) � f(A(x))
>
> Now, say you have an unit vector a and an entire family of functions f. For every unit vector a, there exists a function f. This function f now takes some argument A. That is you have something like
>
> 2) � �f_a(A)
>
> It is clear that 1) and 2) are completely different mathematical objects. Hence, you can prove something for 1) that you cannot prove for 2).
Sure.
> Now the quantum mechanical correlation coefficient defines for each chosen axis q an own probabability measure P_a for events A. That is, you have something like P_a(A).
OK.
> However, in your proof you give an inequality for an object like P(A (a)).
No. In my notation, what you call P_q(#), I call P(q#). It is the
not the same as P( #(q) ), which indeed wouldn't make sense. Also, I
denote a direction by a capital letter, and the events in question are
the signs of spin measurements, so # = "+" or "-". Hence P(A+) and so
on. I could call it P(A_+) but in this simple case the underscore is
not needed.
My notation is convenient to extend to multi-event cases; e.g. P(A+
and B-) or P(England_rain and Florida_sun). Hence it is better to
think of my P() as a probability operator rather than a function.
> Bell's inequalities can only be shown, if another condition is applied,
> namley that of statistical independence of A and B.
>
> 7) P_ab(Aup and Bdown)=P_ab(Aup)*P_Ab(Bdown)
> 8) P_ab(Adown and Bup)=P_ab(Adown)*P_Ab(Bup)
I assume that the P_Ab (with A upper case) in the last term of those
equations is supposed to be P_ab.
What about the EPR correlations?
P_ab(Aup) = P_ab(Bdown) = 50% for any ab
If a=b (or really close to it), then we know that P_ab(Aup and Bdown)
= 50% (or really close) and not 25% (or close to that). So obviously
you are wrong.
Best wishes,
Jack
Ah Ha
> For example, in arXiv:0904.4259v2 equation (148) illustrates a group
> homomorphism between a Hilbert space (as an additive group) and S^3
> (as
> the group of unit quaternions, or equivalently SU(2)). He even makes
> it
> sound like an isomorphism, though he doesn't use that word. Now,
> that's
> my reading of the surrounding text. Perhaps my reading is wrong, but
> as
> I understand his statement, it cannot be correct. Simply put, a
> Hilbert
> space is commutative under addition, while SU(2) is not a commutative
> group, therefore no isomorphism can exist between them. In my mind,
> this kind of error casts serious doubt on the construction in Joy's
> framework, which follows (148), of an arbitrarily entangle state in an
> arbitrary Hilbert space. Could you or Joy clarify this point?
Hi Igor,
I will leave this one to Joy. It may be a serious problem. At the
very least, it is confusing and needs rewriting. I did not intend to
endorse the mathematics of everything he has written, especially as I
have only read a portion of his writing, but only the basic idea of
lifting a problem in a finite dimensional space into an infinite
dimensional space with a noncommutative product, and then interpreting
solutions in the original space. There are various ways to achieve
this, e.g., exterior algebra, Clifford algebra, von Neumann algebra,
C-star algebra, NCQ, and it all comes down to the all important
question of physical realism. I think I understand what Joy is trying
to do mathematically, but I could be superimposing some of my own
thoughts to fill in gaps. As far as Bell goes, I had to take at
face value his claim that his process produces results that match
experiments seen in quantum mechanics because this is out of my realm
of expertise. I am glad to see that he is now participating in this
discussion, and these questions can be sorted out.
Jenny
Benjamin Schulz
> > Bell's inequalities can only be shown, if another condition is applied,
> > namley that of statistical independence of A and B.
>
> > 7) P_ab(Aup and Bdown)=P_ab(Aup)*P_ab(Bdown)
> > 8) P_ab(Adown and Bup)=P_ab(Adown)*P_ab(Bup)
>
> I assume that the P_Ab (with A upper case) in the last term of those
> equations is supposed to be P_ab.
>
> What about the EPR correlations?
>
They violate Bell's inequalities! Hence, one of the assumptions needed
to derive the latter can not hold in nature.
> P_ab(Aup) = P_ab(Bdown) = 50% for any ab
>
> If a=b (or really close to it), then we know that P_ab(Aup and Bdown)
> = 50% (or really close) and not 25% (or close to that). So obviously
> you are wrong.
No. I'm right. Since what I'm saying is:
You only can derive Bell's ineqaulities with conditions 3), 4), 5), 6)
and 7) and 8). Now Bell''s inequalities are violated in nature.
Hence, it suffices to violate one or more of the conditions 3), 4),
5), 6); 7) and 8).
Since, as you wrote:
> P_ab(Aup) = P_ab(Bdown) = 50% for any ab
indeed, the assumptions
> > 7) P_ab(Aup and Bdown)=P_ab(Aup)*P_ab(Bdown)
> > 8) P_ab(Adown and Bup)=P_ab(Adown)*P_ab(Bup)
are violated by quantum mechanics.
But the violation of 7) and 8) does not have anything to do with
signaling, as I tried to explain to you in my previous post.
A violation of the no-signaling condition would only exist if any of
the conditions 3), 4), 5), 6) would be violated.
That is, if
3) P_ab(Aup)=P_ab2(Aup)
4) P_ab(Adown)=P_ab2(Adown)
and
5) P_ab(Bup)=P_a2b(Aup)
6) P_ab(Bdown)=P_a2b(Bdown)
would not hold.
But such a thing was never observed.
Again, one absolutely needs the assumptions 7) and 8) to derive Bell's
inequalities.
Therefore, one can violate Bell's inequalities, if only 7) or 8) are
violated, whereas 3),4) 5) 6) may still hold.
Yet the violation of only 7) and 8) does only forbid theories, where
the event measured at the detectors is equivalent to another event at
the source.
Again, of course, the assumptions 7) and 8) are violated by quantum
mechanics. And Hence, Bell's inequality is violated by quantum
mechanics.
But the violation of 7) and 8) does not forbid all hidden variable
theories that don't incorporate instantaneous signalling.
To forbid hidden variable theories that do not incorporate signalling,
one would have to violate 3), 4) ,5), 6), but those conditions hold in
nature.
With best wishes, Benjamin
Joy Christian
>
> All of this fuss could have been avoided if you simply calculated the
> expected number of counts in each combination of output channels
> (i.e., ++, +-, -+, and --) in your original paper, and from there
> calculated the correlation. �Since these counts are measured in
> experiments, your model must be able to predict them if it is
> complete. �Have you done this calculation anywhere?
>
These calculations are not easy. They involve interplay between
the points of a unit 3-sphere and its equator (a 2-sphere). Since
3-sphere is a highly non-trivial space, the calculations are likewise
non-trivial. But Clifford or geometric algebra allows you to do the
same calculations easily. The price of course is the unfamiliarity
of the language I use. But to me this is only a psychological price,
not a physical or mathematical one. In any case, to answer your
question: No, so far I have not published the explicit calculations
anywhere. But this may change in the future.
Joy Christian
Ilja
> Yet the violation of only 7) and 8) does only forbid theories, where
> the event measured at the detectors is equivalent to another event at
> the source.
>
> Again, of course, the assumptions 7) and 8) are violated by quantum
> mechanics. And Hence, Bell's inequality is violated by quantum
> mechanics.
>
> But the violation of 7) and 8) does not forbid all hidden variable
> theories that don't incorporate instantaneous signalling.
If (7) or (8) are violated, the variables are not complete. That's
the EPR part of the argument.
In a complete realistic theory we do not have unexplained
correlations. We accept a theory as complete only if
all correlations are explained by signalling or common causes,
and if the common causes are included into the hidden parameters,
the correlations disappear.
IIja
Jack
re: Joy Christian's error
On 25 Sep, 15:40, Jack <jackmal...@yahoo.com> wrote:
> The end results of the experiment are just binary values. You could
> label them +/-, or cat/dog, or whatever. They are just macroscopic
> results and are in no way describable the way you are saying. The
> only way things could be more complicated is if there is more than one
> actual outcome, or in other words, in a Many-Worlds Interpretation
> (which can indeed be local).
On Sep 28, 1:23 pm, Jim Black <trams...@yahoo.com> wrote:
> All of this fuss could have been avoided if you simply calculated the
> expected number of counts in each combination of output channels
> (i.e., ++, +-, -+, and --) in your original paper, and from there
> calculated the correlation. Since these counts are measured in
> experiments, your model must be able to predict them if it is
> complete. Have you done this calculation anywhere?
On Sep 28, 1:26 pm, Igor Khavkine <igor...@gmail.com> wrote:
> (2) If I am an experimentalist, I can construct an apparatus that will
> measure the spin of a particle along a given axis and display the
> words "up" or "down" to show the outcome. I set up two of these
> measurement devices and send entangled particles at them. After many
> repetitions, I've collected the frequencies of the occurrence of each
> of four possible outcomes: (up,up), (up,down), (down,up) and
> (down,down). In principle, I could stop here without doing any more
> data analysis, or I could go on to compute a Bell-type correlation. In
> the parts of your papers that I've looked at, I've only seen your
> compute expectations and correlations. How does your probability
> framework yield the four probabilities for each possible experimental
> outcome?
On Sep 28, 2:08 pm, Ilja <ilja.schmel...@googlemail.com> wrote:
> But a realistic model of something of course needs some connection to
> the something it is supposed to explain. In this case, what one has to
> explain are the observable measurement results (detector clicked or
> not - two possible values at the two places A,B in {-1,1}) in
> dependence of the parameters the experimenters can fix by their free
> decision (the orientation of the measurement devices a,b).
>
> One is not free to replace the observations to be explained by something else.
These quotes are indeed all making the same obvious point.
The derivation of Bell's inequality is stated solely in terms of
measurable outcomes (which are binary outcomes; the experimenter does
not record else) and the corresponding outcomes that would have been
measured if the experimenter had chosen a different orientation
setting for his measurement device.
It does not depend on the nature of the hidden variables at all. What
it shows is that if the single real outcome for particle 1 does not
depend on the orientation chosen for the distant detector measuring
particle 2, and vice versa, then the inequality must be obeyed.
There is no loophole. Only the MWI evades this because it has no
single outcome, it has multiple real outcomes with experimenters in
parrallel universes getting different results and acting differently
based on that, as with Shrodinger's cat.
Joy Christian has not put his 'local model' in terms of concrete
probabilities for measured outcomes because as Bell's theorem proves
that would be mathematically impossible.
Let me be explicit. This refers to an experiment like the one
discussed on my site
http://onqm.blogspot.com/2009/07/simple-proof-of-bells-theorem.html
Let (A+ & B-) mean that the result would be + if measured along
direction A and would be - if measured along direction B, and so on.
Let P(A+ & B-) be the probability that the hidden variables were such
that those would be the results.
The following inequality must hold since more general cases are at
least as probable as less general ones:
#1) P(A+ & B-) = P(A+ & B- & C+) + P(A+ & B- & C-) ≤ P(C+ & B-) + P(A+ & C-)
It is not possible to measure Alice's particle along more than one
direction, but Bob can help us do the next best thing; because of the
EPR correlations, measuring his particle should reveal the opposite of
what result Alice's particle would have given. Thus the above
inequality is equivalent to
#2) P(A+ ; B+) ≤ P(C+ ; B+) + P(A+ ; C+)
Now, suppose Christian's model violates the above inequality #2 (as it
must, if it is to reprodice the QM prodictions). Then it would have
to also violate #1. But #1 is just a consequence of the fact that
adding more possibilities to a set can not reduce the probability of
that set, it could only increase it.
In Sakuri's book, the inequality is derived in a slightly different
way that may be helpful here. He explicitly lists the eight possible
combinations of outcomes that a set of hidden variables could
determine for the three possible directions A, B, and C. e.g. A+,B+,C-
is one.
With several runs of the experiment, we can list how many times each
of the eight possibilities occurred on average. That can obviously be
done for any model. Can Christian do it?
Bell's inequality then is due to two of the eight combinations being
on the left hand side, while those two as well as two more are on the
right hand side.
Benjamin Schulz
> On 1 Okt., 10:59, Benjamin Schulz <benjamin_sch...@gmx.de> wrote:
>> Yet the violation of only 7) and 8) does only forbid theories, where
>> the event measured at the detectors is equivalent to another event at
>> the source.
>>
>> Again, of course, the assumptions 7) and 8) are violated by quantum
>> mechanics. And Hence, Bell's inequality is violated by quantum
>> mechanics.
>>
>> But the violation of 7) and 8) does not forbid all hidden variable
>> theories that don't incorporate instantaneous signalling.
>
> If (7) or (8) are violated, the variables are not complete. That's
> the EPR part of the argument.
You can easily violate 7) and 8) with a stochastic hidden variable
theory, which does not incorporate signalling, that is, a theory where
the conditions 3), 4), 5) and 6) hold.
This is shown, for example in
E. Nelson, The Locality Problem in Stochastic Mechanics, in: New
Techniques and Ideas in Quantum Measurement Theory, edited by D.
Greenberger, Ann. N. Y. Acad. Sci. 480, 533 (1986)
and in
B. Schulz, Ann. Phys. (Berlin) 18, 231 (2009).
> In a complete realistic theory we do not have unexplained
> correlations. We accept a theory as complete only if
> all correlations are explained by signalling or common causes,
All what the conditions 7) and 8) imply is that the events at the
detectors are equivalent to another event at the source.
All theories, where such an equivalence does not exist automatically
violate 7) and 8).
In fact, it is easy, to violate 7) and 8) even with a deterministic
common cause model.
An example for such a model is given in in section 6.2 of
B. Schulz, Ann. Phys. (Berlin) 18, 231 (2009)
http://www3.interscience.wiley.com/journal/119493682/abstract
Joy Christian
>
> For example, in arXiv:0904.4259v2 equation (148) illustrates a group
> homomorphism between a Hilbert space (as an additive group) and S^3
> (as
> the group of unit quaternions, or equivalently SU(2)). He even makes
> it
> sound like an isomorphism, though he doesn't use that word. Now,
> that's
> my reading of the surrounding text. Perhaps my reading is wrong, but
> as
> I understand his statement, it cannot be correct. Simply put, a
> Hilbert
> space is commutative under addition, while SU(2) is not a commutative
> this kind of error casts serious doubt on the construction in Joy's
> framework, which follows (148), of an arbitrarily entangle state in an
> arbitrary Hilbert space. Could you or Joy clarify this point?
>
Hello Igor,
Thank you for your thoughtful questions. I reread the text surrounding
eq. (148). I am not in any sense suggesting an isomorphism between
H and SU(2). As you note, there can be no isomorphism between the
two spaces. And no isomorphism (or even homomorphism) is needed
for the main point I am discussing there. Eq. (148) is a part of a
long
discussion where what I am illustrating is how to infer a topological
space for the EPR elements of reality. Now the latter concept itself
is rather vague. All we have to go on is the criterion provided by
EPR.
It is then very difficult to guess what the topological space would be
for a general---potentially highly non-trivial---entangled state. So
all
I am discussing in that section is how to infer this space in general.
I think the text would be more clear if I changed the parenthetical
remark after (148) from "viewed as an additive group" to "apart from
non-commutativity." I will see what I can do to make the text
clearer.
>
> (1) Does your model for 2 spin-1/2 particles differ from the
> corresponding quantum mechanical one? Specifically, you claim that
> your
> model can reproduce quantum mechanical correlations, but does your
> model
> have any features that are detectable by an experimentalist that are
> different from the corresponding quantum model?
>
Not that I know of. It is designed not to. However, the model does
predict violations of Bell inequalities for classical systems, thereby
contradicting the Bell literature. I have proposed an experiment to
test this. You can find it here: http://arxiv.org/abs/0806.3078
> (2) If I am an experimentalist, I can construct an apparatus that will
> measure the spin of a particle along a given axis and display the
> words
> "up" or "down" to show the outcome. I set up two of these measurement
> devices and send entangled particles at them. After many repetitions,
> I've collected the frequencies of the occurrence of each of four
> possible outcomes: (up,up), (up,down), (down,up) and (down,down). In
> principle, I could stop here without doing any more data analysis, or
> I
> could go on to compute a Bell-type correlation. In the parts of your
> papers that I've looked at, I've only seen your compute expectations
> and
> correlations. How does your probability framework yield the four
> probabilities for each possible experimental outcome?
>
My papers are formulated within the hidden variable program set up
by von Neumann in the 1930's. According to his ideas, no matter
which model of physics one is concerned with--the quantum model,
the hidden variables model, or any other--for theoretical purposes all
one needs to consider are the expectation values of the observables
measured in various states of the system. So the scenario you
discribed can also be reformulated as expectation values of some
observables. The hidden variable program is not supposed to be a
practical program. It is only supposed to make the philosophical
observation that a realistic explanation of the quantum phenommena
(local or otherwise) is always possible.
Joy Christian
Jack
> On 29 Sep., 09:54, Jack <jackmal...@yahoo.com> wrote:
> > What about the EPR correlations?
>
> They violate Bell's inequalities! Hence, one of the assumptions needed to derive the latter can not hold in nature.
That is simply false. If it were true, then Bell's work would not
have been needed; EPR would have done the work already.
In a proof of Bell's theorem such as mine (and it is a fairly standard
proof, closely based on the one in Sakuri's textbook), one assumes
that the EPR correlations hold. These are the correlations between
Alice's and Bob's results when their measurement directions are the
same.
One then derives a Bell's inequality based on correlations between
measurements in 3 different directions. Using the EPR correlations is
necessary in order to derive the inequality.
There is no problem with finding a local hidden variable model that
obeys the EPR correlations. The inequality then limits the multi-
direction correlations not to exceed a linear function of the angles.
For example, a classical model might be one in which the two particles
have opposite angular momenta (in an unknown direction which is the
hidden variable), and give result up or down based on the sign of the
angular momentum component. Then the EPR correlations hold: if you
measure particle 1 and particle 2 in the same diection, you will get
opposite results. They are certainly _not_ statistically
independent. This example obeys the linear function of the angles.
One then notes that the actual QM predictions for the multi-direction
correlations violate the resulting inequality.
best wishes,
Jack
Joy Christian
>
> These quotes are indeed all making the same obvious point.
>
And that shows that none of the authors of these quotes have
understood my work, or (in some cases) even read my papers.
I am not trying to demonstrate a loophole in Bell's theorem.
I am saying that Bell's theorem is simply wrong. Therefore,
it cannot---and does not---have relevance for any future
theory of physics. My argument has two independent parts.
To begin with, it is vital to recognize that Bell's theorem
has foundational significance only within the context of
the EPR argument. But the EPR argument is actually a
logically impeccable theorem, which proves that quantum
mechanics is an incomplete theory of nature. On the other
hand, Bell's theorem purports to show that the premises
of EPR are inconsistent (cf., the first paragraph of Bell's
famous paper). But for this claim to be true, Bell must
first adapt the EPR premises correctly within his own
demonstration. So my first observation is that the very
first equation of Bell's famous paper is incompatible
with the EPR premises---i.e., with their criteria of locality,
reality, and completeness. This becomes evident when
one looks at these criteria collectively---not individually
as is usually done---within the coherence of the EPR
argument. Now an inequality derived using a faulty
assumption cannot possibly have relevance for the
question of local realism. Therefore, just as
von Neumann's theorem could not rule out all hidden
variable theories because of its faulty assumption, Bell's
theorem cannot---and does not---rule out a local-realistic
theory of physics. End of the story!
Nevertheless, since Bell's theorem has played such a
central role in foundations of physics, it is important
to produce explicit counterexamples. They constitute the
second part of my argument against Bell's theorem. In
that context, let me first stress that there is absolutely
no operational distinction between my A and B and Bell's
A and B. They are both sets of pure binary numbers, +1
or -1, with a definite number for a given hidden variable
lambda, but of course with different topology.
Bell's with incorrect topology, and mine with correct
topology. Just as in Bell's case, given lambda and
an experimental direction a, my model indeed predicts
a definite binary outcome --- +1 or -1. The only
difference is that my A and B take their values on a
unit 2-sphere---thereby correctly representing the EPR
elements of reality---whereas Bell's A and B take their
values on a subset of the real line---thereby incorrectly
(or fictitiously) representing the EPR elements of reality.
Indeed, EPR elements of reality are not lined up as a real
line. They are organized as points of a unit 2-sphere.
So, to sum up: (1) all of my models produce operationally
identical results---in ALL respects---to those predicted by
quantum mechanics; (2) they are completely consistent with
the EPR criteria of locality, reality, and completeness;
and (3) they quite rigorously respect Bell's own criterion of
locality (or factorizability). Thus they are truly local and
realistic models of QM, claimed by Bell to be impossible.
Further details can be found at:
http://arxiv.org/abs/0904.4259
Joy Christian
Benjamin Schulz
> On Oct 1, 1:59� am, Benjamin Schulz <benjamin_sch...@gmx.de> wrote:
>> On 29 Sep., 09:54, Jack <jackmal...@yahoo.com> wrote:
>>> What about the EPR correlations?
>> They violate Bell's inequalities! Hence, one of the assumptions needed to derive the latter can not hold in nature.
>
> That is simply false. If it were true, then Bell's work would not
> have been needed; EPR would have done the work already.
This comment makes clear that you do not have even read Bell's own
article.
Please read Bell's article before you say that I would be wrong.
Bell's article can be downloaded here:
> http://www.ffn.ub.es/luisnavarro/clasicos_2.htm
Bell derives an inequality based on an assumption which he calls "local
realistic". Bell then finds that the singlett state from quantum
mechanics violates this inequality. Hence, one of the assumptions that
is needed to derive Bell's inequality cannot hold in nature. This is
just, what Bell himself writes.
In equation (2) on page 2 of the above pdf, Bell himself writes the
no-signalling condition combined with the assumption of statistical
independence of the measurement results.
In section IV on page 3, Bell then derives an inequality based on the
assumptions of equation (2) and the observed correlations.
Bell then concludes that the assumptions of equation (2) cannot hold,
since Bell's inequality is violated by the quantum mechanicsl
expectation value.
Please read the original articles before you are calling someone wrong.
Finally:
In your proof, you do not need the assumption of statistical
independence of the measurement results. You do not need it, since you
are using the features that are provided by a single probability measure.
Unfortunately, your proof is for a wrong probability space, which is
different from the one given by quantum mechanics.
However, one can show, that if the assumption of statistical
independence is made, one can define a bijective mapping between the
probability measure from quantum mechanics and the measure which is used
in your proof. Then, one can use your proof to derive Bell's inequalities.
Unfortunately, it is often not mentioned in basic literature that the
proof in your blog needs additional assumptions, if one aims to
translate your inequality (which has nothing to do with quantum
mechanics) into the correct probability space that is given by the
quantum mechanical correlation coefficient.
With best wishes,
Benjamin
Ilja
> Ilja schrieb:
>
> > On 1 Okt., 10:59, Benjamin Schulz <benjamin_sch...@gmx.de> wrote:
> >> Yet the violation of only 7) and 8) does only forbid theories, where
> >> the event measured at the detectors is equivalent to another event at
> >> the source.
>
> >> Again, of course, the assumptions 7) and 8) are violated by quantum
> >> mechanics. And Hence, Bell's inequality is violated by quantum
> >> mechanics.
>
> >> But the violation of 7) and 8) does not forbid all hidden variable
> >> theories that don't incorporate instantaneous signalling.
>
> > If (7) or (8) are violated, the variables are not complete. That's
> > the EPR part of the argument.
>
> You can easily violate 7) and 8) with a stochastic hidden variable
> theory, which does not incorporate signalling, that is, a theory where
> the conditions 3), 4), 5) and 6) hold.
>
> This is shown, for example in
>
> E. Nelson, The Locality Problem in Stochastic Mechanics, in: New
> Techniques and Ideas in Quantum Measurement Theory, edited by D.
> Greenberger, Ann. N. Y. Acad. Sci. 480, 533 (1986)
>
> and in
>
> B. Schulz, Ann. Phys. (Berlin) 18, 231 (2009).
You can reach this only if you ignore some
common causes. This can be easily done in some
stochastic limit, where the information about the
particular instances is forgotten.
But you cannot obtain violations of BI in a complete
realistic description. A theory without output indepedence
is not complete, it does not explain the output
dependence. That's the EPR part of Bell's argument.
> > In a complete realistic theory we do not have unexplained
> > correlations. We accept a theory as complete only if
> > all correlations are explained by signalling or common causes,
>
> All what the conditions 7) and 8) imply is that the events at the
> detectors are equivalent to another event at the source.
>
> All theories, where such an equivalence does not exist automatically
> violate 7) and 8).
>
> In fact, it is easy, to violate 7) and 8) even with a deterministic
> common cause model.
As I say, simply ignore the common cause and you obtain
output dependence.
Benjamin Schulz
> On 2 Okt., 20:38, Benjamin Schulz <Benjamin_Sch...@gmx.de> wrote:
> > Ilja schrieb:
>
> You can reach this only if you ignore some
> common causes. This can be easily done in some
> stochastic limit, where the information about the
> particular instances is forgotten.
This is wrong. As is shown in section 6.2 of
B. Schulz, Ann. Phys. (Berlin) 18, 231 (2009).
For example, say you have random numbers in the range of -100 to 100.
Say, you feed each random number into two pseudo random number
generators of the same type. Let their output also be in the range of
-100 to 100.
Since the pseudo random number generators are initialised with the
same starting number, their n-th output will coincide at everytime.
Now define two mappings. They map the output of each pseudo random
number generator to a two valued set. This two valued set contains the
elements up and down.
The mapping associated with one pseudo random number generator gives
up, if the value of the n-th number is >=0 and down if the value of
the n-th number is <0.
The mapping associated with the other pseudo random number generator
gives up, if the value of the n-th number is <0 and down if the value
of the n-th number is >=0.
Since the two pseudo random number generators are always fed with
coinciding starting values, their outoput numbers will also coincide.
The associated mappings are such, that the n-th up and down value that
are produced, are always opposite.
Now it maybe that the generators are initialised with a starting value
x_1>100.
Then, the first generator gives an up result at startup. Since the
pseudo random number generators are deterministic, their n-th values
are determined by x_1. Now define an index n_1. the starting value
x_1 maybe such, that an x_n_1-th number produced by the two generators
is also above 100. Then you will also get an up result in the two
valued set for the pseudo random number x_n_1
You may have another starting value x_2>100. Then, the first generator
also gives an up rssult at startup. Now define an index n_2, where n
is the same as before. The starting value x_2 maybe such, that the
x_n_2-th number produced by the two generators is also above 100. Then
you will also get an up result in the two valued set for the pseudo
random number x_n_2.
You may hava a third starting value x_3>100. Then, the first
generator also gives an up rssult at startup. Now define an index n_3,
where n is the same as before..
However, the starting value x_3 maybe such, that the x_n_3-th number
produced by the two generators is below 100. Then you will get a down
result in the two valued set for the pseudo random number x_n_3.
In other words, the up result at the beginning does not determine the
down result at the end. Thereby, if you would compute the
probabilities of the n-th outcomes in the two valued set, you would
find that they are not equivalent to the probabilities of the starting
outcomes in the two valued set. Nevertheless this model is
deterministic and produces exact anti correlations without
signalling.
Hence, passive locality automatically fails.
This contradicts your statement.
(Perhaps this contradiction is the reason why certain Bohmians at the
munich university have not much published on their theory recently. I
notice that they even stopped lecturing on this. Perhaps some
acknowledge now that a theory which permits instantaneous signalling
is untenable and one has to use alternative theories. However, I hear
that they are lecturing on stochastic theories now.......)
With Best Wishes,
Benjamin
>
> > In fact, it is easy, to violate 7) and 8) even with a deterministic
> > common cause model.
>
> As I say, simply ignore the common cause and you obtain
> output dependence.
However,
Jim Black
Okay, so how about the expectation value of A^2, i.e. the square of the
outcome at one detector. Since the possible outcomes are +/- 1, we don't
even need to do the experiment to see that the answer should be +1. But if
we work it out according to your model, following the example of Eq. 19 in
quant-ph/0703179 with another a in place of the b, we get -1. So even if
all we care about are expectation values, your use of bivectors where one
would normally use the numbers +/- 1 can yield wrong answers.
--
Jim E. Black (domain in headers)
How to filter out stupid arguments in 40tude Dialog:
!markread,ignore From "Name" +"<email address>"
[X] Watch/Ignore works on subthreads
Ilja
> On 3 Okt., 13:31, Ilja <ilja.schmel...@googlemail.com> wrote:
> > On 2 Okt., 20:38, Benjamin Schulz <Benjamin_Sch...@gmx.de> wrote:
> > > Ilja schrieb:
>
> > You can reach this only if you ignore some
> > common causes. This can be easily done in some
> > stochastic limit, where the information about the
> > particular instances is forgotten.
>
> This is wrong. As is shown in section 6.2 of
> B. Schulz, Ann. Phys. (Berlin) 18, 231 (2009).
Can you mail me something computer-readable
(pdf or tex)? ilja.sc...@gmail.com.
> In other words, the up result at the beginning does not determine the
> down result at the end. Thereby, if you would compute the
> probabilities of the n-th outcomes in the two valued set, you would
> find that they are not equivalent to the probabilities of the starting
> outcomes in the two valued set. Nevertheless this model is
> deterministic and produces exact anti correlations without
> signalling.
I don't get the point. The common causes of the
equality of the outcomes are the equal initial values
you feed the pseudo-random numbers. Forget about
these common causes, and you obtain some output
dependence.
And if this would be your point, it is also not visible
how Bell's inequality could be violated by such a
construction.
Joy Christian
>
> Okay, so how about the expectation value of A^2, i.e. the square of the
> outcome at one detector. Since the possible outcomes are +/- 1, we don't
> even need to do the experiment to see that the answer should be +1. But if
> we work it out according to your model, following the example of Eq. 19 in
> quant-ph/0703179 with another a in place of the b, we get -1. So even if
> all we care about are expectation values, your use of bivectors where one
Once again you are comparing apples with
oranges and concluding that bananas are blue.
For any given quantum mechanical observable one
can always find its dispersion-free counterpart
(a multi-vector in the case of my model) such that
the corresponding expectation value is reproduced.
I am sure you will figure out what the correct
multi-vector is for the scenario you have described.
(Hint: bivectors represent binary rotations and hence
two successive bivectors amount to a sign-flip, but
being multi-vectors bivectors are also invertible).
Joy Christian
Benjamin Schulz
> On 4 Okt., 08:32, Benjamin Schulz <benjamin_sch...@gmx.de> wrote:
>
> > On 3 Okt., 13:31, Ilja <ilja.schmel...@googlemail.com> wrote:
> > > On 2 Okt., 20:38, Benjamin Schulz <Benjamin_Sch...@gmx.de> wrote:
> > > > Ilja schrieb:
>
> > > You can reach this only if you ignore some
> > > common causes. This can be easily done in some
> > > stochastic limit, where the information about the
> > > particular instances is forgotten.
>
> > This is wrong. As is shown in section 6.2 of
> > B. Schulz, Ann. Phys. (Berlin) 18, 231 (2009).
>
> Can you mail me something computer-readable
>
> > In other words, the up result at the beginning does not determine the
> > down result at the end. Thereby, if you would compute the
> > probabilities of the n-th outcomes in the two valued set, you would
> > find that they are not equivalent to the probabilities of the starting
> > outcomes in the two valued set. Nevertheless this model is
> > deterministic and produces exact anti correlations without
> > signalling.
>
> I don't get the point. The common causes of the
> equality of the outcomes are the equal initial values
> you feed the pseudo-random numbers.
In the two valued sets that are determined by the pseudo random
numbers, you have outcome dependence.
Of course for the pseudo random numbers in the range between -100 and
100, you have outcome independence.
However, you can call those numbers hidden variables. Since they are
hidden you don't have access to them and you cannot even measure
them.
In an EPR experiment, the accessible values are the particle spins.
And this two valued set, if it is computed by the procedure above, is
outcome dependent.
This way one can violate bell's inequality even in a deterministic
comon cause model.
Ilja
Does anybody doubt that the quantum predictions themself have
outcome dependence?
The Bell theorem is about the _possibility_ of local hidden variables.
In your example, this is not questioned, and I have also never
questioned that hidden variables may be unobservable.
> This way one can violate bell's inequality even in a deterministic
> comon cause model.
No, one cannot.
Your example in no way leaves the theories covered by Bell's
theorem. The pseudo-random numbers are the lambda of
Bell's theorem.
Benjamin Schulz
One can. In
B. Schulz, Ann. Phys. (Berlin) 18, 231 (2009)
it is demonstrated, that exaclty this simple model leads to the
quantum mechanical singlett state.
Ilja wrote:
> Your example in no way leaves the theories covered by Bell's
> theorem.
It does. Please read the publications referenced. The particle
trajectories associated with this model are shown to obey the two
particle pauli equation for the quantum mechanical singlett state.
The ranfom generators play the role of a stochastic force field which
is locally causal. However, to incorporate the response of the
particles to the Stern-Gerlach magnets of an EPR experiment, one has
to introduce additinal force fields. Yet they are acting on only one
of the spatially separated trajectories. This corresponds to the fact
that without any interaction term, there is no contact between the two
Hilbert spaces of an entangled quantum system.
Hence, one has a model which does not incorporate any signalling. Yet
it gives the correct statistics of the quantum mechanical singlett
state.
Please read the referenced publications.
In this case
B. Schulz, Ann. Phys. (Berlin) 18, 231 (2009)
before you are calling someone wrong.
Lester Welch
>
> Please read the referenced publications.
>
> In this case
>
> B. Schulz, Ann. Phys. (Berlin) 18, 231 (2009)
>
An electronic version is available at:
Benjamin Schulz
This version however, was first submittet only for priority. Therefore
it has been frequently updated.
The latest version already includes the changes of an erratum and
addendum to B. Schulz, Ann. Phys. (Berlin) 18, 231 (2009). which is
currently under review in Annalen der Physik. I assume that the erratum
and addendum will appear in Ann.Phys in the early 2010.
Jack
> Please read Bell's article before you say that I would be wrong.
> Bell's article can be downloaded here:
> >http://www.ffn.ub.es/luisnavarro/clasicos_2.htm
>
> no-signalling condition combined with the assumption of statistical
> independence of the measurement results.
The equation (2) you refer to in Bell 1964 is for the expectation
value (called P(a,b)) of the product of the two results, where each
particle gives a result of +1 or -1. (So hbar/2 is factored out.) a,b
are the measurement directions. A is the result for the first
particle, B that for the second. lambda refers to the hidden
variables, which are distributed with probability density rho. The
equation is
P(a,b) = integral_d_lambda rho(lambda) A(a,lambda) B(b,lambda)
Just to make the point clearer, I will rewrite this assuming discrete
hidden variables i. Also I will write expectation values using
brackets.
<AB>_a,b = sum_i rho(i) A(a,i) B(b,i)
Now, does that express statistical independence of A and B? Quite the
opposite! It is a sum of terms, obviously accounting for
correlations.
For example, in a sum like A(a,0) B(b,0) + A(a,1) B(b,1), correlations
between A and B (due to dependence of both on the second argument in
parentheses) obviously affect the sum.
Of course if we did have statistical independence of A and B we would
have a _product form_ for the expectation value:
<AB>_a,b = <A>_a,b <B>_a,b
which is quite different. Ben, you were confused by the fact that a
product of A and B does appear in the equation, but it is under the
integral sign!
Bell's expectation value is not a product form. He makes no
statistical independence assumption of A and B for his hidden
variable model. Nor, I'll wager, has anyone ever made that assumption
besides Ben Schulz.
> Unfortunately, your proof is for a wrong probability space, which is
> different from the one given by quantum mechanics.
I already explained to you why that assumption of yours is wrong. You
misunderstood my notation.
> Unfortunately, it is often not mentioned in basic literature that the
> proof in your blog needs additional assumptions, if one aims to
> translate your inequality (which has nothing to do with quantum
> mechanics) into the correct probability space that is given by the
> quantum mechanical correlation coefficient.
It is "not mentioned" because your claim is false. My proof is
complete.
I hope you do understand, this time.
Jack
Jim Black
No, I'm not going to try to guess how you would have me do this
computation. As the model-builder, it is your job to provide an objective
prescription for how to calculate any observable quantity, without having
to use intuition or make guesses. If following the same procedure as in
quant-ph/0703179 is not correct, feel free to refer me to where you
describe the correct procedure.
Benjamin Schulz
> On Oct 3, 5:54 am, Benjamin Schulz <Benjamin_Sch...@gmx.de> wrote:
>> Please read Bell's article before you say that I would be wrong.
>> Bell's article can be downloaded here:
>>> http://www.ffn.ub.es/luisnavarro/clasicos_2.htm
>> In equation (2) on page 2 of the above pdf, Bell himself writes the
>> no-signalling condition combined with the assumption of statistical
>> independence of the measurement results.
>
> The equation (2) you refer to in Bell 1964 is for the expectation
> value (called P(a,b)) of the product of the two results, where each
> particle gives a result of +1 or -1. (So hbar/2 is factored out.) a,b
> are the measurement directions. A is the result for the first
> particle, B that for the second. lambda refers to the hidden
> variables, which are distributed with probability density rho. The
> equation is
>
> P(a,b) = integral_d_lambda rho(lambda) A(a,lambda) B(b,lambda)
>
At this we agree.
> Just to make the point clearer, I will rewrite this assuming discrete
> hidden variables i. Also I will write expectation values using
> brackets.
>
> <AB>_a,b = sum_i rho(i) A(a,i) B(b,i)
>
> Now, does that express statistical independence of A and B?
This is still what a mathematician calls statistical independence since
probability measures are additive and the corresponding equations are
often sums of terms.
> which is quite different. Ben, you were confused by the fact that a
> product of A and B does appear in the equation, but it is under the
> integral sign!
No. I'm just citing the standard book
"Michael Redhead,I ncompleteness, Nonlocality, and Realism: A
Prolegomenon to the Philosophy of Quantum Mechanics (Clarendon
Paperbacks) (Paperback)"
> http://www.amazon.com/Incompleteness-Nonlocality-Realism-Prolegomenon-Philosophy/dp/0198242387
Some of this is available via google books:
This volume is frequently cited in the field and generally accepted as
one of the clearest accounts of Bell's inequalities.
On p. 98 of the above mentioned book the following is shown:
The equation under the integral sign in Bell's paper describes in fact
two conditions: Namely statistical independence of the outcomes and
setting independence,
With Best Wishes,
Benjamin
Joy Christian
> No, I'm not going to try to guess how you would have me do this
> computation. As the model-builder, it is your job to provide an objective
> prescription for how to calculate any observable quantity, without having
> to use intuition or make guesses. If following the same procedure as in
> quant-ph/0703179 is not correct, feel free to refer me to where you
> describe the correct procedure.
Fine.
The model discussed in http://arxiv.org/abs/quant-ph/0703179
is for the standard EPR correlations for two different
observers. You, on the other hand, are considering an entirely
different physical scenario. You want to know what would be
the outcome when two (or a sequence of) spin measurements
are performed at one and the same detector. Bell himself
considered such a scenario within his own local model for
spin. He constructed a different observable for it from the
one he constructed for the EPR correlations, as these present
a different physical scenario. Subsequently, the scenario
was also discussed at greater length by Clauser, within his
own extended hidden variable model. I too have discussed
the scenario at some length within my own hidden variable
framework. You can find my treatment of it in Section V,
page 10, of this paper: http://arxiv.org/abs/0707.1333
Joy Christian
Benjamin Schulz
>
> P(a,b) = integral_d_lambda rho(lambda) A(a,lambda) B(b,lambda)
>
>
> opposite! It is a sum of terms, obviously accounting for
> correlations.
Of course does this mean statistical independence.
>
> Bell's expectation value is not a product form. He makes no
> statistical independence assumption of A and B for his hidden
> variable model. Nor, I'll wager, has anyone ever made that assumption
> besides Ben Schulz.
Well, here is another peer reviewed article, where it is shown that
Bell's locality condition contains of two assumptions. Namely setting
independence and outcome independence:
http://dx.doi.org/10.1119/1.15059
> L. E. Ballentine
> Department of Physics, Simon Fraser University, Burnaby, British Columbia V5A1S6 Canada
>
> Jon P. Jarrett
> Department of Philosophy, Harvard University, Cambridge, Massachusetts, 02138
They write in the abstract:
> It is shown, however, that the locality principle needed for Bell's theorem is stronger than the simple locality that is needed to satisfy the demands of relativity and that quantum mechanics satisfies the latter.
>
> The stronger locality principle is equivalent to the conjunction of simple locality and predictive completeness, and it is the latter principle that fails.
>
> The notion of predictive completeness is weaker than, and is implied by, the completeness criterion of Einstein, Podolsky, and Rosen.
Again, please read the litherature before you are calling someone wrong.
What Ballentine and Jarret are calling "predictive completeness" is just
the condition that I described as "the events at the detectors are
determined by the events at the source". And it is this condition which
fails.
I also pointed you towards the article of Princeton mathematician E. Nelson:
> The Locality Problem in Stochastic Mechanics
> EDWARD NELSON 1
> 1 Department of Mathematics Princeton University Princeton, New Jersey 08544
http://www3.interscience.wiley.com/journal/119493682/abstract
which says the same.
Generally, it is not a good idea to say that proofs by well known
probabilists, that is mathematicians who work professionally on
probability theory at Princeton university, are wrong. Especially if you
do not have read those articles.
In the well cited standard book:
> "Michael Redhead,I ncompleteness, Nonlocality, and Realism: A
> Prolegomenon to the Philosophy of Quantum Mechanics (Clarendon
> Paperbacks) (Paperback)"
>
>> > http://www.amazon.com/Incompleteness-Nonlocality-Realism-Prolegomenon-Philosophy/dp/0198242387
You can find the same statement (that Bell's inequality involves the
assumptions of statistical independence and setting independence) on p. 98.
Finally, there is a lecture of a mathematician, that gives a detailed
analysis of the proof you have written in your blog. And it is found
that your proof is wrong.
The mathematician W. G. Faris, who was a student of Nelson at Princeton
(see his publications http://math.arizona.edu/~faris/publications.html )
published his lecture notes in an appendix of a popular book
> "Probability in quantum mechanics," appendix to David Wick, The Infamous Boundary: Seven Decades of Controversy in Quantum Physics , Birkh "auser, Boston, 1995.
He published this in a popular book, because the error in your proof is
trivial enough that only someone would make it who does not have heard
graduate level courses on probability theory.
In the appendix, Faris, also explains which assumptions are needed to
show Bell's theorem with full mathematical rigor. And they are indeed
statistical independence of the outcomes and setting independence.
Jim Black
> The model discussed in http://arxiv.org/abs/quant-ph/0703179
> is for the standard EPR correlations for two different
> observers. You, on the other hand, are considering an entirely
> different physical scenario. You want to know what would be
> the outcome when two (or a sequence of) spin measurements
> are performed at one and the same detector.
No, this is the exact same experimental setup. (And thus discussion of
alternate experiments is irrelevant.) I just want to take the one
measurement, square it, and compute the expectation value of the square
according to your model. How do I do that?
I'm trying to be open-minded here. It's already clear that in your model,
you have to pretend that the +/- 1 values the experimenter is recording,
multiplying, and averaging are actually bivectors. This is flat-out wrong,
but let's bear with it and see if it gives us the right answers. And it
seems that in your view, "gives us the right answers" means "gives us the
right expectation values." That's acceptable, since if you can compute
arbitrary expectation values, you can compute arbitrary probabilities. A
probability is just the expectation value of a projection operator.
So the question is, can your model accurately predict the expectation
values of arbitrary functions of measurable quantities? As a basic
consistency check, I tried to calculate the expectation value of the square
of one measurement result. The correct answer is trivially +1. Following
your method, I obtained -1. If my attempt to follow your method was wrong,
then how would you do the calculation?
Joy Christian
>
> No, this is the exact same experimental setup.
I disagree. What you are considering is a different
experiment, and hence requires a different beable.
Read Bohr (1949) and Bell (1965) again. My model,
just like Bell's own, is a *contextual* local hidden
variable model. Different physical question means
different *context*, and that in turn means that
one has to consider different beables in general.
> I just want to take the one
> measurement, square it, and compute the expectation value of the square
> according to your model. �How do I do that?
>
> As a basic
> consistency check, I tried to calculate the expectation value of the square
> of one measurement result. �The correct answer is trivially +1. �Following
> your method, I obtained -1. �If my attempt to follow your method was wrong,
> then how would you do the calculation?
>
This is how I would do the calculation informally:
< A^2 > = < A* A > = < (- mu.a) (+ mu.a) > = +1 ,
where * represents the "reverse" in the language of
geometric algebra. This is correct as far as it
goes, but for a full understanding of how things
work in my framework I again urge you to look at
Section V, page 10, of http://arxiv.org/abs/0707.1333
See especially the derivation of eq.(40), which
is for a more general case than what you have been
considering. Put p = a in eq.(40), and you will have
your answer --- namely, +1.
Joy Christian
Jim Black
> On Oct 14, 11:39�am, Jim Black <fmla...@organization.edu> wrote:
>
>>
>> No, this is the exact same experimental setup.
>
> I disagree. What you are considering is a different
> experiment, and hence requires a different beable.
> Read Bohr (1949) and Bell (1965) again. My model,
> just like Bell's own, is a *contextual* local hidden
> variable model. Different physical question means
> different *context*, and that in turn means that
> one has to consider different beables in general.
It is not yours to disagree. I was posing the problem, and in the problem
I was posing, the experimental setup and the measurements performed are
specified to be the same as in the Bell's inequality experiment. Only the
calculations done by the experimenter to arrive at an expectation value are
different. If you disagree, then there has been a failure in
communication.
Either that, or you are considering the calculations done by the
experimenter to be part of the context, which I find conceivable but quite
bizarre. But more importantly, it would disprove the claim that the model
is local, since the calculations might be done in the distant future.
> This is how I would do the calculation informally:
>
> < A^2 > = < A* A > = < (- mu.a) (+ mu.a) > = +1 ,
>
> where * represents the "reverse" in the language of
> geometric algebra.
Given that the experimental setup is the same, what is the reason for
taking the reverse in this calculation but not in the calculation of the
expectation value of AB?
> This is correct as far as it
> goes, but for a full understanding of how things
> work in my framework I again urge you to look at
> Section V, page 10, of http://arxiv.org/abs/0707.1333
> See especially the derivation of eq.(40), which
> is for a more general case than what you have been
> considering. Put p = a in eq.(40), and you will have
> your answer --- namely, +1.
I did look at it, but it appears to apply to a different experimental
setup, in which two measurements are taken, and then multiplied. This is
not what I am asking about.
Joy Christian
>
> It is not yours to disagree. �I was posing the problem, and in the problem
> I was posing, the experimental setup and the measurements performed are
> specified to be the same as in the Bell's inequality experiment. �Only the
> calculations done by the experimenter to arrive at an expectation value are
> different. �If you disagree, then there has been a failure in
> communication.
>
The experimental setup is NOT the same in your question
and the question asked by EPR and Bell. Read Bell�s 1964
paper again, especially the paragraph around his equations
(4) to (7). He carefully---and separately---discusses the
case of repeated measurements made at a single detector.
Then read Bell�s �Reply to Critics� (1975). Let me quote
from his reply to make things clear. Concerning the EPR-Bell
setup he says this: �We are not at all concerned with
sequences of measurements on a given particle, or of pairs of
measurements on a given pair of particles. We are concerned
with experiments in which for each pair the �spin� of each
particle is measured *once only* (my emphasis).�
It should be clear then that what you are asking is a
different question from the question one is concerned with
in an EPR-Bell experiment. EPR and Bell are concerned with
the expectation value < AB >, but you are asking how
the expectation value < A^2 > is calculated within my
framework. These are two separate physical questions, and
they both have two separate answers within my framework,
which is a framework for a locally *contextual* hidden
variable theory. For the question you are asking, I have
already showed you how to perform the calculation
informally. Here it is again:
< A^2 > = < A* A > = < (- mu.a) (+ mu.a) > = +1 .
>
> Given that the experimental setup is the same, what is the reason for
> taking the reverse in this calculation but not in the calculation of the
> expectation value of AB?
>
> I did look at it, but it appears to apply to a different experimental
> setup, in which two measurements are taken, and then multiplied. �This is
> not what I am asking about.
>
The bottom line is not �why reverse� or �why not reverse�.
�Reverse� is simply how geometric algebra does the job
correctly. No; the bottom line is the expectation value
< AB > is equal to - a.b in one experimental question, and
the expectation value < A^2 > is equal to +1 in an entirely
different experimental question. Apples and Oranges.
Joy Christian
Jim Black
> On Oct 15, 12:53 am, Jim Black <fmla...@organization.edu> wrote:
>
> >
> > It is not yours to disagree. I was posing the problem, and in the problem
> > I was posing, the experimental setup and the measurements performed are
> > calculations done by the experimenter to arrive at an expectation value are
> > communication.
> >
>
> The experimental setup is NOT the same in your question
> and the question asked by EPR and Bell.
I'm sensing a pattern here. Given an experimental question, instead
of showing how to answer the question using your model, you attempt to
change the question. I specify that you take one measurement and
square it; you insist I take two measurements. Bell specifies that
you express the measured quantities as +/- 1 and then compute the
product of their expectation value; you insist that the measured
quantities be expressed as bivectors.
You will never understand Bell's theorem until you learn to follow an
operational description of an experiment. Bell's theorem is not about
the expectation value of some abstract quantum mechanical operator; it
is about the expectation value of the product of two measured
quantities, whose values can only be +1 or -1, never a bivector.
--
Jim E. Black
Joy Christian
>
> Bell's theorem is not about
> the expectation value of some abstract quantum mechanical operator; it
> is about the expectation value of the product of two measured
> quantities, whose values can only be +1 or -1, never a bivector.
>
You are absolutely right. Bell�s theorem is not about
two vectors---or even about one vector. It is not about
the expectation value of some abstract quantum mechanical
operator. It is about the expectation value of the product of
two measured quantities, whose values can only be +1 or -1.
Bell�s theorem is all about measurement results. Sadly for
Bell and his followers, however, physics is not about
measurement results. Measurement results are about physics.
Joy Christian