Disproof of ''Disproof of Bell's Theorem by Clifford Algebra Valued Local Variables''

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by  Tung Ten Yong

Disproof of “Disproof of Bell’s Theorem by Clifford Algebra Valued Local Variables”
∗
Tung Ten Yong
A key mathematical error is pointed out regarding the paper ”Disproof of Bell’s Theorem by
Clifford Algebra Valued Local Variables” by Joy Christian.
ORIGINAL COMMENT measure, and then take the product. But this will then
yield correlation functions that obey Bell’s inequalities.
In a series of papers ([1], [2] and [3]), Joy Christian Another possible wayis to firstperformsome operations
challenges the mainstream view by showing that a local betweenthe three multivectorsandthen map the result-
hidden variable model can violate the Bell’s inequalities. ing quantity (which need not be a scalar) into a scalar.
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Thenovelfeatureofthemodelisthatinsteadoftheusual In this case however one needs to justify what kind of
0
0 real number, the complete state of quantum systems is operations are good operations, i.e. such that the scalar
2 described by geometric entities calledtrivectors,and the that is finally obtained can be taken to depict the corre-
spin observables are described by bivectors. Both are lationbetweenthetwobivectors. SinceChristian’spaper
n
a elements in geometric algebra. did not do so, his argument is based on unjustifiable as-
J The model and the relevantgeometric algebra notions sumptions.

2 will not be introduced here [4], but will only point out It should be noted that the error pointed out here is
a quite elementary mathematical error that invalidates different from those pointed out by Philippe Grangier in
h] the whole argument: The expectation value in the equa- [8], where he did not question the validity of the corre-
p tion (19) of paper [1] is not a valid expectation value at lation functions in Christian’s paper, but instead refute
- all. An expectation value function is a function with a the paper’s use of bivectors to represent measurement
t
n real codomain [5], but the codomain of the function in results. While his refutation is more on the conceptual
a equation (19): side, the error pointed out in this paper is much more
u
elementary and decisive.
q
The firstcommentonChristian’spaper,[9]by Marcin
[ (µ·a)(µ·b)dρ(µ) (1)
Z Pawlowski,isalsodifferentfromthecurrentpaper. That
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comment seems to say that in Clifford-algebra-valued
v is not the space of scalar, but the larger space of mul-
hidden variable theory it is unable to derive Bell’s in-
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tivectors. This is seen clearly from the equation (17) of
3 equalities. This is not true since they are indeed deriv-
the paper:
6 able, as is explicitly shown in [2].
1
. (µ·a)(µ·b)=−a·b−µ·(a×b) (2)
2
1 UPDATED COMMENT: REPLY TO
The secondtermonrighthandsideis abivector,asseen
7 ARXIV:QUANT-PH/0703244V9
from the equation (15) of the paper. Therefore the ex-
0
: pectation value (1) above is a (directed) integration of
v In order to avoid further misunderstandings, this new
the sum of scalar and bivector. The result is not (just)
Xi a scalar. In fact, this can be clearly seen from equation section will spell out as clearly as possible the relevant
mathematical error made in Joy Christian’s papers, be-
r (18)ofthepaper,thecorrectanswerofwhichshouldbea
a fore replying to his quant-ph/0703244v9.
nullvector(cancellingoftwovectorsisstill avector,not
a) The expectation value in his theory is a func-
a scalar. An elementary fact about vector space.)[6] [7].
tion with codomain the total geometric algebra of 3-
As a consequence, the correlations calculated from such
‘expectation functions’ do not have the usual statistical dimensional Euclidean space, G3 (notation see reference
4),notthegrade-0subspace(scalar)ofit,ascanbeseen
meanings,andthereforetheirviolationsofBell’sinequal-
from the way he constructs the function. Therefore his
ities is meaningless and do not have the significance as
‘expectationvalue’is actuallya linearcombinationofel-
claimed by the author.
ements from subspaces of different grades, i.e. a linear
Inordertoobtainavalidcorrelationfunctionfromthe
combination of some scalar, vector, bivector and trivec-
two bivectorsin (1)above,we should havea map f such
tor. For example, codomain of the function in Equation
that :
(18) is grade-1 subspace (vector space) of G3, while that
f :(µ·a,µ·b,dρ(µ))→R (3) inEquation(19)isthespacespanbygrade-1andgrade-3
(trivector) subspaces of G3.
before summation. However, there is no unique way to If this point is understood, then the error is obvious
do so. The simplest way is by letting each µ·a, µ·b andno furthurargumentsneedbe given. Butif onecan-
mapped into scalars, and dρ(µ) mapped into a scalar not see this, then furthur clarifications can be given as
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follows: that is not obvious and is one that needs to be justified,
b) If the expectation value is a valid one (a function whichJoyChristiandoesnotdoandsimplyassumesthat
with scalar codomain), then it must do so whether the they already are. Whatever a localrealist’s goals are,he
distribution is isotropic or not. When the distribution is needs to use legitimate statistical functions and proba-
not isotropic, the expectation value in his papers will in bilistic measures. One cannot simply use non-statistical
general contain a non-zero grade-1 component (vector). functionstoviolateBell’sinequalitiesandthenclaimthat
If the distribution is isotropic as assumed in his papers, this disproves Bell’s claims.
the errorstillpersists(but is veryeasyto be overlooked) If this is understood and agreed upon, then it is baf-
becausenowthegrade-1elementisazeroelementofthat fling why he still does not see the problem. In fact this
subspace (i.e. a zero vector). Thus the reason to point is already clear from his statement: ‘the word “scalar”
outthe non-isotropiccaseismerelyto showmoreclearly withinthisframeworksimplymeansamultivector whose
the codomain of the function. all but one components are strictly vanishing.’(emphasis
Noneofhisrepliesanswerstheobjectionsmadeabove. by me). This immediately means that all his ‘statistical
c)Butiftheaboveistooconvolutedthenjustfocuson functions’ (expectation, correlation) are actually multi-
the term in the third line in Equation (19). This should vectors and subjects to my objection above. (His com-
be a trivector and not a scalar, because it is the conse- ment on my remark about zero vector and zero scalar is
quence of directed integration (with trivector measure) not directly related to my objection, so my reply is put
ofascalar−a·b. Soevenifonecanignorethevanishing in a footnote [10].)
terms we still don’t get a scalar out of the function. In Lastly, it is in fact his mistake that he takes the third
factthisalreadyshowsthatthe trivectorvaluedmeasure line ofhis Equation(19),a trivectormeasureintegration
is nota validprobabilitymeasureatall,because they do ofascalartobeascalar. Additionisclosedwithin grade-
not normalized to a scalar unity. 3 subspace of G3 (an elementary fact about Geometric
Now, regarding his reply (in quant-ph/0703244v9) on algebra): summation of trivectors gives a trivector. So
null vector or zero vector, he totally misses my point. his use of trivector measure as probabilistic measure is
By a null vector of course I meant a vector with zero totally erroneous (It is urged that Joy Christian should
magnitude, which is just what he mean by zero vector. explicitly point out the error of this instead of avoiding
As he also said in that reply, a zero vector is a vector to give any answers, which only shows that he does not
with all components zero value. But this is exactly my want to admit the elementary mistakes that he made).
objection: it is not a scalar!
UPDATED COMMENT: REPLY TO
UPDATED COMMENT: REPLY TO ARXIV:QUANT-PH/0703244V11
ARXIV:QUANT-PH/0703244V10
This will be my final comment to Joy Christian’s pa-
The preprint quant-ph/0703244v10 avoids answering pers and his replies.
to my main objection (a) above (in fact, it avoids an- It seems that from his reply he yet again failed to no-
swering all of the objections above). He seems to con- tice, and continues to misunderstand, the central objec-
tinue to missunderstand my objections and still think tion to his model. Even if the mathematical theory of
that the main objection hinges on whether the distribu- probability can be formulated in a more generally way
tionisisotropicornot. Letmeemphasizeagainthatthis such that the codomain of functions is not necessarily
isnotso. Themainproblemisthathisprobabilisticfunc- real number, this is really unrelated to the main objec-
tionsarenotourusualnumberedprobabilisticfunctions. tion. First, the relevant functions in such formulations
The fact that Clifford algebra blurrs the distinction be- will eventually have to be mapped into real numbers if
tweenscalarsandvectorspreciselyimpliesthattheycan- one is to connect with experimental results. In fact such
notbeusedinprobabilistictheoryinthewaythatheem- generalized ‘statistical functions’ do not has the direct
ploys them (none of my sentence says that such blurring meaningastheir originalandmorelimited counterparts,
is an “error”). The factthat in suchframeworkitmakes andhastobeinterpretedbeforecanbeunderstoodinthe
no sense to speak of a scalar codomain implies precisely usualsense. Second,Bell’stheoremisbasedontheusual
that such framework cannot be employed here (at least probability theory, the significance and meaning that we
in the way that he does it). That the algebra blurrs the usually attaches to the theorem relies on the use of it,
distinction cannot be taken to mean that we are now al- because in experiments testing the inequalities we only
lowed to have a theory that will give, eg. a multivector encountersscalarfrequenciesandrecordsnumberedmea-
correlation function, a vectorial average value etc. We surementresults. ItiswithinsuchcontextsthatIclaimed
need to know what such functions means before we can that his ‘statistical functions’ are not valid (i.e. they are
understand them in the way we understand our usual notvalidfunctions tobe compareddirectlywiththe cor-
number valued statistical functions. This is something responding functions obtained from experiment results,
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whichissomethingthatvonNeumannandSegaldidnot [2] “Disproof of Bell’s Theorem: Reply to Critics”, Joy
do). Christian, quant-ph/0703244
Therefore it is of no use that one has a model that do [3] “Disproof of Bell’s Theorem: Further Consolidations”,
Joy Christian, quant-ph/0707.1333
nothavea strictlyscalaroutput ifonewantsto compare
[4] See, eg. C. Doran, A.Lasenby, Geometric Algebra for
this to the usual Bell’s inequalities, and one cannot at-
Physicists, Cambridge University Press (2003).
tach the same meaning to the violation of its version of
[5] See, eg. A. Gut, Probability: A Graduate Course,
‘Bell’sinequalities’astotheviolationbyascalar-valued- Springer (2007).
observablehiddenvariabletheory. Justificationisneeded [6] RegardingEquation(19),thiscanclearlybeseenbylet-
if one wants to claimthat it does has the same meaning. tingvectorsaandbtobeorthogonaltoeachother.Then
Besides the above conceptual problem, there is also we will have − µ·(a×b)dρ(µ),where insidetheinte-
grationisthepRroductofabivectorandatrivector(mea-
mathematical error. In arXiv:quant-ph/0703244v11, he
sure), which is just a vector. In thecase of isotropic dis-
claimsthathis modelgivesfunctions (linearfunctionals)
tributionassumedinthepaper,wegetanullvector,not
with codomain effectively the real [−1,1]. But from his
a zero scalar.
equationsitisobviousthatthefunctionals,curiously,can [7] It would be question begging if the distribution is re-
onlyobtainthezeromultivector,andcannotobtainother quiredtobeisotropicinordertoreproducethequantum
scalarvalueswithin the domain. This isnotsocuriousif correlation results.
one is reminded that the correctcodomain that one gets [8] “ ‘Disproof of Bell’s theorem’: more critics”, Philippe
Grangier, arXiv:0707.2223
forthesefunctionals(whenonevariesthedistribution)is
[9] “Comment on: Disproof of Bell’s Theorem by Clifford
the non-scalar linear subspace of G3 (see (a) above). So
Algebra Valued Local Variables”, Marcin Pawlowski,
his (36) and (37) are both wrong and very misleading.
quant-ph/0703218
Lastly, there is no error in my comment that his di- [10] Herehemadeadoubleerrorandalsomisunderstoodmy
rected measure is not the usual valid probability mea- sentence. First, there is of course a difference between a
sure,since the former normalizesto a trivector(as inhis zero element in grade-1 subspace and a zero element in
equation (39)) but the latter normalizes to a scalar one. grade-0 subspace. They are identity elements in the re-
spectivesubspacesrelativetoadditionswithinthespaces.
Secondly, when he mentions ‘zero scalar’ he is actually
referring to the identity element of the total G3. There
is a conceptual difference here. In this context, my re-
∗ mark was actually referring to the identity elements of
Electronic address: [email protected]
the subspaces.
[1] “Disproof of Bells Theorem by Clifford Algebra Valued
Local Variables”, Joy Christian, quant-ph/0703179