Disproof of Joy Christian's "Disproof of Bell's theorem"

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by  Florin Moldoveanu

Disproof of Joy Christian’s “Disproof of Bell’s theorem”
Florin Moldoveanu∗
Committee for Philosophy and the Sciences, University of Maryland, College Park, MD 20742
Four critical elementary mathematical mistakes in Joy Christian’s counterexample to Bell’s the-
orem are presented. Consequently, Joy Christian’s hidden variable model cannot reproduce any
quantummechanicsresultsandcannotbeusedasacounterexampletoBell’s theorem. Themathe-
maticalinvestigationisfollowedbyashortdiscussionaboutthepossibilitytoconstructotherhidden
variable theories.
Introduction ism. In order to translate the model to computer code,
1
1 all results had to be systematically double checked and
0 Inaseriesofpapers[1–8],JoyChristianclaimstohave this led to the big surprise of finding out the elementary
2 but critical mathematical mistakes presented below.
constructed a local and realistic hidden variable theory

g abletoreproducequantummechanics’correlationsinthe
u caseofanEPR-Bohmexperimentwithspinonehalfpar-
A Error 1: losing part of the correlation result by
ticles. Joy Christian’s model was received with serious
incorrect averaging
criticisms [9], [10], [11], [12], [13], but no single analysis
8
2 could provide a decisive argument against it. Interest-
In his first paper [1], Joy Christian introduces his hid-
ingly, the first comment [9] came very close to uncover
denvariableµasarandomhandednessofthebasisforhis
] the first mathematical problem discussed below. In his
h reply[2]JoyChristianstated: “Withhindsight,however, geometric algebra basis. If e1,e2,e3 are a set of fixed
p orthonormal vectors, the hid{den varia}ble µ is defined as
itwouldhavebeenperhapsbetterhadInotleftoutasan
-
t exercise anexplicit derivationof the CHSH inequality in follows
n
Ref.[1]. Letme,therefore,trytorectifythispedagogical
a µ, I = e1e2e3 = e1 e2 e3 (1)
u defficiency here.”. This reply unfortunately managed to ± ± ± ∧ ∧
q discourageotherpeople fromtryingto checkthe validity
(see Eq. 14 of Ref. [1])
[ of the mathematical results which are actually very easy
ThenJoyChristianproceedsoncomputingtheClifford
1 to find. product of two bivectors µ a, µ b obtaining in Eq. 17:
v Here are some key statements from subsequentcritics’ · ·
5 replies showing that the mathematical correctness was
3 not the main focus of the criticisms: (µ a)(µ b)= a b a b (2)
5 · · − · − ∧
“themodelformallymanagestoreproducesomequan-
0 which is correct (please note that the plus and minus
tum theoretical expectation values correctly” [10]
.
9 factors in the two µ’s cancel each other out and the ex-
“and here we will assume that the content of these
0 pression is equivalent with (I a)(I b)). However, the
1 papers is correct” [11] last line of Eq. 17 is incorrect.·By a w· ell-known identity
1 “ThefirstcommentonChristian’spaper,[9]byMarcin
(Hodge duality):
: Pawlowski,isalsodifferentfromthecurrentpaper. That
v
comment seems to say that in Clifford-algebra-valued
i a b=I (a b) (3)
X hidden variable theory it is unable to derive Bell’s in- ∧ · ×
r equalities. This is not true since they are indeed deriv- itisnoweasytoseethatthelastlineofEq.17isincorrect
a able, as is explicitly shown in [2].” [12] whenµ= I. BecauseI isincorrectlyreplacedbyµand
−
”By providing an explicit factorizable model, Joy gainsanillegalminussignwhenµ= I,thisleadstoan
−
Christian’s example only disproves the importance of incorrectcancelingofthe a b termwhenaveragingover
∧
Bell’s theorem as an argument against contextual hid- all µ’s in the oriented vector manifold 3.
V
den variable theories.” [13]. In Ref. [9], Pawlowski criticized Christian’s proposal
In the meantime, a new challenge to Joy Christian’s for the presence of CliffordAlgebra valued observablesif
workcamefrominformalphysicsblogs: ifJoyChristian’s wearetogetascalarintheRHSoftheCHSHinequality.
claim of a local realistic theory able to reproduce quan- Joy Christian responded [2] by stating that the average
tum correlations is right, then a computer simulation on on 3removesthea belement. However,Eq.3ofRef.[2]
V ∧
a classical computer would be possible. The author of is incorrect just like the last line of Eq. 17 in Ref. [1]
thispaperproceedtodojustthat: modelJoyChristian’s Even without spelling in detail the error, it is easy
theory on a computer to clarify its claims of local real- to see that the exterior product term should not vanish
on any handedness average because handedness is just
a paper conventionon how to consistently make compu-
tations. For example one can apply the same incorect
∗Electronicaddress: [email protected] argumentto complex numbers because there is the same
2
freedom to choose the sign of √ 1 based on the two di- are even and odd, integrating on a manifold 3 where
− V
mensional coordinate handedness in this case. Then one µ changes signs evenly, only one of the two equations
can compute the averageof let’s say z =3+2i for a fair can be zero. Expanding the (µ a)(µ b) term and using
· ·
coinrandomdistributionofhandednessandarriveatthe Hodgedualityitfollowsthatisotropicallyweightedaver-
incorrect answer: <z >=3 instead of <z >=z. ages of non-scalar part of correlations and measurement
One advertised strength of geometric algebra is the outcomes cannot be both zero.
ability to make computations in a coordinate-free fash- By the prior error we already know Eq. 19 is incor-
ion. If breakingup an object in its components and per- rect and the statement that Eq. 18 and Eq. 19 cannot
forming an average results in elimination of some com- be both right is not a surprise. The new content of this
ponents, then we are guaranteed that the operation is error is that fixing Eq. 19 by any hypothetical general-
mathematically illegal. ization of the manifold 3 breaks Eq. 18. Both Eqs. 18
V
Thesamemistakeispresentintheminimalistpaper[7] and19areneededtoberightifthemodelistoreproduce
inEq.4. Givenafixedbivectorbasis β ,β ,β ,Eq.3of experimentalresults. Thereforethehandednessmistakes
i j k
{ }
Ref. [7] is correct. The corresponding λ-dependent basis conclusively rule out both Joy’s model and all its poten-
product however should be the same as Eq. 3 of Ref. [7] tial Clifford algebra generalizations.
because β (λ)β (λ) = β β λ2 = β β and not gain an
i j i j i j
illegal λ term for the cross product.
Sohowitwaspossibletohavesuchanelementarymis- Error 3: Illegal limit for a bivector equation
take undetected? Most of Joy Christian’s papers suffer
fromaconventionambiguity: insomecasesthecomputa-
InRef.[8],JoyChristianattemptstoimplicitlyanswer
tionsaredoneusingtheµ= I conventionwithI arising
± Holman’s criticisms [10] that the final answer in Eq. 19
from a fixed basis, while in others the computations are
of Ref. [1] has a wrong sign and that the outcome of
done using the µ=I convention (indefinite Hodge dual-
the experiments is always the same resulting in perfect
ity)whichmeansthatI isthe currenttrivectoranddoes
correlations. Joy Christian first seems to agree that the
not arise from a fixed basis. Illegally mixing the con-
outcomeforanypairsofexperimentalresultsisthesame,
ventions during one computation yields the supposedly
but then tries to prove the opposite in an explanation
agreement with quantum mechanics.
marred by mathematical mistakes.
Another way this mistake can arise can be seen in
The discussion of this problem takes place around
Eqs. 23 and 24 of Ref. [6]. In there Eq. 23 is correct
Eqs. 42-46 of that paper. Citing Joy Christian: “Fur-
and Eq. 24 is derived by switching I to I for a change
− thermore, we have taken the randomness µ = +I or I
of handedness. It is true that changing handedness (or −
shared by Alice and Bob to be the initial orientation (or
equivalent performing a reflection) changes the sign of
handedness) of the entire physical space, or equivalently
the pseudoscalar I, but what is incorrect in Eq. 24 is
that of a 3-sphere. Consequently, once µ is given as an
that a b is a pseudovector who should change sign as
× initial state, the polarizations along all directions cho-
well. The corrected Eq. 24 should be:
sen by Alice and Bob would have the same value, because
µ completely fixes the sense of bivectors µ n belonging
( I a)( I b)= a b ( I)( (a b)) (4) ·
− · − · − · − − − × to S2 S3 , regardless of direction.” In other words,
⊂
Thesamemechanismforproducingtheerrorprobably the correlation of the experimental outcome is always
occurred in deriving Eq. 4 of Ref. [7] due to an incor- +1 contradictingquantum mechanics predictions. Up to
rect replacement of β(λ) in Eq. 3 without appropriately this point Joy Christian is correct.
switching the sign of the Levi-Civita pseudotensor. Still, Joy Christian continues: “However, and this is
Also there are two additional physics objections as an important point, the polarization (+µ a) observed
·
well. First, associating a hidden variable to an abstract by Alice is measured with respect to the analyzer ( I
− ·
computation convention is completely unphysical. Sec- a), whereas the polarization (+µ b) observed by Bob is
·
ond, by doing it so, the theory predicts the same corre- measured with respect to the analyzer (+I b).”
·
lation regardless of the spin state of the particles in the It will be shown below in this and next sec-
EPR-Bohm experiment. tion that the mathematical arguments supporting this
(in an attempt to produce both the minus cosine
correlation and all the four experimental outcomes
Error 2: Isotropically weighted averages of (+,+),( , ),(+, ),( ,+)) are mathematically incor-
− − − −
non-scalar part of correlations and measurement rect.
outcomes cannot be both zero Firstlet us note that the changein signbetweenAlice
and Bob is illegal because they both use the same kind
Specifically, Eq. 18 and Eq. 19 of paper [1] cannot be of apparatus during measurement and swapping them
both right. Let us count how many factors of µ are in should not change anything. Therefore we could stop
Eq. 18 and Eq. 19. In Eq. 18 there are two factors of µ, the analysis here because Joy Christian’s model is obvi-
onefromµ nandtheotherfromdρ(µ). Eq.19hasthree ously wrong as it does not respect this basic symmetry.
·
factors of µ from µ a, µ b, and dρ(µ). As the factors Howeverlet’s follow along Joy Christian’s argumentand
· ·
3
discover where the mistakes occur. then Joy Christian’s incorrect argument is as follows:
To solve the problem, Joy Christian considers two al-
mostparallelvectorsinsteadofoneateachdetector: aa′ cosΩ+BˆsinΩ=
′ ′ ′
for Alice and bb for Bob. The two vectors form a bivec- R(ab)=R(aa )R(a b)=
tor and starting from an aligned vector configuration (a (cos(ǫΩ)+Bˆsin(ǫΩ))(cos((1 ǫ)Ω)+Bˆsin((1 ǫ)Ω))=
alignedwithbanda′ alignedwithb′)thegoalistomove − −
′ cos(ǫΩ)cos((1 ǫ)Ω) sin(ǫΩ)sin((1 ǫ)Ω)+
the second pair (bb ) at Bob’s location in the final de- − − −
tector position using a rotor. In general, a rotor can be Baa′(sin(ǫΩ)cos((1−ǫ)Ω)+cos(ǫΩ)sin((1−ǫ)Ω))=
expressed as follows: cosΩ
(10)
R=cosΩ+BˆsinΩ (5)
The last equality comes from taking the exact limit
with Bˆ a unit bivector defined as: a=a′ which makes Baa′ vanish due to a∧a=0. Com-
paring the first and last rows, it is clear that there is
m n
Bˆ = ∧ (6) no discontinuity even when a is strictly a′ and the limit
sin Ω
result is illegal.
where m and n are two unit vectors and Ω is the angle It is also easy to see the mistake another way. Just
between them. In the case of Eq. 45 of Ref. [8], the unit compareEq.46 with the definition ofthe rotorfollowing
bivector corresponds to an axial vector c of unit norm: Eq.45. InlinetwoofEq.46the termfollowingthe rotor
computes to λandthe finalanswerinEq.46up to the
′ −
a a λfactorshouldbe the entirevalue ofthe rotorandnot
c= × (7) −
a a′ just its cosine part.
| × |
Thenattheendoftherotation,afterapplyingasimple
trigonometric identity composing bivectors (multiplying Error 4: Incorrect parallel transport rotor direction
′
two exponential expressions corresponding to the a a
∧
and the a b bivectors) Joy Christian takes the limit In the problem above Joy Christian attempts to elim-
′ ∧
a a and claims that this makes the bivector com- inate a bivector by an illegal limit. Taking the limit cor-
→ ′
ponent of the final result zero as a a becomes zero. rectly still results in the wrong result because the first
∧
The endresult is that only the cosine factor survivesthe line of Eq. 46 should be equal with the last line of the
operation therefore a rotation by “parallel transport” in same equation. The error is in using the incorrect rotor
a “twisted manifold” allows to recover the cosine of the to perform the parallel transport. In geometric algebra
angle between Alice and Bob in their correlation as pre- any object G transforms under a rotation by a rotor R
dicted by quantum mechanics. as G R†GR with R = ab and R† = ba. The angle
→
Thelimitoperationaboveismathematicallyillegal. To of rotation is double the angle between vectors a and b.
seewhy,recallthattheaxialvectorcisofunitnormand Thisformulaiscompletelygeneralandworksforscalars,
assuchitisnormalizedbythesineoftheanglesbetween vectors, bivectors, pseudoscalars, or any of their linear
′ ′
a and a . As a approaches a, the wedge product goes combinations. Let us try to apply this general formula
to zero as sine of the angle, but the denominator goes to Eq. 46. Rab reads:
to zero by the same sine of the angle factor. As such
R =exp (I c)θ /2 =cos(θ /2)+(I c)sin(θ /2)
the two sines cancel each other and the bivector main- ab ab ab ab
{ · } ·
(11)
tainsitsmagnitude. Thisisnothingbutarestatementof
and the correct computation in Eq. 46 is:
thegeometricalgebrafactthatthemagnitudeofabivec-
tor does not depend on the shape of the parallelepiped
defining it. But maybe there is a discontinuity at the ′
limit when a=a′ and the bivector Bˆ does become zero. limb′→b[(+I·b)(+µ·b )]=−λ
We c′an se′e that this is not the case as follows. Suppose =lima′→a{Ra†b[(+I·a)(+µ·a′)]Rab}
ave,cato,rbc, ibs aaruenfiotuvrecutnoirtovretchtoogrosnianltthoethsaismpelapnlea:ne, and =lima′→a{Ra†b[(−λ)(exp{(I·c)θaa′})]Rab}=
′ ′ =lima′→a{Ra†bRab[(−λ)(exp{(I·c)θaa′})]}=
c= aa×aa′ = aa′ ×bb (8) =lima′→a[(+I ·a)(+µ·a′)]=−λ
(12)
| × | | × |
′
let ǫΩ be the angle between a and a , and (1 ǫ)Ω the The final result after parallel transport is still λ be-
angle between a′ and b. If Bˆ is the unit bivect−or causeλisascalar. Thefourthlineintheequation−above
comes from the fact that vector c commutes with itself.
′ ′
a a a b There is another way to understand why the final re-
Bˆ =Baa′ =Ba′b =Ic= sin∧(ǫΩ) = sin(1∧ ǫΩ) (9) sult was not changed even when the limit is not taken.
−
4
A bivector is an oriented surface characterized only by Bell’s theoremby counterexample. Another erroris that
direction, magnitude, and sense of rotation. The vec- computingEq.16ofRef.[5]yieldsEq.3andnotEq.15as
′ ′
tors a, a , b, b are in the same plane and the bivector claimed: “we believe the experiment will vindicate pre-
′ ′
[(+I a)(+µ a )]and[(+I b)(+µ b )]areactuallyiden- diction(15)andrefuteprediction(3).”. Thiswasproven
· · · ·
ticalbecausetheyhavethesameorientation,magnitude, bothanalyticallyandbycomputersimulation. Sincethis
′ ′
and sense of rotation. Rotating the pair aa into bb by impacts only a particular claim of a paper and not the
the angle θ preserves the orientation, magnitude, and viability of the whole research program it was not pre-
ab
sense of rotation. sented here. (A similar computer simulation was carried
The second mistake in Eq. 46 (when computing the out earlier by Stephen Lee [14], but the source code was
limit correctly) comes from applying incorrectly the law not made publicly available.) After those mistakes were
of rotation for rotors. This formula is different than the found the systematic checking of Joy Christian’s mathe-
generalmultivectorformulaanditappliesonlyforrotors. matical claims was stopped.
SpecificallythemistakeinJoyChristian’spaperisinthe
JoyChristian’smodelisnotcorrect,butcanBell’sthe-
orientationofhisrotor : insteadofrotatingaroundthe
orem be invalidated by another non-commutative “be-
R ′
vector c (or equivalently a a ), the correct direction is
ables” theory? Two theorems by Clifton [15] answer
′ ′ ×
(a a ) (b b ). This is the correct direction because this in the negative for non-contextual hidden variable
× × × ′
wearetryingtorotateana a bivectorwithorientation theories and for relativistic quantum field theories with
′ ′ ∧ ′
a a into a b b bivector with orientation b b and bounded energy. During computer simulations, several
× ∧ ×
the rotationneeds to align the two directions. Therefore other non-commutative models which correctly realize
′ ′
the correctrotationdirection is (a a ) (b b ). With the minus cosine correlations were discovered. However,
× × ×
this correct direction one can prove that the rotor law thesimulationalsoshowedthatanytransitionfromnon-
of rotation and the multivector law of rotation produce commutative beables to discrete experimental outcomes
the same result(for example one canuse a tedious brute destroys this correlation and yields the classical correla-
force expansion of the two formulae and show they are tions as expected from Bell’s theorem. It is therefore es-
identical in components). Applied to the discussion in sentialforahiddenvariablemodeltopredictbothquan-
′ ′
the paper,since(a a )isparallelwith(b b )the rotor tum correlations and the experimental outcomes. Any
× ×
reduces to identity in this case. commutative beable hidden variable theory is ruled out
Computed correctly with the right limit and the right byBell’stheorem. Anynon-commutativenon-contextual
rotor, we can now see that the outcome at Alice’s and beable hidden variable theory is ruled out by Clifton’s
Bob’s detectors is always the same and the results are analysis. The only remaining way out is to construct a
completely correlatedcontradictingquantummechanics. non-commutativecontextual beable hidden variable the-
The reason is that the outcome results are nothing but ory. But contextual hidden variable theories are not re-
the negative of the local bivectors’ magnitude - a fixed ally considered physical theories as no experimental evi-
value regardless of direction. Holman’s analysis [10] is denceeverbackedthemout. ItisdebatableifBell’stheo-
therefore proven correct: “Because µ is a local deter- remisimportanttoruleoutsomecontextualhiddenvari-
ministic hidden variable, its value cannot depend on the able theories as well, or only non-contextual ones. Bell’s
choicemade bythe experimenter. Ifthis value is leftun- theorem is not the only result ruling out non-contextual
changedbythefirstmeasurement,performingthesecond hidden variable theories, but only Bell’s result is robust
measurement in the e -direction would result in “spin enough (because involves an inequality) to be put to an
x
up”inthisdirectionwithcertainty,incontradictionwith experimental test.
theusualquantumpredictions.”. Flippingofthesignson
Alice’s and Bob’s experimental outcomes is also without
merit.
Acknowledgement
Conclusion
I want to thank Cristi Stoica for his help in com-
Any of the four mathematical mistakes presented puter modeling of Joy Christian’s proposed experiment
above can reject Joy Christian’s claims of a disproval of in Ref. [5] and for useful discussions.
[1] “Disproof of Bell’s Theorem by Clifford Algebra Val- [3] “Disproof of Bell’s Theorem: Further Consolidations”,
ued Local Variables”, Joy Christian, arXiv:quant- Joy Christian, arXiv:quant-ph/0707.1333
ph/0703179 [4] “DisproofsofBell,GHZ,andHardyTypeTheoremsand
[2] “Disproof of Bell’s Theorem: Reply to Critics”, Joy theIllusionofEntanglement”,arXiv:quant-ph/0904.4259
Christian,arXiv:quant-ph/0703244 [5] “Can Bell’s Prescription for Physical Reality Be
5
Considered Complete?”, Joy Christian, arXiv:quant- rem”, Marc Holman, arXiv:quant-ph/0704.2038
ph/0806.3078 [11] ““Disproof of Bell’s Theorem” : more critics”, Philippe
[6] “What Really Sets the Upper Bound on Quantum Cor- Grangier, arXiv:quant-ph/0707.2223
relations?”, Joy Christian, arXiv:quant-ph:1101.1958 [12] “Disproof of “Disproof of Bell’s Theorem by Clifford
[7] “Disproof of Bell’s Theorem”, Joy Christian, Algebra Valued Local Variables””, Tung Ten Yong,
arXiv:quant-ph/1103.1879 arXiv:quant-ph/0712.1637
[8] “Restoring Local Causality and Objective Reality to [13] “Comments on “Disproof of Bell’s Theorem””, Florin
the Entangled Photons”, Joy Christian, arXiv:quant- Moldoveanu, arXiv:quant-ph/1107.1007
ph/1106.0748 [14] “A simulation of the two hemispheres Bell experiment”,
[9] “Comment on: Disproof of Bell’s Theorem by Clifford S. Lee, http://quantropy.org/13/1/bell.pdf
Algebra Valued Local Variables”, Marcin Pawlowski, [15] “Beables in algebraic quantum mechanics”, R. Clifton,
arXiv:quant-ph/0703218 arXiv:quant-ph/9711009v1
[10] “Non-Viability of a Counter-Argument to Bell’s Theo-