Bell inequality and
common causal explanation
in algebraic quantum field theory
Gábor Hofer-Szabó
Research Centre for the Humanities, Budapest
Péter Vecsernyés
Wigner Research Centre for Physics, Budapest
– p. 1
Project
Question: What is the relation between
the Bell inequalities
and the common causal explanation of correlations
in algebraic quantum field theory (AQFT)?
– p. 2
Project
Question: What is the relation between
the Bell inequalities
and the common causal explanation of correlations
in algebraic quantum field theory (AQFT)?
Classical case: Common cause =⇒ Bell inequality
– p. 3
Project
Question: What is the relation between
the Bell inequalities
and the common causal explanation of correlations
in algebraic quantum field theory (AQFT)?
Classical case: Common cause =⇒ Bell inequality
↓
Quantum case:
Bell inequality
– p. 4
Project
Question: What is the relation between
the Bell inequalities
and the common causal explanation of correlations
in algebraic quantum field theory (AQFT)?
Classical case: Common cause =⇒ Bell inequality
↓
Quantum case:
?
↓
=⇒ Bell inequality
– p. 5
Project
I. Classical common causal explanation
II. Nonclassical common causal explanation
III. One correlation: Common Cause Principles in AQFT
IV. More correlations: joint common causal explanation in
AQFT
– p. 6
Reichenbachian common cause
Classical probability space: (Σ, p)
Positive correlation: A, B ∈ Σ
p(AB) > p(A) p(B)
Reichenbachian common cause: C ∈ Σ
p(AB|C) = p(A|C)p(B|C)
p(AB|C) = p(A|C)p(B|C)
p(A|C) > p(A|C)
p(B|C) > p(B|C)
– p. 7
Common cause system
Correlation: A, B ∈ Σ
p(AB) 6= p(A)p(B)
Common cause system (CCS): partition {Ck }k∈K in Σ
p(AB|Ck ) = p(A|Ck ) p(B|Ck )
Common cause: CCS of size 2.
– p. 8
Localization of the common cause
1111
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– p. 9
Localization of the common cause
111111
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A
B
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Ck
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Weak common cause system
– p. 10
Localization of the common cause
111111
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Ck
00000000000000000
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00000000000000000
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Common cause system
– p. 11
Localization of the common cause
1111
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Ck
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111111111
Strong common cause system
– p. 12
Localization of the common cause
VA and VB : localization of A and B .
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VA
VB
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Weak past:
Common past:
Strong past:
VA
VB
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VA
VB
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wpast(VA , VB ) := I− (VA ) ∪ I− (VB )
cpast(VA , VB ) := I− (VA ) ∩ I− (VB )
spast(VA , VB ) := ∩x∈VA ∪VB I− (x)
– p. 13
Joint common cause system
Correlations: Am , Bn ∈ Σ (m ∈ M, n ∈ N )
p(Am Bn ) 6= p(Am ) p(Bn )
Joint CCS: partition {Ck }k∈K in Σ
p(Am Bn |Ck ) = p(Am |Ck ) p(Bn |Ck )
– p. 14
Conditional probabilities
Measurement outcomes: Am , Bn ∈ Σ (m ∈ M, n ∈ N )
Measurement choices: am , bn ∈ Σ (m ∈ M, n ∈ N ) also
localized in VA and VB
Conditional correlations:
p(Am Bn |am bn ) 6= p(Am |am ) p(Bn |bn )
– p. 15
Common causal explanation
Local, non-conspriratorial joint common causal
explanation: a partition {Ck } in Σ
p(Am Bn |am bn Ck ) = p(Am |am bn Ck ) p(Bn |am bn Ck )
(screening-off)
p(Am |am bn Ck ) = p(Am |am Ck )
(locality)
p(Bn |am bn Ck ) = p(Bn |bn Ck )
(locality)
p(am bn Ck ) = p(am bn ) p(Ck )
(no-conspiracy)
{Ck } can be localized in any of the three different pasts
– p. 16
Motivation by Markov condition
A
a
B
C
b
Markov condition =⇒ screening-off, locality and
no-conspiracy
– p. 17
Clauser–Horne inequality
Joint CCS
Locality =⇒ CH inequality
No-conspiracy
−1 6 p(A1 B1 |a1 b1 ) + p(A1 B2 |a1 b2 ) + p(A2 B1 |a2 b1 )
−p(A2 B2 |a2 b2 ) − p(A1 |a1 ) − p(B1 |b1 ) 6 0
(which is equivalent to the CHSH inequality.)
– p. 18
EPR correlations
Conditional probabilities:
p(Am |am ), p(Bn |bn ), p(Am Bn |am bn )
(m, n = 1, 2)
CH inequalitity is violated.
Therefore: no common causal explanation for EPR.
– p. 19
Classical ontology
”Bell inequalities are relations between conditional
probabilities valid under the locality assumption.”
(Gisin, 2009)
– p. 20
Quantum ontology
Event space: von Neumann lattice
Events: projections
Probability: quantum state
CH inequality:
−1 6 φ (A1 B1 + A1 B2 + A2 B1 − A2 B2 − A1 − B1 ) 6 0
– p. 21
Quantum ontology
Event space: von Neumann lattice
Events: projections
Probability: quantum state
CH inequality:
−1 6 φ (A1 B1 + A1 B2 + A2 B1 − A2 B2 − A1 − B1 ) 6 0
Then let us take the quantum ontology seriously!
Common cause:
1. What is that?
2. How it relates to the CH inequality?
– p. 22
Non-classical common cause system
Non-classical probability space: (P(N ), φ)
Correlation: A, B ∈ P(N )
φ(AB) 6= φ(A)φ(B)
(Non-classical) CCS: partition {Ck }k∈K in P(N )
φ(Ck ACk ) φ(Ck BCk )
φ(Ck ABCk )
=
φ(Ck )
φ(Ck )
φ(Ck )
– p. 23
Non-classical common cause system
Non-classical probability space: (P(N ), φ)
Correlation: A, B ∈ P(N )
φ(AB) 6= φ(A)φ(B)
(Non-classical) CCS: partition {Ck }k∈K in P(N )
φ(Ck ACk ) φ(Ck BCk )
φ(Ck ABCk )
=
φ(Ck )
φ(Ck )
φ(Ck )
Commuting / Noncommuting CCS: {Ck }k∈K is
commuting / not commuting with A and B
Nontrivial CCS: Ck 6≤ A, A⊥ , B or B ⊥ for some k ∈ K
– p. 24
Joint common cause system
Set of correlations: Am , Bn ∈ P(N )
φ(Am Bn ) 6= φ(Am )φ(Bn )
Joint CCS: partition {Ck }k∈K in P(N )
φ(Ck Am Bn Ck )
φ(Ck Am Ck ) φ(Ck Bn Ck )
=
φ(Ck )
φ(Ck )
φ(Ck )
A subtle point:
Joint CCS = local, non-conspiratorial joint CCS
– p. 25
Clauser–Horne inequality
Commuting joint CCS
(Locality) =⇒ CH inequality
(No-conspiracy)
−1 6 φ (A1 B1 + A1 B2 + A2 B1 − A2 B2 − A1 − B1 ) 6 0
EPR has no commutative common causal explanation
– p. 26
Noncommuting common causes
But why to demand commutativity between a cause and
its effects?
Standard QM: operators do not commute with their time
translates:
Harmonic oscillator: x(t) ≡ U (t)−1 xU (t)
i~
x(t), x ψ0 = −
sin (~ωt)ψ0 6≡ 0
mω
– p. 27
Noncommuting common causes
Noncommuting joint CCS
(Locality) =⇒
6
CH inequality
(No-conspiracy)
Question: Can a set of correlations violating the CH
inequality have a noncommuting joint common causal
explanation in AQFT?
– p. 28
Noncommuting common causes
An easier question: Can one correlation have a
common causal explanation in AQFT? (Rédei 1997)
Common Cause Principle (CCP): If there is a
correlation between two events and there is no direct
causal (or logical) connection between them, then there
always exists a common cause of the correlation.
– p. 29
Algebraic quantum field theory
Poincaré covariant AQFT:
1111
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Quantum Ising model:
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– p. 30
Common Cause Principles in AQFT
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B
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Ck
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Weak (Commutative/Noncommutative) CCP
– p. 31
Common Cause Principles in AQFT
111111
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B
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Ck
00000000000000000
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00000000000000000
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(Commutative/Noncommutative) CCP
– p. 32
Common Cause Principles in AQFT
1111
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Ck
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Strong (Commutative/Noncommutative) CCP
– p. 33
Common Cause Principles in AQFT
111111
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A
B
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Ck
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Proposition: The Weak Commutative CCP holds in
Poincaré covariant AQFT (Rédei, Summers, 2002).
Question: What about other AQFTs?
– p. 34
Common Cause Principles in AQFT
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Proposition: The Weak Commutative CCP does not
hold in quantum Ising model (Hofer-Szabó, Vecsernyés,
2012a).
Question: What about abandoning commutativity?
– p. 35
Common Cause Principles in AQFT
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Proposition: The Weak Noncommutative CCP holds in
quantum Ising model (Hofer-Szabó, Vecsernyés, 2012b).
– p. 36
Joint common cause system in AQFT
Original question: Can a set of correlations violating
the CH inequality have a noncommuting joint common
causal explanation in AQFT?
– p. 37
Correlations violating CH
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Bn
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→n
−
−
→
m
Am = A( a ), Bn = B( b ): four projections (m, n = 1, 2)
→n
−
→
−
m
a , b : Bell directions
ρs : singlet state
Maximal violation of the CH (CHSH) inequality.
– p. 38
Noncommuting common causes
... after some calculation ...
ρC
=
1 + λU− 21 U 21
1+λ
1−λ ′
1
1
c1 (U− 2 + U 2 ) +
c1 (U− 12 − U 21 )
+
2
2
1+λ
c2 (U0 − U− 12 U0 U 21 ) − λc2 (U−1 U0 U1 + U−1 U− 21 U0 U 21 U1 )
+
2
1−λ ′
+
c (U0 + U− 12 U0 U 21 )
2 2
1−λ ′
1+λ
1
1
c3 i(U− 2 U0 − U0 U 2 ) +
c3 i(U− 12 U0 + U0 U 21 )
+
2
2
+λc1 c2 (U−1 U− 21 U0 U1 + U−1 U0 U 21 U1 )
+λc22 (−U−1 U1 + U−1 U− 21 U 21 U1 )
+λc2 c3 i(U−1 U− 21 U1 − U−1 U 12 U1 ).
Answer: Yes.
– p. 39
Joint common cause system in AQFT
1111
0000
0000
1111
00001111
1111
0000
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
00000
11111
0000
1111
0000
1111
A
00000
11111
Bn
m
0000
1111
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
0000
1111
0000
1111
0000
1111
00000
11111
C1111
0000
1111
0000
00000
11111
0000
1111
00000
11111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
0000
00001111
1111
0000
1111
1111
0000
0000
00001111
1111
0000
1111
Proposition: (Hofer-Szabó, Vecsernyés, 2012c)
There is a noncommuting common cause {C, C ⊥ } of the
correlations {(Am , Bn )}; and it can be localized in the
shaded region.
– p. 40
Conclusion
Classical case: Common cause =⇒ Bell inequality
↓
Quantum case:
Bell inequality
– p. 41
Conclusion
Classical case: Common cause =⇒ Bell inequality
↓
↓
Quantum case: Common cause =⇒
6
Bell inequality
The violation of the Bell inequality in AQFT does not
exclude a set of correlations to have a joint common
causal explanation if commutativity is abandoned.
– p. 42
Remarks
In the noncommutative case the theorem of total
probability does not hold. (No ’Hempelian’ explanation.)
Are the (Strong/Weak) Noncommutative Joint CCPs
valid in AQFT?
What are the ontological consequences of applying
noncommutative common causes?
– p. 43
References
N. Gisin, Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle, (The
Western Ontario Series in Philosophy of Science, 73, III / 1, 125-138, 2009).
G. Hofer-Szabó, P. Vecsernyés, ”Reichenbach’s Common Cause Principle in AQFT with
locally finite degrees of freedom,” Found. Phys., 42, 241-255 (2012a).
G. Hofer-Szabó, P. Vecsernyés, ”Noncommutative Common Cause Principles in algebraic
quantum field theory,” Phil. of Sci., (submitted) (2012b).
G. Hofer-Szabó, P. Vecsernyés, ”Noncommuting local common causes for correlations
violating the Clauser–Horne inequality,” Jour. of Math. Phys., (forthcoming) (2012c).
V.F. Müller and P. Vecsernyés, ”The phase structure of G-spin models”, to be published
F. Nill and K. Szlachányi, ”Quantum chains of Hopf algebras with quantum double
cosymmetry” Commun. Math. Phys., 187 159-200 (1997).
M. Rédei, ”Reichenbach’s Common Cause Principle and quantum field theory,” Found.
Phys., 27, 1309–1321 (1997).
M. Rédei and J. S. Summers, ”Local primitive causality and the Common Cause Principle in
quantum field theory,” Found. Phys., 32, 335-355 (2002).
H. Reichenbach, The Direction of Time, (University of California Press, Los Angeles, 1956).
K. Szlachányi and P. Vecsernyés, ”Quantum symmetry and braid group statistics in G-spin
models” Commun. Math. Phys., 156, 127-168 (1993).
– p. 44
Quantum Ising model
O−2m O−1m O0m O1m Om2
m
m
m
m
O−3/2
O−1/2
O1/2
O3/2
Minimal double cones: Oim
– p. 45
Quantum Ising model
O−2m O−1m O0m O1m Om2
m
m
m
m
O−3/2
O−1/2
O1/2
O3/2
Double cones: Oi,j , smallest double cone containing
Oim and Ojm
– p. 46
Quantum Ising model
1111
0000
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
+ 2t
1111
0000
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
+ 3t
11111
00000
00000
11111
00000
11111
m11111
m
m
m
m
00000
11111
O−200000
O
O
O
O−1
0000
1111
00000
11111
0
1
2
0000
1111
00000
11111
0000
1111
m
m
m
m
00000
11111
0000
00000
11111
O−3/2 O−1/2 1111
O1/2 O3/2
0000
1111
0000
1111
0000
1111
0000
1111
Net: Km , by integer time translation
– p. 47
‘One-point’ algebras
1111
0000
0000
1111
m
0000
1111
0000
1111
O
0000
1111
0
0000
1111
0000
1111
0000
1111
Linear basis: 1, U0
Minimal projections: P = 12 (1 ± U0 )
Commutation relations:
(
Ui Uj =
−Uj Ui , if |i − j| =
Uj Ui , otherwise
1
2
– p. 48
‘Two-point’ algebras
1111
0000
0000
1111
m
0000
1111
00000
11111
0000
1111
O
00000
11111
0000
1111
0
00000
11111
0000
1111
m
00000
11111
0000
1111
00000
11111
0000
1111
O
00000
11111
1/2
00000
11111
00000
11111
00000
11111
Linear basis: 1, U0 , U 1 , iU0 U 1
2
Minimal projections: P =
2
1
2
−
→
1+ n ·U ,
−
→
n ∈ R3
– p. 49
Dynamics
1111
0000
0000
1111
00000
11111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
00000
11111
0000
1111
00000
11111
0000
000001111
11111
0000
1111
Dynamics: automorphisms of A (Müller, Vecsernyés
2012)
Local primitive causality holds.
– p. 50
Dynamics
1111
0000
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
Dynamics: automorphisms of A (Müller, Vecsernyés
2012)
Local primitive causality holds.
– p. 51
Correlations violating CH
1111
0000
00000
11111
0000
1111
00000
11111
0000
1111
00000
11111
A
00000
11111
0000
1111
Bn
0000
1111
00000
m 11111
00000
11111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
00000
11111
0000
1111
→n
−
−
→
m
Am = A( a ), Bn = B( b ): four projections (m, n = 1, 2)
ρs : singlet state
– p. 52
Correlations violating CH
Directions:
b2
a1
b1
y
a2
x
maximally violating of the CH inequality ...
– p. 53
Correlations violating CHSH
... or, equivalently, the CHSH inequality:
φ U1 (V1 + V2 ) + U2 (V1 − V2 ) 6 2
where
Um := 2Am − 1
Vn := 2Bn − 1
– p. 54
Correlations violating CHSH
Question: Can these four correlations have a
noncommutative joint common causal explanation?
– p. 55
Localization of the common cause
1111
0000
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
00000
11111
A
00000
11111
00000
11111
0000
1111
B11111
0000
1111
0000
m 1111
00000
11111
n
0000
1111
0000
1111
0000
1111
00000
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
00000
11111
0000
1111
11111
00000
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
00000
11111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
C
00000
11111
00000
11111
00000
11111
0000
1111
0000
1111
k
0000
1111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
00000
11111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
00000
11111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
00000
11111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
00001111
1111
00001111
00001111
0000
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
00001111
1111
00001111
00001111
0000
00000
11111
00000
11111
Weak joint common cause system
– p. 56
Localization of the common cause
1111
0000
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
00000
11111
A
00000
11111
00000
11111
0000
1111
Bn
0000
1111
0000
m 1111
00000
11111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
000001111
11111
00000
11111
0000
11111
00000
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
00000
11111
C
00000
11111
00000
11111
0000
1111
k
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
0000
1111
00000
11111
00001111
1111
00001111
0000
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
00001111
1111
00001111
0000
00000
11111
Joint common cause system
– p. 57
Localization of the common cause
1111
0000
0000
1111
0000
1111
0000
1111
0000
1111
0000
1111
A
00000
11111
0000
1111
Bn
0000
1111
0000
m 1111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
11111
00000
00000
11111
00000
11111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
C
00000
11111
k
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
00000
11111
0000
1111
0000
00001111
1111
0000
1111
0000
1111
0000
00001111
1111
0000
1111
Strong joint common cause system
– p. 58
Localization of the common cause
0000
1111
11111
00000
0000
1111
00000
11111
00000
11111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
0000
1111
00000
11111
A
B
m
n
00000
11111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
00000
11111
0000
1111
1111
0000
0000
00000
11111
0000
00000
11111
0000
00001111
1111
0000
1111
000001111
11111
0000
1111
000001111
11111
0000
1111
0000
1111
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
{
}
C
00000
11111
0000
1111
00000
11111
0000
1111
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
0000
1111
00000
11111
0000
1111
k
00000
11111
0000
00000
11111
0000
0000
00000
11111
0000
000001111
11111
0000
1111
000001111
11111
00001111
1111
0000
1111
000001111
11111
0000
1111
Weak joint common cause system: one needs only local
primitivity and isotony (no dynamics)
– p. 59
Localization of the common cause
11111
00000
0000
000001111
11111
0000
1111
00000
11111
0000
00000
11111
m 1111
n
0000
1111
00000
11111
0000
1111
00000
11111
A
B
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
00000
11111
0000 11111
1111
00000
1111
0000
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
0000
1111
00000
11111
00000
11111
t
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
0000
1111
00000
11111
00000
11111
0000
1111
0000
1111
0000
1111
00000
11111
{
}
C
00000
11111
0000
1111
0000
1111
00000
11111
00000
11111
k 11111
00000
11111
0000
1111
0000
1111
00000
00000
11111
00000
11111
0000
1111
0000
1111
00000
11111
00000
11111
000001111
11111
00001111
0000
00000
11111
00000
11111
(Strong) joint common cause system: one needs also
dynamics
– p. 60
Bell inequality in AQFT
A and B : two mutually commuting C ∗ -subalgebras of C
Bell operator for (A, B): R, an element of the set
1
B(A, B) ≡
A1 (B1 + B2 ) + A1 (B1 − B2 ) 2
Ai = A∗i ∈ A; Bi = Bi∗ ∈ B; −1 6 Ai , Bi 6 1
– p. 61
Bell inequality in AQFT
Bell correlation coefficient of a state φ:
β(φ, A, B) ≡ sup |φ(R)| R ∈ B(A, B)
The Bell inequality is violated if
|β(φ, A, B)| > 1
– p. 62
Mathematical results
Proposition: If A and B are C ∗ -algebras then there are
some states violating the Bell inequality for A ⊗ B iff both
A and B are non-abelian (Bacciagaluppi, 1994).
Going over to von Neumann algebras ... (Landau
1987)
Adding further constraints ... (Summer-Werner, 1988;
Halvorson, Clifton, 2000)
The above theorems apply in ”typical” AQFTs ...
– p. 63
Joint common cause system
Joint CCS = local, non-conspiratorial joint CCS
Proof:
Rewriting both the classical and the non-classical local,
non-conspiratorial joint CCS in an indexical form.
’Translating’ quantum probabilities into classical
conditional probabilities by the Kolmogorovian
Censorship Hypothesis.
– p. 64
Non-classical joint common cause system
Correlation:
φ(Am Bn ) 6= φ(Am ) φ(Bn )
Indexical notation:
(φ ◦ Ec )(XCk )
φ(Ck XCk )
φCk (X) :=
=
.
φ(Ck )
φ(Ck )
Non-classical, local, non-conspiratorial joint CCS:
φCk (Am Bn ) = φCk (Am ) φCk (Bn )
φCk (Am ) = φCk (Am Bn ) + φCk (Am Bn⊥ )
φCk (Bn ) = φCk (Am Bn ) + φCk (A⊥
m Bn )
φCk (1) = 1.
– p. 65
Kolmogorovian Censorship Hypothesis
Let (N , P(N ), φ) be a non-classical probability space. Let Γ be a countable set
of non-commuting selfadjoint operators in N . For every Q ∈ Γ, let P(Q) be a
maximal Abelian sublattice of P(N ) containing all the spectral projections of
Q. Finally, let a map p0 : Γ → [0, 1] be such that
X
p0 (Q) = 1,
p0 (Q) > 0.
Q∈Γ
Then there exists a classical probability space (Ω, Σ, p) such that for every
Q
projection X Q in any P(Q) there exist events Xcl
and xQ
cl in Σ such that
Q
Xcl
⊂ xQ
cl
R
xQ
cl ∩ xcl = 0, if Q 6= R
p(xQ
cl ) = p0 (Q)
Q Q
φ(X Q ) = p(Xcl
|xcl )
– p. 66
Classical joint common cause system
Correlation:
p(Am ∧ Bn | am ∧ bn ) 6= p(Am |am ) p(Bn |bn )
Indexical notation:
p(X ∧ Ck |x)
pCk (X|x) :=
.
p(Ck )
Classical, local, non-conspiratorial joint CCS:
pCk (Am ∧ Bn |am ∧ bn ) = pCk (Am |am ∧ bn ) pCk (Bn |am ∧ bn ),
pCk (Am |am ∧ bn ) = pCk (Am |am ∧ bn′ ),
pCk (Bn |am ∧ bn ) = pCk (Bn |am′ ∧ bn ),
pCk (Ω|am ∧ bn ) = 1.
– p. 67
Quantum Ising model
O−1, 1
O1, 2
m
m
m
m
m
O−2 O−1 O0 O1 O2
m
m
m
m
O−3/2 O−1/2 O1/2 O3/2
m , poset of double cones based
Cauchy surface net: KCS
on the Cauchy surface
– p. 68
‘One-point’ algebras
1111
0000
0000
1111
m
0000
1111
0000
1111
O
00000
11111
0000
1111
0000
1111
0
00000
11111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
00000
11111
0000
1111
00000
11111
0000
1111
00000
11111
0000
000001111
11111
0000
1111
Linear basis: 1, U0
Minimal projections: P = 12 (1 ± U0 )
Commutation relations:
(
Ui Uj =
−Uj Ui , if |i − j| =
Uj Ui , otherwise
1
2
– p. 69
‘One-point’ algebras
111111111
00000
0000
1111
0000
00000
11111
0000
1111
0000
1111
00000
11111
0000
0000
1111
m 1111
00000
11111
0000
1111
0000
1111
00000
11111
O
0000
1111
0000
1111
00000
11111
0
0000
1111
0000
1111
00000
11111
0000
1111
0000
1111
00000
11111
0000
00001111
000001111
11111
Linear basis: 1, U0
Minimal projections: P = 12 (1 ± U0 )
Commutation relations:
(
Ui Uj =
−Uj Ui , if |i − j| =
Uj Ui , otherwise
1
2
– p. 70
‘Three-point’ algebras
1111
0000
0000
1111
m
0000
1111
00000
11111
0000
1111
O
00000
11111
00000
11111
0000
1111
0
00000
11111
00000
11111
0000
1111
m11111
m
00000
11111
00000
0000
1111
00000
11111
00000
11111
0000
1111
O
O
00000
11111
00000
−1/211111
1/2
00000
11111
00000
11111
00000
11111
00000
11111
00000
11111
00000
11111
Linear basis:
1, U− 1 , U0 , U 1 , iU− 1 U0 , iU0 U 1 , U− 1 U 1 , U− 1 U0 U 1
2
2
2
2
2
2
2
2
→
Minimal projections: P = P (−
n ),
−
→
n ∈ R3
→
→
Two dimensional projections: P = P (−
n,−
n ′ ),
→
−
→
n,−
n ′ ∈ R3
– p. 71