## Local primitive causality and the common cause principle in quantum field theory (2002)

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Venue: | FOUND. PHYS |

Citations: | 9 - 5 self |

### BibTeX

@ARTICLE{Rédei02localprimitive,

author = {Miklós Rédei and Stephen J. Summers},

title = {Local primitive causality and the common cause principle in quantum field theory },

journal = {FOUND. PHYS},

year = {2002},

volume = {32},

number = {3},

pages = {335--355}

}

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### Abstract

If {A(V)} is a net of local von Neumann algebras satisfying standard axioms of algebraic relativistic quantum field theory and V 1 and V 2 are spacelike separated spacetime regions, then the system (A(V 1), A(V 2), f) is said to satisfy the Weak Reichenbach’s Common Cause Principle iff for every pair of projections A ¥ A(V 1), B ¥ A(V 2) correlated in the normal state f there exists a projection C belonging to a von Neumann algebra associated with a spacetime region V contained in the union of the backward light cones of V 1 and V 2 and disjoint from both V 1 and V 2, a projection having the properties of a Reichenbachian common cause of the correlation between A and B. It is shown that if the net has the local primitive causality property then every local system (A(V 1), A(V 2), f) with a locally normal and locally faithful state f and suitable bounded V 1 and V 2 satisfies the

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Citation Context ...restrictive. We formulate our final assumption. For a convex spacetime region V , let V ′′ = (V ′ ) ′ denote the causal completion (also called causal closure and causal hull in the literature) 2 cf. =-=[16]-=- for a description of types of von Neumann algebras 4of V . One notes that every light ray running through any given point in V ′′ must intersect V . One should also note that V = V ′′ for every doub... |

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Citation Context ...s are always signs of causal relations. More precisely, this position is typically formulated in the form of what has become known as Reichenbach’s Common Cause Principle. This principle asserts (cf. =-=[24]-=-) that if two events A and B are correlated, then the correlation between A and B is either due to a direct causal influence connecting A and B, or there is a third event C which is a common cause of ... |

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Citation Context ...(X|Y ) denotes here the conditional probability of X on condition Y , and it is assumed that none of the probabilities p(X), (X = A,B,C) is equal to zero. The above definition is due to Reichenbach ( =-=[22]-=-, Section 19). We wish to extend this definition to the setting of AQFT. To do this, we first define a notion of common cause of a correlation in a noncommutative measure space (P(N),φ) with a non-dis... |

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Citation Context ...ies the Weak Reichenbach’s Common Cause Principle. 1 Introduction An operationally motivated and mathematically powerful approach to quantum field theory is algebraic quantum field theory (AQFT) (cf. =-=[7]-=-). Although in its axiomatic nature ∗ e-mail: redei@ludens.elte.hu † e-mail: sjs@math.ufl.edu 1it includes models which have no physical significance at all, it does subsume all physically interestin... |

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Citation Context ...Neumann algebra containing both N 1, and N 2. 8 Only normal states are considered here, since they have the continuity property which generalizes the property of s-additivity in the classical context =-=(21)-=-—see below. 9 In point of fact, Halvorson and Clifton required of N1 and N 2 that they be of infinite type, not the more restrictive type III, but this is the version which we shall employ below. See,... |

61 |
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Citation Context ... defined by b(f, N 1, N 2) — sup 1 2 f(X 1(Y 1+Y 2)+X 2(Y 1 −Y 2)) (2) where the supremum in (2) is taken over all self-adjoint X i ¥ N 1, Y j ¥ N 2 with norm less than or equal to 1. It can be shown =-=(6, 32)-=- that b(f, N 1, N 2) [ `2. The Clauser–Holt– Shimony–Horne version of Bell’s inequality in this notation reads: b(f, N 1, N 2) [ 1 (3) 7 If N1, N 2 do not mutually commute, then C*- independence is st... |

44 |
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Citation Context ...nvex subsets of M, such as the wedges discussed immediately below. 5 For the origin and a detailed analysis of the interrelation of these and other notions of statistical independence, see the review =-=(30)-=- and Chap. 11 in Ref. 26—for more recent results, see Refs. 7 and 15. 6 See precise statements in Ref. 30.342 Rédei and Summers A pair (A1, A2) of such algebras is called C*-independent if for any st... |

43 | Scaling algebras and renormalization group in algebraic quantum field theory
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Citation Context ...f have a nontrivial scaling limit, either in the sense of Fredenhagen (9) (which assumes the existence of Wightman field operators locally affiliated to the net 3 )orinthe sense of Buchholz and Verch =-=(4)-=- (which is a bit more technically involved, but which makes no appeal to affiliated Wightman fields and is more general). Heuristically, if one considers the dilations on M, dl(x)=lx, l >0, and if one... |

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Citation Context ...(5)–(8)) hold: f(A N B N C) f(C) f(A N B N C +) N C) =f(A f(C) f(C +) =f( N C +) f(C +) f(A N C) f(C) > f(A N C +) f(C +) f(B N C) f(C) > f(B N C +) f(C +) f(B N C) f(C) f(B N C + ) f(C + ) (10) (11) =-=(12)-=- (13) Definition 3 is a natural specification in a noncommutative probability space (P(N), f) of the classical notion of common cause. The only deviation from Definition 2 is that the commutativity of... |

37 |
Postulates of quantum field theory
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Citation Context ...8)) hold: f(A N B N C) f(C) f(A N B N C +) N C) =f(A f(C) f(C +) =f( N C +) f(C +) f(A N C) f(C) > f(A N C +) f(C +) f(B N C) f(C) > f(B N C +) f(C +) f(B N C) f(C) f(B N C + ) f(C + ) (10) (11) (12) =-=(13)-=- Definition 3 is a natural specification in a noncommutative probability space (P(N), f) of the classical notion of common cause. The only deviation from Definition 2 is that the commutativity of the ... |

35 |
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Citation Context ..., however, can be said of the352 Rédei and Summers location of the specific common cause C displayed in the proof of Proposition 3. Define V ˜ 1 and V ˜ 2 by V˜ 1 — (BLC(V1) 5 V)0(BLC(V1) 5 BLC(V2)) =-=(20)-=- V˜ 2 — (BLC(V2) 5 V)0(BLC(V1) 5 BLC(V2)) (21) (i.e., V ˜ 1 and V ˜ 2 are the parts of V that are in the backward light cones of V 1 and V2, respectively, but do not intersect with the part of V which... |

35 |
Quantum Logic in Algebraic Approach
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(Show Context)
Citation Context ...ically independent if A N B ] 0 for any projections 0 ] A ¥ N1, 0 ] B ¥ N2.IfN1, N2 is a mutually commuting pair of von Neumann algebras, then C*-independence and logical independence are equivalent. =-=(26)-=- 7 In light of our preceding remarks, we conclude: Lemma 1. Assumptions (i)–(v) entail that the pair (A(V1), A(V2)) is logically independent for any spacelike separated double cones V1, V2. Let V 1 an... |

33 |
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Citation Context ... Y j ¥ N 2 with norm less than or equal to 1. It can be shown (6, 32) that b(f, N 1, N 2) [ `2. The Clauser–Holt– Shimony–Horne version of Bell’s inequality in this notation reads: b(f, N 1, N 2) [ 1 =-=(3)-=- 7 If N1, N 2 do not mutually commute, then C*- independence is strictly weaker than logical independence. (15)Local Primitive Causality in Quantum Field Theory 343 and a state f for which b(f, N 1, ... |

22 |
Maximal violation of Bell’s inequalities for algebras of observables in tangent spacetime regions. Annales de l’Institut Henri Poincaré – Physique théorique
- Summers, Werner
- 1988
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Citation Context ...ation of Bell’s inequalities (cf. Refs. 32–34). Indeed, if V 1 and V 2 are tangent spacelike separated wedges or double cones, there is maximal violation of Bell’s inequalities in every normal state. =-=(34)-=- We refer the interested reader to those papers. To keep the discussion as simple as possible, we shall only discuss the recent results established by Halvorson and Clifton. 9 The symbol N 1 K N 2 den... |

22 |
Maximal violation of Bell’s inequalities is generic in quantum field theory
- Summers, Werner
- 1987
(Show Context)
Citation Context ...B), then also φ((1 − A) ∧ (1 − B)) > φ(1 − A)φ(1 − B). □ There are many situations in which β(φ, A(V1), A(V2)) = √ 2, in other words, where there is maximal violation of Bell’s inequalities (cf. [27] =-=[28]-=- [29]). Indeed, if V1 and V2 are tangent spacelike separated wedges or double cones, there is maximal violation of Bell’s inequalities in every normal state [29]. We refer the interested reader to tho... |

22 | Independence of local algebras in quantum field theory
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(Show Context)
Citation Context ...ble cones are C*-independent, since, as pointed out above, they form a mutually commuting pair of algebras satisfying the Schlieder property, which in this context is equivalent with C*-independence. =-=(28)-=- Two von Neumann subalgebras N1, N2 of the von Neumann algebra (23, 24) N are called logically independent if A N B ] 0 for any projections 0 ] A ¥ N1, 0 ] B ¥ N2.IfN1, N2 is a mutually commuting pair... |

18 | On the statistical independence of algebras of observables
- Florig, Summers
(Show Context)
Citation Context ...on 2. C ¥ W is a common cause of the correlation (4) if the following (independent) conditions hold: p(A N B | C)=p(A | C) p(B | C) (5) p(A N B|C + )=p(A | C + ) p(B | C + ) (6) p(A | C) > p(A | C +) =-=(7)-=- p(B | C) > p(B | C +) (8) where p(X | Y) denotes here the conditional probability of X on condition Y, and it is assumed that none of the probabilities p(X), (X=A, B, C) is equal to zero. The above d... |

16 |
On the modular structure of local algebras of observables
- Fredenhagen
- 1985
(Show Context)
Citation Context ...s in the vacuum representation will be locally faithful.340 Rédei and Summers We shall also assume that the net {A(V)} and state f have a nontrivial scaling limit, either in the sense of Fredenhagen =-=(9)-=- (which assumes the existence of Wightman field operators locally affiliated to the net 3 )orinthe sense of Buchholz and Verch (4) (which is a bit more technically involved, but which makes no appeal ... |

16 | Generic Bell correlation between arbitrary local algebras in quantum field theory
- Halvorson, Clifton
(Show Context)
Citation Context ...pe, not the more restrictive type III, but this is the version which we shall employ below. See, e.g., Ref. 21 for a description of types of von Neumann algebras.344 Rédei and Summers Proposition 2. =-=(14)-=- If (N1, N2) is a pair of commuting type III von Neumann algebras acting on the Hilbert space H and having the Schlieder property, then the set of unit vectors which induce Bell correlated states on N... |

16 | On Reichenbach’s common cause principle and Reichenbach’s notion of common cause”. British Journal for the Philosophy of Science 50
- Hofer-Szabo, Redei, et al.
- 1999
(Show Context)
Citation Context ... of Lemma 3. Note first that if A and B are correlated, then one must have 1−f(A K B) > 0. If, on the contrary, one had f(1)=1= f(A K B), then since 1=A K B+A + N B + , one would have f(A + N B + )=0 =-=(18)-=- On the other hand, since A and B are correlated, it follows that f(A + N B + )>f(A + ) f(B + ) 12 This means that if N1 … N1 1 and N 2 … N1 2, then the logical independence of the pair (N1 1, N1 2) i... |

16 |
Lyapunov theorems for operator algebras
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(Show Context)
Citation Context ...es V1, V2. Let V 1 and V 2 be two spacelike separated spacetime regions and A ¥ A(V 1) and B ¥ A(V 2) be two projections. If f is a state on A(V 1 2 V 2), then it can happen that f(A N B) > f(A) f(B) =-=(1)-=- If (1) is the case, then we say that there is superluminal (or spacelike) correlation between A and B in the state f. We now explain why such correlations are common when assumptions (i)–(v) hold. Th... |

14 | Reichenbach’s common cause principle and quantum correlations
- Rédei
- 2002
(Show Context)
Citation Context ...tained in the intersection of the backward light cones of V1 and V2. This requirement and the resulting notion of Reichenbachian common cause in AQFT (Definition 4) was formulated in a previous paper =-=(25)-=- (see also Chap. 8 in Ref. 26), together with the problem of whether AQFT complies with Reichenbach’s Common Cause Principle, as described. This problem is still open; the present paper does not settl... |

14 | On the causal structure of Minkowski space-time - Thomas, Wichmann - 1997 |

14 |
Bell’s inequalities and quantum field theory
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Citation Context ... defined by b(f, N 1, N 2) — sup 1 2 f(X 1(Y 1+Y 2)+X 2(Y 1 −Y 2)) (2) where the supremum in (2) is taken over all self-adjoint X i ¥ N 1, Y j ¥ N 2 with norm less than or equal to 1. It can be shown =-=(6, 32)-=- that b(f, N 1, N 2) [ `2. The Clauser–Holt– Shimony–Horne version of Bell’s inequality in this notation reads: b(f, N 1, N 2) [ 1 (3) 7 If N1, N 2 do not mutually commute, then C*- independence is st... |

12 |
On the Vacuum State in Quantum Field Theory
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Citation Context ...n require infinite energy, it is evident that (convex combinations of ) states induced by such analytic vectors include all of the physically interesting states in this representation. Proposition 1. =-=(2)-=- Under the assumptions (i)–(v), for any nonempty open region V, the set of vectors A(V) F is dense in H, for all vectors F which are analytic for the energy. Note that assumption (ii) entails that suc... |

9 | Local Quantum Theory - Haag - 1996 |

9 |
Fraassen. The charybdis of realism: Epistemological implicaltions of Bell's inequality
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- 1982
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Citation Context ...use of the correlation (4) if the following (independent) conditions hold: p(A N B | C)=p(A | C) p(B | C) (5) p(A N B|C + )=p(A | C + ) p(B | C + ) (6) p(A | C) > p(A | C +) (7) p(B | C) > p(B | C +) =-=(8)-=- where p(X | Y) denotes here the conditional probability of X on condition Y, and it is assumed that none of the probabilities p(X), (X=A, B, C) is equal to zero. The above definition is due to Reiche... |

9 | The vacuum violates Bell’s inequalities, Phys - Summers, Werner - 1985 |

7 | Logically independent von Neumann lattices
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(Show Context)
Citation Context ...m a mutually commuting pair of algebras satisfying the Schlieder property, which in this context is equivalent with C*-independence. (28) Two von Neumann subalgebras N1, N2 of the von Neumann algebra =-=(23, 24)-=- N are called logically independent if A N B ] 0 for any projections 0 ] A ¥ N1, 0 ] B ¥ N2.IfN1, N2 is a mutually commuting pair of von Neumann algebras, then C*-independence and logical independence... |

7 |
Statistical independence of operator algebras
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(Show Context)
Citation Context ...lt– Shimony–Horne version of Bell’s inequality in this notation reads: b(f, N 1, N 2) [ 1 (3) 7 If N1, N 2 do not mutually commute, then C*- independence is strictly weaker than logical independence. =-=(15)-=-Local Primitive Causality in Quantum Field Theory 343 and a state f for which b(f, N 1, N 2)>1 is called Bell correlated. It is known (32) that if N 1 or N 2 is abelian, or if f is a product state ac... |

6 |
The connexion of duality and causality properties for generalized free fields
- Garber
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(Show Context)
Citation Context ...hich does not follow from assumptions (i)–(vi). Indeed, there exist nets associated with certain generalized free fields which satisfy conditions (i)–(vi) but which violate local primitive causality. =-=(10)-=- However, this condition has been verified in many concrete models. For further insight into the content of local primitive causality, see the discussion directly after the proof of Proposition 3 belo... |

6 | Common cause completability of classical and quantum probability spaces
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(Show Context)
Citation Context ...A N B) — b>0. Elementary algebraic calculation shows that the inequality is equivalent to the inequality But and f(A N B) − f(A) f(B) < f(A N B) 1−f(A K B) f(A) f(B) > [f(A)+f(B) − f(A N B)] f(A N B) =-=(19)-=- f(A) f(B)=f(A)[b+f(A N B)] [f(A)+f(B) − f(A N B)] f(A N B)=[f(A)+b] f(A N B) Since f(A) b > bf(A N B), inequality (19) follows. Therefore, the right hand side of (17)—and hence the value of f(C)—is s... |

6 |
Causality between preparation and registration processes in relativistic quantum theory
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- 1983
(Show Context)
Citation Context ...ace here? It is not widely understood that an “observable” A does not represent a unique measuring apparatus in some fixed laboratory, but rather represents an equivalence class of such apparata (cf. =-=[17]-=-). Consider two such idealized apparata X,Y such that φ(X) = φ(Y ) for all (idealized) states φ admitted in the theory (the set of such states contains as a subset — at least in principle — all states... |

6 | Logical independence in quantum logic
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(Show Context)
Citation Context ...m a mutually commuting pair of algebras satisfying the Schlieder property, which in this context is equivalent with C*-independence. (28) Two von Neumann subalgebras N1, N2 of the von Neumann algebra =-=(23, 24)-=- N are called logically independent if A N B ] 0 for any projections 0 ] A ¥ N1, 0 ] B ¥ N2.IfN1, N2 is a mutually commuting pair of von Neumann algebras, then C*-independence and logical independence... |

6 |
The vacuum violates Bell’s inequalities
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(Show Context)
Citation Context ... precise statements in [25] 5 If N1, N2 do not mutually commute, then C ∗ –independence is strictly weaker than logical independence [10]. 5von Neumann algebra N and let φ be a state on N. Following =-=[26]-=-, the Bell correlation β(φ, N1, N2) between the algebras N1, N2 in state φ is defined by β(φ, N1, N2) ≡ sup 1 2 φ(X1(Y1 + Y2) + X2(Y1 − Y2)) , (2) where the supremum in (2) is taken over all self–adjo... |

4 | On the independence of local algebras in quantum theory - Summers - 1990 |

4 | Maximal violation of Bell’s inequalities is generic - Summers, Werner - 1987 |

3 |
The formal existence and uniqueness of the Reichenbachian common cause on Hilbert lattices
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(Show Context)
Citation Context ...s N1, N2 which are logically independent. Let A ¥ N1 and B ¥ N2 be projections satisfying (15). Then a sufficient condition for C to satisfy (10)–(13) is that the following two conditions hold: C<ANB =-=(16)-=- f(A N B) − f(A) f(B) f(C)= 1−f(A K B) (17) Lemma 4. Let N be a type III von Neumann algebra on a separable Hilbert space H, and let f be a faithful normal state on N. Then for every projection A ¥ P(... |

3 |
A space-time approach to the Bell inequality, in Philosophical Consequences of Quantum Theory
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- 1989
(Show Context)
Citation Context ... the events A and B are said to be (positively) correlated. Definition 2. C ¥ W is a common cause of the correlation (4) if the following (independent) conditions hold: p(A N B | C)=p(A | C) p(B | C) =-=(5)-=- p(A N B|C + )=p(A | C + ) p(B | C + ) (6) p(A | C) > p(A | C +) (7) p(B | C) > p(B | C +) (8) where p(X | Y) denotes here the conditional probability of X on condition Y, and it is assumed that none ... |

3 |
Reichenbach’s common cause definition on Hilbert lattices
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- 1998
(Show Context)
Citation Context ...et A ¥ N1 and B ¥ N2 be projections satisfying (15). Then a sufficient condition for C to satisfy (10)–(13) is that the following two conditions hold: C<ANB (16) f(A N B) − f(A) f(B) f(C)= 1−f(A K B) =-=(17)-=- Lemma 4. Let N be a type III von Neumann algebra on a separable Hilbert space H, and let f be a faithful normal state on N. Then for every projection A ¥ P(N) and every positive real number 0<r<f(A) ... |

3 | The Direction of Time (University of California - Reichenbach - 1956 |

2 | On the vacuum state in quantum theory - Borchers - 1965 |

2 |
Reichenbachian common cause closed event structures (in preparation
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(Show Context)
Citation Context ...s to (5)–(8)) hold: f(A N B N C) f(C) f(A N B N C +) N C) =f(A f(C) f(C +) =f( N C +) f(C +) f(A N C) f(C) > f(A N C +) f(C +) f(B N C) f(C) > f(B N C +) f(C +) f(B N C) f(C) f(B N C + ) f(C + ) (10) =-=(11)-=- (12) (13) Definition 3 is a natural specification in a noncommutative probability space (P(N), f) of the classical notion of common cause. The only deviation from Definition 2 is that the commutativi... |

2 | Independence of local algebras in quantum theory - Roos - 1970 |

2 | Bell's inequalities and quantum theory - Summers, Werner - 1987 |

1 |
Statistical independence of operator algebras, Annales de l'Institut Henri Poincare { Physique theorique
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(Show Context)
Citation Context ...the 3 For the origin and a detailed analysis of the interrelation of these and other notions of statistical independence, see the review [25] and Chapter 11 in [21] — for more recent results, see [3] =-=[10]-=-. 4 see precise statements in [25] 5 If N1, N2 do not mutually commute, then C ∗ –independence is strictly weaker than logical independence [10]. 5von Neumann algebra N and let φ be a state on N. Fol... |

1 | Reichenbach's common cause de on Hilbert lattices - Hofer-Szabo - 1998 |

1 | Bell's inequalities and quantum theory,I, General setting - Summers, Werner - 1987 |

1 | A space{time approach to the Bell inequality, in: Philosophical Consequences of Quantum Theory, edited by - Butter - 1989 |