How local are local operations in local quantum field theory?

doi:10.1016/j.shpsb.2010.09.001

Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics

Volume 41, Issue 4, November 2010, Pages 346-353

https://doi.org/10.1016/j.shpsb.2010.09.001 Get rights and content

Abstract

A notion called operational C⁎-separability of local C*-algebras ( $A (V_{1})$ and $A (V_{2})$ ) associated with spacelike separated spacetime regions V₁ and V₂ in a net of local observable algebras satisfying the standard axioms of local, algebraic relativistic quantum field theory is defined in terms of operations (completely positive unit preserving linear maps) on the local algebras $A (V_{1})$ and $A (V_{2})$ . Operational C*-separability is interpreted as a “no-signaling” condition formulated for general operations, for which a straightforward no-signaling theorem is shown not to hold. By linking operational C*-separability of $(A (V_{1}), A (V_{2}))$ to the recently introduced (Rédei & Summers, forthcoming) operational C*-independence of $(A (V_{1}), A (V_{2}))$ it is shown that operational C*-separability typically holds for the pair $(A (V_{1}), A (V_{2}))$ if V₁ and V₂ are strictly spacelike separated double cone regions. The status in local, algebraic relativistic quantum field theory of a natural strengthening of operational C*-separability, i.e. operational W*-separability, is discussed and open problems about the relation of operational separability and operational independence are formulated.

Section snippets

Aim of paper, its motivation and overview of the main claims

The aim of the paper is to investigate to what extent the quantum mechanical notion of operation is compatible with the concepts of locality and causality, where “locality” and “causality” are understood as expressed by the features of the net of local algebras of observables ${A (V), V \subset M}$ specified in local (algebraic, relativistic) quantum field theory (AQFT) and where “operation” means a completely positive (CP) unit preserving linear map defined on the algebra of local observables. In this

Some notions of algebraic quantum mechanics

Throughout the paper $A$ denotes a unital C*-algebra, $A_{1}, A_{2}$ are assumed to be C*-subalgebras of $A$ (with common unit). $N$ denotes a von Neumann algebra; algebras $N_{1}, N_{2}$ are assumed to be von Neumann subalgebras of $N$ (with common unit). A C*-algebra $A$ is hyperfinite if there exist a series of finite dimensional full matrix algebras M_n (n=1,2,…) such that $\cup_{n} M_{n}$ is dense in $A$ . A W*-algebra $N$ is hyperfinite (or approximately finite dimensional) if there exist a series of finite dimensional full matrix

Algebraic quantum field theory

In AQFT, observables, interpreted as selfadjoint parts of C*-algebras, are assumed to be localized in regions V of the Minkowski spacetime M. The basic object in the mathematical model of a quantum field is thus the association of a C*-algebra $A (V)$ to (open, bounded) regions V of M. ${A (V), V \subset M}$ is called the net of algebras of local observables. The net is specified by imposing on it physically motivated postulates. Below we list these postulates.

(i)
Isotony: $A (V_{1})$ is a C*-subalgebra (with common

Operational C*-separability

In what follows, V₁, V₂ and V are assumed to be open bounded spacetime regions, with V₁ and V₂ spacelike separated and $V_{1}, V_{2} \subseteq V$ . Let T be an operation on $A (V)$ and $ϕ$ be a state on $A (V)$ . Then $(A (V), A (V_{1}), A (V_{2}), ϕ, T)$ is called a local system.

Given such a local system, let $ϕ_{1}$ and $ϕ_{2}$ be the restrictions of $ϕ$ to $A (V_{1})$ and $A (V_{2})$ , respectively. Suppose T₁ is an operation on $A (V_{1})$ . Carrying out this operation changes the state $ϕ_{1}$ into $T_{1}^{*} ϕ_{1}$ . By the requirement of isotony $A (V_{1})$ is a subalgebra of $A (V)$ , so

Operational C-separability and operational C-independence

The next definition formulates the idea of operational C*-independence, this definition was proposed in Rédei and Summers (forthcoming):

Definition 4

A pair $(A_{1}, A_{2})$ of C*-subalgebras of C*-algebra $A$ is operationally C*-independent in $A$ if any two operations on $A_{1}$ and $A_{2}$ , respectively, have a joint extension to an operation on $A$ ; i.e. if for any two completely positive unit preserving maps $T_{1} : A_{1} \to A_{1}$ $T_{2} : A_{2} \to A_{2}$ there exists a completely positive unit preserving map $T : A \to A$ such that $T (X) = T_{1} (X) for all X \in A_{1}$ $T (Y) = T_{2} (Y) for$

Operational W*-separability

In the category of von Neumann algebras both states and operations can have additional continuity properties: One can consider normal states and normal operations and define operational W*-separability naturally:

Definition 6

A pair $(N_{1}, N_{2})$ of W*-subalgebras of W*-algebra $N$ is operationally W*-separable in $N$ if every normal operation T₁ on $A_{1}$ that has an extension to a normal operation on $N$ also has an extension to a normal operation on $N$ which is the identity map on $N_{2}$ , and every normal operation T₂ on $A_{2}$

Closing comments

We have seen (Proposition 6) that the local commutativity (Einstein causality or microcausality) postulate in AQFT does not exclude violation of operational separatedness; in other words, a straightforward no-signaling theorem does not hold for general operations. As indicated in the introductory section, one can react to this situation of “locality violation” by operations in two ways: (i) to declare unphysical all operations T for which the system is not operationally separated, or (ii) to

Acknowledgements

Miklos Redei's work is supported in part by the Hungarian Scientific Research Found (OTKA), contract no. K68043. Giovanni Valente's work is supported by the National Science Foundation (NSF), Grant no. 0749856.

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How local are local operations in local quantum field theory?

Abstract

Section snippets

Aim of paper, its motivation and overview of the main claims

Some notions of algebraic quantum mechanics

Algebraic quantum field theory

Operational C*-separability

Operational C*-separability and operational C*-independence

Operational W*-separability

Closing comments

Acknowledgements

Journal of Functional Analysis

Studies in History and Philosophy of Modern Physics

Studies in History and Philosophy of Modern Physics

Subalgebras of C*-algebras

Acta Mathematica

The universal structure of local algebras

Communications in Mathematical Physics

Classification of injective factors. Cases II1, II∞, IIIλ, λ≠1

The Annals of Mathematics

Non-commutative geometry

Operational C-separability and operational C-independence

Classification of injective factors. Cases II₁, ${II}_{\infty}$ , ${III}_{λ}$ , $λ \neq 1$