The axiomatic formulation of an Algebraic Relativistic Quantum Field Theory leads naturally -via the Haag-Kastler axioms- to the fact that operators representing observables generate a von Neumann algebra localized in a space-time region in which said observations are made. I present and explain the axioms, as well as the mathematical preliminaries needed to understand them and apply them to the so-called causal diamond and Fermi's Two-Atom System. The latter shows the importance of formulating axioms concerning causality that try to make compatible the requirement from Special Relativity according to which no causal influence can travel faster than the speed of light and the quantum mechanical theory of measurement and preparability of states. For a relativistic field-theoretic setting, these concepts must be re-formulated, and this reasoning suggests that the operator algebra must be a type III von Neumann factor. I also introduce some philosophical problems concerning our usual notion of causality from Hume to contemporary theories of counterfactual causation and show how the algebraic structure of the theory is deeply related to the causal one and how this theory can be extended if we change the way we think about causality, measurements, and local preparability of states.The axiomatic formulation of an Algebraic Relativistic Quantum Field Theory leads naturally -via the Haag-Kastler axioms- to the fact that operators representing observables generate a von Neumann algebra localized in a space-time region in which said observations are made. I present and explain the axioms, as well as the mathematical preliminaries needed to understand them and apply them to the so-called causal diamond and Fermi's Two-Atom System. The latter shows the importance of formulating axioms concerning causality that try to make compatible the requirement from Special Relativity according to which no causal influence can travel faster than the speed of light and the quantum mechanical theory of measurement and preparability of states. For a relativistic field-theoretic setting, these concepts must be re-formulated, and this reasoning suggests that the operator algebra must be a type III von Neumann factor. I also introduce some philosophical problems concerning our usual notion of causality from Hume to contemporary theories of counterfactual causation and show how the algebraic structure of the theory is deeply related to the causal one and how this theory can be extended if we change the way we think about causality, measurements, and local preparability of states.

Jan 1, 20190 citations

The Algebra of Observables

Jean-Louis Basdevant

Graduate texts in physicsJan 1, 20230 citations

The C* Axioms and the Phase Space Fomalism of Quantum Mechanics

Franklin E. Schroeck

In 1972, “Algebraic Methods in Statistical Mechanics and Quantum Field Theory” was published by Gerard Emch [8], giving the axioms for a physical system in an algebraic setting using the language of Irving Segal [19], and then continuing to obtain the C*-algebraic formalism for a physical system. Of these axioms, only the fifth contained an assumption that was questionable in its physical content. Bearing in mind that for each observable A and state φ , one obtains a distribution of values for the observed results of measurement, we have: Axiom 5: For any element A in the set of observables $\mathfrak{A}$ and any non-negative integer n , there is at least one element, denoted A n , in $\mathfrak{A}$ such that (i) the set of dispersion-free states for A n is contained in the set of dispersion-free states for A, (ii) < φ ; A n > = < φ ; A > n for all φ in the set of dispersion-free states for A . (Here < φ ; B > is the expectation of B in state φ .)In 1972, “Algebraic Methods in Statistical Mechanics and Quantum Field Theory” was published by Gerard Emch [8], giving the axioms for a physical system in an algebraic setting using the language of Irving Segal [19], and then continuing to obtain the C*-algebraic formalism for a physical system. Of these axioms, only the fifth contained an assumption that was questionable in its physical content. Bearing in mind that for each observable A and state φ , one obtains a distribution of values for the observed results of measurement, we have: Axiom 5: For any element A in the set of observables $\mathfrak{A}$ and any non-negative integer n , there is at least one element, denoted A n , in $\mathfrak{A}$ such that (i) the set of dispersion-free states for A n is contained in the set of dispersion-free states for A, (ii) < φ ; A n > = < φ ; A > n for all φ in the set of dispersion-free states for A . (Here < φ ; B > is the expectation of B in state φ .)

Hindustan Book AgencyJan 1, 20073 citations

The Algebra of Observables

M. D. Maia

Springer eBooksJan 1, 20110 citations

The Axioms of the Theory of Local Observables

Lecture notes in physics. M, MonographsNov 28, 20080 citations

Operator Algebraic Aspects for Sufficiency

Makato Tsukada

Sufficiency is one of the most important concepts in mathematical statistics. In the measure theoretic context, the sufficiency aspect is specified with a measurable space. This chapter discusses an axiom of set theory, that is, existence of measurable cardinality. It provides a specific definition of sufficiency in the view of operator algebras and applies it to the theory of Gibbs states on countable sets. The chapter discusses the set of all countably additive bounded complex-valued measures. It also discusses isometric isomorphism in the context of sufficiency. The chapter provides a list of experiments and observables for describing the sufficiency concept.Sufficiency is one of the most important concepts in mathematical statistics. In the measure theoretic context, the sufficiency aspect is specified with a measurable space. This chapter discusses an axiom of set theory, that is, existence of measurable cardinality. It provides a specific definition of sufficiency in the view of operator algebras and applies it to the theory of Gibbs states on countable sets. The chapter discusses the set of all countably additive bounded complex-valued measures. It also discusses isometric isomorphism in the context of sufficiency. The chapter provides a list of experiments and observables for describing the sufficiency concept.

CRC Press eBooksSep 17, 20200 citations

\mathfrak{A}

and any non-negative integer n , there is at least one element, denoted A n , in

\mathfrak{A}

such that (i) the set of dispersion-free states for A n is contained in the set of dispersion-free states for A, (ii) < φ ; A n > = < φ ; A > n for all φ in the set of dispersion-free states for A . (Here < φ ; B > is the expectation of B in state φ .)