Einstein Meets von Neumann: Locality and Operational Independence in Algebraic Quantum Field Theory

Local Operations and Completely Positive Maps in Algebraic Quantum Field Theory

“…Recently, in algebraic quantum field theory, Rédei [15,17] captured the idea of the locality principle by the notion of operational separability (Definition 10), which had been introduced by Rédei and Valente [19]. The reason why he adopts the formalism of algebraic quantum field theory is that Einstein [3] says that physical things are conceived of as being arranged in a space-time continuum, and that observables in algebraic quantum field theory are 'explicitly regarded as localized in regions of the space-time continuum' [15, p.1045].…”

A categorial approach to relativistic locality

“…It should be clear now how the notion of quantum field theory as a covariant functor captures the crucial components of what could be called a "field theoretical paradigm", which was informally articulated by Einstein in his critique of standard, non-relativistic quantum mechanics of finite degrees of freedom (see [5] and [6] for a more detailed discussion of this historical aspect from the perspective of the less, general, non-categorially formulated algebraic approach to quantum field theory): Physical systems represented by the observables that one can measure on them are always considered as "located somewhere" in spacetime, and their association with particular spacetime regions is in harmony with the causal structure of spacetimes in the spirit of the theory of (general) relativity. In short: Categorial quantum field theory is a mathematically precise general specification of the field theoretical paradigm, no matter whether the spacetimes are flat or not.…”

“…an extension of T 0 from F(ψ)F(M ) to a morphism on F(M ) may not exist. This is the case, for instance, if one takes the operations (completely positive, unit preserving linear maps, see section 7) as morphisms: Operations defined on sub-C * -algebras of C * -algebras need not be extendable from the subalgebra to the superalgebra [33], and this complicates assessment of the status of operational independence in quantum field theory (see [34], [6], [32], [35] for further discussion of this point. )…”

“…Second, as Howard argued by reconstructing the gradual changes in Einstein's views about quantum mechanics and "relativistic locality" in the years 1935-1949 [3], [4], Einstein's worry about ordinary quantum mechanics of finite degrees of freedom was not so much about completeness of the theory; rather, the worry was about the theory not being field theoretical: Einstein thought that quantum mechanics did not fit into a field theoretical paradigm (see [5] and [6] for further details of the analysis of Einstein's views from this perspective). Thus it is natural to ask if (relativistic) quantum field theory itself is "relativistically local"?…”