Abstract
We generalize the operadic approach to algebraic quantum field theory (arXiv:1709.08657) to a broader class of field theories whose observables on a spacetime are algebras over any single-colored operad. A novel feature of our framework is that it gives rise to adjunctions between different types of field theories. As an interesting example, we study an adjunction whose left adjoint describes the quantization of linear field theories. We also analyze homotopical properties of the linear quantization adjunction for chain complex valued field theories, which leads to a homotopically meaningful quantization prescription for linear gauge theories.
Highlights
Introduction and summaryAlgebraic quantum field theory is a well-established and successful framework to axiomatize and investigate quantum field theories on the Minkowski spacetime and on more general Lorentzian manifolds, see e.g. [11,25] for overviews
In locally covariant algebraic quantum field theory [13,18], C = Loc is the category of globally hyperbolic Lorentzian manifolds with morphisms given by causal isometric embeddings and the physical axioms are Einstein causality and the time-slice axiom
Einstein causality demands that any two observables, i.e. elements of the algebras assigned by A, that are causally disjoint commute with each other, which encodes the idea that two measurements in causally disjoint spacetime regions do not influence each other
Summary
Algebraic quantum field theory is a well-established and successful framework to axiomatize and investigate quantum field theories on the Minkowski spacetime and on more general Lorentzian manifolds, see e.g. [11,25] for overviews. 4, we harness this functorial behavior in order to study adjunctions between the categories of field theories corresponding to different C and P(r1,r2) This includes generalizations of the time-slicification and local-to-global adjunctions from [8], which have already found interesting applications to quantum field theory on spacetimes with boundaries [4]. A novel feature of our framework, which is not captured by [8], is a second kind of functorial assignment P(r1,r2) → P(r1,r2) of our colored operads to bipointed single-colored operads This results in adjunctions between the categories of field theories of different types. We conclude by analyzing in some detail the interplay between our (derived) linear quantization functor and suitable homotopical generalizations of the time-slice axiom and localto-global property of gauge theories
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