Algebraic field theory operads and linear quantization

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 singlecolored 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.Keywords: algebraic quantum field theory, locally covariant quantum field theory, colored operads, universal constructions, gauge theory, model categories MSC 2010: 81Txx, 18D50, 18G55 F (G)/(r 1 = r 2 ) (2.5) in Op C (M). Example 2.6. Consider for the moment M = Set. The associative operad As ∈ Op { * } (Set) is the single-colored operad (i.e. C = { * } is a singleton) presented by the following generators and relations: We define the set of generators of arity n by G(n) :=      {η} , for n = 0 , {µ} , for n = 2 , ∅ , else , (2.6) for all n ≥ 0. The generator µ in arity 2 is interpreted as a multiplication operation and the generator η in arity 0 as a unit element. To implement associativity and left/right unitality of these operations, we consider R(n) :=      {λ, ρ} , for n = 1 , {a} , for n = 3 , ∅ , else , (2.7)

Coloring Operads for Algebraic Field Theory

“…We again omit the proof, referring to []. Note that this is a refinement of the previous lemma , and the proof is similar: we have to additionally note that our taking the quotient exactly enforces Einstein causality.…”

“…For the proof we refer to []. Here, we will note that an algebra A over assigns to a color an object , which is how the corresponding functor acts on objects.…”

“…Extra assumptions are necessary to show that maps to W ‐constant theories and that preserves W ‐constant field theories (including once more that ). We again refer for [] for details.…”

Higher Structures in Algebraic Quantum Field Theory

“…Definition ) has to be adapted in order to be homotopically meaningful. Following, we say that is homotopy j ‐local if the corresponding component of the derived counit is a weak equivalence. It is easy to prove that the derived extension of every is homotopy j ‐local.…”

“…Our concept of W ‐constancy from Corollary has to be adapted in order to be homotopically meaningful. Following, we say that is homotopy W ‐constant if the corresponding component of the derived unit is a weak equivalence. Note that homotopy W ‐constancy can be interpreted as a homotopy theoretic generalization of the time‐slice axiom.…”