A Cohomological Perspective on Algebraic Quantum Field Theory

We provide a direct combinatorial formula for the interacting star product in perturbative Algebraic Quantum Field Theory (pAQFT). Our expression is non-perturbative in the coupling constant and is well defined on regular observables if the interaction is also regular. Under the same assumptions, we also show that the quantum Møller operator exists non-perturbatively (in the coupling constant) provided that the classical Møller operator can be constructed exactly. This provides a first step towards understanding how pAQFT can be formulated such that the only formal parameter is , while the coupling constant can be treated as a number.In the introductory part of the paper, apart from reviewing the framework, we make precise several statements present in the pAQFT literature and recast these in the language of (formal) deformation quantization. Finally, we use our formalism to streamline the proof of perturbative agreement provided by Drago, Hack, and Pinamonti and to generalize some of the results obtained in that work to the case of a non-linear interaction. The space of field configurationsWe start with a globally hyperbolic spacetime M = (M, g). Next we introduce E, the space of field configurations. The choice of E specifies what kind of objects our model describes (e.g., scalar fields, gauge fields, etc.). Definition 2.1. The configuration space E on the fixed spacetime M = (M, g) is realized as the space of smooth sections Γ(E → M) of some vector bundle E π − → M over M.

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“…We also want to see how the present results are compatible with the algebraic adiabatic limit [3,20] and the general framework proposed in [24].…”

Section: Discussionsupporting

confidence: 72%

“…Next, one takes the algebraic adiabatic limit to deal with the IR problem. For more details see [19,40] and [24] for an alternative formulation.…”

Section: The Formal S-matrix and Møller Operatorsmentioning

confidence: 99%

The star product in interacting quantum field theory

Hawkins

Rejzner

2020

Lett Math Phys

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“…the differential graded algebras arising in the BRST/BV formalism. We refer to [Hol08,FR12,FR13] for concrete constructions within the BRST/BV formalism in algebraic quantum field theory, to [BSW19] for the relevant model categorical perspective and to [Haw18] for a related deformation theoretic point of view.…”

Section: Towards the Quantization Of Linear Gauge Theoriesmentioning

confidence: 99%

Algebraic field theory operads and linear quantization

Bruinsma

Schenkel

2019

Lett Math Phys

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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)

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“…Though a large portion of the interest in non-commutative geometry stems from its connections with physics, see [5][6][7]. A. Connes largely made these connections through the cyclic cohomology theory of [8], a generalized de Rham cohomology theory for noncommutative spaces, which closely tied through the Connes complex to one of the central tools of non-commutative geometry and the central object of study of this paper, namely Hochschild (co)homology.…”

Section: Introductionmentioning

confidence: 99%

Lower-Estimates on the Hochschild (Co)Homological Dimension of Commutative Algebras and Applications to Smooth Affine Schemes and Quasi-Free Algebras

Kratsios

2021

Mathematics

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The Hochschild cohomological dimension of any commutative k-algebra is lower-bounded by the least-upper bound of the flat-dimension difference and its global dimension. Our result is used to show that for a smooth affine scheme X satisfying Pointcaré duality, there must exist a vector bundle with section M and suitable n which the module of algebraic differential n-forms Ωn(X,M). Further restricting the notion of smoothness, we use our result to show that most k-algebras fail to be smooth in the quasi-free sense. This consequence, extends the currently known results, which are restricted to the case where k=C.

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A Cohomological Perspective on Algebraic Quantum Field Theory

Cited by 5 publications

References 36 publications

The star product in interacting quantum field theory

The star product in interacting quantum field theory

Algebraic field theory operads and linear quantization

Lower-Estimates on the Hochschild (Co)Homological Dimension of Commutative Algebras and Applications to Smooth Affine Schemes and Quasi-Free Algebras

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