The algebra of Wick polynomials of a scalar field on a Riemannian manifold

Abstract

On a connected, oriented, smooth Riemannian manifold without boundary we consider a real scalar field whose dynamics is ruled by E, a second order elliptic partial differential operator of Laplace type. Using the functional formalism and working within the framework of algebraic quantum field theory and of the principle of general local covariance, first we construct the algebra of locally covariant observables in terms of equivariant sections of a bundle of smooth, regular polynomial functionals over the affine space of the parametrices associated to E. Subsequently, adapting to the case in hand a strategy first introduced by Hollands and Wald in a Lorentzian setting, we prove the existence of Wick powers of the underlying field, extending the procedure to smooth, local and polynomial functionals and discussing in the process the regularization ambiguities of such procedure. Subsequently we endow the space of Wick powers with an algebra structure, dubbed E-product, which plays in a Riemannian setting the same rôle of the time ordered product for field theories on globally hyperbolic spacetimes. In particular we prove the existence of the Eproduct and we discuss both its properties and the renormalization ambiguities in the underlying procedure. As last step we extend the whole analysis to observables admitting derivatives of the field configurations and we discuss the quantum Møller operator which is used to investigate interacting models at a perturbative level.

Ricci Flow from the Renormalization of Nonlinear Sigma Models in the Framework of Euclidean Algebraic Quantum Field Theory

“…As pointed out in [Za15] the PPA is important in our setting because, among other things, it ensures that the renormalization group flow technique we applied in Section 3 does not depend on the splitting L = L free + L int . A complete discussion of the PPA is out not within the scopes of this paper -for a complete discussion in the Riemannian setting see [DDR19]. In the present appendix we provide a brief resumé of the content of the PPA, proving that there exists a family of Wick powers as per definition 39 which fulfils itcf.…”

“…An application of these ideas has already been studied in [FR12,FR13] in the context of gauge theories. The discussion of such scenario is behind the scopes of this paper and it is postponed to a future work [DDR19], see also [Kel09,Kel10] and [Da14].…”

“…As already highlighted in Remark 10, one can consider the coinciding point limit An extension procedure should be applied to define the product · among local polynomial functionals in the same spirit of [BDF09, HW02, Kel09]. We will refrain from describing such a procedure here, see however [DDR19].…”

“…In particular E s is elliptic for all s. Notice that, for the sake of simplicity, we are assuming that E s − E ∈ Γ c (S ⊗2 ψ * T * M ) is a differential operator of order at most 1. This is actually enough for our setting see however [DDR19,HW05] for completeness. Here the subscript reg denotes the algebra generated by regular local functional, namely those with smooth functional derivatives of all orders.…”

“…Remark 62: A direct computation shows that the PPA is satisfied if and only if equation (79) holds true for n = 1cf. [DDR19]. Moreover, on account of the lack of renormalization ambiguities for m ≥ 2cf.…”