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# The up-to-second-order contributions to the two-point (Wightman) function 〈 ϕ I ( x ) ϕ I ( y ) 〉 Ω of the interacting field with potential λ 4 ϕ ( x ) 4 + μ ( x ) 2 ϕ ( x ) 2 . We omit the labels of the external vertices after the first line using the convention that the left external vertex is always the x-vertex.

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We develop a renormalisation scheme for time--ordered products in interacting
field theories on curved spacetimes which consists of an analytic
regularisation of Feynman amplitudes and a minimal subtraction of the resulting
pole parts. This scheme is directly applicable to spacetimes with Lorentzian
signature, manifestly generally covariant, invari...

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... Picture of the one-loop contribution T V φ 2 −V φ 2 at the linear order in . The propagator ∆F (y−x)is represented by a non-oriented line because it is symmetric in the exchange of x ↔ y, while the two-point function ∆+(y − x) by an arrow from y to x[GHP16]. ...

The linearization of semiclassical theories of gravity is investigated in a toy model, consisting of a quantum scalar field in interaction with a second classical scalar field which plays the role of a classical background. This toy model mimics also the evolution induced by semiclassical Einstein equations, such as the one which describes the early universe in the cosmological case. The equations governing the dynamics of linear perturbations around simple exact solutions of this toy model are analyzed by constructing the corresponding retarded fundamental solutions, and by discussing the corresponding initial value problem. It is shown that, if the quantum field which drives the back-reaction to the classical background is massive, then there are choices of the renormalization parameters for which the linear perturbations with compact spatial support decay polynomially in time for large times, thus indicating stability of the underlying semiclassical solution.

... In this setting, the Epstein-Glaser method has been extended [BF00, HW01,HW02] and some of its aspects were studied and improved [BW14,GB03,GBL03,FHS10]. Other schemes were translated to configuration space [DFKR14] and formulated for non-trivial geometries [GHP16,DZ17]. Also the BPHZ scheme was modified [Zav90] and related to configuration space by either Fourier transformation of Epstein-Glaser renormalization [GB03,GBL03,Pra99] or by Fourier transformation of the graph weights as functions for quantum electrodynamics [Ste00]. ...

The notion of normal products, a generalization of Wick products, is derived with respect to BPHZ renormalization formulated entirely in configuration space. If inserted into time-ordered products, they admit the limit of coinciding field operators, which constitute the normal product. The derivation requires the introduction of Zimmermann identities, which relate field monomials or renormalization parts with differing subtraction degree. Furthermore, we calculate the action of wave operators on elementary fields inserted into time-ordered products using the properties of normal products.

... Furthermore, the axioms admit a classification of the renormalization ambiguities, thus the conditions imposed on the equivalence of other prescriptions to the Epstein-Glaser scheme. Subsequently, methods have been developed for Mellin-Barnes regularization [Hol13] (requiring specific spacetimes), dimensional regularization on flat configuration space [BG72,tHV72,DFKR14] and analytic regularization on curved spacetimes [Spe71,GHP16]. It is worth noting that all of them resolve the combinatorial structure with forest formula. ...

A configuration space version of BPHZ renormalization is proved in the realm of perturbative algebraic quantum field theory. All arguments are formulated entirely in configuration space so that the range of application is extended to analytic spacetimes. Further the relation to the momentum space method is established. In the course of that, it is necessary to study the limit of constant coupling.

... Defining the forest formula for the R-operation, any sequence of integrations leads to finite values. Indeed, the forest formula can also be applied in other renormalization schemes when one deals with weighted Feynman graphs [Hol13,DFKR14,GHP15] and, depending on the chosen regularization and subtraction, can be proved in a different fashion in comparison to Zimmermann's approach [Col86]. In [CK00], Connes and Kreimer established that the deeper mathematical structure of weighted graphs and their singularities can be found in the realm of Hopf algebras. ...

The concept of BPHZ renormalization is translated into configuration space. A new version of the convergence theorem by means of Zimmermann's forest formula is proved and a sufficient condition for the existence of the constant coupling limit is derived in the new setting.

The Principle of Perturbative Agreement, as introduced by Hollands and Wald,
is a renormalisation condition in quantum field theory on curved spacetimes.
This principle states that the perturbative and exact constructions of a field
theoretic model given by the sum of a free and an exactly tractable interaction
Lagrangean should agree. We develop a proof of the validity of this principle
in the case of scalar fields and quadratic interactions which differs from the
one of Hollands and Wald and treats all such interactions on an equal footing.
Thereby we profit from the observation that, in the case of quadratic
interactions, the composition of the inverse classical M{\o}ller map and the
quantum M{\o}ller map is a contraction exponential of a particular type.
Afterwards, we prove a generalisation of the Principle of Perturbative
Agreement and show that considering a quadratic contribution of a general
interaction either as part of the free theory or as part of the perturbation
gives equivalent results. Motivated by the thermal mass idea, we use our
findings in order to extend the construction of massive interacting thermal
equilibrium states in Minkowski spacetime developed by Fredenhagen and Lindner
to the massless case. In passing, we also prove a property of the construction
of Fredenhagen and Lindner which was conjectured by these authors.