Local Solutions of RG Flow Equations from the Nash–Moser Theorem

Abstract

We prove local existence of solutions of a functional renormalisation group equation for the effective action of an interacting quantum field theory, when a suitable local potential approximation is considered. To obtain this equation in a Lorentzian setting a quantum state for the theory is selected and a regulator consisting in a mass is added to the action. The flow equation for mass rescalings is then studied using the renown Nash–Moser theorem.

1 Introduction

In his historical essay on the development of the Standard Model [41], Weinberg recalls how Oppenheimer used to grumble that renormalization was just a way to sweep infinities under the rug. In the last 70 years, from a collection of heuristic procedures to extract finite predictions from the infinities arising in Quantum Field Theory (QFT), the Renormalization Group (RG) has turned into a fundamental, organising principle of modern physics, describing the emergence of macroscopic phenomena from the interactions of microscopic degrees of freedom [43, 44].

The Functional Renormalization Group (FRG) represents one of the modern implementations of the RG [4, 14, 32, 42]. In this approach, correlation functions depends on the energy scale k through the insertion of a regulator term that usually acts as a momentum-dependent mass $Q_k$, suppressing modes with frequencies smaller than k. The most convenient way of describing the effective theory at some scale k is through the effective average action $\Gamma _k$, which interpolates between the classical, microscopic action I for $k\rightarrow \infty$, that is, when all quantum fluctuations are suppressed, and the full quantum action $\Gamma$ for $k \rightarrow 0$. The equation governing the flow of the effective average action under changes in the scale k, in Euclidean spaces, is the Wetterich equation [27, 42].

The FRG has been successfully applied to many different physical situations, although mainly in Euclidean spaces: from condensed matter systems to high-energy physics, most notably QCD, (see for example [14] and references therein) and quantum gravity [33, 36,37,38,39], where it represents the principal tool of investigation of the asymptotic safety scenario. In fact, thanks to the structure of the Wetterich equation, the FRG admits non-perturbative approximation schemes that can go beyond usual perturbation theory, allowing the study of strongly coupled systems and perturbatively non-renormalisable theories such as quantum gravity.

The approximation schemes in the FRG usually start from an ansatz for the effective average action, based on different expansions of $\Gamma _k$ into polynomials of the fields and their derivatives; the most used is the Derivative Expansion (DE), based on the number of spatial derivatives of the fields. The lowest order approximation in DE is the Local Potential Approximation (LPA), in which $\Gamma _k$ contains only an effective potential term $U_k(\phi )$, with no derivatives of the fields, and a kinetic term corresponding to the classical one.

According to the RG philosophy, every possible interaction term is admitted in principle in the effective average action. The FRG flow reflects this behaviour in its mathematical structure, which can be written as

$$\begin{aligned} \partial _k \Gamma _k = \frac{i}{2}\langle \partial _k q_k, (\Gamma _k^{(2)} -q_k)^{-1} \rangle , \end{aligned}$$

where $Q_k = - \frac{1}{2} \int q_k \chi ^2 \textrm{d}\mu _x$ is the regulator term acting as an artificial mass for the field $\chi$, and $q_k$ is its integral kernel. The pairing is the standard pairing of bi-distributions over the spacetime ${\mathcal {M}}$ namely on ${\mathcal {M}} \times {\mathcal {M}}$. In Euclidean spacetimes, where Euclidean invariance selects a natural notion of vacuum, (since the Euler–Lagrange equations for the action I are elliptic, rather than hyperbolic), the inverse $(\Gamma _k^{(2)} -q_k)^{-1}$ is unique. However, due to the appearance of the inverse of $\Gamma _k^{(2)}$ on the r.h.s, independently from the initial datum for $\Gamma _k$, the flow will always produce additional interaction terms in the effective average action. This becomes evident in a perturbative setting, where the inverse is constructed as a Neumann series and, thus, any power of some field polynomial could in principle appear on the right hand side of the equation.

Only the artificial truncation of $\Gamma _k$ in a polynomial expression of finite order prevents the generation of infinite terms. Therefore, a standard approach in the literature is to simply truncate the expansion of $\Gamma _k$ in a finite number of terms, neglecting higher order contributions generated along the RG flow [27].

A simple example where this mechanism happens is the interacting scalar field theory. To study its RG flow, in the LPA, one starts from an Ansatz for the effective average action of the form $\Gamma _k = \int c_0 \partial \overline{\phi } \partial \phi + \sum _{i=1}^N c_i (\overline{\phi }\phi )^{i} \text {d}\mu _x$ for the scalar field $\varphi$. Inserting the Ansatz in the Wetterich equation an infinite series appears at the r.h.s., due to the inversion of the quantum wave operator $\Gamma _k^{(2)} - q_k$. Thanks to the truncation of the series up to order N, and assuming $\phi$ to be constant, it is possible to equate the contributions proportional to $(\overline{\phi }\phi )^i$ on both sides of the equation. Hence, the Wetterich equation reduces to a system of coupled, partial differential equations for the coefficients $c_i$ of the field functionals, given in terms of $\beta -$functions of the theory. However, in this way, because of the truncation, only approximate solutions are obtained. Furthermore, little control on the quality of the approximation scheme, compared to the full theory space, is possible.

Mathematically, the problem of the generation of every possible term along the flow is connected with the problem of loss of derivatives: intuitively, since the r.h.s of the Wetterich depends on the inverse of the second derivative of the effective average action, a Green operator (a fundamental solution) for the Wetterich equation will also depend on the second derivative $\Gamma ^{(2)}_k$. It follows that, if the Wetterich equation is an operator acting on some space of $C^n$ functions of the fields, its solutions will generally be only $C^{n-2}-$regular, losing two derivatives. Due to the loss of derivatives, standard iterative procedures to produce solutions in suitable Banach spaces fail to converge.

Recently, together with Drago and Rejzner we developed a new approach to the FRG, based on the methods of perturbative Algebraic Quantum Field Theory (pAQFT) [5, 6, 16, 35], and later extended to the case of gauge theories [9, 10]. The approach is fully Lorentzian, and allows for a generalization of the Wetterich equation to generic states and curved backgrounds, where a natural notion of vacuum is usually not at disposal. These Lorentzian RG flow equations exhibit a state dependent flow, and are based on a Hadamard regularisation of the UV divergences, instead of a regularisation based on a momentum-dependent regulator $Q_k$ and its derivative. The Hadamard regularisation is made possible by selecting the class of Hadamard states as background states for the free theory. Both these features are shared with a similar approach developed recently for scalar fields in cosmological spacetimes [2]. Moreover, the regulator term is chosen as an artificial mass term, without momentum dependence: such a regulator term is particularly useful to preserve unitarity of the $S-$matrix and Lorentz invariance of the flow, and does not modify the structure of the propagator. In the literature, this regulator term has been called Callan-Symanzik-type cut-off, and the respective RG flow equation is also known as functional Callan-Symanzik equations [1].

The main difference of the Lorentzian case, compared to the Euclidean case, is that the quantum wave operator $\Gamma _k^{(2)} - q_k$ is, typically, a hyperbolic differential operator, instead of an elliptic one. This implies in particular that there exists an infinite family of fundamental solutions, and therefore the inverse appearing at the r.h.s. of the Wetterich equation (also called interacting propagator) is in the best case not uniquely determined. To fix a choice for the inverse, the idea used in [9] is to select a state for the free theory, providing a background reference state. To do so, we decompose the effective average action into

$$\begin{aligned} \Gamma _k + Q_k(\phi ):= I_0(\phi ) + U_k(\phi ), \end{aligned}$$

where $I_0$ is the free (quadratic) part of the classical action and $U_k$, called the effective interaction, contains all possible quantum corrections. The above decomposition allows to rewrite the quantum wave operator as $\Gamma _k^{(2)} - q_k = P_0 + {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}$ where $P_0 = I^{(2)}_0$ is the wave operator for the free theory.

The RG flow equation in Lorentzian spacetimes then takes the expression of a functional differential equation for the effective interaction $U_k$, depending on: a fixed background geometry ${\mathcal {M}}$; the smooth part w of the two-point function of the reference state $\omega$ for the free theory; the advanced and retarded propagators $\Delta _{A,R}^U$ for the wave operator $P_0 + {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}$; and some initial value and boundary conditions. We derive its form in (18) and we recall here its expression,

$$\begin{aligned} \partial _k U_k = - \frac{1}{2} \int _{{\mathcal {M}}} \partial _k q_k(x) (1 -\Delta _R^U {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}) w (1 - {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}\Delta _A^U )(x,x) \textrm{d}\mu _x. \end{aligned}$$

(1)

Equation (1) is non-perturbative in the coupling constant, as the Wetterich equation in Euclidean space, and so it allows for non-perturbative approximation schemes.

In this paper, we take a step further in clarifying the mathematical structure of the RG flow equations, and we prove that, with possibly a non-polynomial effective potential $U_k$ that contains no derivatives of the Dirac delta or of the field, the RG flow equations admit a local solution for suitably regular initial conditions.

Once the existence of the exact solution is established, we can compare it with the solutions of the truncated Wetterich equation mentioned above. This is actually an approximation of the exact solution; in fact, the difference between the exact solutions and the solution of the truncated equation at order N is always at least of order $N+1$ in powers of $U_k$, because this is the order of the error provided by the truncation.

In order to prove the existence of local solutions for the RG flow (18), we need to choose an appropriate approximation. Inspired by Euclidean FRG approaches, as a first step towards more general results we choose to approximate $U_k$ with the Local Potential Approximation, defined in Eq. (19), as a local function of the field $\phi$ containing no derivatives:

$$\begin{aligned} U_k(\phi ) = \int _{{\mathcal {M}}} u(\phi (x), k) f(x) \textrm{d}\mu _x, \qquad U_k^{(2)}(\phi )(x,y) = \partial _\phi ^2 u(\phi , k) f(x) \delta (x,y). \end{aligned}$$

f is a compactly supported smooth function, equal to 1 on large regions of the spacetime. We further assume that the field $\phi$ is constant over the whole space, so that $\partial ^2_\phi u$ is function of k and $\phi$ only. Although the LPA does not take into account spacetime fluctuations of the fields, it is still non-trivial since $U_k$ can be any smooth non-polynomial function of the field $\phi$.

Within this approximation, the r.h.s. of the RG flow equation can be written in terms of the map given in (21), which we recall here:

$$\begin{aligned} G_k(\partial _\phi ^2u):=- \frac{1}{2 \left| \!\left| \!\left| f\right| \!\right| \!\right| _1} \int _{{\mathcal {M}}} \partial _k q_k(x) \left\{ (1-\partial _\phi ^2 u \Delta _R^u f) \otimes (1-\partial _\phi ^2 u \Delta _R^u f)(w)(x,x) \right\} \textrm{d}\mu _x. \end{aligned}$$

The RG flow equation reduces to an equation for $u(\phi ,k)$. Thus, we are interested in studying the existence of solutions for the following problem,

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _k u = G_k(\partial _\phi ^2u), \\ u(\phi , a) = \psi , \\ \left. u\right| _{\partial X \times [a,b]} = \beta \ \end{array}\right. } \end{aligned}$$

(2)

where $\psi$ and $\beta$ are known functions which characterise respectively the initial and boundary conditions of the problem.

The LPA greatly simplifies the RG flow equation, which now is an equation for u, as a function of k and $\phi$. However, the LPA does not remove the problem of the loss of derivatives.

The main result of this paper is the proof of local existence of solutions of this problem, which is given below in Theorem 4.13. The proof is an application of the renown Nash–Moser Theorem.

Nash provided a beautiful theorem to prove local existence of solutions of non-linear partial differential equations in spaces of smooth functions, which are particularly suited to deal with the problem of loss of derivatives [30]. The theory was first developed in the context of isometric embeddings of Riemannian manifolds by Nash, and then further generalised by Moser [28, 29].

Hamilton [19] provided a particularly natural setting for the theorem in the space of tame Fréchet spaces. Hamilton used his formulation of the Nash–Moser to prove short-time existence and uniqueness of the Ricci flow [20]. Shortly later, DeTurck exploited the invariance of the Ricci flow under the infinite-dimensional diffeomorphism group to provide a simpler proof of the short-time existence of solutions of the Ricci flow. The proof is based on the Bianchi identity associated with diffeomorphism invariance, together with an appropriate gauge-fixing procedure, to obtain a strictly parabolic flow equation equivalent, modulo gauge transformations, to the Ricci flow. Existence of local solutions then follow by standard arguments for parabolic equations [12].

While here we use Hamilton’s formulation of the Nash–Moser theorem, it would be interesting to identify an infinite-dimensional symmetry group for the RG flow equation, analogous to the diffeomorphism group of the Ricci flow. In this case, it would be possible to use DeTurck’s trick to provide a simpler proof of short-time existence of solutions of the RG flow equations. It would be particularly suggestive to apply DeTurck’s procedure to gauge theories, where the effective average action is invariant under a gauge group encoded in the Slavnov-Taylor identities [10]. We leave this comment as a suggestion for future works.

We recall the basic definitions and the formulation of the Nash–Moser theorem we are using in this paper in the Section A.1 of the Appendix.

1.1 Strategy and summary of results

In order to prove the main theorem of this paper, using Hamilton’s formulation of Nash–Moser theorem, the RG flow equation needs to satisfy a number of assumptions. First of all, it must be cast in the form of a suitable map acting on a tame Fréchet space. Requiring that $\phi$ and k are limited in some compact interval, and that $u(\phi ,k)$ is a smooth function, is sufficient for $u(\phi ,k)$ to be an element of a tame Fréchet space. Secondly, the operator $\mathcal{R}\mathcal{G}: u \in F_0 \rightarrow F$ defining the RG flow equation (see Def. (27)) must be a smooth tame map between tame Fréchet spaces. In order to be tame, the RG operator must satisfy some estimates on its seminorms. Assuming that u lies in some neighbourhood of 0 (by requiring that a suitable seminorm of u, given below in Eq. (23), is $\Vert u\Vert _4 < A$ for sufficiently small A), it is possible to prove these estimates using the Grönwall lemma, since the normal-ordered interacting propagator $G_k(\partial ^2_\phi u)$ satisfies a recursive integral inequality.

Then, the linearisation of the RG operator must be an invertible smooth tame operator, and its inverse must be tame smooth. In the LPA, the linearisation $L = D \mathcal{R}\mathcal{G}$ takes the form of a parabolic equation, analogous to a heat equation with a $k,\phi -$dependent heat conductivity $\sigma$. The inverse of linear parabolic equations is known [17] (see also [11, 18]), and the inverse of the linearised RG operator can be constructed from the heat kernel. Once the inverse of the linearised RG operator is known, it is possible to prove that it is tame smooth.

All these results are presented and proved in Propositions 4.2, 4.7, 4.8, and 4.9. These are used to prove our main result, Theorem 4.13, on the existence of local solutions of the RG flow which we report here for completeness in a compact form.

Theorem 1.1

The RG operator admits a unique family of tame smooth local inverses, and unique local solutions of the RG flow equations exist.

The Theorem provides a proof of existence of local solution, namely a local potential which solves the RG flow equation for the mean field configuration $\phi$ in some compact space X and for k in some interval [a, b]. For the Theorem to hold, the initial conditions at $k=a$ and the boundary conditions at $\partial X$ are required to be small in the sense stated in Proposition 4.9, see also Proposition 4.6. We observe that these requirements are not restrictive and can always be fulfilled for given initial values and boundary conditions by considering a sufficiently small interval [a, b].

The material is organised as follows. We start with a review of the derivation of the RG flow equations, to set the notation and clarify the underlying framework, in Sect. 2. This section closely follows the presentation in [9]. In Sect. 3, we clarify the structure of the RG flow equation, expanding on some points the derivation presented in [9], and we show how to compute the interacting propagator $G_k$ from the free propagators and the underlying state for the free theory, to get the RG flow equations as closed differential equations for the effective interaction $U_k$.

In the main part, starting from Sect. 4, we prove our main theorem. First of all, we define the LPA and identify the appropriate tame Fréchet spaces in which we want to solve the equation. We then proceed proving the main propositions 4.2, 4.7, 4.8, and 4.9. In theorem 4.13 we state our main result on the existence of local solutions of the RG flow equations, which follows immediately from the propositions.

2 Functional RG Flow Equations

2.1 Generating functionals

We recall here the main steps in the derivation of the RG flow equation in the form given in Eq. (1), as presented in Ref. [9]. We refer to the paper [9] for further details.

The methods used to represent the objects we are working with are those proper of pAQFT [5,6,7,8, 21, 22, 35]. In this framework field observables are seen as functional over smooth field configurations. The quantum properties manifest themselves in the various products used to multiply those objects and in the involutions used to construct positive elements. In this way one obtains a $*$-algebra ${\mathcal {A}}$ of field observables. Expectation values of observables are obtained testing elements of ${\mathcal {A}}$ over a positive, normalised linear functional $\omega$. We refer to [35] for full details on the quantization procedure.

We work with a globally hyperbolic spacetime ${\mathcal {M}}$, which is a smooth, oriented, and time oriented manifold, equipped with Lorentzian metric $\eta$ which makes ${\mathcal {M}}$ globally hyperbolic. For simplicity we shall assume that ${\mathcal {M}}$ is stationary and ultra-static. This restriction is required in order to have good control on the regularity of the advanced and retarded propagators; however, the estimates that we prove are known to hold in more general spacetimes, such as de Sitter.

We start with the action of a quantum field theory propagating on ${\mathcal {M}}$. For simplicity we discuss here the scalar case. The lagrangian density of the theory is

$$\begin{aligned} {\mathcal {L}}(\chi ) = {\mathcal {L}}_0(\chi )+{\mathcal {L}}_I(\chi ) = -\frac{1}{2} (g^{-1}(\partial \chi , \partial \chi ) + \xi \chi ^2 R + m^2 \chi ^2) + {\mathcal {L}}_I(\chi ) \end{aligned}$$

where $\chi$ is a field configuration and ${\mathcal {L}}_0(\chi )$ is the Lagrangian density of the free theory, quadratic in the fields. The free theory can be quantised, providing a $*-$algebra of observables for which states are known to exist. ${\mathcal {L}}_I$ is the interaction Lagrangian, the contribution to ${\mathcal {L}}$ which is more than quadratic in the fields. We denote by I and $I_0$ the bare action associated to the Lagrangian density ${\mathcal {L}}$ and ${\mathcal {L}}_0$ respectively, and by

$$\begin{aligned} V(\chi ):= \int _{{\mathcal {M}}} \lambda {\mathcal {L}}_I(\chi ) f \textrm{d}\mu _x \end{aligned}$$

the interacting action. f is a cut-off function, which is equal to 1 in the region of the spacetime ${\mathcal {M}}$ where we are analysing our theory, and it is inserted to make the integral over the manifold finite. Similar regulators needs to be considered also in I and $I_0$. Since a direct construction of the observables of the interacting theory is not available, we use perturbative methods to represent interacting fields over the free theory. Interacting fields are thus represented as formal power series in the parameter $\lambda$, which governs the non-linear coupling of the theory. The coefficients of the formal power series are elements of the free theory. See [6, 7, 21,22,23] for further details. States are thus linear functionals on the algebra of the free theory.

We now need to introduce the generating functionals for correlation functions, which are the starting point of most treatments of the functional Renormalization Group (fRG) [4, 31, 33, 37,38,39].

In this framework, the generating functional of correlation functions depends on the state of the free theory, and it is defined as

$$\begin{aligned} Z(j):= \omega \left( S(V)^{-1} \star S(V+J) \right) , \end{aligned}$$

where $J = \int j(x) \chi (x) \textrm{d}x$. $V=I-I_0$ is the interaction action, and S(V) is the time ordered exponential of V. The $\star -$product represents the quantum, non-commutative product in the algebra of the free theory constructed from $I_0$, and $\omega$ is the state in which we are interested to evaluate correlation functions.

The functional derivatives of Z(j) for vanishing sources gives the interacting, time-ordered correlation functions of the interacting fields,

$$\begin{aligned} {Z^{(n)}(j)}\Bigg |_{j=0} = i^n \omega \left( S(V)^{-1} \star (S(V) \cdot _T \chi (x_1) \cdot _T... \cdot _T \chi (x_n) )\right) . \end{aligned}$$

We refer to [9] for a discussion of the relation to the standard approach present in the physics literature. However, we observe that the standard approach may be recovered only for states $\omega$ for which the Gell-Mann-Low formula holds, namely when the star product above factorises in the product of expectation values. Equilibrium states at finite temperature or states on curved backgrounds do not have this property.

The functional Renormalization Group approach works by deforming the underlying theory with an artificial mass scale k. In standard treatments, the procedure consists in the addition of a non-local regulator, quadratic in the fields, to the bare action I, acting as a scale-dependent mass term. Although a non-local, momentum dependent regulator implements a Wilsonian renormalization flow, (i.e., a genuine coarse-graining procedure in which field modes with increasing frequency are progressively integrated out), at least in Euclidean settings, in the case of Lorentzian signature it appears less favourable, due to its non-local nature in position [9]. In particular, it can spoil the unitarity of the $S-$matrix and it can introduce artificial poles in the propagator. For this reason, in [9] we chose to use a local regulator term

$$\begin{aligned} Q_k(\chi ) = - \frac{1}{2} \int \textrm{d}\mu _x q_k(x) T \chi ^2(x) . \end{aligned}$$

Notice that the time-ordering operator $T\chi ^2$ naturally introduces a normal-ordering prescription, so that $T \chi ^2$ is actually finite.

This regulator function will act as an artificial mass term in the correlation functions. Therefore, although it does not regularise UV divergences, it regularises IR divergences. In turn, it does not spoil the unitarity of the $S-$matrix and does not alter the structure of the propagator. Moreover, in [9] it was proven that, in the limit of infinite mass, $k\rightarrow \infty$, the Feynman propagator reduces to zero and quantum effects are completely suppressed. We then see that with the introduction of such a term, we can picture the flow of correlation functions under changes of the scale k, from large scales to the vanishing limit $k \rightarrow 0$, as a flow from the classical theory to the quantum one. This in turn justifies the terminology of renormalization group flow. Finally, notice that local regulators of this type were already introduced in [1], where the RG flow equations with a local regulator have been called functional Callan-Symanzik equations, and appear as a special case of the Wetterich equation [42] in the case of local regulators instead of non-local ones. More recently, local regulators have been used in the Lorentzian setting in [15], where they are used to study the flow of the graviton spectral functions, and in [25, 26] in the context of renormalization of thermal field theories.

We thus deform the generating functional Z, defining a $k-$dependent generating functional

$$\begin{aligned} Z_k(j) = \omega \left( S(V)^{-1} \star S(V+J + Q_k) \right) . \end{aligned}$$

From the definition of the regularised generating functional $Z_k$, the steps to define the effective action are standard: we first define the (regularised) generating functional for the connected correlation functions as

$$\begin{aligned} e^{i W_k(j)}:= Z_k(j) . \end{aligned}$$

The first derivative of $W_k$ defines the classical field $\phi$ as a function of j,

$$\begin{aligned} \phi _j(x):= W^{(1)}_k(j)(x) = \frac{\delta W_k(j)}{\delta j(x)} = e^{-iW_k} \omega \left( S(V)^{-1} \star S(V+Q_k+J) \cdot _T \chi (x) \right) = \langle \chi \rangle \, \end{aligned}$$

while the second derivative is proportional to the connected, interacting Feynman propagator

$$\begin{aligned} - i W^{(2)}_k(j) = \langle \chi (x) \cdot _T \chi (y) \rangle - \phi _j(x) \phi _j(y) . \end{aligned}$$

(3)

In the above relations, we introduced the angle brackets to denote the weighted expectation value of an interacting operator F, for non-vanishing sources and regulator:

$$\begin{aligned} \langle F \rangle := e^{-iW_k} \omega \left( S(V)^{-1} \star S(V+Q_k+J) \cdot _T F \right) \end{aligned}$$

The relation between $\phi$ and j can be inverted, giving

$$\begin{aligned} - j_\phi (x) = (P_0 \phi )(x) +Q^{(1)}_k(\phi )(x) +\langle T V^{(1)}(\chi ) \rangle , \end{aligned}$$

(4)

which can be solved perturbatively at all orders [9]. Thanks to this inversion, we can define the Legendre transform of $W_k$ as

$$\begin{aligned} {\tilde{\Gamma }}_k(\phi ):= W_k(j_\phi ) - J_\phi (\phi ), \end{aligned}$$

(5)

satisfying

$$\begin{aligned} {\tilde{\Gamma }}^{(1)}_k(\phi ) = - j_\phi . \end{aligned}$$

The second derivative of ${\tilde{\Gamma }}_k$ and $W_k$ are related by the standard formula between Legendre transforms,

$$\begin{aligned} ( \Gamma _k^{(2)} - q_k) (x,z) W^{(2)}_k(z,y) = - \delta (x,y) . \end{aligned}$$

(6)

Therefore, $W^{(2)}_k$ is a propagator for the quantum wave operator $\Gamma _k^{(2)} - q_k$.

Finally, the effective average action is defined subtracting the classical term $Q_k(\phi )$ from the Legendre transform of $W_k$:

$$\begin{aligned} \Gamma _k(\phi ):= {\tilde{\Gamma }}_k(\phi ) - Q_k(\phi ) = W_k(j_\phi ) - J_\phi (\phi ) - Q_k(\phi ) . \end{aligned}$$

2.2 RG flow equations

The RG flow equations govern the flow of the effective average action under scaling of the parameter k. We now remember the main steps in their derivation.

The first step is computing the $k-$derivative of $Z_k$, which is straightforward from its definition

$$\begin{aligned} \partial _k Z_k(j) = i \omega \left( S(V)^{-1} \star S(V+Q_k + J) \cdot _T \partial _k Q_k \right) . \end{aligned}$$

The k-derivative of $W_k$ follows immediately,

$$\begin{aligned} \partial _k W_k(j) = \langle \partial _k Q_k \rangle \, \end{aligned}$$

(7)

where we recall that

$$\begin{aligned} \partial _kQ_k(\chi ) = - \frac{1}{2} \int \textrm{d}x \partial _k q_k(x) T \chi ^2(x) . \end{aligned}$$

Notice that, thanks to normal ordering introduced by the $T-$products of local observables, the flow equation is UV finite.

The contribution $\langle T \chi ^2 \rangle$ in $\langle \partial _k Q_k \rangle$ can be obtained as

$$\begin{aligned} \langle T \chi ^2(x) \rangle = \lim _{y \rightarrow x} \left( \langle \chi (x) \cdot _T \chi (y) \rangle - {\tilde{H}}_F(x,y)\right) , \end{aligned}$$

where the counterterms ${{\tilde{H}}}_F(x,y)$, arising from the expectation value of a normal-ordered quantity, implements normal ordering in the interacting theory and make the expression finite.

Recalling (3), we can rewrite (7) as (the Lorentzian generalization of ) the Polchinski equation [34].

$$\begin{aligned} \partial _k W_k= - \frac{1}{2} \lim _{y \rightarrow x} \int \textrm{d}x \partial _k q_k(x) \left[ -i W^{(2)}_k(x,y) + \phi (x) \phi (y) - {{\tilde{H}}}_F(x,y) \right] . \end{aligned}$$

(8)

Notice that $-i W^{(2)}_k$ is the propagator of the interacting theory. In the case of fundamental solutions of free hyperbolic equations, the counter-terms ${\tilde{H}}_F$ necessary to implement normal ordering are well known, and are given in terms of suitable Hadamard parametrix $H_F$ (see e.g. [24]). This normal-ordering procedure is known as Hadamard subtraction, or point-splitting regularisation. We refer to [24] for further details.

Recalling (5), the Polchinski equation (8) can now be written as the RG flow equation for $\Gamma _k$ with a self-consistency relation

$$\begin{aligned} \partial _k \Gamma _k= & {} -\frac{1}{2} \int _x \partial _k q_k(x):G_k(x,x):_{{{\tilde{H}}}_F} \end{aligned}$$

(9)

$$\begin{aligned} (\Gamma ^{(2)}_k - q_k) G_k= & {} i \delta , \end{aligned}$$

(10)

while the normal-ordering prescription is given by

$$\begin{aligned}:G_k(x,y):_{{{\tilde{H}}}_F} = G_k(x,y) - {{\tilde{H}}}_F(x,y) \end{aligned}$$

and $G_k(x,y) = -i W^{(2)}_k(x,y)$.

In the following section we will gain more insight in the r.h.s of the RG flow equation. In particular, we will use the fact that $G_k$ is a fundamental solution of $(\Gamma ^{(2)}_k - q_k)$ and ${\tilde{H}}_F$ is a fundamental solution of $(\Gamma ^{(2)}_k - q_k)$ up to known smooth terms. Hence, $:G_k:$ is a bi-solution of the equation of motion up to known smooth terms. Moreover, in the regions of spacetime $J^-({\mathcal {O}})$ where $V\rightarrow 0$ and $q_k\rightarrow 0$, $G_k$ reduces to $\omega _2+i\Delta _A$, the Feynman propagator of the free theory, where $\omega _2$ is the two-point function of the free theory. Similarly, ${\tilde{H}}_F$ reduces to $H_F$, the Hadamard parametrix of the free theory. These observations allow to obtain an explicit form of $:G_k:(x,y)$ in terms of the effective average action and the smooth part of the state for the free theory, $w = \omega _2+i\Delta _A-H_F$, by means of the classical Møller maps [13].

3 Quantum Equations of Motion and RG Flow for the Effective Interaction

Recalling that, by definition,

$$\begin{aligned} \frac{\delta }{\delta \phi (x)} \left( \Gamma _k + Q_k \right) = - j_\phi (x), \end{aligned}$$

equation (4) can be recast in the quantum equation of motion (QEOM)

$$\begin{aligned} \frac{\delta }{\delta \phi (x)} \left( \Gamma _k + Q_k \right) = P_0 \phi (x) + \langle V^{(1)}(x) + Q_k^{(1)}(x) \rangle . \end{aligned}$$

(11)

The above relation can also be re-expressed as a Dyson-Schwinger equation, since $P_0 \phi = \langle \frac{\delta I_0}{\delta \chi } \rangle$, so that

$$\begin{aligned} \frac{\delta \Gamma _k}{\delta \phi (x)} = \langle \frac{\delta I}{\delta \chi (x)} \rangle . \end{aligned}$$

(12)

The QEOM suggests to decompose the effective average action into

$$\begin{aligned} \Gamma _k^{(2)}(\phi ) - q_k:= P_0 + U^{(2)}_k(\phi ), \end{aligned}$$

(13)

where $P_0:= I^{(2)}_0$ is the Green hyperbolic operator defined by the free action, while the effective interaction $U_k(\phi )$ is defined by the relation

$$\begin{aligned} U_k^{(1)}(x) = \langle V^{(1)}(x) + Q_k^{(1)}(x) \rangle . \end{aligned}$$

(14)

The effective interaction includes all the quantum corrections to the interaction V, and it can be seen as a non-perturbative definition for the sum of perturbative Feynman diagrams. In its perturbative expansion, the effective interaction contains non-localities and possibly higher-derivative terms.

We call the operator $P_0 + {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}$ the quantum wave operator. In terms of the effective interaction, the relation between the second derivative of the effective average action and $W^{(2)}_k$ reads

$$\begin{aligned} (P_0 + {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}) W^{(2)}_k = - 1 . \end{aligned}$$

(15)

In the following, we will assume that, despite quantum corrections, the quantum wave operator $P_0 + {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}$ remains Green hyperbolic; that is, it admits unique advanced and retarded propagators such that

$$\begin{aligned} (P_0 + {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}) \Delta ^U_{A,R}(f) = f \ \text {and} \ {{\,\mathrm{\text {supp}}\,}}\Delta ^U_{A}(f) \subset J^-({{\,\mathrm{\text {supp}}\,}}f) \, {{\,\mathrm{\text {supp}}\,}}\Delta ^U_{R}(f) \subset J^+({{\,\mathrm{\text {supp}}\,}}f). \end{aligned}$$

There is a standard procedure to intertwine the free and the quantum wave operators $P_0$ and $P_0 + {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}$, see [13]. In fact, consider the operator $(1 - \Delta _{R}^U {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}})$ applied to any function f; we have

$$\begin{aligned} (P_0 + {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}) (1 - \Delta _R^U {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}) f = P_0 f. \end{aligned}$$

It follows that the operators $(1- \Delta _{A,R}^U {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}})$ intertwine between the free and quantum wave operators. We call $(1- {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}\Delta _{A}^U)$ and $(1- \Delta _{R}^U {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}})$ the advanced/retarded Møller operators.

We can now rewrite the Møller operators in terms of the propagators for the free theory and the effective interaction. We start from the defining property of $\Delta _{A,R}^U$, that they are fundamental solutions of the QEOM:

$$\begin{aligned} (P_0 + {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}) \Delta _{A,R}^U = 1 . \end{aligned}$$

It follows that

$$\begin{aligned} P_0 (1 + \Delta _{A,R} {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}) \Delta _{A,R}^U = 1, \end{aligned}$$

and so the following recursive formula for $\Delta _{A,R}^U$ holds:

$$\begin{aligned} \Delta ^U_{A,R} = \Delta _{A,R} (1- {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}\Delta ^U_{A,R}). \end{aligned}$$

(16)

The last relation can be used to obtain a series representation of $\Delta ^U_{A,R}$ in terms of powers of ${{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}$. Thanks to the Møller operators, we can write solutions and propagators of the quantum wave operator $P_0 + {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}$ in terms of the solutions and propagators of the free theory and of the effective interaction ${{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}$.

In what follows, we compute the regularised propagator $:G_k:_{{{\tilde{H}}}_F}$ for the interacting theory. We recall in particular that the regularised propagator $:G_k:_{{{\tilde{H}}}_F}$ is a solution for the quantum wave operator $( \Gamma _k^{(2)} - q_k):G_k:_{{{\tilde{H}}}_F} =0$ up to known smooth terms. Furthermore, denoting by ${\mathcal {O}}\subset {\mathcal {M}}$ the support of V and of $q_k$, which is a compact set because of the cut-off functions used in their construction, it holds by causality that

$$\begin{aligned}:G_k(x,y): = \Delta _F(x,y) - H_F(x,y) = w(x,y) \, \qquad \forall x,y \in {\mathcal {M}}\setminus J^+({\mathcal {O}}). \end{aligned}$$

Since $:G_k:$ is a bi-solution of the QEOM, and it reduces to w in the past of the supports of $V, q_k$, and j, we arrive at

$$\begin{aligned} :G_k: = (1 - \Delta _R^U U^{(2)}_k ) w (1 - U^{(2)}_k \Delta _A^U ). \end{aligned}$$

(17)

Finally, thanks to the above expression, we conclude that the RG flow equations can be rewritten as differential equations for the effective interaction $U_k$ as

$$\begin{aligned} \partial _k U_k = - \frac{1}{2} \int _{{\mathcal {M}}} \partial _k q_k(x) (1 -\Delta _R^U {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}) w (1 - {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}\Delta _A^U ) \textrm{d}\mu _x . \end{aligned}$$

(18)

In the next section we discuss existence and uniqueness of local solutions of this equation.

4 Existence of Local Solutions

4.1 Local potential approximation

In this section, we would like to prove an existence theorem for local solutions of the RG flow equations. In order to do so, we restrict our attention to the Local Potential Approximation: in this approximation, the effective interaction is a local functional which does not contain derivatives of the fields. Furthermore, we consider the case in which the classical field $\phi$ is constant throughout spacetime.

More precisely, the Local Potential Approximation (LPA) assumes that the effective interaction and its second functional derivative are given by a local potential,

$$\begin{aligned} U_k(\phi ) = \int _{\mathcal {M}} u(\phi (x), k) f(x) \text {d} \mu _x \ ,\quad U_k^{(2)}(\phi )(x,y) = \partial _\phi ^2 u(\phi (x), k) f(x) \delta (x,y),\nonumber \\ \end{aligned}$$

(19)

where $f\in C^{\infty }_0({\mathcal {O}})$ is an adiabatic cutoff ($f\ge 1$ and $f=1$ on the relevant part of the spacetime we are working with), which is inserted to keep the theory infrared finite, and ${\mathcal {O}} \in {\mathcal {M}}$ is a compact region in the space-time containing the support of f.

We further recall that the background spacetime ${\mathcal {M}}$ is ultra-static. This assumption simplifies the explicit form of the retarded and advanced propagators for the free theory $\Delta _{A,R}$, and it allows for simple estimates of their norms. However, these estimates can be easily generalised to static spacetimes, and are known to hold in some special cases, such as de Sitter space.

Finally, in the simplest approximation, we choose the field $\phi$ to be a constant throughout the space-time, so that $u(\phi , k)$ and $\partial _\phi ^2 u(\phi , k)$ are constant in space. The arbitrary function $u(\phi ,k)$ and its second field derivative $\partial ^2_\phi u$ thus determine the effective potential, and so the effective average action. The projection of the RG flow equation on the constant field configurations corresponds to the LPA employed in the physics literature [14]. The effective potential $U_k$ differs from the standard one (see e.g. Ref. [40]) only by the volume integral of the cutoff function f, in place of the more usual (typically divergent) volume of the spacetime.

In the limit where $V\rightarrow 0$ the effective potential reduces to $Q_k$ and u reduces to $-q_k\phi ^2/2$. We shall take this into account in fixing the initial conditions for u.

Thanks to this approximation, the second derivative of the effective potential ${{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}$ appearing in the QEOM reduces to a perturbation of the free wave operator $P_0$ with a smooth external potential that has compact support, and in the limit where $f\rightarrow 1$ on ${\mathcal {M}}$ the potential reduces to a mass perturbation. It follows that many techniques of the generalised principle of perturbative agreement [13] become readily available.

In particular, it is known that the interacting advanced and retarded propagators $\Delta _{A,R}^U$ are given by the free propagators $\Delta _{A,R}$ associated to $P_0$, with a mass modified by the external potential. In particular the recursive relations given in (16) permits to analyse analytically how $G_k$ depends on u.

By the LPA, the RG flow equation (18) becomes a partial differential equation for $u(\phi ,k)$. Thus, we are interested in studying the existence and uniqueness of solutions of the problem associated with the RG flow equation (18), supplemented with suitable boundary conditions and a set of initial values explicitly given in terms of the functions $\psi$ and $\beta$ as:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _k u = G_k(\partial _\phi ^2u), \\ u(\phi , a) = \psi , \\ \left. u\right| _{\partial X \times [a,b]} = \beta . \end{array}\right. } \end{aligned}$$

(20)

where the function $G_k$ is defined as

$$\begin{aligned} G_k(\partial _\phi ^2u):=- \frac{1}{2 \left| \!\left| \!\left| f\right| \!\right| \!\right| _1} \int _{{\mathcal {M}}} \textrm{d}\mu _x \partial _k q_k(x) \left\{ (1-\partial _\phi ^2 u \Delta _R^u f) \otimes (1-\partial _\phi ^2 u \Delta _R^u f)(w)(x,x) \right\} . \nonumber \\ \end{aligned}$$

(21)

$w \in C^{\infty }({{\mathcal {M}}}^2)$ is a given symmetric smooth function (the smooth part of the chosen background state); $f\in C^\infty _0({\mathcal {M}})$ is the positive cutoff function used in U and $\left| \!\left| \!\left| f\right| \!\right| \!\right| _1$ is the $L^1$ norm of f computed with respect to the standard measure on ${\mathcal {M}}$; $q_k$ is the integral kernel of the adiabatic regulator $Q_k$, which is assumed to be smooth and with compact support in x. $\Delta _{R}^u:C^{\infty }_0({\mathcal {M}})\rightarrow C^\infty ({\mathcal {M}})$ is the retarded fundamental solutions of $(P_0+ f \partial _\phi ^2 u) g=0,$ which coincides with $\Delta _R^U$ used in other parts of the paper; it exists and it is unique because $P_0+ f \partial _\phi ^2 u$ is a Green-hyperbolic operator [3]. Furthermore, in the integrand in (21) f is seen as a multiplicative operator which maps $C^{\infty }({\mathcal {M}})\rightarrow C^{\infty }_0({\mathcal {M}}),$ and 1 is the identity map in $C^{\infty }({\mathcal {M}})$. Notice that $\partial _\phi ^2 u$ is constant with respect to $P_0$. Furthermore, thanks to the support properties of f we have that $O:=(1-\partial _\phi ^2 u \Delta _R^u f) \otimes (1-\partial _\phi ^2 u \Delta _R^u f)$ is a linear operator on $C^{\infty }({\mathcal {M}}\times {\mathcal {M}})$ to itself. Since w is smooth on ${\mathcal {M}}$ the evaluation of Ow on (x, x) can be easily taken and the integral over ${\mathcal {M}}$ is finite because $q_k$ is of compact support.

To keep the analysis of this part as simple as possible, we finally assume that the regulator term takes the expression

$$\begin{aligned} q_k(x):= (k_0^2+\epsilon k k_0) f(x), \end{aligned}$$

(22)

where f is the same spacetime cutoff used in $U_k$, $k_0$ is a fixed parameter that has the dimensions of a mass, and $\epsilon$ is a dimensionless parameter. This choice corresponds effectively to a linearisation of a regulator quadratic in the RG scale k, standard in the literature, around an arbitrary reference RG scale $k_0$. With this choice, $\partial _k{q_k}$ is independent on k. We furthermore observe that the contribution proportional to $k_0^2$ is constant in k and it can always be reabsorbed in a redefinition of the mass of the free theory. Many other choices, like the more usual $q_k(x) = k^2 f(x)$ can be brought to the same case using $k^2$ in the equation in place of $k_0^2+\epsilon k k_0$.

The function u in (21) is a smooth function on compact spaces, and therefore the tame Fréchet space we are working with is $F=C^{\infty }(X\times [a,b])$, where X is a compact space in ${\mathbb {R}}$ which contains all possible values of $\phi$ and k is in the positive interval $[a,b] \subset {\mathbb {R}}^+$, because the sign of k is always assumed to be positive.

This space is Fréchet with seminorms

$$\begin{aligned} \left\| u\right\| _n = \sum _j^n \sum _{|\alpha | = j }\sup _{\phi , k} \left| D^\alpha u(\phi , k) \right| , \end{aligned}$$

(23)

where $\alpha \in {\mathbb {N}}\times {\mathbb {N}}$ is a multi-index, and thus the derivatives $D^\alpha$ are taken both in $\phi$ and k. The space F is tame because it is the space of smooth functions over a compact space [19].

To have uniqueness of the solution of (20) we need to provide suitable boundary conditions and to prescribe initial values. We thus assume that

$$\begin{aligned} u(\phi ,a)=\psi , \qquad \left. u\right| _{\partial X \times [a,b]} = \beta \end{aligned}$$

(24)

where $\psi$ is a given smooth function on X and $\beta$ is a given smooth function on $\partial X \times [a,b]$ compatible with $\psi$. To impose the initial values and the boundary conditions we introduce the tame Fréchet subspace of F

$$\begin{aligned} F_0:= \{ u\in F \,|\, u(\phi ,a)=0,\, \left. u\right| _{\partial X \times [a,b]} = 0 \}. \end{aligned}$$

The solution ${\tilde{u}}$ of (20) we are looking for is then of the form

$$\begin{aligned} {\tilde{u}} = u_b+u,\qquad u\in F_0 \end{aligned}$$

(25)

where $u_b$ is a given element of F selected in such a way that it satisfies the boundary conditions and respects the initial values given in (24).

We also further assume that $\partial ^2_\phi u$ and its second derivatives lie in a suitably small neighbourhood of 0, that is, $\left\| u\right\| _4 \le A$ for some positive constant A.

To prove existence of local solutions of the RG flow equations, we make use of Nash–Moser theorem in Hamilton’s formulation. To do so, we need to prove the validity of the strong assumptions of the theorem. We already remarked that u lives in a suitable tame Fréchet space $F_0$. The RG flow equations, acting on u, determine a RG operator $\mathcal{R}\mathcal{G}: {\mathcal {U}} \subset F_0 \rightarrow F$, given below in Definition 4.1. To use the Nash–Moser theorem, we further need to prove that i) the RG operator acting on u, is a tame smooth operator; ii) that its linearisation $D\mathcal{R}\mathcal{G}(u): F_0 \rightarrow F$ is tame smooth as well; iii) that the linearisation of the RG operator admits a unique inverse $D\mathcal{R}\mathcal{G}^{-1}(u):F\rightarrow F_0$ for every $u\in U$, and that the inverse is tame smooth. We will prove each of these assumptions in the following propositions. Since we can prove the assumptions of Nash–Moser theorem, it follows that the RG operator admits a local inverse. The solution of the RG flow equations is then determined as the unique solution of the equation [19]

$$\begin{aligned} \frac{d}{dt} u_t = - c D\mathcal{R}\mathcal{G}^{-1}(S_t u_t) S_t \left( \mathcal{R}\mathcal{G}(u_t) \right) \end{aligned}$$

(26)

with a given $u_0=0$. In this equation c is a positive fixed arbitrary constant and $S_t$ is a smoothing operator, see e.g. [19].

If $\mathcal{R}\mathcal{G}$ is a smooth tame map, if $D\mathcal{R}\mathcal{G}(u)$ admits an unique inverse for every u in a suitable subset of $F_0$, and if the inverse $D\mathcal{R}\mathcal{G}^{-1}$ is also tame, a unique solution of equation (26) exists for all t such that the limit of the sequence of approximated solutions converges to a solution of the RG flow equations, $\lim _{t\rightarrow \infty } u_t = u_\infty$ is such that $\mathcal{R}\mathcal{G}(u_\infty )=0$, as it is proved in [19].

4.2 The RG operator is tame smooth

Following the strategy presented in the last section, we start with a formal definition.

Definition 4.1

(RG operator) Let $u_b\in F$ be such that it satisfies the initial values and the boundary conditions given in (24). The RG operator $\mathcal{R}\mathcal{G}: {\mathcal {U}} \subset F_0 \rightarrow F$ is defined as

$$\begin{aligned} \mathcal{R}\mathcal{G}: u \mapsto \mathcal{R}\mathcal{G}(u):= \partial _k (u+u_b) - G_k(\partial _\phi ^2 (u+u_b)) \end{aligned}$$

(27)

where $G_k$ is given in (21).

As a first step in the proof of existence of local solutions, we want to prove that the RG operator is of the right class to apply the Nash–Moser theorem, i.e., it is tame smooth. In order to prove it, we start considering $G_k$ in (21). We observe that since with $q_k$ given in (22) $\partial _k q_k$ is constant in $(\phi ,k)$, we have that ${G}_k$ depends on $(\phi ,k)$ only through $\partial _\phi ^2 {\tilde{u}}$, where we recall that ${{\tilde{u}}} = u + u_b$. Consider now $G_k$, written as

$$\begin{aligned} G_k(\partial _{\phi }^2 {\tilde{u}}) = -\frac{1}{2\left| \!\left| \!\left| f\right| \!\right| \!\right| _1} \int _{{\mathcal {M}}} \textrm{d}\mu _x \partial _k q_k(x):{G}_k:(x,x) \end{aligned}$$

for ${\tilde{u}}\in F$. We analyse how $:{G}_k:(x,x)$ depends on $\partial _\phi ^2 {\tilde{u}}$. Notice that $:{G}_k:(x,y)$ can be given explicitly as

$$\begin{aligned}:{G}_k:(x,y):= \int \textrm{d}\mu _{z_1} \textrm{d}\mu _{z_2} (\delta - \Delta ^U_{R} {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}})(x,z_1) w(z_1,z_2) (\delta - {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}\Delta ^U_{A} )(z_2, y) \ \end{aligned}$$

where $\delta$ is the Dirac delta function (the integral kernel of the identity). Recalling that $(1 - \Delta ^U_{R} {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}})\circ (1 + \Delta _R {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}) =1$, using the recursive relations given in (16), we obtain a recursive formula for $:{G}_k:(x,y)$:

$$\begin{aligned} :{G}_k:(x,y) = {\tilde{w}}(x,y) - \int \textrm{d}\mu _z \Delta _R {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}(x,z):{G}_k:(z,y), \end{aligned}$$

(28)

where

$$\begin{aligned} {\tilde{w}}(x,y):= \int \textrm{d}\mu _z w(x,z) (\delta -{{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}\Delta _A^U )(z,y). \end{aligned}$$

This recursive relation can be used to get estimates of $:G_k(x,y):$, for x, y contained in some compact region of the spacetime ${\mathcal {M}}$. First of all, we can prove the following Lemma, on estimates of the retarded propagator $\Delta _R^U g$ acting on some compactly supported smooth function g.

Lemma 4.1

Let ${\mathcal {M}}$ be a ultra-static spacetime and let t be a time function. Let ${\tilde{u}}\in F$, and consider

$$\begin{aligned} h= \Delta ^U_R g \end{aligned}$$

where g is a compactly supported smooth function on ${\mathcal {M}}$. It then holds that h is a past-compact smooth function with compact support on every Cauchy surface $\Sigma$. Moreover, recalling (16), writing h as

$$\begin{aligned} \begin{aligned} h= (1-\Delta ^U_R U^{(2)}) \xi , \end{aligned} \end{aligned}$$

where $\xi = \Delta _R g$, the following estimates hold:

$$\begin{aligned} \begin{aligned} \left| \!\left| \!\left| h\right| \!\right| \!\right| ^{t}_\infty \le c {\left| \!\left| \!\left| h\right| \!\right| \!\right| _{2,2}^{t}} \le c {\left| \!\left| \!\left| \xi \right| \!\right| \!\right| _{2,2}^{t}} e^{C |\partial _{\phi }^2 {\tilde{u}}| } \le c e^{C \left\| {\tilde{u}}\right\| _2 } {\left| \!\left| \!\left| \xi \right| \!\right| \!\right| _{2,2}^{t}} \end{aligned} \end{aligned}$$

(29)

and

$$\begin{aligned} \left| \!\left| \!\left| h\right| \!\right| \!\right| ^{t}_\infty \le c {\left| \!\left| \!\left| h\right| \!\right| \!\right| _{2,2}^{t}} \le c e^{C \left\| {\tilde{u}}\right\| _2 } \int _{-\infty }^t \textrm{d}\tau (t-\tau ) {\left| \!\left| \!\left| g\right| \!\right| \!\right| _{2,2}^{\tau }} \le {\tilde{C}}e^{C \left\| {\tilde{u}}\right\| _2 } \sup _{\tau \le t} {\left| \!\left| \!\left| g\right| \!\right| \!\right| _{2,2}^{\tau }} . \nonumber \\ \end{aligned}$$

(30)

In the above inequalities, $C>0$ is a positive constant, which depends on the support of f in U but not on ${\tilde{u}}$; similarly, ${\tilde{C}}>0$ depends only on the support of g and c is positive and does not depend on U. Furthermore, ${\left| \!\left| \!\left| \cdot \right| \!\right| \!\right| _{2,2}^{t}}$ is the norm on the Sobolev space $W_{2,2}(\Sigma _t)$ and $\left| \!\left| \!\left| \cdot \right| \!\right| \!\right| _\alpha ^{t}$ is the norm on $L^{\alpha }(\Sigma _t)$ where $\Sigma _t=\{x \in {\mathcal {M}}| t(x)=t\}$ is the Cauchy surface at fixed time t.

Proof

We recall that both $\Delta _R$ and $\Delta ^U_R$ map past-compact smooth functions to past-compact smooth functions, hence both $\xi = \Delta _R g$ and $h=\Delta ^U_R g$ are smooth and past-compact. We also recall that

$$\begin{aligned} \Delta ^U_R = \Delta _R (1-{{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}\Delta ^U_R ) = (1-\Delta ^U_R {{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}})\Delta _R . \end{aligned}$$

Since ${{\,\mathrm{{\textit{U}}^{(2)}_{\textit{k}}}\,}}= f \partial _\phi ^2 {\tilde{u}}$, where f is a smooth compactly supported function and $\partial _\phi ^2 {\tilde{u}}$ is constant on ${\mathcal {M}}$, the following recursive relation holds

$$\begin{aligned} \begin{aligned} h = \Delta _R g - \Delta _R {{\,\mathrm {{\textit{U}}^{(2)}_{\textit{k}}}\,}}h = \xi - \Delta _R {{\,\mathrm {{\textit{U}}^{(2)}_{\textit{k}}}\,}}h. \end{aligned} \end{aligned}$$

Now, let D be the (positive) Laplace operator on $\Sigma _t$ constructed with the induced metric on $\Sigma _t$, and define $\omega =\sqrt{D+m^2}$ as the square root of the positive operator $D+m^2$. Hence

$$\begin{aligned} \begin{aligned} h(t,{{\textbf {x}}}) = \xi (t,{{\textbf {x}}}) - \partial _\phi ^2 {\tilde{u}} \int _{-\infty }^{t} \text {d}\tau \frac{ \sin {(\omega (t-\tau ))} }{\omega } (f h) (\tau ,{{\textbf {x}}} ) . \end{aligned} \end{aligned}$$

We thus have

$$\begin{aligned} \begin{aligned} \left| \!\left| \!\left| h\right| \!\right| \!\right| ^t_{2} \le \left| \!\left| \!\left| \xi \right| \!\right| \!\right| ^t_{2} + |\partial _\phi ^2 {\tilde{u}} | \int _{-\infty }^{t} \text {d}\tau (t-\tau ) \Vert f\Vert ^\tau _{\infty } \left| \!\left| \!\left| h\right| \!\right| \!\right| ^\tau _2, \end{aligned} \end{aligned}$$

or, passing to the Sobolev norm $\left| \!\left| \!\left| h\right| \!\right| \!\right| _{2,2}^t= \left| \!\left| \!\left| h\right| \!\right| \!\right| _2^t+ \left| \!\left| \!\left| D h\right| \!\right| \!\right| ^t_2$, we have

$$\begin{aligned} \begin{aligned} \left| \!\left| \!\left| h\right| \!\right| \!\right| ^t_{2,2}&\le \left| \!\left| \!\left| \xi \right| \!\right| \!\right| ^t_{2,2} + |\partial _\phi ^2 {\tilde{u}} | \int _{-\infty }^{t} \text {d}\tau (t-\tau ) \left| \!\left| \!\left| f h\right| \!\right| \!\right| ^\tau _{2,2} \\ {}&\le \left| \!\left| \!\left| \xi \right| \!\right| \!\right| ^t_{2,2} + |\partial _\phi ^2 {\tilde{u}} | \int _{-\infty }^{t} \text {d}\tau (t-\tau ) (\Vert f\Vert ^\tau _{\infty } + \sup _{i} 2 \Vert \partial _i f\Vert ^\tau _{\infty } + \Vert D f\Vert ^\tau _{\infty }) \left| \!\left| \!\left| h\right| \!\right| \!\right| ^\tau _{2,2} \\ {}&\le \left| \!\left| \!\left| \xi \right| \!\right| \!\right| ^t_{2,2} + C |\partial _\phi ^2 {\tilde{u}} | \int _{a}^t \text {d}\tau \left| \!\left| \!\left| h\right| \!\right| \!\right| ^\tau _{2,2} \end{aligned} \end{aligned}$$

where $a=\inf _{x\in {{\,\mathrm{\text {supp}}\,}}f}\{ t(x) \}$ and for a suitable positive constant C independent on $\partial _\phi ^2 {\tilde{u}}$. C is in fact finite because f is smooth and with compact support on ${\mathcal {M}}$.

Applying the Grönwall Lemma in integrated form to the previous inequality we obtain

$$\begin{aligned} \begin{aligned} \left| \!\left| \!\left| h\right| \!\right| \!\right| ^t_{2,2} \le \left| \!\left| \!\left| \xi \right| \!\right| \!\right| ^t_{2,2} e^{C |\partial _\phi ^2 {\tilde{u}}| }. \end{aligned} \end{aligned}$$

To conclude the proof of the first inequality (29), we observe that $\Sigma _t$ is a three dimensional space, and so by standard arguments we have

$$\begin{aligned} \left| \!\left| \!\left| h\right| \!\right| \!\right| ^t_\infty \le \left| \!\left| \!\left| \hat{h}\right| \!\right| \!\right| ^t_1 \le \left| \!\left| \!\left| (1+D){h}\right| \!\right| \!\right| _2 \left| \!\left| \!\left| (1+D)^{-1}\right| \!\right| \!\right| _2 \le c \left| \!\left| \!\left| {h}\right| \!\right| \!\right| _{2,2}. \end{aligned}$$

where the $\left| \!\left| \!\left| (1+D)^{-1}\right| \!\right| \!\right| _2$ is the L-2 norm of $(1+{D})^{-1}$. To prove (30) we use (29) for $\xi = \Delta _R g$. Recalling that

$$\begin{aligned} \begin{aligned} \xi (t,{{\textbf {x}}}) = \partial _\phi ^2 {\tilde{u}} \int _{-\infty }^{t} \text {d}\tau \frac{ \sin {(\omega (t-\tau ))} }{\omega } (g) (\tau ,{{\textbf {x}}} ). \end{aligned} \end{aligned}$$

and taking the Sobolev norms we have

$$\begin{aligned} \left| \!\left| \!\left| \Delta _R g\right| \!\right| \!\right| _{2,2}^t \le \int _{-\infty }^t \textrm{d}\tau (t-\tau ) \left| \!\left| \!\left| g\right| \!\right| \!\right| _{2,2}^{\tau }. \end{aligned}$$

$\square$

Starting from the above analysis and the previous Lemma, we can prove that the RG operator is tame smooth.

Proposition 4.2

Assume that ${\mathcal {U}}\subset F_0$ is a small neighbourhood of 0 so that for $u\in {\mathcal {U}}$, $\left\| u \right\| _2 < A$ for some constant A. Then the RG operator is a smooth tame map.

Proof

We start considering ${\tilde{u}}=u_b+u$ for $u\in F_0$ and for a given $u_b$ which satisfies (24) so that ${\tilde{u}}\in F$ and it satisfies the prescribed initial values and boundary conditions. We recall that from (27)

$$\begin{aligned} \mathcal{R}\mathcal{G}(u) = \partial _k (u+u_b) - G_k(\partial _\phi ^2 (u+u_b)) \end{aligned}$$

where $G_k$ is given in (21). To prove that $\mathcal{R}\mathcal{G}$ is tame smooth we just need to prove that $G_k$ is tame smooth for ${\tilde{u}}\in u_b+{\mathcal {U}}$. We have actually the following lemma

Lemma 4.3

The functional $G_k$ is a smooth function of $\partial _\phi ^2{\tilde{u}}$. Furthermore, it is tame smooth for ${\tilde{u}}\in u_b+{\mathcal {U}}$.

Proof

We observe that $\partial _k q_k$ is constant on $X\times [a,b]$; hence, recalling the definition of $G_k$ given in (21), we have that $G_k(\partial _\phi ^2{\tilde{u}})$ as a function on $X\times [a,b]$ depends on $(\phi ,k)$ only through $\partial _\phi ^2{\tilde{u}}$, that is, $G_k(\partial _\phi ^2 {\tilde{u}})(\phi ,k)=G_k(\partial _\phi ^2 {\tilde{u}}(\phi ,k))$. We also observe that $G_k(\partial _\phi ^2{\tilde{u}})$ depends smoothly on ${\tilde{u}}\in F$. Actually, the $n-$th order functional derivative of ${\tilde{G}}_k({\tilde{u}})=G_k(\partial _\phi ^2 {\tilde{u}})$ with respect to ${\tilde{u}}$ can be explicitly computed and it is well defined for every n; in fact, it is given by

$$\begin{aligned} \begin{aligned}&{\tilde{G}}_k^{(n)}(v_1,\dots ,v_n) = \frac{(-1)^{n+1}}{\left| \!\left| \!\left| f\right| \!\right| \!\right| _1} n! \sum _{l=0}^n \int _{{\mathcal {M}}}\textrm{d}\mu _x \partial _kq_k(x) \cdot \\&\qquad \left\{ (\Delta ^U_R f)^l \otimes (\Delta ^U_R f)^{n-l} \circ (1-\partial _\phi ^2 {\tilde{u}} \Delta _R^U f) \otimes (1-\partial _\phi ^2 {\tilde{u}} \Delta _R^U f)(w)(x,x) \right\} \prod _{j=1}^n\partial _\phi ^2 v_j\\&\quad =: A_n({\tilde{u}})\prod _{j=1}^n\partial _\phi ^2 v_j . \end{aligned} \nonumber \\ \end{aligned}$$

(31)

In the last formula, f in $\Delta ^U_R f$ is a multiplicative operator, and $A_n({\tilde{u}})$ are suitable functionals of ${\tilde{u}}$. Notice that both the cutoff functions f and $q_k$ have compact support. w is a smooth function on ${\mathcal {M}}^2$. Hence for every ${\tilde{u}}\in F$ the integral which defines $A_n$ can always be taken and it gives a finite bounded result. We thus have that $G_k$ is a smooth function of $\partial _\phi ^2 {\tilde{u}}$.

To prove that $G_k(\partial _\phi ^2{\tilde{u}})$ is also tame, we proceed as follows. We recall that $\Vert u\Vert _2< \Vert u \Vert _4 < A$, and that $G_k$ depends on $\phi$ and k only through $\partial _\phi ^2{{\tilde{u}}}$, because $\partial _k q_k = f$. By direct inspection, we have that

$$\begin{aligned} \Vert G_k\Vert _n < \Vert A_0({\tilde{u}})\Vert _0 + \sum _{p=1}^n \sum _{l=1}^p \Vert A_l({\tilde{u}})\Vert _0 \Vert (\partial _\phi ^2 {\tilde{u}})^l \Vert _{p-l} . \end{aligned}$$

(32)

To estimate $\Vert (\partial _\phi ^2 {\tilde{u}})^l \Vert _{p-l}$, we use Leibniz rule together with an interpolating argument (See Corollary 2.2.2 in [19]), stating that, for every $f,g\in F$,

$$\begin{aligned} \Vert f \Vert _n\Vert g\Vert _m\le C (\Vert f\Vert _{n+m}\Vert g\Vert _0+\Vert f\Vert _0 \Vert g\Vert _{n+m}) . \end{aligned}$$

Hence, by Leibniz rule, we have that

$$\begin{aligned} \Vert \partial _\phi ^2{\tilde{u}}^l\Vert _{r}\le C \sum _{R=(r_1,\dots , r_l), |R|=r} \prod _{i=1}^l \Vert \partial _\phi ^2{\tilde{u}}\Vert _{r_i} \le C' \Vert \partial _\phi ^2{\tilde{u}}\Vert _r\Vert \partial _\phi ^2{\tilde{u}}\Vert _0^{l-1} . \end{aligned}$$

Using this in (32) we get

$$\begin{aligned} \Vert G_k\Vert _n \ {}&< C\left( \Vert A_0({\tilde{u}})\Vert _0 + \sum _{p=1}^n \sum _{l=1}^p \Vert A_l({\tilde{u}})\Vert _0 \Vert (\partial _\phi ^2 {\tilde{u}})\Vert _0^{l-1} \Vert {\tilde{u}} \Vert _{p+2} \right) \\&< C\left( \Vert A_0({\tilde{u}})\Vert _0 + \sum _{p=1}^n \sum _{l=1}^p \Vert A_l({\tilde{u}})\Vert _0 \Vert {\tilde{u}}\Vert _2^{l-1} \Vert {\tilde{u}} \Vert _{p+2} \right) \\&< C\left( 1+ \Vert {\tilde{u}}\Vert _{n+2}\right) \ , \end{aligned}$$

where in the last step we used the fact that $\Vert A_l({\tilde{u}})\Vert _0\le C(1+\Vert {\tilde{u}}\Vert _2)$. This last inequality is proved in the following Lemma 4.4. $\square$

Lemma 4.4

Consider the functionals $A_l({\tilde{u}})$ for ${\tilde{u}}\in F$ given in (31). If $\Vert {\tilde{u}}\Vert _2< A$, it holds that

$$\begin{aligned} \Vert A_l({\tilde{u}})\Vert _0\le C(1+\Vert {\tilde{u}}\Vert _2). \end{aligned}$$

Proof

To prove this result we observe that both $\partial _k q_k$ and f are smooth compactly supported functions on ${\mathcal {M}}$. The integral present in (31) is thus taken on a compact region, even if w is a smooth function supported in general everywhere on ${\mathcal {M}}^2$. Now, we need to estimate the action of each $\Delta ^U_R f$ and of $(1-\partial _\phi ^2 {\tilde{u}} \Delta _R^U f)$ by means of Lemma 4.1.

Actually, Lemma 4.1 implies that if g is a smooth past-compact function, the following estimates hold:

$$\begin{aligned} \left| \!\left| \!\left| (\Delta ^U_R f)^{n} g\right| \!\right| \!\right| ^t_{2,2} \le \sup _{\tau<t} \left| \!\left| \!\left| (\Delta ^U_R f)^{n-1} g\right| \!\right| \!\right| ^\tau _{2,2} {\tilde{C}}e^{C \Vert {\tilde{u}}\Vert _2} \le \sup _{\tau <t} \left| \!\left| \!\left| g\right| \!\right| \!\right| ^\tau _{2,2} {\tilde{C}}^n e^{n C \Vert {\tilde{u}}\Vert _2}, \end{aligned}$$

where the constant ${\tilde{C}}$ depends on f. Similarly,

$$\begin{aligned} \left| \!\left| \!\left| (1-\partial _\phi ^2 {\tilde{u}} \Delta _R^U f) g\right| \!\right| \!\right| ^t_{2,2}\le ce^{C \Vert {\tilde{u}}\Vert _2} . \end{aligned}$$

We now use these estimates in

$$\begin{aligned} \begin{aligned} a_{l_1,l_2}(x,y):= \left\{ (\Delta ^U_R f)^{l_1} \otimes (\Delta ^U_R f)^{l_2} \circ (1-\partial _\phi ^2 {\tilde{u}} \Delta _R^U f) \otimes (1-\partial _\phi ^2 {\tilde{u}} \Delta _R^U f)(\theta w \theta )(x,y) \right\} , \end{aligned} \end{aligned}$$

for $l_1+l_2 = n$, and where $\theta$ is a smooth compactly supported function which is equal to 1 in a region which contains the support of $q_k$ and f. Thanks to this choice, we can replace w in $G_k$ with $\theta w \theta$, getting

$$\begin{aligned} \sup _{x\in {{\,\mathrm{\text {supp}}\,}}f} | a_{l_1,l_2}(x,x)| \le \sup _{x,y\in {{\,\mathrm{\text {supp}}\,}}f} | a_{l_1,l_2}(x,y)| \le \sup _{t_x,t_y \in {{\,\mathrm{\text {supp}}\,}}f} \left| \!\left| \!\left| \theta w \theta \right| \!\right| \!\right| ^{(t_x,t_y)}_{4,2} c^2 {\tilde{C}}^n e^{(n+2) C \Vert {\tilde{u}}\Vert _2} \end{aligned}$$

where $\left| \!\left| \!\left| \cdot \right| \!\right| \!\right| ^{(t_x,t_y)}_{4,2}$ is the Sobolev norm for functions defined on $\Sigma _{t_x}\times \Sigma _{t_y}$. Using this estimate sufficiently many times in $A_l$, and recalling that $e^{ C \Vert {\tilde{u}}\Vert _2} \le C_1 (1+\Vert {\tilde{u}} \Vert _2)$ for a sufficiently large $C_1$ because $\Vert {\tilde{u}}\Vert _2 < A$, we have the thesis. $\square$

With this results, we can thus conclude the proof recalling that the linear combinations of smooth tame functionals is tame smooth. $\square$

4.3 The linearisation of the RG operator is tame smooth

The first derivative of the RG operator defines the linearised RG operator $L(u) v = D\mathcal{R}\mathcal{G}(u)v$: by direct inspection, it is given by the linear operator

$$\begin{aligned} D\mathcal{R}\mathcal{G}(u)v = \partial _k v - \sigma \partial ^2_\phi v, \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} \sigma (u)&:= \frac{1}{\left| \!\left| \!\left| f\right| \!\right| \!\right| _1} \int _{{\mathcal {M}}^2} \textrm{d}\mu _x\textrm{d}\mu _y \,\partial _k q_k(x) \Delta _R^U(x,y)f(y) \\&\quad \times \left\{ (1-\partial _\phi ^2 u \Delta _R^u f) \otimes (1-\partial _\phi ^2 u \Delta _R^u f)(w)(y,x) \right\} . \end{aligned} \nonumber \\ \end{aligned}$$

(33)

The function $\sigma$ as a function on $X\times [a,b]$ depends on $\phi$ through $\partial _\phi ^2 u$ and on k through $\partial _k q_k$ and $\partial _\phi ^2 u$. With the choice of $q_k$ given in (22) $\partial _k q_k$ is constant in k, and the only way in which $\sigma$ depends on $(\phi ,k)$ is through u.

Definition 4.2

Let $u_b\in F$ be a function which satisfies the initial values and boundary conditions given in (24), and let ${\mathcal {U}}$ be a neighbourhood of 0 in $F_0$. The linearised RG operator is defined as the map

$$\begin{aligned} L:(u_b+{\mathcal {U}})\times F_0\rightarrow F \end{aligned}$$

$$\begin{aligned} L(u)f:= \partial _k g - \sigma (u)\partial _\phi ^2 g, \end{aligned}$$

where $\sigma$ is the map defined in (33).

The following Proposition specifies some of the properties of $\sigma$ that will be useful in the analysis of L(u).

Proposition 4.5

The function $\sigma (u)$ is tame smooth.

Proof

The function $\sigma$ is linear in $q_k$ and $q_k$ is a smooth function of k: actually recalling (22) $q_k= k_0(\epsilon k +k_0) f(x)$, and so $\partial _k q_k$ is constant in $(\phi ,k)$. Hence, $\sigma$ depends on k and on $\phi$ only through u. Furthermore, the $n-$th order functional derivative $\sigma$ with respect to $\partial _\phi ^2 u$ is always well-defined because it equals the $n+1$ order functional derivative of $G_k$ with respect to $\partial _\phi ^2 u$, and we already proved in Lemma (4.3) that $G_k$ is a smooth function of $\partial _\phi ^2u$. Furthermore, $\sigma$ is a smooth function and it is tame with respect to u because it is related to the functional derivative of $G_k$, which is tame smooth as proven in Lemma 4.3. Hence $\sigma$ is tame smooth. $\square$

The next proposition shows that, by a suitable choice of smooth functions w (or, equivalently, by suitable choices of states), the assumptions that: i) $\sigma$ is larger than some positive constant c, and ii) that $\left\| u\right\| _2 \le A$ is in some small neighbourhood of 0, hold.

Proposition 4.6

If the boundary conditions given in (24) are such that $\Vert \beta \Vert _2+\Vert \psi \Vert _2 < \epsilon$ for a sufficiently small $\epsilon$ and if $u_b$ in (25) is chosen to be such that $\Vert u_b\Vert _2\le \epsilon$, then for certain choices of the function $w\in C^{\infty }({\mathcal {M}}^2)$, it exists a neighbourhood ${\mathcal {U}}\subset F_0$ such that for every $u\in {\mathcal {U}}$, $\sigma (u_b+u)\ge c>0$ and $\Vert u\Vert _2< A=\epsilon$.

Proof

We recall that

$$\begin{aligned} \sigma (0) = \frac{1}{\left| \!\left| \!\left| f\right| \!\right| \!\right| _1} \int _{{\mathcal {M}}^2} \textrm{d}\mu _x\textrm{d}\mu _y \,\partial _k q_k(x) f(y) \Delta _R(x,y)(w)(y,x). \end{aligned}$$

$\sigma (0)$ is linear in w and it cannot be identically 0 for every w, hence it is possible to choose a w such that $\sigma (0)\ge (2\epsilon C + c)>0$, where $C > \sup _{\lambda \in [0,1]} \Vert \sigma ^{(1)}(\lambda (u+u_b))\Vert _0$. Moreover, $\sigma$ depends smoothly on u. We can choose $u_b$ so that $\Vert u_b\Vert _2\le (\Vert \beta \Vert _2+\Vert \psi \Vert _2 ) < \epsilon$ and we can choose a sufficiently small ${\mathcal {U}} \subset F_0$ such that every $u\in {\mathcal {U}}$ is such that $\Vert u\Vert _2<\epsilon$. Hence, the smoothness of $\sigma (u)$ implies that

$$\begin{aligned} \begin{aligned} \sigma (u)&= \sigma (0) + \int _0^1\textrm{d}\lambda \, \frac{d}{d\lambda }\sigma (\lambda (u+u_b)) \\&\ge \sigma (0) - \sup _{\lambda } \Vert \sigma ^{(1)}(\lambda (u+u_b))(u+u_b)\Vert _0 . \end{aligned} \end{aligned}$$

(34)

We notice that $\sigma ^{(1)}$ is related to $G_k^{(2)}$ and it can be given in terms of the functions $A_n$ with $n=2$ defined in (31). More explicitly, it takes the form

$$\begin{aligned} \begin{aligned}&\sigma ^{(1)}({\tilde{u}})(v) = \frac{(-1)^{3}}{\left| \!\left| \!\left| f\right| \!\right| \!\right| _1} 2 \sum _{l=0}^2 \int _{{\mathcal {M}}}\textrm{d}\mu _x \partial _kq_k(x) \cdot \\&\qquad \left\{ (\Delta ^U_R f)^l \otimes (\Delta ^U_R f)^{2-l} \circ (1-\partial _\phi ^2 {\tilde{u}} \Delta _R^U f) \otimes (1-\partial _\phi ^2 {\tilde{u}} \Delta _R^U f)(w)(x,x) \right\} \partial _\phi ^2 v\\&\quad = A_2({\tilde{u}})\partial _\phi ^2 v. \end{aligned} \nonumber \\ \end{aligned}$$

(35)

Thanks to the estimate given in Lemma 4.4, we have that

$$\begin{aligned} \begin{aligned} \Vert \sigma ^{(1)}(\lambda (u_b+u))(u_b+u)\Vert _0&\le \Vert A_2\Vert _0 \Vert (u_b+u)\Vert _0 \\&\le C'(1+\Vert u_b+u\Vert _2) \Vert (u_b+u)\Vert _0 \\&\le C'' \Vert u_b+u\Vert _2 \end{aligned} \end{aligned}$$

for suitable constants $C'$ and $C''$ depending on A. Using this estimate in (34), and recalling the choices we made for w in $\sigma (0)$, we obtain that for a suitable $c'$

$$\begin{aligned} \sigma (u) \ge \sigma (0) - \sup _{\lambda } \Vert \sigma ^{(1)}(\lambda (u+u_b))(u+u_b)\Vert _0 \ge c' >0, \end{aligned}$$

thus concluding the proof. $\square$

Remark

Notice that thanks to Proposition 4.6, $\sigma$ can be chosen to be positive. In applications to physics, when w is obtained as the smooth part of the two-point function of a quantum state, it is not obvious that the choices necessary to have $\sigma$ positive can be made. In spite of this problem we observed in [9] that this is the case in many physically sensible states, also thanks to the freedom in the split of the smooth part from the singular one present in any Hadamard two-point function. This freedom is related to the ordinary renormalization freedom when coinciding point limits are taken.

Now we can prove the proposition

Proposition 4.7

The linearisation of the RG operator

$$\begin{aligned} L(u) v = \partial _k v - \sigma \partial _\phi ^2 v \end{aligned}$$

is tame smooth.

Proof

Since L acts as a second order linear differential operator, its $n-$th order seminorm is controlled by the $n+2-$th order seminorm of v. Using the Lebiniz rule and an interpolating argument (see e.g. in Corollary 2.2.2 in [19])

$$\begin{aligned} \left\| L(u)v\right\| _n \le \left\| v\right\| _{n+1} + C\left( \Vert \sigma \Vert _0 \left\| v\right\| _{n+2}+\Vert \sigma \Vert _{n+2} \left\| v\right\| _{0} \right) \end{aligned}$$

where C is a constant. $\sigma$ is tame smooth and the composition of tame smooth maps is tame smooth, and thus $L:({\mathcal {U}})\times F_0 \rightarrow F$ is tame smooth. $\square$

4.4 The linearisation of the RG operator is invertible, and the inverse is tame smooth

If $\sigma \ge c>0$ on $X\times [a,b]$, the linearised RG operator L(u) on $X\times [a,b]$ has the form of a parabolic equation. The existence and uniqueness of an inverse which satisfies the chosen boundary conditions

$$\begin{aligned} E(g)(\phi ,a) = 0,\qquad E(g)|_{\partial X\times [a,b]}=0, \qquad g\in C^\infty _0(X\times [a,b]) \end{aligned}$$

is known [17]. Furthermore, by an application of the maximum principle, it is possible to prove that E is continuous with respect to the uniform norm; see e.g. Section 3 of Chapter 2 in [17]. We collect these results in the following Proposition.

Proposition 4.8

Consider the linearised RG operator L. Assume that $\sigma (u)$ is positive for every $u\in {\mathcal {U}}\subset F$. Then, it exists an unique inverse $E:F \rightarrow F_0$ which is compatible with the initial and boundary conditions, thus satisfying

$$\begin{aligned} E(L(g))=L(E(g))=g, \qquad g\in F_0 . \end{aligned}$$

Moreover, the inverse is continuous with respect to the uniform norm. More precisely, it exists a positive constant $C>0$ such that

$$\begin{aligned} \Vert E(g) \Vert _0 < C \Vert g\Vert _0. \end{aligned}$$

We now pass to analyse the regularity of E, which is a necessary condition to apply the Nash–Moser Theorem.

Proposition 4.9

Consider the case where $\sigma \ge c > 0$, let $u\in {\mathcal {U}}\subset F_0$ such that $\Vert u\Vert _4\le A$, and assume that $\sup _{i\in \{\phi ,k\}} |D_i \log \sigma (u)|< \epsilon$ with a sufficiently small $\epsilon$. The inverse E of the linearized RG operator L is tame smooth.

Proof

We first observe that L(u) depends on u only through $\sigma$. Furthermore, $\sigma$ is a tame map of u. The composition of tame maps is tame, and so, to prove the statement, it suffices to study how L depends on $\sigma$. To this end, with a little abuse of notation in this proof we shall denote L(u) by $L(\sigma (u))$ and we estimate how L depends on $\sigma$. Consider $L(\sigma )(v)=g$. We look for an estimate which permits to control the higher derivative of v with those of g. We start with two Lemmas.

Lemma 4.10

Under the hypothesis of Proposition 4.9, the following estimate holds.

$$\begin{aligned} \Vert v\Vert _1 < C \left( \Vert g\Vert _1 + \Vert \sigma \Vert _1 \Vert g\Vert _0 \right) \end{aligned}$$

Proof

The uniform continuity of E stated in Proposition 4.8 implies that if $L(\sigma )v=g$,

$$\begin{aligned} \Vert v \Vert _0 < C \Vert g \Vert _0. \end{aligned}$$

We apply this continuity result to Dv where $D\in \{\partial _\phi , \partial _k\}$. We have

$$\begin{aligned} \Vert D v\Vert _0 < C \Vert L(\sigma ) D v\Vert _0. \end{aligned}$$

We observe that

$$\begin{aligned} \begin{aligned} L(\sigma )Dv&= D L(\sigma )v - D(\sigma ) \partial _\phi ^2 v \\&= D L(\sigma )v + \frac{D(\sigma )}{\sigma }\left( L(\sigma )v - \partial _k v \right) \end{aligned} \end{aligned}$$

(36)

Hence, the uniform continuity of E and the fact that $\sigma \ge c>0$ implies that

$$\begin{aligned} \Vert Dv\Vert _0&< C \left( \Vert D L(\sigma ) v\Vert _0 + \Vert D \log (\sigma ) \Vert _0 (\Vert L(\sigma ) v\Vert _0 + \Vert v\Vert _1 ) \right) \\&< C \left( \Vert D g\Vert _0 + \Vert D\log (\sigma ) \Vert _0 (\Vert g\Vert _0 + \Vert v\Vert _1 ) \right) \end{aligned}$$

considering all possible D, using the uniform continuity of E and the fact that $1/\sigma > 1/c'$, we obtain

$$\begin{aligned} \Vert v\Vert _1&\le \left( \Vert v\Vert _0 + \sum _{D\in \{\partial _\phi ,\partial _k\}} \Vert Dv\Vert _0\right) < C \left( \Vert g\Vert _1 + \Vert \sigma \Vert _1 \Vert g\Vert _0 + \sup _D\Vert D \log (\sigma ) \Vert _0 \Vert v\Vert _1 \right) \end{aligned}$$

hence

$$\begin{aligned} (1-C \sup _D\Vert D \log (\sigma ) \Vert _0)\Vert v\Vert _1 < C \left( \Vert g\Vert _1 + \Vert \sigma \Vert _1 \Vert g\Vert _0 \right) . \end{aligned}$$

Notice that by hypothesis, $\sup _D\Vert D \log (\sigma ) \Vert _0 \le \epsilon$, hence, if $\epsilon$ is chosen sufficiently small

$$\begin{aligned} (1-C \sup _D\Vert D \log (\sigma ) \Vert _0)\ge c'>0 \end{aligned}$$

(37)

and

$$\begin{aligned} \Vert v\Vert _1&< \frac{1}{(1-C \sup _D\Vert D \sigma \Vert _0)} C \left( \Vert g\Vert _1 + \Vert \sigma \Vert _1 \Vert g\Vert _0 \right) \end{aligned}$$

from which the thesis follows. $\square$

Lemma 4.11

Under the hypothesis of Proposition 4.9, it holds that for every n

$$\begin{aligned} \Vert v\Vert _n < C\left( \Vert g\Vert _n + \Vert g\Vert _0 \Vert \sigma \Vert _{n+1} \right) . \end{aligned}$$

(38)

Proof

We prove it by induction. The case $n=1$ follows from Lemma 4.10 and the standard property $\Vert \sigma \Vert _1 \le \Vert \sigma \Vert _2$. We assume now that inequality (38) holds up n. To prove that it holds also for the case $n+1$ we apply it to Dv where $D\in \{\partial _\phi , \partial _k\}$. We have

$$\begin{aligned} \Vert D v\Vert _n < C\left( \Vert L(\sigma )Dv \Vert _n + \Vert L(\sigma )Dv\Vert _0 \Vert \sigma \Vert _{n+1} \right) . \end{aligned}$$

(39)

Recalling (36), by Leibniz rule, the interpolating argument (Corollary 2.2.2 in [19]) and the fact that $\sigma \ge c'>0$ we have

$$\begin{aligned} \Vert L(\sigma )Dv \Vert _n&< \Vert g\Vert _{n+1} + \Vert {D(\log \sigma )} g\Vert _n + \Vert D(\log \sigma ) \partial _k v \Vert _n \\&< \Vert g\Vert _{n+1} + C\left( \Vert \log {\sigma }\Vert _0 \Vert g\Vert _{n+1} + \Vert \log {\sigma }\Vert _{n+1} \Vert g\Vert _0 + \Vert \frac{D(\sigma )}{\sigma }\Vert _0\Vert \partial _k v \Vert _n \right. \\&\quad + \left. \Vert \frac{D(\sigma )}{\sigma }\Vert _n \Vert \partial _k v \Vert _0 \right) \\&< C\left( \bigg (1+\Vert \log {\sigma }\Vert _0 \right) \Vert g\Vert _{n+1} + \Vert \log {\sigma }\Vert _{n+1} \Vert g\Vert _0 + \Vert D(\log \sigma )\Vert _0 \Vert v \Vert _{n+1} \\&\quad + \Vert \frac{D(\sigma )}{\sigma }\Vert _n \Vert \partial _k v \Vert _0 \bigg ) \end{aligned}$$

From the last inequality, using the results of Lemma 4.10, it thus follows that

$$\begin{aligned} \begin{aligned} \Vert L(\sigma )Dv \Vert _n&< \left( 1+\Vert \log {\sigma }\Vert _0 \right) \Vert g\Vert _{n+1} + \Vert \log {\sigma }\Vert _{n+1} \Vert g\Vert _0 \\&\quad + \Vert D(\log \sigma )\Vert _0 \Vert v \Vert _{n+1} + \Vert \frac{D(\sigma )}{\sigma }\Vert _n (\Vert g\Vert _1+ \Vert \sigma \Vert _1\Vert g\Vert _0) \end{aligned} \end{aligned}$$

(40)

Furthermore from (36) and Lemma 4.10, we can prove that

$$\begin{aligned} \begin{aligned} \Vert L(\sigma )Dv\Vert _0&< \Vert D L(\sigma ) v\Vert _0 + \Vert D\sigma \Vert _0 (\Vert L(\sigma ) v\Vert _0 + \Vert v\Vert _1 ) \\&< \Vert D g\Vert _0 + \Vert D\sigma \Vert _0 (\Vert g\Vert _0 + \Vert v\Vert _1 ) \\&< \Vert g\Vert _1 + \Vert \sigma \Vert _1 ((1+\Vert \sigma \Vert _1)\Vert g\Vert _0 + \Vert g\Vert _1) . \end{aligned} \end{aligned}$$

(41)

Hence, combining the two inequalities (40) and (41) in (39)

$$\begin{aligned} (1-C \Vert D(\log (\sigma ))\Vert _0) \Vert v\Vert _{n+1}< & {} C\left[ \left( 1+\Vert \log {\sigma }\Vert _0 \right) \Vert g\Vert _{n+1} + \Vert \log {\sigma }\Vert _{n+1} \Vert g\Vert _0 + \right. \\{} & {} \left. + \Vert D(\log {\sigma })\Vert _n (\Vert g\Vert _1+ \Vert \sigma \Vert _1\Vert g\Vert _0) + \right. \\{} & {} \left. +\left( \Vert g\Vert _1 +\Vert \sigma \Vert _1 (\Vert g\Vert _0 + \Vert \sigma \Vert _1\Vert g\Vert _0 + \Vert g\Vert _1) \right) \Vert \sigma \Vert _{n+1} \right] \end{aligned}$$

Notice that, as stated in (37), $(1-C \Vert D(\log (\sigma ))\Vert _0)>0$, and so

$$\begin{aligned} \Vert v\Vert _{n+1}< & {} C\left[ \left( \left( 1+\Vert \log {\sigma }\Vert _0 \right) \Vert g\Vert _{n+1} + \Vert \log {\sigma }\Vert _{n+1} \Vert g\Vert _0 + \right. \right. \\{} & {} \left. \left. + \Vert D(\log \sigma )\Vert _n (\Vert g\Vert _1+ \Vert \sigma \Vert _1\Vert g\Vert _0) \right) + \right. \\{} & {} \left. +\left( \Vert g\Vert _1 + \Vert \sigma \Vert _1 ((1 + \Vert \sigma \Vert _1)\Vert g\Vert _0 + \Vert g\Vert _1 ) \right) \Vert \sigma \Vert _{ n+1} \right] . \end{aligned}$$

By the interpolating argument, it holds that $\Vert g\Vert _1\Vert h\Vert _{n} \le C(\Vert g\Vert _0\Vert h\Vert _{n+1}+\Vert g\Vert _{n+1}\Vert h\Vert _{0})$. Moreover, we have $\Vert \sigma \Vert _1 <A$ from which it follows that $\Vert D\log \sigma \Vert _n \le C \Vert \sigma \Vert _{n+1}$, and thus we obtain

$$\begin{aligned} \Vert v\Vert _{n+1} < C\left( \Vert g\Vert _{n+1}+ \Vert g\Vert _0 \Vert \sigma \Vert _{n+2} \right) \end{aligned}$$

and the thesis is proved. $\square$

Estimates of Lemma 4.11 implies that Ef is a tame map of $\sigma$ and f. The map $\sigma (u)$ is a smooth tame function of u. The composition of tame maps is tame, and so we have the result. $\square$

To prove that E is tame smooth we made two assumptions for $\sigma$: first, that $\sigma >c$, and second, that $\partial _{i} \log (\sigma ) <\epsilon '$ for small $\epsilon '$. We have already seen in Proposition 4.6 that $u_b$ and ${\mathcal {U}}$ can be chosen in such a way that, for every $u\in {\mathcal {U}}$, $\sigma (u_b+u)>c$. We now want to prove that the second requirement also holds.

Proposition 4.12

Let $\epsilon '>0$ and consider the initial conditions given in (24), and let $u_b$ in (25) be such that $\Vert u_b\Vert _3\le A$. If [a, b] is such that $b-a$ together with A are sufficiently small, it holds that

$$\begin{aligned} |\partial _{i} \log (\sigma )| < \epsilon ' \end{aligned}$$

for every $u\in {\mathcal {U}}$, recalling that $\Vert u\Vert _4<A$.

Proof

Let D be either $\partial _\phi$ or $\partial _k$, and notice that $D\log \sigma = D\sigma /\sigma$. In proposition 4.6 we have shown that there are choices of w for which $1/\sigma < 1/c$. We now observe that

$$\begin{aligned} D \sigma (\phi ,k) = D \sigma (\phi ,a) + \int _{a}^k \partial _{\kappa }D\sigma (\phi ,\kappa ) \textrm{d}\kappa . \end{aligned}$$

Therefore, since both $\sigma$ and $D(\sigma )$ are smooth, we have that

$$\begin{aligned} \Vert D \sigma (\phi ,k)\Vert _0 \le \Vert D \sigma (\phi ,a)\Vert _0 + (b-a) \Vert \sigma \Vert _2 < C ( A + (b-a) (1+A)), \end{aligned}$$

where we used the fact that $\sigma$ is tame, and in particular $\Vert \sigma \Vert _{2} \le C(1+\Vert u\Vert _4)\le C(1+A)$. Furthermore, $\sigma (\phi ,a)$ depends on $\phi$ and a through $u_b+u$; hence, in view of the continuity of $\sigma$,

$$\begin{aligned} | D \sigma (\phi ,a)| \le C| D\partial _{\phi }^2u_b(\phi ,a) |\le \Vert u_b\Vert _3\le A \end{aligned}$$

where we used the fact that $u=0$ at $k=a$ and the fact that we can chose $u_b$ in such a way that $\Vert u_b\Vert _3<A$. Therefore,

$$\begin{aligned} |D \log \sigma | = \frac{|D \sigma (\phi ,k)|}{\sigma }\le \frac{C}{c} ( A + (b-a) (1+A)) \le \epsilon ', \end{aligned}$$

where we have chosen both A and $b-a$ sufficiently small to make the last inequality valid. $\square$

Remark

We recall that in $u_b$ there is a contribution which is $-q_k \phi ^2$. For $q_k$ given in (22) it is in general not possible to make the choice $\Vert q_k \phi ^2\Vert _3\le A$ for small A because of the constant contribution $k_0^2 f$ in (22), while the other corrections can be made small with judicious choices of the chosen parameters. However, as observed above, such a contribution can always be reabsorbed in the mass of the free theory present in $P_0$.

Theorem 4.13

Under the hypothesis of Proposition 4.9, the RG operator admits a unique family of tame smooth local inverses, and unique local solutions of the RG flow equations exist.

Proof

The proof is a direct application of the Nash–Moser theorem [19], which can be applied thanks to the results of Propositions 4.2, 4.7, 4.8, and 4.9. Actually, it follows from the Nash–Moser theorem that the RG operator admits a unique family of tame smooth local inverses. This guarantees the existence of local solutions of the RG flow equations. $\square$

Appendix A.

1.1 Nash–Moser theorem in the Hamilton formulation

In the formulation of Hamilton, the Nash–Moser theorem is given for elements in a suitable tame Fréchet space. Mainly to fix notation, we recall here some basic definitions and the Theorem we shall use to get the main result presented in this paper.

Definition A.1

A seminorm on a vector space F is a function $\left\| \cdot \right\| : F \rightarrow {\mathbb {R}}$ such that, $\forall \ f, \ g \in F$ and $\forall c \in {\mathbb {R}}$: (i) $\left\| f\right\| \ge 0$; (ii) $\left\| f+g\right\| \le \left\| f\right\| + \left\| g\right\|$; (iii) $\left\| cf\right\| = \left| c\right| \left\| f\right\|$. A collection of seminorms $\{ \left\| \cdot \right\| _n \}_{n \in {\mathbb {N}}}$ defines a unique topology such that a sequence $f_i \rightarrow f \Leftrightarrow \left\| f_i - f\right\| _n \rightarrow 0 \ \forall n \in {\mathbb {N}}$. A locally convex topological vector space is a vector space with a topology arising from a collection of seminorms. The topology is called Hausdorff if $f=0$ when $\left\| f\right\| _n = 0 \ \forall n$. The topology is called metrizable if the family $\{ \left\| \cdot \right\| _n \}_n$ is countable, and the space F is complete if every Cauchy sequence converges. A Fréchet space is a complete Hausdorff metrizable locally convex topological vector space, and a graded Fréchet space has a collection of seminorms that are increasing in strength, so that $\left\| f\right\| _n \le \left\| f\right\| _{n+1} \ \forall n$.

Definition A.2

A graded space F is tame if, given the space $\Sigma (B)$ of exponentially decreasing sequences in some Banach space B, it is possible to find two linear maps $L: F \rightarrow \Sigma (B)$, $M: \Sigma (B) \rightarrow F$, such that $ML: F \rightarrow F$ is the identity

$$\begin{aligned} F \rightarrow ^L \Sigma (B) \rightarrow ^M F . \end{aligned}$$

(42)

Consider two graded spaces F and G, and a map $P: {\mathcal {U}} \subset F \rightarrow G$ from an open subset ${\mathcal {U}}$ of F to ${\mathcal {G}}$. The map P is tame of degree r and base b if it is continuous and satisfies

$$\begin{aligned} \left\| P(f)\right\| _n \le C ( 1 + \left\| f\right\| _{n+r}) \end{aligned}$$

for all f in the neighbourhood of each $f_0 \in {\mathcal {U}}$, for all $n \ge b$, and with a constant C that may depend on n.

In this setting, we can make use of the following, classic theorem on the inverse function problem:

Theorem A.1

(Nash–Moser theorem in Hamilton’s formulation) Consider a smooth tame map $P: {\mathcal {U}} \subset F \rightarrow G$ between two tame Fréchet spaces F and G. Suppose that

• (i) the linear map $DP(u) v = f$ obtained as the first functional derivative of P has unique inverse $E(u)f = v \ \forall \ u \in {\mathcal {U}}$ and all $f\in G$, and

• (ii) the inverse map $E: {\mathcal {U}} \times G \rightarrow F$ is smooth tame.

Then P is locally invertible and $P^{-1}$ is a smooth tame map.