I.INTRODUCTION

The Kadanoff–Wilson point of view on quantum field theory is a central idea in modern physics. It dictates that one should develop the fluctuation of quantum or statistical fields along a scale decomposition associated to the tunable precision of the observation device. In this framework a random Euclidean field is described by a stochastic process \((\varphi_a)_{a \geqslant 0}\) where \(a\) parametrizes a scale of observation with characteristic length \(1 / a\) and \(\varphi_a\) is the corresponding observation of the field, which must be thought of as containing fluctuations with spatial scales \(\gtrsim 1 / a\). It is natural to assume that, as we gradually increase the resolution of our measuring devices, the resulting measurements vary continuously. Therefore we will postulate that the stochastic process \(\varphi_a\) is a pathwise continuous function of \(a\). It is also reasonable to assume that \(\varphi_a\) contains all the information gathered by observations with lesser precisions, which implies that it is a Markov process along the scale parameter, and we assume further that \(\varphi_a \rightarrow \varphi_{\infty}\) as \(a \rightarrow \infty\). We denote by \(\mathscr{F} = (\mathscr{F}_a)_{a \geqslant 0}\) the filtration generated by \((\varphi_a)_{a \geqslant 0}\) and by \(\mathbb{E}_b [\hspace{0.17em} \cdot \hspace{0.17em}]\) the operator of conditional expectation given \(\mathscr{F}_b\), for any \(b \geqslant 0\), with \(\mathbb{E}=\mathbb{E}_0\) the full expectation. We assume that \(\varphi_0 = 0\) whenever the fields take value in a vector space; otherwise we take \(\varphi_0\) to be a fixed default (classical) configuration (e.g. a background field). We say that a functional \(\varphi \mapsto F (\varphi)\) of the field \(\varphi\) is supported in \(U \subseteq \mathbb{R}^d\) if \(F (\varphi) = F (\psi)\) for any field \(\psi\) such that \(\varphi = \psi\) in \(U\). For \(\varepsilon > 0\) let \(U_{\varepsilon} := \{ x \in \mathbb{R}^d : d (x, U) < \varepsilon \}\) be the \(\varepsilon\)-enlargement of \(U\).

The definition of the \(a^{- 1}\)-enlargement is a matter of convention. An observable \(O\) that has a germ

for some finite \(k\) and some function \(\mathcal{O}_a\), is local. Germs serve to parametrize the space of local observables via concrete functionals of the fields.

One owes to Itô the fundamental insight that continuous diffusions can be constructed via stochastic differential equations [25]. As such, since we postulated that the scale description \((\varphi_a)_a\) of a random field is a continuous Markov process, it is completely described via its infinitesimal rate of change

which comes in two parts: a drift \(B^{\varphi}_a \mathrm{d} a\) and a martingale part \(\mathrm{d} M^{\varphi}_a\). The drift component \(B^{\varphi}_a\) is the part of the rate of change which can be predicted from the large scale observations, while the “innovation” process \(M\) is a martingale w.r.t. this filtration, it models the additional information gained by augmenting the resolution. Recall a continuous martingale is described via its quadratic variation process \(\langle M \rangle_a\) [22].

Enters Wilson. We inject in this standard framework the spatial random structure of Euclidean quantum fields by imposing that the dynamics is completely described by ‘local’ coefficients. We do so by postulating that there exists a local observable field \((f_a)_{a \geqslant 0}\) which models the microscopic force and that, inspired by the Kadanoff–Wilson block averaging procedure, the drift \(B^{\varphi}_a\) at scale \(a\) is determined by an averaging of the force field \(f_a\) over a region of size \(1 / a\). Therefore we introduce a “block averaging” operator \(C_a = C_a (\varphi_a)\), which could be random and depend on \(\varphi_a\). We require \(C_a\) to have support on a ball of radius \(1 / a\), that \(a \mapsto C_a (\psi)\) varies smoothly for all \(\psi\), that \(C_a\) be symmetric and positive definite and such that \(C_{\infty} = 1\). For example, one can take

where \(x \in \mathbb{R}^d\) and \(\chi\) is a smooth, radially symmetric, positive definite function of unit integral. In particular note that \(C_0 = 0\) on all sufficiently nice functions \(h\). Since the drift will be integrated along the scales, the local averaging has to be done in such a way as to not over-count the contributions of the microscopic force. We denote by \(\dot{C}_a := \partial_a C_a\) the scale-derivative of this averaging and let \(B^{\varphi}_a := \dot{C}_a f_a\). This fixes the previsible part of the stochastic dynamics as a function of the microscopic force. To complete our description we need to specify also the quadratic variation of the martingale part. Using the same principles we can assume

for a microscopic positive ‘‘diffusivity” \((\sigma^2_a)_a\) which is a local observable field of positive scalars. In this way the local diffusion on scales is determined by the datum of two local observable fields \((f_a, \sigma^2_a)_a\) and a family of averaging operators \((C_a)_a\). In particular, by standard results there exists, possibly on an extended probability space, a cylindrical \(\mathscr{F}\)-Brownian motion

such that

where \(\sigma_b := (\sigma^2_b)^{1 / 2}_b\).

The main goal of this paper is to propose the hypothesis that Euclidean quantum field theories can be identified with Wilson–Itô fields. This provides a new framework, independent of the path-integral formalism, to study Euclidean quantum fields.

Dyson–Schwinger equations and martingale problems – The law of the process \((\varphi_a)_{a \geqslant 0}\) is determined by a martingale problem: for all sufficiently nice scale-dependent test functions \(F (a, \psi)\), the process

is an \(\mathscr{F}\)-martingale where

is the generator of the Wilson–Itô diffusion. Here we denote by \(\mathrm{D} := \delta / \delta \varphi\) the gradient of a functional \(\varphi \mapsto F (\varphi)\) with respect to the field \(\varphi\) taken wrt. the \(L^2\) norm and by \(\operatorname{Tr}\) the trace operator.

II.FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS

Some observables can be obtained via the conditional expectation of a function of \(\varphi_{\infty}\), i.e. let \(O_a =\mathbb{E}_a [F (\varphi_{\infty})]\). This kind of martingale is called closed, i.e. it admits a “terminal value” \(O_{\infty} = F (\varphi_{\infty})\) from which it can be reconstructed. In the case when \(F\) is a linear functional we have

assuming suitable integrability conditions here and below. Using the Wilson–Itô dynamics, and since \((f_a)_a\) is a martingale, we have

where \(C_{\infty, a} := C_{\infty} - C_a\). In general we are not allowed to form non-linear local functions \(F (\varphi_{\infty})\) of the distribution \(\varphi_{\infty}\): the locality condition for an observable is non-trivial and usually requires renormalization. As a consequence, we do not expect non-linear local observables to be closed martingales. Ignoring for the moment this difficulty, we note that when \(F (\varphi_{\infty})\) is well-defined the observables \((O_a)_a\) are closed martingales and they satisfy a backward stochastic differential equations (BSDEs) [19]

for a pair of adapted processes \((O_a, Z^O_a)\) with terminal condition \(O_{\infty} = F (\varphi_{\infty})\).

A procedure to construct local observables starts with some approximate local observable given by a function \(\mathring{O}^{a_0} (\varphi_{\infty})\) localized at scale \(a_0^{- 1}\) and setting

One can then study the convergence of the family of the non-local observables \((O^{a_0}_a)_{0 \leqslant a \leqslant a_0}\) as \(a_0 \rightarrow \infty\). Provided the functions \(\mathring{O}^{a_0} (\varphi_{\infty})\) contains appropriate (diverging) renormalizations one is able to show that the observables \(O^{a_0}\) converge to a local observable as \(a_0 \rightarrow \infty\).

This approach leads naturally to the analysis of a general class of forward-backward stochastic differential equation (FBSDEs) of the form

for some approximately local functionals

Recall \(\dot{C}_a\) may depend on \(\phi_a\) in a general setting. Note also that we use the letter \(\phi\) for these “approximate” dynamics while we use \(\varphi\) for a local dynamics. These FBSDE are not proper Wilson–Itô diffusions, since their coefficients are non-local. BSDEs and FBSDEs are well studied in the mathematical literature (see e.g. [19]) and we dispose also of numerical methods to approximate their solutions. In relation to the numerical aspects, note that the formalism allows us to replace \(\mathbb{R}^d\) by a finite discrete lattice \((\varepsilon \mathbb{Z} \cap [- L, L])^d\) with mesh \(\varepsilon\).

ALinear-like force

To give a first example consider the case of an approximate force \(F\) with a linear component and a constant diffusivity

for some positive constant \(\alpha\), some positive linear operator \(A\) and some additional force component \(h (\phi_{\infty})\). For a local operator \(A\) the linear functional \(A \phi_{\infty}\) is always well-defined in the space of distributions, so it defines a local force field. We assume here that the averaging operators \(C_a\) are deterministic, field-independent and commute with \(A\). Think of the case where \(A = m^2 - \Delta\), \(h = 0\) and \(C_a\) is given by Eq. (

1

Solving for \(\mathbb{E}_a [\phi_{\infty}]\) and letting

we have \(\psi_{\infty} = \phi_{\infty}\) and

where \(\dot{Q}_a := {\color{black}{\partial_a}} (A^{- 1} (1 + \alpha C_{\infty, a} A)^{- 1})\). This computation shows that the linear component in the force can always be integrated and gives rise to a modified FBSDE where the local averaging operator \(\dot{C}_a\) has been replaced by the operator \(\dot{Q}_a\). Note in particular that the Gaussian field

has covariance \(Q_a {\color{black}{- Q_0}}\) and that \(X_{\infty}^Q\) has covariance \(\alpha (1 + \alpha A)^{- 1}\). One gets back the operator \(A^{- 1}\) in the large \(\alpha\) limit, in which case \(X_{\infty}^Q\) is a massive GFF in the model situation. We can invert the transformation and go from a FBSDE of the form (

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for all \(a \geqslant 0\).

BGradient diffusions

Assume further that \(A\) is a symmetric operator and the additional force component \(h\) is given by an effective UV-regularized potential \(V_{\infty}\) as

and let

Then \((V_a)_a\) is the solution of the Polchinski flow equation [21]

with terminal condition \(V_{\infty}\) for \(a \rightarrow \infty\). Letting \(Z_a := \exp (V_a (\psi_a) - V_0 (0))\) and using Itô formula, we have that \(Z_a\) is a positive martingale with \(Z_0 = 1\) and moreover that, under the probability measure \(\mathbb{Q}\) defined on \(\mathscr{F}\)\(_a\) by \(\mathrm{d} \mathbb{Q} := Z_a \mathrm{d} \mathbb{P}\), the process \(\psi_a\) is a martingale with deterministic quadratic variation \(\mathrm{d} \langle \psi \rangle_a = \dot{Q}_a \mathrm{d} a\), so \(\psi_{\infty}\) is under \(\mathbb{Q}\) a GFF. Note also that it follows from (

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for any function \(G\) and any \(a \geqslant 0\). Here we used that \(e^{- V_a (X^Q_a)}\) is a martingale. This implies that the law \(\nu_{\infty}\) of the random field \(\psi_{\infty} = \phi_{\infty}\) is given by

where \(\mu^{Q_{\infty}}\) denotes the Gaussian of covariance \(Q_{\infty, 0}\).

This shows that the class of Wilson–Itô fields comprise as a particular case the Euclidean quantum fields (

10

which we call Polchinski FBSDEs, since they describe the Polchinski semigroup [7, 8]. Even in this potential framework where there is a formal link with the well-known Polchinski flow equation, the FBSDE (

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COne-dimensional gradient diffusions

Take \(d = 1\) and let

\(x \in \mathbb{R}\), where \(v'\) is the derivative of a function \(v : \mathbb{R} \rightarrow \mathbb{R}\) smooth and bounded from below and \(\rho : \mathbb{R} \rightarrow \mathbb{R}_+\) is a compactly supported function of the space variable (an IR cutoff). The formulation (

6

From this equation we see that \(\psi_{\infty}\) is actually comparable in regularity to the \(d = 1\) massive GFF \(X_{\infty}\) with covariance \((1 + A)^{- 1}\), i.e.

which is a Hölder continuous function for which the mild formulation (

12

and this shows that \(\psi_{\infty}\) is a (time-inhomogeneous) Markovian diffusion in the space parameter since \(X_{\infty}\) can be described also as the Ornstein–Uhlenbeck process, solution of the stochastic differential equation

where \((B_x)_{x \in \mathbb{R}}\) is a two-sided Brownian motion. Note that this is a Markov process in the space variable. Moreover if we take \(\rho \rightarrow 1\), under suitable assumptions, the dynamics will converge to a time-homogeneous space-Markovian diffusion described by the WIDE (recall that \(\varphi_{\infty} = \psi_{\infty}\))

along the scale variable \(a \geqslant 0\), and, at the same time, it is also described by the stochastic differential equation

along the space variable \(x \in \mathbb{R}\), where \(\Psi : \mathbb{R} \rightarrow \mathbb{R}_+\) is the ground state of the Hamiltonian with potential \(\phi^2 / 2 + v (\phi)\) (see e.g. [18]).

In absence of the IR cutoff \(\rho\), the random field \(\psi_{\infty}\) is not absolutely continuous wrt. \(X_{\infty}\) and therefore the path-integral formulation (

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This example shows that a local gradient Wilson–Itô dynamics gives rise to a Markov process in the space variable. It is natural to conjecture that this is a general feature of (a wide class of) higher-dimensional Wilson–Itô diffusions.

DVariational formulation

Eq. (

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with an adapted test field \((v_a)_a\) and integrate both in scale and in the probability space to get the equation

Eq. (

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over all adapted controls \((u_a)_{a \geqslant 0}\), where

is the controlled process.

ERigorous results

Variants of Eq. (

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III.PROPERTIES OF WILSON–ITÔ DIFFUSIONS

ACoherent germs

Let \(\mathring{O}\) be a germ for an observable \(O\). For \(0 \leqslant b \leqslant a\), one has by Itô's formula

where \(\mathscr{L}_c\) is the generator (

3

Therefore \(R^O\), and hence \(O\) itself, is completely determined by the germ \(\mathring{O}\) provided the integral

converges absolutely. A germ which has this property is called a coherent germ; its associated observable is determined from it. Note that Eq. (

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for the pair of adapted processes \((R^O_a, Z^O_a)_a\).

The effective force itself is an observable. For simplicity we assume that the diffusivity is taken constant \(\sigma_a^2 = 1\), similar considerations otherwise apply to it. Assume that the force has a germ \(\mathring{f}_a (\varphi_a)\) and a remainder \(R^f\) which then, due to (

16

where we introduced the operator

Similarly to the general case of Eq. (

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A basic requirement for Eq. (

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for some scale \(a_0\). If equation (

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Assume now we are given an observable \(O\) which has a germ \(\mathring{O}\). Under proper assumptions on \(\mathring{O}\), and for a choice of \(\mathring{f}\) that ensure an appropriate strong decay of \(R^f_a\) as \(a \rightarrow \infty\), the coherence relation

is a consequence of the relation

A family \(\mathring{O}\) that satisfies the estimate (

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These considerations lead to the following strategy for constructing the law of a random field \(\varphi\). Associate to each force germ \((\mathring{f}_c)_c\) and each scale parameter \(a_0 > 0\) the solution to the FBSDEs

for \(0 \leqslant b \leqslant a_0\) with mixed initial/final conditions

Now we need to find \((\mathring{f_b} )_{b \geqslant 0}\) such that the coupled system (

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BLink with factorization algebras

Definition

1

Since \(\operatorname{Tr} \dot{C}_a^{1 / 2} \sigma_a^2 \dot{C}_a^{1 / 2} \mathrm{D} \mathring{O}^{(1)}_a \mathrm{D} \mathring{O}^{(2)}_a = 0\), for \(a \geqslant a_0\), because of the support condition, one has

therefore provided the germs are approximately coherent and we impose sufficient decay on \(\mathring{\mathscr{L}_a} \mathring{O}^{(i)}_a\) and moderate growth conditions on \(\mathring{O}^{(i)}_a\), it is possible to guarantee that \(\mathring{\mathscr{L}_a} \mathring{O}^{(12)}_a\) is integrable and therefore \(\mathring{O}^{(12)}_a\) is itself an approximately coherent germ which defines a unique observable \(O^{(12)} =: O^{(1)} \ast O^{(2)}\). This gives on a subspace of observables a natural pre-factorization algebra structure, as defined in Costello–Gwilliam [13, 14].

IV.GAUGE THEORIES

In this section we assume that the field \(\varphi\) is a connection on a principal bundle over \(\mathbb{R}^d\), with finite dimensional compact structure group G and Lie algebra \(\mathfrak{g}\). Recall that the space of connections is affine with \(\mathfrak{g}\)-valued \(1\)-forms as underlying vector space\(\). The gauge group \(\mathfrak{}\)consists of G-valued functions and acts on connections by \(g \cdot \varphi =\operatorname{Ad}_g \varphi - (\mathrm{d} g) g^{- 1}\) and forms by \(g \cdot f =\operatorname{Ad}_g f\). We assume moreover that the force field is a gauge covariant function of the underlying field \(f_a = f_a (\varphi_a)\) (taking values in the space of \(1\)-forms) :

for every \(g \in \mathfrak{G}\) and connection \(\psi\). Given a connection \(\varphi\), denote by \(h^{xy} (\varphi)\) the \(\varphi\)-holonomy along the geodesic from \(x\) to \(y\). Recall that

Let \(\chi_a (x, y)\) be a symmetric function of \((x, y)\). We define a map \(\dot{C}_a^{1 / 2} (\varphi)\) acting on \(1\)-forms by

The operator \(\dot{C}_a^{1 / 2}\) is symmetric and \(\dot{C}_a (\varphi) := (\dot{C}_a^{1 / 2} (\varphi))^2\) is symmetric and non-negative. By (

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The tangent space of the gauge orbit at a given connection \(\varphi\) is spanned by the elements of the form \(\mathrm{d}_{\varphi} h\), where \(\mathrm{d}_{\varphi}\) is the \(\varphi\)-covariant derivative and \(h\) an arbitrary (smooth enough) \(\mathfrak{g}\)-valued\(\) \(0\)-form,

Two scale-dependent families of connections \((\varphi_a)_{a \geqslant 0}\) and \((\varphi'_a)_{a \geqslant 0}\) are said to be gauge equivalent if there exists a scale-dependent family of gauge transforms \((g_a)_{a \geqslant 0}\) such that \(\varphi'_a = g_a \cdot \varphi_a\) for all \(a \geqslant 0\). We note that if \((\varphi_a)_{a \geqslant 0}\) is a solution of the Wilson–Itô equation (

2

This is a particular case of a more general situation.

The proof proceeds by solving the ordinary differential equation \(\dot{g}_a g_a^{- 1} = h_a\) and remarking that \(\varphi^{[h]}_a := g_a \cdot \varphi^{(h) }_a\) solves (

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As Parisi–Wu stochastic quantization scheme [20] this approach allows to quantize gauge theories without path integrals and their associated ghosts or BRS symmetries. Adding terms of the form \(\mathrm{d}_{\varphi_a} h_a\) in the drift allows us to perform gauge-fixing in analogy to the Zwanziger–DeTurck–Sadun trick [15]. Moreover it points to a covariant formulation of the flow equation for the effective force \(f_a\)

with terminal condition \(f_{\infty}\). This is similar to Polchinski equation (

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V.CONCLUSIONS

We introduced Wilson–Itô fields, a novel class of random fields described by “local interactions”. Our point of view is inscribed in the general idea of stochastic quantization [15] initiated by Parisi–Wu [20], which replaces the use of functional integrals with stochastic partial differential equations for the Euclidean fields. This topic has recently witnessed a renewed interest since SPDEs can be used to study rigorously Euclidean fields: see e.g. [9, 10, 11] for non-Abelian gauge theories, [2, 1] for \(\Phi^4_3\) on Riemmanian manifolds and also [17] for a partial verification of the Osterwalder–Schrader axioms in \(\Phi^4_3\). However, at variance with classical stochastic quantization methods, our approach does not require to introduce fictious additional parameters and depends on a physically relevant scale of observation, motivated by the Kadanoff–Wilson picture of renormalization and by the mathematical framework for Euclidean QFT via effective theories, see e.g. [12, 24, 23]. Wilson–Itô fields have a natural intrinsic definition, independent of cutoff procedures and it seems interesting to pursue further their study, e.g. investigate spatial Markov properties, reflection positivity, uniqueness/non-uniqueness of solutions, numerical approximations, description of scale-invariant and conformal invariant fields.