A stochastic analysis of subcritical Euclidean fermionic field theories

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A stochastic analysis of subcritical

Euclidean fermionic ﬁeld theories

FRANCESCO DEVECCHI

Department of Mathematics, University of Pavia,

Via Adolfo Ferrata 5 27100, Pavia, Italy

Email: francescocarlo.devecchi@unipv.it

LUCA FRESTA

Institute for Applied Mathematics &

Hausdorﬀ Center for Mathematics, University of Bonn,

Endenicher Allee 60 53115, Bonn, Germany

Email: fresta@iam.uni-bonn.de

MASSIMILIANO GUBINELLI

Mathematical Institute, University of Oxford,

Woodstock Road OX2 6GG Oxford, United Kingdom

Email: gubinelli@maths.ox.ac.uk

Abstract

Building on previous work on the stochastic analysis for Grassmann random variables, we intro-

duce a forward-backward stochastic diﬀerential equation (FBSDE) which provides a stochastic

quantisation of Grassmann measures. Our method is inspired by the so-called continuous renor-

malisation group, but avoids the technical diﬃculties encountered in the direct study of the ﬂow

equation for the eﬀective potentials. As an application, we construct a family of weakly coupled

subcritical Euclidean fermionic ﬁeld theories and prove exponential decay of correlations.

Keywords: stochastic quantisation, renormalisation group, non-commutative probability, Euc-

lidean quantum ﬁeld theory

A.M.S. subject classiﬁcation: 60H30,81T16

Table of contents

1 Introduction ............................................................ 2

1.1Methodologyandmainresults .............................................. 3

1.2 Comparisonwiththeliterature .............................................. 6

2 Grassmann stochastic analysis ................................................ 10

2.1 Grassmannprobability .................................................. 10

2.2 Processes and conditional expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 GrassmannBrownianmartingales ........................................... 13

3 The FBSDE approach ..................................................... 14

3.1 Finite-dimensionalSDEs ................................................. 14

3.2 Flowundertheconditionalexpectation ........................................ 17

3.3 FBSDEforthemodel .................................................. 21

3.4 Sobolev spaces and covariance estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Solution in the subcritical regime .............................................. 29

4.1 SpacesofGrassmannmonomials ............................................ 29

4.2 Thetruncatedﬂowequation ............................................... 33

4.3 Global solution of the FBSDE without cutoﬀs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.4 Exponential decay and short-distance divergence of correlations . . . . . . . . . . . . . . . . . . . . . . . . . 43

Appendix A Proof of Theorem 2.20 .............................................. 50

Bibliography ............................................................ 54

1 Introduction

Let (ℳ,𝜔)be an algebraic probability space, i.e. a unital C∗-algebra ℳendowed with a norm-

alised, positive linear functional 𝜔:ℳ→ℂand let (𝜓(f))f∈hbe a centred Grassmann Gaussian

ﬁeld indexed by the Hilbert space h≔L2(ℝd;ℂ2)⊕L2(ℝd;ℂ2)with covariance

𝜔(𝜓(f)𝜓(g))=⟨Θf,Gg⟩, f,g∈h,

where Θis the anti-unitary involution on hgiven by Θ(f1⊕f2)≔(f2

¯⊕f1

¯)and Gthe bounded

operator

G≔(1−Δ)𝛾−d/2⊕−(1−Δ)𝛾−d/2 (1.1)

Δbeing the Laplacian on L2(ℝd;ℂ2)and 0<𝛾<d/2. The ﬁeld 𝜓is said to be Grassmann

because it satisﬁes anticommutation relations

𝜓(f)𝜓(g)+𝜓(g)𝜓(f)=0, ∀f,g∈h,

hence the need for the algebraic, i.e. non-commutative, setting (ℳ,𝜔)[Mey93,Par92,Bia95].

One can think of 𝜓as a massive Grassmann Gaussian free ﬁeld: the fractional exponent in the

covariance is there only to model the scaling behaviour one would expect for arbitrary (non-

integer) dimension.

We want to prove the existence and cluster properties of a class of Grassmann measures that

are constructed as Gibbsian perturbations of the Grassmann Gaussian ﬁeld 𝜓on ℝd: namely, we

want to make sense of the continuous linear functional 𝜔Von the algebra 𝒜⊆ℳof polynomials

of 𝜓, and given by 𝜔V(A)≔𝜔(AeV(𝜓))

𝜔(eV(𝜓)),A∈𝒜, (1.2)

which we call the Gibbsian Grassmann measure with covariance Gand interaction potential

V(𝜓). In particular, we will take V(𝜓)as a Gross–Neveu-type interaction of the form [GN74]

V(𝜓)≔ℝdλ

4(𝜓x)4+𝜇(𝜓x)2dx,

for some constants λ,𝜇∈ℝand with 𝜓x=𝜓(𝛿x)and (𝜓x)4,(𝜓x)2suitable quartic and quadratic

homogeneous polynomial of the ﬁeld 𝜓evaluated at a point x∈ℝd. Due to singularities in the

covariance Gthe random variable 𝜓xcannot really be deﬁned and the expression of the inter-

action potential Vis at best a suggestive notation for a more concrete family of approximations

which avoid both the small-scale (or ultraviolet, or UV) singularities implicit in considering local

monomials and the large-scale (or infrared, or IR) singularities relative to the computation of the

integral over all ℝd.

Gibbsian Grassmann measures appear naturally in the study of interacting Euclidean fermionic

quantum ﬁeld theories, see, e.g., [GJ87] for a review, but also in the study of condensed-matter

systems [GM10,GMP17] and as an emergent description in statistical mechanics, see [Mas08].

Although the Grassmann measures in (1.2) do not describe any relevant quantum ﬁeld theory

nor statistical mechanics system, they are a class of Gross–Neveu-type toy models that exhibit

the same mathematical challenges of subcritical (that is, super-renormalisable) Euclidean ﬁeld

theories. They are therefore important from a methodological perspective and should be con-

sidered in view of the stochastic analysis in the broader context of super-renormalisable (possibly

supersymmetric) ﬁeld theories with bosons and fermions.

2 SECTION 1

For a rigorous deﬁnition of the measure 𝜔Vwe need to regularise the IR and UV singularities

and then perform a limit to remove the regularisation parameters. A popular choice is to restrict

the randomness in the problem to a ﬁnite number of degrees of freedom by working on a ﬁnite

set of space points. Let L∈ℕ,𝜀∈2−ℕand introduce the d-dimensional toroidal lattice 𝕋L,𝜀

d≔

((𝜀ℤ)/(Lℤ))dand a suitable 𝕋L,𝜀

d-lattice regularisation1.1 GL,𝜀of the continuum covariance Gin

eq. (1.1). Moreover, let (𝜓x

L,𝜀)x∈𝕋L,𝜀

dbe the Grassmann Gaussian ﬁeld with covariance GL,𝜀 (and

with the same involution Θ) and discretise the interaction potential as

VL,𝜀(𝜓L,𝜀)≔𝕋L,𝜀

dλ

4(𝜓x

L,𝜀)4+𝜇𝜀(λ)(𝜓x

L,𝜀)2dx,(1.3)

where ∫𝕋L,𝜀

d⋅dx≔𝜀d∑x∈𝕋L,𝜀

dis a normalised counting measure on 𝕋L,𝜀

dand where we choose

𝜇𝜀(λ)depending on 𝜀,λin order to achieve a non-trivial limit as 𝜀→0. Due to the singularities

in the covariance Gwe expect to need to consider a diverging family (𝜇𝜀(λ))𝜀>0 in order to com-

pensate for the divergences introduced by the non-linear factor eVL,𝜀(𝜓) in the averages. To the

regularised potential there correspond regularised measures 𝜔VL,𝜀, so that our problem becomes

the analysis of the weak limit of the families 𝜔VL,𝜀𝜀>0,L>0:

𝜔V(A)≔ lim

L→∞lim

𝜀→0𝜔VL,𝜀(A)

≔ lim

L→∞lim

𝜀→0 𝜔AeVL,𝜀(𝜓L,𝜀)

𝜔eVL,𝜀(𝜓L,𝜀),A∈𝒜. (1.4)

1.1 Methodology and main results

The point of our work is mainly methodological: to construct the Grassmann measure 𝜔Vwe use

a novel approach based on a non-commutative forward–backward stochastic diﬀerential equation

(FBSDE), which provides a stochastic quantisation of the measure of interest. Our approach

applies to any subcritical Grassmann measure, including the equivalents of the well-known 𝜑2

and 𝜑3

4bosonic (or commutative) theories.

To describe in detail our strategy, we introduce a diﬀerentiable interpolation (Gt

L,𝜀)t⩾0 such

that

L,𝜀=0, G∞

L,𝜀=GL,𝜀, Θ(Gt

L,𝜀)∗Θ=−Gt

L,𝜀.

This interpolation should also suppress momenta larger than 2s, so that Gt

L,𝜀 is a bounded oper-

ator, uniformly in Land 𝜀, if t<∞. In (ℳ,𝜔)we introduce a Grassmann Brownian martingale

(GBM) as the family (Xt

L,𝜀)t⩾0 of centred Grassmann Gaussian random ﬁelds indexed by the

Hilbert space1.2 hL,𝜀≔L2(𝕋L,𝜀

d;ℂ2)⊕L2(𝕋L,𝜀

d;ℂ2)and with covariance Gt

L,𝜀,

𝜔(Xt

L,𝜀(f)Xs

L,𝜀(g))=⟨Θf,Gt∧s

L,𝜀g⟩hL,𝜀.

Note that the X∞

L,𝜀 is distributed in law as the Gaussian free ﬁeld 𝜓L,𝜀. See Section 2for the

detailed deﬁnitions and properties of these objects. From a stochastic perspective, the crucial

point to note is that the parameter tis not a physical nor a stochastic time. Rather, following the

core idea of RG techniques, it is a continuous ﬂow parameter associated with the scale decompos-

ition of the problem, which we choose in such a way that ultraviolet scales correspond to t→∞.

1.1. Because of technical reasons, we will avoid simply replacing Δby the lattice Laplacian ΔL,𝜀 on L2𝕋L,𝜀

d;ℂ2, see Deﬁni-

tion 3.16 in Section 3.3.

1.2. The space L2𝕋L,𝜀

d;ℂ2is associated with 𝜀dtimes the counting measure: if f,g∈L2𝕋L,𝜀

d;ℂ2we write ⟨f,g⟩L2(𝕋L,𝜀

d;ℂ2)=

∑i∑x∈𝕋L,𝜀

d𝜀dfi(x)gi(x).

INTRODUCTION 3

As we have already announced, we would like to study the interacting measure (1.4) via

stochastic quantisation, that is, by identifying it as the marginal law of a suitable stochastic pro-

cess coupled with the Grassmann Gaussian ﬁeld. In particular, we will prove that for any nice

enough function Pthe following identity holds true

𝜔V(P(X∞

L,𝜀))=𝜔(P(Ψ∞

L,𝜀)), (1.5)

provided that the process (Ψs

L,𝜀)ssolves the following forward-backward stochastic diﬀerential

equation (FBSDE) on [0,∞]

dΨs

L,𝜀=−G

˙s

L,𝜀𝜔s(DVL,𝜀(Ψ∞

L,𝜀))ds+dXs

L,𝜀, Ψ0

L,𝜀=0, (1.6)

where DVL,𝜀 denotes the functional derivative of VL,𝜀 and where 𝜔t(⋅) denotes the conditional

expectation with respect to a ﬁltration such that (Xt

L,𝜀)tis adapted, see Section 2.

It is important to note that the drift term in (1.6) satisﬁes the identity

𝜔s(DVL,𝜀(Ψ∞

L,𝜀))=DVs

L,𝜀(Ψs

L,𝜀),

where VtL,𝜀 is the solution of the Hamilton–Jacobi–Bellman (HJB) ﬂow equation

∂tVtL,𝜀+1

2DG

˙t

L,𝜀

2VtL,𝜀+1

2(DVt)G

˙t

L,𝜀

2,=0, (1.7)

where DG

˙t

L,𝜀

2is the functional Laplacian and where (⋅)G

˙t

L,𝜀

2is a suitable quadratic form, associated

with the operator G

˙t

L,𝜀. The typical approach in the physics and mathematical physics literature

is based on the study of (1.7), see below. However, we do not want to make this substitution: the

FBSDE is a powerful tool that allow us to truncate the ﬂow equation (1.7) for the eﬀective force

𝜔s(DVL,𝜀(Ψ∞

L,𝜀)) in a suitable way, and to control the remainder due to the truncation for any

subcritical theory. In fact, we decompose the drift term as

𝜔s(DVL,𝜀(Ψ∞

L,𝜀))=Fs

L,𝜀(Ψs

L,𝜀)+Rs

L,𝜀,

where the “remainder process” Rs

L,𝜀 solves the following self-consistent equation, for s>0

L,𝜀 =s

∞𝜔s(ℋr[Fr

L,𝜀](Ψr

L,𝜀))dr+s

∞𝜔s(DFr

L,𝜀(Ψr

L,𝜀)⋅G

˙r

L,𝜀Rr

L,𝜀)dr,(1.8)

with

ℋr[Fr

L,𝜀]≔∂rFr

L,𝜀+1

2DG

˙r

L,𝜀

2Fr

L,𝜀+DFr

L,𝜀⋅G

˙r

L,𝜀Fr

L,𝜀.

We have now the freedom to choose Fs

L,𝜀 in any convenient way. If we let Fs

L,𝜀 be the solution

of the HJB ﬂow equation, then ℋr[Fr

L,𝜀]vanishes and so does Rs

L,𝜀. More generally, we can

take Fs

L,𝜀 to be the solution of a simpler ﬂow equation, provided that we can control eq. (1.8). In

fact, once we have good estimates for Fs

L,𝜀, we can prove the existence and uniqueness of a pair

(ΨL,𝜀,RL,𝜀)solving eqs. (1.6) and (1.8) by a ﬁxed-point argument, for which λsmall is needed.

To provide a precise statement, for a,𝜂⩾0 we introduce the following topology on ﬁelds1.3

on ℝ+×ℝd, that is, maps 𝜓:ℝ+×ℝd→ℳ4,(t,x)↦(𝜓t,x,𝜇)𝜇∈{↓,↑}×{±}:

‖𝜓‖a,𝜂≔sup

𝜇sup

(t,x)∈ℝ+×ℝd𝜚𝜂(x)2−at‖𝜓t,x,𝜇‖. (1.9)

1.3. If ﬁelds are deﬁned on ℝ+×𝕋L,𝜀

d, they can be suitably extended to ℝ+×ℝd, see Section 3.3.

4 SECTION 1

where 𝜚𝜂(x)≔(1+|x|2)−𝜂/2. Note that when t∈ℝ+is the scale parameter described above, −a

measures the Hölder regularity in space of the ﬁeld, in a suitable weighted L∞topology. We

also remark that, in order to carry out the ﬁxed-point argument described above, we also need to

control the derivatives of ΨL,𝜀, see Section 4.3 for details; for the moment (1.9) suﬃces to state

the theorem below. On ﬁelds on ℝd, we consider the weighted Besov spaces Bp,q,𝜂

a(ℝd;ℳ4),

see, e.g., [Ama19,BCD11] for the deﬁnition of Besov spaces and [ABDG20] for their use in the

analysis of fermionic systems, where 𝜂∈ℝ indicates integration with weight 𝜚𝜂; we denote the

Besov norms simply by ‖⋅‖Bp,q,𝜂

a. If 𝜂=0, as usual, we abridge the notation to Bp,q

aand ‖⋅‖Bp,q

Theorem 1.1. Let d ∈ℕ,𝛾<min{d/4,1}. Then, there exist 𝜆0=𝜆0(d,𝜆)and a function 𝜇∞

𝜀:

ℝ+→ℝ such that if 𝜆⩽𝜆0and 𝜇𝜀(𝜆)=𝜇∞

𝜀(𝜆) then for any L ∈ℕ and 𝜀∈2−ℕif V L,𝜀 is

chosen as in (1.3), the FBSDE in (1.6) has unique global solution ΨL,𝜀. The sequence (ΨL,𝜀)L,𝜀

is bounded in the ‖⋅‖𝛾,0 topology and Cauchy in the ‖⋅‖𝛾+𝜃,𝜂 topology for 𝜂,𝜃>0small enough.

Denoting by Ψthe limit we have

‖Ψ−ΨL,𝜀‖𝛾+𝜃,𝜂≲𝜃,𝜂𝜀𝜃+L−𝜂.(1.10)

Furthermore, the sequence (Ψ∞

L,𝜀)L,𝜀 is bounded in B∞,∞

−𝛾−𝜌(ℝd;ℳ4)for any 𝜌>0and Cauchy in

B∞,∞,𝜂

−𝛾−𝜌(ℝd;ℳ4)for 𝜌,𝜂>0small enough. Denoting the limit Ψ∞, for 𝜃>𝜌>0small enough

we have ‖Ψ∞−Ψ∞

L,𝜀‖B∞,∞,𝜂

−𝛾−𝜃≲𝜃,𝜂(𝜀𝜃−𝜌+L−𝜂). (1.11)

Remark 1.2. The constraint 𝛾<1 is for technical reasons and can be removed provided that

further “counter-terms” are included in the potential, e.g., of the form ∫𝕋L,𝜀

dv𝜀(λ)(∂xXt,x

L,𝜀)2dx,

where ∂xis the lattice derivative.

Theorem 1.1 allows us to identify the algebra of observables 𝒜on which the measures 𝜔VL,𝜀

can be deﬁned via the stochastic quantisation equation (1.5). This is the polynomial algebra

generated by the Grassmann Gaussian free ﬁeld 𝜓

𝒜≔span1,𝜓(f1)⋅⋅⋅𝜓(fn)|n∈ℕ,(fi)i=1

n⊆B1,1,0−

𝛾+, (1.12)

see Section 4.3, where B1,1,0−

𝛾+≔⋃a>𝛾,𝜂<0 B1,1,𝜂

a. Since B1,1,𝜂

a⊂B1,1,𝜂′

a′for a⩾a′and 𝜂⩽𝜂′, if f∈

B1,1,0−

𝛾+, with abuse of notation we write ‖f‖B1,1,0−

𝛾+≔‖f‖B1,1,𝜂

¯, where a

¯and 𝜂

¯are such that f∈B1,1,𝜂

but f∉B1,1,𝜂

¯−

¯+. We prove existence, cluster properties and short-distance divergence of the weak

limit of 𝜔VL,𝜀 on 𝒜.

Theorem 1.3. Under the same assumption of Theorem 1.1, for any L ∈ℕ,𝜀∈2−ℕ,λ⩽λ0, the

measures 𝜔VL,𝜀 exist and have a weak limit 𝜔Von 𝒜, as 𝜀→0and L →∞. Furthermore:

1. Consider m1,m2∈ℕ, and let ((f(i,k))k=1

mi)i=1,2 ⊆B1,1,0−

𝛾+(ℝd;ℂ4). Let Cov𝜔V(A;B)≔

𝜔V(AB)−𝜔V(A)𝜔V(B)and let 𝜓be the Grassmann Gaussian field with covariance (1.1) .

Then, letting Di≔⋃k=1

misupp(f(i,k)), for some universal c >0we have

Cov𝜔V

(



k=1

m1𝜓(f(1,k));

k=1

m2𝜓(f(2,k))

)

≲d,𝛾,λ

(



i=1

2

k=1

mi‖f(i,k)‖B1,1,0−

𝛾+

)

e−cdist(D1,D2).

2. Fix some 𝜒∈Cc

∞(ℝd)such that ‖𝜒‖L1=1 and for any 𝜀>0 and x ∈ℝddeﬁne 𝜒x

𝜀≔

𝜀−d𝜒(𝜀−1(⋅−x)). Then, for any x ≠y∈ℝdand 𝜀>0small enough,

|𝜔V(𝜓(𝜒x

𝜀)𝜓(𝜒y

𝜀))−𝜔(𝜓(𝜒x

𝜀)𝜓(𝜒y

𝜀))|≲𝜆(𝜀∨|x−y|)−2𝛾.

INTRODUCTION 5

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AstochasticanalysisofsubcriticalEuclideanfermionicﬁeldtheoriesFRANCESCODEVECCHIDepartmentofMathematics,UniversityofPavia,ViaAdolfoFerrata527100,Pavia,ItalyEmail:francescocarlo.devecchi@unipv.itLUCAFRESTAInstituteforAppliedMathematics&HausdorﬀCenterforMathematics,UniversityofBonn,EndenicherAllee60531...

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A stochastic analysis of subcritical Euclidean fermionic field theories

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