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Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 909 (2016) 880–920

www.elsevier.com/locate/nuclphysb

The analytic renormalization group

Frank Ferrari

a,b,∗

Service de Physique Théorique et Mathématique, Université libre de Bruxelles (ULB) and

International Solvay Institutes, Campus de la Plaine, CP 231, B-1050 Bruxelles, Belgique

Theoretical Physics Department, CERN, CH-1211 Genève, Suisse

Received 3

March 2016; received in revised form 1 June 2016; accepted 3 June 2016

Available

online 8 June 2016

Editor: Hubert

Saleur

Abstract

Finite

temperature Euclidean two-point functions in quantum mechanics or quantum ﬁeld theory are

characterized by a discrete set of Fourier coefﬁcients G

, k ∈ Z, associated with the Matsubara frequencies

= 2πk/β. We show that analyticity implies that the coefﬁcients G

must satisfy an inﬁnite number of

model-independent linear equations that we write down explicitly. In particular, we construct “Analytic

Renormalization Group” linear maps A

which, for any choice of cut-off μ, allow to express the low

energy Fourier coefﬁcients for |ν

| <μ(with the possible exception of the zero mode G

), together with

the real-time correlators and spectral functions, in terms of the high energy Fourier coefﬁcients for |ν

| ≥ μ.

Operating a simple numerical algorithm, we show that the exact universal linear constraints on G

can be

used to systematically improve any random approximate data set obtained, for example, from Monte-Carlo

simulations. Our results are illustrated on several explicit examples.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

Correspondence to: Service de Physique Théorique et Mathématique, Université libre de Bruxelles (ULB) and

International Solvay Institutes, Campus de la Plaine, CP 231, B-1050 Bruxelles, Belgique.

E-mail

address: frank.ferrari@cern.ch.

http://dx.doi.org/10.1016/j.nuclphysb.2016.06.003

The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

F. Ferrari / Nuclear Physics B 909 (2016) 880–920 881

1. General presentation

Consider the space M of arbitrary two-point functions between bosonic operators A and B in

Quantum Mechanics or Quantum Field Theory, at ﬁnite temperature T = 1/β.

As is well-known

and will be reviewed in details in Section 2, the space M can be presented in several equivalent

ways. One can consider various real-time two-point functions (advanced, retarded, time-ordered,

etc.), which turn out to be all related to each other, since their Fourier transforms can be expressed

in terms of a unique spectral function ρ(

ω). Alternatively, one can work with the Euclidean-time

two-point function G(τ ). By the KMS condition, G is periodic and can be expanded in Fourier

series,

G(τ ) =



k∈Z

−iν

, (1.1)

where the Matsubara frequencies are deﬁned by

= 2πk/β. (1.2)

We shall often refer to the set of Fourier coefﬁcients (G

)

k∈Z

as the “data” which encodes the

two-point function. In a generic strongly coupled quantum mechanical model, this data can only

be computed numerically, using Monte-Carlo numerical simulations. Analytic non-perturbative

methods exist only in rare occasions.

By Carlson’s theorem [2], the real-time and Euclidean-time points of view are equivalent:

the continuous spectral function ρ(ω) can be expressed in terms of the discrete set of Fourier

coefﬁcients G

and vice-versa, under some very general assumptions that are valid in all known

interesting physical theories.

The map between the real-time and the Euclidean-time formalism

is quite interesting and will be discussed very explicitly below.

The tw

o-point functions must satisfy general well-known constraints that follow straightfor-

wardly from the deﬁnitions and the spectral decomposition, see Section 2. For example, on top of

being β-periodic, G(τ) is analytic except at the points τ =kβ , k ∈ Z, where it is discontinuous if

A and B do not commute. This implies in particular that G

= O(1/k) at large |k|. The Fourier

coefﬁcients also satisfy reality and positivity constraints depending on the reality properties of A

and B. We shall call F the real vector space of β-periodic functions satisfying all these standard

model-independent constraints.

One of the main goal of the present wo

rk is to show that M is a linear subspace of F of inﬁ-

nite codimension. This may come as a surprise. It means that a typical set of Fourier coefﬁcients

)

k∈Z

satisfying all the usual constraints is actually inconsistent! Our central result is to show

that the Fourier coefﬁcients must always obey an inﬁnite set of universal, model-independent,

linear equations. For reasons that will become clear below, we call these equations “Analytic

Renormalization Group” (ARG) equations. We shall write down these equations ve

ry explicitly

in Section 3 and use them extensively in Sections 4 and 5.

We focus on the case of bosonic operators in the present paper. The case of fermionic operators can be discussed

along the same lines, with minor and straightforward modiﬁcations.

See e.g. [1] for a recent example from which the investigations presented in this paper originated.

In Quantum Field Theory, two-point functions of local operators do not in general satisfy the hypothesis of Carlson’s

theorem, due to the usual UV divergences at coinciding points. These divergences are governed by the Operator Product

Expansion which, in asymptotically free theories, can be reliably computed in perturbation theory. This problem is

handled in a standard way: one either considers smeared versions of the local operators or, more generally, one subtracts

explicitly the diverging piece in the correlator using the OPE.

882 F. Ferrari / Nuclear Physics B 909 (2016) 880–920

A spectacular concrete application of the existence of the ARG equations is as follows. Sup-

pose that we have at our disposal an approximate data set (G

)

k∈Z

, obtained, in a strongly

coupled model of interest, by using Monte-Carlo simulations, possibly combined with pertur-

bation theory at high energies. This data set corresponds to a point G

∈ F , randomly chosen

in a small neighborhood of the exact, but unknown, result G

∈ M . The point G

will never

belong to M , because a random approximate data set always violate the ARG equations (actu-

ally, this violation is always massive, even if the precision of the data is excellent; see below).

It is then possible to use the ARG equations to systematically improve the approximate data, by

suitably projecting G

onto M . An explicit algorithm implementing this idea will be presented

and tested in Section 5. In spite of its simplicity, our algorithm is able to improve the accuracy of

typical random approximate data by a factor of 2 to 4! The startling feature is that the procedure

is totally model-independent and can be applied straightforw

ardly to Monte-Carlo data in any

quantum mechanical system.

The fundamental ingredient at the basis of the ARG equations is analyticity, which is itself a

consequence of causality

. The fact that analyticity yields non-trivial constraints on real-time cor-

relation functions is of course well-known. For example, the famous Plemelj–Kramers–Kronig

identities relate the real and imaginary parts of the Fourier transform of the retarded tw

o-point

function,

Re ˜χ

(ω) =

+∞



−∞

Im ˜χ

(ω



)



−ω

dω



, Im ˜χ

(ω) =−

+∞



−∞

Re ˜χ

(ω



)



−ω

dω



. (1.3)

The starting point of our work is the very same analyticity property at the basis of (1.3), but we

use it in a more sophisticated way to derive the inﬁnite set of linear constraints that the Fourier

coefﬁcients G

must satisfy.

An interesting aspect of the construction is to mak

e an unexpected link, valid in any quantum

mechanical system, between the concept of Renormalization Group (RG) and analyticity. The

fundamental idea of the RG is to describe the physics below a certain RG scale μ in terms of

a Wilsonian action S

that takes into account the physics at scales greater than μ. When μ is

lowered, the action S

ﬂows to a natural description of the physics at low energies. This ﬂow is

constrained by the obvious fact that physics is independent of the arbitrary RG scale. This yields

the RG ﬂow equations. The Analytic version of the RG that we ﬁnd (the ARG) can be described

as follows. To an arbitrary RG scale μ, we associate the inte

ger k

> 0 (that we also call the RG

scale by abuse of language) such that

−1

≤ μ<ν

. (1.4)

We also introduce a strictly positive integer δ, that we call the “index” of the ARG. The ARG

states that there exist linear maps

,δ

and A

−

,δ

allowing to express the low energy Fourier

coefﬁcients, for 1 ≤ k<k

and −k

<k≤−1 respectively, in terms of the high energy Fourier

coefﬁcients G

+δq

and G

−k

−δq

, q ∈ N, respectively:



,...,G

−1



,δ



+δ

+2δ

+3δ

,...



, (1.5)



−1

−2

,...,G

−k



−

,δ



−k

−δ

−k

−2δ

−k

−3δ

,...



. (1.6)

Note that the zero mode G

plays a special role since it cannot be obtained, in general, from the

ARG. This subtlety is related to the phenomenon of Bose–Einstein condensation (see e.g. [1]).

F. Ferrari / Nuclear Physics B 909 (2016) 880–920 883

The above ARG maps A

,δ

are not the most general one can build. The most general maps

allow to reconstruct the analytic continuations of the Fourier transforms of the retarded and

advanced two-point functions, ˜χ

(z) and ˜χ

(z), in the upper and lower half complex plane re-

spectively. This, in particular, automatically provides explicit maps between the real-time and

the Euclidean-time formalism. The general relations are of the form

˜χ

(ω +i) = A

,δ

(ω, )



+δ

+2δ

+3δ

,...



, (1.7)

˜χ

(ω −i) = A

−

,δ

(ω, )



−k

−δ

−k

−2δ

−k

−3δ

,...



, (1.8)

for any 0 < ≤ μ. Explicit formulas for A

,δ

(ω, ) are given in Section 3. From (1.7) and

(1.8), one can straightforwardly obtain similar ARG maps expressing the real-time correlators

(t) and χ

(t), the spectral function, or any other real-time two-point function in terms of the

Euclidean Fourier coefﬁcients above any RG scale k

. The ARG maps (1.5) and (1.6) can also

be easily obtained from (1.7) and (1.8). Full details will be given below.

The lo

w energy Fourier coefﬁcients G

, for |k| <k

, obviously do not depend on the arbitrary

choice of RG scale k

and index δ. On the other hand, the right-hand sides of the equations (1.5)

and (1.6) depend explicitly, and non-trivially, on k

and δ. As usual, this yields renormalization

group equations. The same remark applies to the equations (1.7) and (1.8). More generally, by

abuse of language, we call “ARG equation” any universal linear relation between the Fourier

coefﬁcients like (1.5) or (1.6).

Our results rely in an absolutely crucial wa

y on a generalization of mathematical tech-

niques ﬁrst introduced in a remarkable paper by Cuniberti et al. [3].

The aim of [3] was to

provide explicit formulas for the reconstruction of the real-time correlators from the Euclidean-

time correlators, a notoriously difﬁcult and important problem. One particular aspect of our

results is to provide a useful generalization and simpliﬁcation of the reconstruction procedure

of [3].

The plan of the paper is as follo

ws. In Section 2, we review basic facts on two-point func-

tions in quantum mechanics: deﬁnitions of real-time and Euclidean-time correlators, spectral

decompositions and spectral function, the resolvent, analytic properties and Carlson theorem.

Section 3 is devoted to the derivation of the ARG maps and equations. We start in 3.1 e

xplaining

a simple idea at the basis of the ARG. We then present in 3.2 some useful mathematical results

on Laguerre polynomials, Pollaczek polynomials and the relation between them. These results

are used in 3.3 to build the general ARG maps

,δ

(ω, ). We specialize to the case of the

maps

,δ

in 3.4 and to the reconstruction of the real-time correlators from the Euclidean data

in 3.5. This eventually yields a very general multi-parameter continuous family of ARG equa-

tions. In Section 4, we discuss the numerical implementation of the ARG. We use in particular

the example of the damped harmonic oscillator to illustrate our results. In Section 5, we e

xplain

how the ARG equations can be used to systematically improve any given random approximate

data set. This is certainly the newest, most surprising and most central concrete application of

our work. We provide a simple numerical algorithm that we test successfully on several ex-

amples. Finally

, we brieﬂy summarize our results and suggest future directions of research in

Section 6.

See also [4] for a concrete discussion of the construction in [3].

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