Renormalization group limits for Wick polynomials of Gaussian processes

Vol. 13 (1978)

REPORTS

RENORMALIZATION

ON MATHEMATICAL

PHYSICS

No. 1

GROUP LIMlTS FOR WICK POLYNOMIALS OF GAUSSIAN PROCESSES VOLKER ENSS

Department

of Theoretical

Physics, University of Bielefeld, Bielefeld, Federal Republic of Germany (Received February 24, 1977)

We consider sequences of Gaussian stationary generalized stochastic processes, admitting formation of Wick polynomials, which converge to infrared singular limit processes for which the Wick powers are not defined. Nevertheless the suitably renormalized Wick polynomials are convergent, the limit processes form a dense set in the space of superpositions of Gaussian processes, i.e. their characteristic functional has the form 44

= j exp{--f~z<~lpl)~d~(~)

where Y is a probability measure. In the probabilistic version of renormalization group for continuous systems the sequences can be viewed as orbits under RG transformations having a short scale fixed point.

1. Introduction The Wick powers (or multiple Ito integrals as they are called in the mathematical literature) of a generalized Gaussian random process 0 are defined only if the spectral measure of the process obeys certain integrability conditions. The class of these processes is not closed, in particular orbits under renormalization group (RG) transformations may converge to a fixed point which does not belong to this class. Typical examples are the free time-O-field in one space dimension or the free Euclidean field in two dimensions Q,(x). For any mass m > 0 the Wick powers :@,“:(x) are generalized random processes over Y(R) (Y(P) resp.), but for the short distance scaling limit, namely the massless field d>,(x), the Wick powers do not exist. This phenomenon occurs whenever the limiting scale invariant process cannot be defined on the whole Schwartz-space Y but only on those test functions which have a zero of appropriate order at the origin of momentum space. Nevertheless also in this case the short range RG limits exist. In terms of the above examples the Wick power of the massive field converges when the mass goes to zero (if it is suitably renormalized) to a generalized random process. More generally we will introduce a regularization of the infrared singular process such that the suitably renormalized Wick powers converge as the regularization is taken away. The limiting RG fixed point (which is independent of the regularization) turns out to be a superposition of Gaussian processes, on the other hand all these superpositions can be obtained starting t871

88

V. ENSS

from appropriate Wick polynomials. Using a different method Karwowski and Streit [9] have recently treated the Wick square of the first of the above mentioned examples. The setup of our investigations, definitions and auxiliary lemmas follow next, Sections 3 and 6 contain the main results and their proofs, two illustrative examples are given in Section 4, in Section 5 we describe the connection of our results with the probabilistic formulation of the renormalization group in Dobrushin’s work [3]. 2. Setup and preliminaries Let Go be a generalized stationary Gaussian random process over Y,(R’) given by the characteristic functional [4] v E Y,(p).

L(V) = expC--~<&)I,

which is (1)

9JR”)

is the closed subspace of those test functions from Y(R’) which are real functions on Y-dimensional Z-space and have a zero of order Y in momentum space at the origin: -i, . ..p.jv@($)~P’(R”) ifj,>O,j,+ . . . +j, (p-- 1)/2 2 0.

llldll

=

l4ml

yp(n+p+p-?

is a continuous norm on Y,, the test functions where 0 $ supp~(_ii) form a in ~7, w.r.t. this norm. According to Minlos’ theorem [12] there is a canonical tation of this process on the measure space (Y:, 93, dy), where Y: is the dual ,40,, B the o-field generated by the cylinder sets, and q a probability measure, C&,(T)(T) = T(p)

(4) dense set represenspace of such that

for TE 9’:.

For the space of square integrable functionals L2(Y:, .8, dq) we will use the shorthand notation (15~) and I] - II2 for the norm. h(jQ = h(p) is any nonnegative measurable function such that 1

sp-8d’h2(p><

00,

lim h(p) = 1, P-too

0

(5)

and h,(p) Further we define f’(m)

: = h(M).

:= ci hE(p)p-Bdp 0

c co

Vm > 0

(6)

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GROUP

LIMITS

89

which obeys

thus lim f(m) = co.

(7)

m+O

measure on the Bore1 sets of R’ such

denotes the random variable-valued that ([4], Section III, 3.4) _&(d"p)

it obeys (formally) (~,,(d’p)&(d”k)) (.. .) denotes expectation

= 8(5-@G(dS;Z,)p+dpd”k,

w.r.t. the measure dq. We define the regularized process a,,, by

@m(d=

s$@)&(d”p)

s

=

(8)

~~V~,(P)~~VP)

which is again stationary and Gaussian. As m tends to zero the original process is recovered, actually LEMMA

2.1.

lim ]]@~(pi)-@0(~)](2 = 0

for all h, M E 9,.

fTt+O

Proof

As ~1E 9’,, y(p) := pw2’ sup ]$(~)12 is bounded. Iia=P

ll@m(d---Qio(cp)ll~= ~l~(r;)12(~lll(P)-l)2G(d~)p-Bdp < c SUPY(P) j p2r-Bdp+sup(h,(p)-1)2 PZY P 0

1 Ws2)p-‘%p PBY

I~cP)I’~

The first term can be made arbitrarily small by choosing y small because 2r-b > - 1. The second integral converges and for any y > 0: lim sup (h,(p)1)” = 0. n m-+0pay The nth (local) Wick power (or n-fold Ito integral) [7], [3b] of the regularized process CD,,,is the process :@“,: given by :@*:(pl) = (27r)(1-“)v12lG($, + . . . +&) :6,(d”p,) = lim(2~)(1-“1”~2{P)~(& , . . . , i;.) $,,,(d’p,)

. . . &(d”pJ: . . . !&,,(dvpn),

k-+m

(9)

where lim P)L@~,...,A)

=

&A+

... +A)

k+a,

and q&ii,, .,. ,a”) = 0 in an open neighbourhood

of the points

+p’i = jji, i # j, Vk.

V. ENSS

90

(The 2z-factor is the physical convention, it is missing in the mathematical literature.) LEMMA

2.2. (a) G;:(v) E (Lz) for all h, n EN, m > 0, q.~E 9.

(b) :@‘:: is a generalized random process over Y for all m > 0, n E N. (c) II:@;:(y),)112 < M* lllpllll* (f(m))“-l Proof

Vp E 9,,

independent of m.

M

(a)

II:@;:(p),)II: = (2x)“‘-“‘n!

s

I@CpI+ . . .

+>“)I’ IJ hi(pj) G(dQj)pF’dpj j=1

Pi<1 i#

= (2x)‘(‘-“‘n!

1,2,...,n-k

(;)f2’(m)mk

f: kc0

by splitting the ranges of integration If /I > 1:

and using the symmetry of the integrand.

Mk G S~Pb?@)12(c[

p-8dp)‘-k

< 00.

1

4

In the case ,L?= 1 we observe that w(A + *.. +j&-k) :=

lGm+ ***+?“)I2

,szy

i#l, . . ..a-k obeys for some I E N e.g.

+q)-‘,

0 < y@)

< const. [ln(e+q)l’(l

cow.

sup(n+q)21@(ii)12 < 111~ll12. P

=

(10) (11)

To bound Mk one has to calculate successively integrals of the type SG(dQ,)~Pi’dp,

ycP~+7;‘)

1

which obey as functions ofj;’ again the estimate (10). The last of these integrals is evaluated at j’ = 0 thus Mk < m VO
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91

GROUP LIMITS

(c) We have to improve only the estimate of the term with k = n in the first part of the proof for rp E Y,:

1 <

nll[q~IIl~const.c” s p:‘-@& (5 P-Bdp%(P))“-1

= const.‘,l,p,l,~(fP~~-~.

P

We denote by g, the function with L(li)

: = (.f(~))-‘hn(PY(1

Obviously Go(g,,,) is a normalized Il@0(gm)l13

distribution

(12)

Gaussian random variable:

= = <@o(%rl))

The probability

-PI.

for

1 =

all h, m > 0,

(13)

0 *

of Go(g,,,) is independent

of m and of the regularization

dp, (y) = (2n)-1’zexp { - qy2} dy .

h:

(14)

The Wick powers of the random variable @,(g,) (not of the process !) are known to be [13] :@o(g,)k : = He~(@o(gm)) where He,(x) are the Hermite polynomials with Hek(x) = xk- . . . [l I]. Their probability distributions d&y) are the same as those of Hek(x) w.r.t. the measure d,ul(x), they can be calculated explicitly, e.g. %(Y)

d,My) = Q-l)&, d,u,(y) = (2~)-l’~ -__?__ I/l+Y

see (14),

.~-(~+“)/~&l +y)dy,

etc.

(15)

As the linear span of the Hermite polynomials is dense in L2(R, dp,) their probability distributions are weakly dense in the set of all probability measures on the real line. 3. Results and remarks We use the abbreviation !Jqfp)

:=

f

x vi2 n-1 Y(A) :@$:(lp).

( 1

(16)

V. ENSS

92 THEOREM

3.1.

We have

We say that a sequence of generalized random processes converges if all finite dimensional joint probability distributions converge weakly, or equivalently if the characteristic functionals converge pointwise to a characteristic functional. THEOREM3.2. The sequence of processes Y:(v) converges for any regularization h andfor any n E N a$ m -+ 0 to the generalized process given by the characteristic functional L(P)

= SexpI-:y’(pll~>)d~._,(y).

(17)

dpk are defined at the end of Section 2. THEOREM3.3.

The set of processes

is dense in the set of processes given bJ7

JW =

few~--fY<~l~)~d~(l,)

(18)

where 11is an arbitrary probability measure on the real (half-)line. Due to the symmetry of the exponential the limits of even or odd Wick polynomials alone are dense. A canonical representation of these superpositions of Gaussian processes is realized if 9’“: is equipped with the measure d?(T) = f dv(y)MTly),

T E 9:

where 7 is the measure corresponding to GO. With the substitution yz = z we rewrite -

Wd

= jexp{-fz(alp>}Wz), Ud~tpi)

swpdv(4

= 10, ~0).

= ~exp{-8(z+Z)((yl~))dy(z)d~(~) = ~exp(--~z(dp>>)4~4

(4.

Thus such a process is divisible into factors which are again superpositions of Gaussian processes if and only if the measure dv(z) is divisible. E.g.

dp,(z) = &e is the infinitely divisible Gamma distribution

-z~zz-l’z~(z)dz

[IO] which shows that the limit process of

RENORMALIZATION

GROUP

LIMITS

93

a Wick square is infinitely divisible (see [I] for the massive free Euclidean field). Whether these processes can be divided into factors of a different kind remains as an open question, The N-point functions

of these processes (18) vanish for odd N, for even N they are multiples of ((~1~))““. For the nth Wick power (17) they can be easily obtained as wN(qaN)

=~~((~O(9)):~O(gm)"-1:)N>

=

((~O(~))N)<(:~)O(g,)"-l:)N)

(19)

by Lemma 6.4 below. 4. Examples (a) n=2: Mv)

= Sexp{-~~‘(~l~))~~,(y)

(20)

= (I+
IPI-‘&I~(P)I’

and regularization h(p) = (]p ]/l/p’ + 1 )I” this result was obtained by Karwowski and Streit [9]; Buchholz [2] first observed that L,(v) can be represented as superposition of Gaussians. The 2N-point functions (19) of this process are For P E ~“,(R’h

(dpl)

W~JV(@~~~)

and for all (&J)

= 3s

=

WV-

I>! !(~IP)~

((@ok,n))2N> = UN-- I>! !12<~lv~N

< 1 the series W,N(v@2N) = 2

(-l)N[(2N-1)!!]2[(2N)!]-‘(~]~)N

NZIO

is absolutely convergent. (b) n = 3: L,(~)= Sexp{-~y2(~l~)}~~2(y)

(21)

= ~211pll)-‘~2exp where Ilvll” =
with &N

=

<(:@o(&,)2:)2N)

=

2

(-

1,k(2f)

k=O

which grows asymptotically

like a2N x ;(4N-l)!!.

[2(2N-k)-

I]!!

V. ENSS

94

Thus the series co

c

(-- llN

N=O

&N(2N-

l)!!

,,p,,‘N

(2N)!

has zero radius of convergence but it is an asymptotic expansion of &(QI) for small 9 (what means large arguments of D-, J. For higher Wick powers the coefficients aZN grow still faster, therefore the expansion of the characteristic functional into N-point functions converges only for n < 2. 5. Connection with the renormalization

group

In the context of probability theory the concept of renormalization group transformations for continuous systems (analogous to block spin formation for discrete systems) has been explained in detail by Dobrushin [3], see also Jona-Lasinio [8]. We sketch the definitions and facts, relevant for us. A scale transformation 17, of test functions means

and a re-norming transformation Their dual transformations

U,*, T$ on the generalized random process @ are defined by

(u,*@) (9) = @(V,CP), (TS) (pl) = @(T,V). The process @ is said to have a short-scale limit if for some measurable function b(A) lim T&, U,f@ exists in the topology mentioned in Section 3. The limit is a scaling process. A-t0

This means that there is a parameter x such that the process is invariant under the group {T$ U,*} which is called the renormalization group. x is such that b(A) A-” is a slowly varying function [6]. Our process die is the most general stationary Gaussian process with continuous spectral measure, -r < x = i(l -@) < 0. In this framework y; = Tb:,,,,u: ul; where for j!I > 1, b(m) = m for /3 = 1. We have shown in Theorem 3.2 that for Wick powers of certain Gaussian processes the orbits under RG transformations converge to RG-fixed points which are no longer Wick powers but superpositions of Gaussian processes with scaling parameter x = (l-8)/2. The same applies to Wick polynomials (Theorem 3.3) if /I > 1, whereas for p = 1 due to the singular n-dependence of b(m) we modified slightly Dobrushin’s setup to keep all powers and not the highest one alone. The general problem whether the Wick powers of general Gaussian processes have RG limit points is not treated in this paper.

RENORMALIZATION

GROUP LIMITS

95

6. Proofs of the theorems For n = 1 the theorems are proved in Lemma 2.1. For n > 2 we first decompose the function I$@, + . . . +jiJ into a sum of functions on R’“. For any 0 < B < 1 define the projectors Fit =nO(S-pj), j,i~ (192, . . ..Iz}. ifi (22) F := fi 0(S-~i) = FiFk if i#k. j=l

fJ

(1 -Fi)

functions

h

as its support where at least two pi’s are bigger than 6 therefore

gij(al

, . . . , t;.)

with igijl < 1 such that

With the following resolution l=

of the identity

~Fi-(Il_I)F+CgijO(pi_B)D(~))-6) ix1

we

iij

decompose the test functions

=

(23)

WI -ryJ27-y3+y4.

In the following we will restrict our considerations ?(Z) = 0

to those test functions with

ifp<~forsomee>O.

For all 8 < e/n this implies y3 EE0. Set x(k) = (n- 1) sup 1x1=

k

SUP I-ql<

(24)

I%(~+

&-1)/n

then Fjx(k,)

In the range of Fi holds

thus

there exist

= 0

if i # j.

tat 9

V. ENSS

96

Next we will estimate the various terms in the decomposition and show that the asymptotically leading contribution comes from y1 alone. Recall the definition (16) of !Pk . !Pz(yJ (by an obvious extension of our previous notation) is well defined because &,(d’p) is a measure on the Bore1 sets of R'. LEMMA 6.1.

We have lim IlKkfd-

:@0(~)@0(gmY-1:l12

m-PO

for ail h,nEN, Proof:

~JEY~,

0 < d<

= 0

1.

Using that the product is commutative

we have

d

IlA@llf =

Due to the hypercontractive bounds (see e.g. Theorem I.22 of [13]) one knows that boundedness or vanishing of Gaussian random variables in the (L2)-norm implies the same for any (P)-norm with p < co. Applying Hiilder’s inequality and Lemma 2.1 one gets the desired result. n LEMMA 6.2.

for all pl E Y,

We have

obeying

IlKkf~~Nl~ G cow.6

(24), 0 < S < e/n < 1, Vh, m > 0, n 2 2.

Proof: n-l

IlwY2)ll3

G fP

$g

IIt

1

: @k(&

I2 X(Pl))l 2

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97

GROUP LIMITS

= (const.@)?. n

LEMMA6.3.

We have 1l:@~:bJl2

for all h, q~E Y,,

G

Mw-o)"-2

m > 0, n > 2, 0 c 6 < 1.

Proof: il:@~:@J4)112 G

(2rC)Y’“-l)l[:~~:(1~(7ilf =

Ip%:(Jfm + -*a+;i;.>lf3~PI -wxP2-4)lj2,

ny

c

.** +~“)le(p1--)e(pz-6))1/% G(dQj)P~‘dPjh~(P~) X

SII

Pe~UtiltiOOS j=l of{l,...,n} x 16a+

<

n!

... +~.)i2e(Pl-s)e(P2-s)e(Pil-s)e(Pi,-6)

1 G(dQ,>p;‘dpl

S G (dQ2)pZadP2 x P.>d

P*>d

n X

sn j=3

G(dsai>p~'apihi(Pi>

li?)(7;1+**a

+&>I2

Q n!~(“*2)(‘jp-Bdph~cp)rx 0

<

n!T(y) k=O

One shows that M,(S) < co V6 > 0 exactly as in the proof of Lemma 2.2. m To evaluate further :@,,(q~)@~(g,,,)n-~:we will need LEMMA6.4.

We have lim <@o(q+@o(gm)) = 0

m-r0

Proof:

I(@oO@o(97))

Vh, 9 E 9,.

V. ENSS

98

the first term vanishes as m --, 0 for any value of 0 < y < 1, the second term becomes small as y -+ 0 uniformly in m as one can see applying Schwartz’s inequality:

As a consequence of this lemma Qo(g,,,) tends weakly to zero in (L2). LEMMA

6.5.

We have lim Il:@o(pl)@O(g,Y-’

:-@,(rp):

@O(gm)n-l:ll* = 0

m-0

for all h, p E Y,, Proof:

(EJ

n E N.

We use the following identity for Gaussian random variables c”, , Sz with

= 0: = .FL._ .F Q:k._ k. gk-1. .(& &). Y, .Y2. .-1-z..UZ

/I :cDo(gm)“-2: 11: = (n-2)!

and Lemma 6.4 proves this lemma. II

Proof of Theorem 3.1: Combining Lemmas 6.1, 6.2, 6.3 and 6.5 proves the theorem for test functions which vanish in a neighbourhood of 5 in momentum space. By using the uniform bound of Lemma 2.2(c) this result extends by continuity to the II I * I II-closure of this test function space which includes 9,. n Proof of Theorem 3.2: Let 9, e” be two independent

As their joint probability product is

distribution

factorizes

random variables from (L’). the probability distribution of their

Vs.&Z) = s dv,(x)&*(~)e(z-x~ and its characteristic

y)

function is L(E . sl) = 1 e’zdv,.,,(z)

= 1 L(y * S)dvs.(y).

Let again v obey (24) and define gk such that &,(p’) = &,,(Z)~(Q--p). From the proof of Lemma 6.1 we know that d@ = Qo(g, -gL) + 0 in (Lz) as m -+ 0 for any e > 0. This implies weak convergence of the probability distribution dp;, ,,,(y) of : @o(gk)l)k:to dpk(y). 4Do(y) and Go(g6) are independent Gaussian variables, therefore also @Jo(~)and :@o(gh)R: are independent random variables. If we now apply the calculation made above we see that the characteristic functional of Y’: converges pointwise to lim

PLO

Sexp{-fy2~d~:-~,~(~) = Sexp(-fy’(pl~)~d~.-,(y).

RENORMALIZATION

GROUP

LIMITS

argument this extends to yor. AS the limit is obviously functional we have proved the theorem. n Finally Theorem 3.3 follows from the remark following (15).

By a continuity

99 a

characteristic

I am indebted to G. Jona-Lasinio, because my first contact with Dobrushin’s work was in his Cargbe lectures 1976, and to W. Karwowski and L. Streit because their first result in this direction stimulated these investigations. REFERENCES [l] Baumann, K., G. C. Hegerfeldt: J. Math. Phys. 18 (1977), 171. [2] Buchholz, D. : private communication. [3] Dobrushin, R. L. : (a) Avtomodel'nost’ i renorm-gruppa obobshEennych polej and (b) Gaussovskie i podfirlennye Gaussovskitn avtomodel’nye ObobshEennye sluEajnye polya, communicated to us by G. Jona-Lasinio, to be published in Ann. Probability. [4] Gel’fand, I. M., N. Ya. Vilenkin: Generalized fictions, Vol. 4, Academic Press, New York, London 1964. [S] Gradshteyn, I. S., I. M. Ryzhik: Table of integrals, series, andproducts, Academic Press, New York, London 1965, Section 9.2.4. [6] Ibragimov, I. A., Yu. V. Linnik: Independent and stationary sequences of random variables, WoltersNoordhoff, Groningen 1971, Appendix 1. [7] Ito, E.: Jap. .I. of Math. 22 (1952), 63. Modification : Totoki, H.: Ergodic theory, Aarhus Univ., Lecture Note Ser. No. 14, 1970, Section 10. [S] Jona-Lasinio, G.: Lectures at the Summer Institute on Field Theory and Statistical Mechanics, Cargitse, France, July 1976 (preprint n. 32, Istituto di Fisica G. Marconi, Roma 1977, to be published in the proceedings). [9] Karwowski, W., L. Streit: Rep. Math. Whys. 13 (1978), 1. [lo] Lukacs, E.: Characteristic functions, Griffin, London 1970. [ll] Magnus, W., F. Oberhettinger, R. P. Soni: Formulas and theorems for the special functions of mathematical physics, 3rd ed., Springer, Heidelberg, New York 1966, Section V.5.6. [12] Minlos, R. A.: Trudy Mosk. Mat. Obshc’. 8 (1959), 497. [13] Simon, B. : The P(@Jz Euclidean (quantum) field theory, Princeton University Press, Princeton 1974.