phase space structure and short distance behavior of local quantum field theories

Phase space structure and short distance behavior of local quantum fieldtheoriesStephan Mohrdieck Citation: J. Math. Phys. 43, 3565 (2002); doi: 10.1063/1.1486262 View online: http://dx.doi.org/10.1063/1.1486262 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v43/i7 Published by the AIP Publishing LLC. Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors

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JOURNAL OF MATHEMATICAL PHYSICS VOLUME 43, NUMBER 7 JULY 2002

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Phase space structure and short distance behaviorof local quantum field theories

Stephan Mohrdiecka)

Mathematisches Institut, Universita¨t Basel, Rheinsprung 21, CH 4051 Basel, Switzerland

~Received 6 September 2000; accepted for publication 11 April 2002!

In this article a general relation between the short distance structure of quantumfield theories and their phase space properties is exemplified by the simple class ofgeneralized free fields. As is known, theories with decent phase space properties,resulting from a finite or moderately increasing infinite particle spectrum, alwayshave non-trivial scaling~short distance! limits @cf. D. Buchholz and R. Verch, Rev.Math. Phys.10, 775~1998!#. But, whereas in the finite particle case the phase spaceproperties of the limit theories comply with strong nuclearity conditions, they vio-late in the infinite particle case even rather mild compactness assumptions. Theseresults provide further evidence to the effect that relevant information on the shortdistance structure of a theory can be obtained by phase space analysis. ©2002American Institute of Physics.@DOI: 10.1063/1.1486262#

I. INTRODUCTION

In quantum field theory the structure of a physical theory at small distances is an interissue in several respects. It is important for understanding the particle structure at small scwell as the classification of the ultraviolet properties of the theory.

Recently, Buchholz and Verch presented a model independent approach to the investigathese problems in Ref. 8, which is carried out in the framework of algebraic quantum field thThere, they adapted the method of the renormalization group to this framework by introducinotions of scaling algebra and scaling limit.

Before summarizing the ideas of Buchholz and Verch, let us give a brief account oframework of algebraic quantum field theory~see, e.g., Ref. 12!.

In this framework, a quantum field theory is given by the net of algebras of local observA together with a covariant action of the translation group by automorphismsax,xPRd. This is amap

O°A~O!, ~1!

fulfilling certain properties. HereO,Rd5s11 is an open bounded region ind5s11 dimensionalMinkowski-space andA(O) is unital C* -algebra, the algebra generated by all observables wcan be measured inO. TheC* -algebra generated by all the local algebrasA(O) ~asC* -inductivelimit ! is also denoted byA. Furthermore, we impose the following properties:

~1! Locality: ObservablesAPA(O1) and BPA(O2) corresponding to spacelike separatregionsO1 andO2 should commute:

@A,B#50. ~2!

~2! Covariance:The translation groupRd acts on the net by automorphismsaxPAut(A), forxPRd:

ax~A~O!!5A~O1x!. ~3!

a!Electronic mail: [email protected]

35650022-2488/2002/43(7)/3565/10/$19.00 © 2002 American Institute of Physics

2013 to 129.81.226.149. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

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3566 J. Math. Phys., Vol. 43, No. 7, July 2002 Stephan Mohrdieck

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Furthermore, we require the representation of the translation group to be continuous in thetopology, i.e., the mapsx°ax(A) to be continuous in the norm topology, for allAPA.

~3! Vacuum state:In general, a physical state of the theory is a positive, linear, normalfunctional s on the netA. Such a state allows us to get a representationps of the netA on aHilbert spaceHs via GNS-construction. If the corresponding states is invariant under the translation group, there exists a unitary representationUs(x), xPRd, of the translation group onHs

implementing the representationa:

Us~x!ps~A!Us~2x!5ps~ax~A!!, ~4!

for all APA. Let us assume that our theory has a vacuum state, i.e., that there exist a transinvariant states, such that the joint spectrum of the generators of all implementors is containthe forward light cone.

Now, let us briefly summarize the approach of Buchholz and Verch~for details we refer toRef. 8!:

Given a local, covariant net of observablesA, ax,xPRd in d space–time dimensions we construct the scaling netAI ,aI x,xPRd, again a local covariant net, by the following rule: LetAI :R.0

→A be a bounded function. Then we set

AI ~O!ª$AI uAI lPA~lO!, x°aI x~AI ! continuous in norm topology%,

aI x~AI !lªalx~AI l!.

The crucial point is that the scaling net carries a representation of the dilation grouptl , lPR.0. Starting with the vacuum states on the underlying theory, we define itscanonical lift sIon the scaling net:sI (AI )ªs(AI 1). Then one considers the scaled statessI +tl . This net of states onthe scaling net has limit pointssI (0,n), nPJ for l↘0, in the weak* -topology. Here,J is asuitable index set. Via GNS-construction, we get representationsA(0,n), nPJ, of the scaling netAI corresponding to these states. TheA(0,n), nPJ are calledthe scaling limit netsof the under-lying theory.

Buchholz and Verch also provided a classification of the scaling limit. Since we do notit here, we refer to Ref. 9.

Furthermore, it was observed by Buchholz,1 that phase space properties should play anportant role for the short distance structure of a theory. This can heuristically be illustrated bfollowing example, which was more thoroughly investigated by Lutz:15

Assume we have a quantum field theory exhibiting the following behavior: The enemomentum transfer of local observable scales withl2q with q.1 under renormalization grouptransformations, while its localization in space–time scales withl. Since we require Planck’sconstant not to vary with the scale, its scaling limit theory should be a classical theory, i.observables commute.

In this article we consider a generalized free field ind>3 space–time dimensions with finitor moderately increasing infinite particle spectrum. To be more precise, we impose the follcondition on the mass spectrum:

;b.0: (i PI

e2bmi,`, ~5!

where I is either finite orN. It was shown by Buchholz and Verch that these theories hnontrivial scaling limit, cf. Ref. 9. As is well known,10 these theories fulfill the nuclearity condtion.

Our aim is to show that its scaling limit theories exhibit the following phase space propeIf the mass spectrum is finite, all the scaling limit theories fulfill the strong nuclearity condiBut, none of its scaling limit theories complies with the weaker compactness criterion iinfinite case. Heuristically the situation can be understood as follows: In the finite case the s


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3567J. Math. Phys., Vol. 43, No. 7, July 2002 Phase space structure and quantum field theories

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limit should be the massless theory with finite multiplicity, which is known to fulfill the nucleacondition. On the other hand, in the infinite case, the scaling limit theory is expected to contamassless free field with infinite multiplicity, which does not comply with the compactness crion.

Our analysis will be carried out by only investigating the phase space properties ounderlying theory.

The article is organized as follows:In Sec. II we rephrase the mathematical tools for the investigation of phase space prope

quantum field theory. Afterwards, we describe the interaction between phase space properthe structure of the scaling limit.

The models we are dealing with are introduced in Sec. III. There we also indicate ubounds in the scaling limit of the nuclear norms of the phase space maps of our theories.this we show that all the scaling limit theories in the case of a finite mass spectrum fulfinuclearity condition. In Sec. IV, we prove that the scaling limit in the case of an infinite mspectrum has no decent phase space behavior, and it violates the compactness criterioncarried out by considering lower bounds of thee-contents of the phase space maps.

II. THE PHASE SPACE AND THE SCALING LIMIT

As was already pointed out in the Introduction, the phase space properties of the undetheory should~partially! determine its short distance behavior. In fact, a link between propertiethe scaling limit and the phase space properties of a local quantum field theory was establisBuchholz~cf. Ref. 1!. Here, we want to summarize the main results of his article.

Let us start by recalling the main notions for the description of phase space propertiesframework of local quantum field theory~cf. Refs. 1, 2, 4, 17, and 18!.

Let T be an arbitrary bounded linear operator between two Banach spacesE andF.Definition 2.1: (i) Thee-content NT(e) of T is the maximal number (or infinity) of elemen

EiPE1 , such that

; iÞ j : iT~Ei2Ej !i.e. ~6!

(ii) T is called p-nuclear, pPR.0, if there exist sequences enPE* and FnPF, nPN, suchthat

T5 (n51

`

enFn ,

~7!

(n51

`

ienipiFnip,`.

The first sum should converge in the strong topology. We setiuTuipª inf( (n51` ienipiFnip)1/p and

call this the nuclear p-norm. (To be precise, it is only a quasi-norm for0,p,1.! The infimum istaken over all decompositions of T as in Eq. (7).

Furthermore, we setiAipª(TruAup)1/p for a bounded endomorphismA of a Hilbert space, ifthis trace exists. This will be called thepth trace norm of A. For further properties of and relationbetween these concepts, we refer to Refs. 1 and 2.

In the fr amework of local quantum field theory, ‘‘decent’’ phase space properties of a lcovariant netA are described by the compactness criterion or, a sharpened version ofnuclearity criterion~cf. Refs. 13, 10, and 7!: First let us define thephase space mapsQb,O , foreach bounded regionO,Rd andbPR.0:

Qb,O :A~O!→H,~8!

A°e2bHAV.


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Here,H is the Hamiltonian of the theory.Compactness condition:A local covariant netA fulfills the compactness condition, iff the

phase space mapsQb,O are compact, for everyb andO as above.Nuclearity Condition:A local covariant net fulfills thenuclearity condition, iff the phase space

mapsQb,O arep-nuclear, for everyp.0, bounded regionO and sufficiently largeb.0.We should mention that these conditions have a large variety of consequences for the p

interpretation and statistical and thermodynamical properties of the underlying theory. Sincenot need them here, we refer to Refs. 13, 10, 7, 6, 14, and 19.

With these tools we are able to give an account on the interplay between phase spacerties of the underlying theory and the structure of the scaling limit. We only present the reFor the proofs, we refer to Ref. 1, Proposition 4.3 and Theorems 4.5 and 4.6.

Investigating the nuclearp-norms of the phase space maps we can decide whether the sclimit theories fulfill the nuclearity condition or not:

Theorem 2.1: Consider a quantum field theory with p-nuclear phase space mapsQb,O forsome0,p, 1

3. Furthermore, assume thatlim supl↘0iuQlb,lOuip,`. Then we have the following.

The phase space mapsQb,O(0,n) of every scaling limit netA(0,n), nPJ, are q-nuclear, for q

.2p/(223p.) Furthermore, there exists a constant c depending only on pandq, such that

iuQb,O(0,n)uiq<clim sup

l↘0iuQlb,lOuip . ~9!

Using thee-contents the following sufficient and necessary conditions for the validity ofcompactness criterion of the scaling limit theories hold. Here, thee-content ofQb,O will bedenoted byNb,O(e).

Proposition 2.1: All scaling limit theories of a given quantum field theory fulfill the compness criterion, iflim supl↘0Nlb,lO(e),`, for all e.0 and bounded regionsO. A necessarycondition for the compactness criterion is thatlim infl↘0Nlb,lO(e),`, for all e.0 andbounded regionsO.

Remark:By investigating phase space properties it is also possible to decide whethescaling limit theories are trivial~cf. Ref. 1!. Using the techniques presented in this article evennontriviality of the scaling limit of the models under consideration can be shown~cf. Ref. 16!. Wewill not outline this here because of the stronger results in Ref. 9, where the scaling limit nedirectly constructed.

III. THE MODELS AND UPPER BOUNDS

In this section we first give a brief description of our models. Then we show that the narity condition holds in the scaling limit, in the case of a finite mass spectrum. This is doncalculating upper bounds for the nuclearp-norms of the phase space mapsQb,O and investigatingtheir behavior in the limitl↘0.

Let us start by giving the construction of our quantum field theories in the Cauchyformulation.

Let K be the one-particle Hilbert space:

Kª%i PI

L2~Rs,C!, ~10!

where s5d21 is the number of space-dimensions. OnK we have a scalar product^,& in acanonical way. In addition, there is an antilinear involutionJ given by componentwise compleconjugation of the functions inK. Our vacuum Hilbert spaceH will be the symmetric Fock spacover K. Its scalar product will be denoted by~,!.

The creation and annihilation operatorsa* (F) resp.a(F), FPK, act onH. We define the theWeyl operatorsW(F), FPK:


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W~F !ªei (a* (F)1a(F))2, FPK. ~11!

Here, (a* (F)1a(F))2 denotes the self-adjoint operator extending the densely defined~un-bounded! operatora* (F)1a(F). The Weyl operators are subject to the canonical commutarelations:

W~F !W~G!5eiI^G,F&W~F1G!. ~12!

Let P be the momentum operator onL2(Rs,C). Then the one particle HamiltonianvI acts asfollows:

vI Fª%i PI

v i f i , v iªAuPu21mi2. ~13!

By second quantization we get the HamiltonianH onH. It is the generator of the time translationof our quantum field theory.

Let OrPRd be a double cone with BasisOr5$xPRs,uxu,r %,Rs, the ball of radiusr cen-tered at 0 in thet50-plane. Then we define closed subspaces ofK @HereD(Or) denotes the seof test function with support contained inOr!:

Lfmi~Or !ªv i

2 1/2D~Or !i .iL2(Rs,C), ~14!

Lpmi~Or !ªv i

1/2D~Or !i .iL2(Rs,C), ~15!

Lf~Or !ªvI 2 1/2%i PI

D~Or !i .iK5 %

i PI

Lfmi~Or !

i .iK, ~16!

Lp~Or !ªvI 1/2%i PI

D~Or !i .iK5 %

i PI

Lpmi~Or !

i .iK. ~17!

The corresponding projectors are denoted byEfmi(r ), Ep

mi(r ), EI f(r ), resp.EI p(r ). Now we con-sider the real linear subspaceL(Or),K:

L~Or !ª~11J!Lf~Or ! % ~12J!Lp~Or !. ~18!

We define the local von Neumann algebrasA(O), for O as above, to be the von Neumanalgebras generated by all Weyl operators located inO:

A~Or !ª$W~F !u FPL~Or !%9. ~19!

Here the prime denotes the commutant. The local algebras for arbitrary bounded regiodefined by use of the translation operators and additivity:

A~O!ªS øx1O,O

A~x1O! D 9. ~20!

Here,O is a double cone of the shape as above. So,A(O) is the smallest von Neumann algebcontaining all translated algebrasA(x1O).

Before continuing we have to deal with the following rather subtle problem: The net ofNeumann algebras we have constructed so far complies with all of the assumptions madeIntroduction except for the norm continuity of the mapsx°ax(A). Now, one can obtain a net oC* -algebrasO°A(O) fulfilling norm continuity by ‘‘smearing out’’ with test functions. Thelocal algebrasA(O) will be weakly dense in the algebrasA(O). Now, it is nota priori clear how


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the phase space properties behave under this procedure. The following lemma shows thatnot change. Thus, we are allowed to work with the larger von Neumann algebras.

The phase space maps and thee-contents for theory using the weakly dense subalgebrasA(O)are denoted with a tilde.

Lemma 3.1: We have the following.(i) The phase space mapQb,O is p-nuclear, iff Qb,O is p-nuclear and their nuclear p-norms

coincide:

iuQb,Ouip5iuQb,Ouip . ~21!

(ii) For the correspondinge-contents, the following chain of inequalities holds:

Nb,O~e!<Nb,O~e!<Nb,OS e

2D . ~22!

Proof: The proof of~i! is in analogy to the proof of Lemma 2.2. in Ref. 3. It can be foundRef. 16.

~ii ! follows by use of a 3e-type argument from the Kaplansky density theorem and the dnition of e-content. j

Our main result of this section is the following:Theorem 3.1:All scaling limit theories of the generalized free field with finite mass spect

in d>3 space–time dimensions fulfill the nuclearity condition.Proof: In d53,4 space–time dimensions the theorem follows easily from Ref. 9: There

proven that scaling limit of the theories in question is a generalized free field of mass zerofinite multiplicity. By Ref. 5 these theories fulfill the nuclearity condition.

In higher dimensions we use Theorem 2.1 to reduce the statement of the theoremfollowing proposition:

Proposition 3.1:Let O,Or be a double cone contained in a second one with a ball of radr centered at 0 as a basis in the t50-plane.

Then, the following upper bounds hold for the nuclear p-norms of the phase space maps:

iQb,Oip<5 expFC~p,s!S r

b D s

(i PI

e2 ~b/4! pmiG , b<r ,

expFC~p,s!S r

b D @~s21!/2# p

(i PI

e2 ~b/4! pmiG , b.r .

Remark:One easily sees that these upper bounds forlr ,lb diverge in the limitl↘0, if themass spectrum is infinite.

Proof of the proposition:Analogously to the proof of the theorem in the Appendix of Ref.and the proofs of Theorem 2.1 and Lemma 2.2 in Ref. 5~see also Ref. 16, Lemma 4.2.1!, we get

iuQb,Ouip<expH 2

p S (n51

`1

nI uEI f~r !e2bvI unI

p

p

1 (n51

`1

nI uEI p~r !e2bvI unI

p

pD J .

Now, we have the direct sum decompositionEI f(r )5 % i PIEfmi(r ), a corresponding one forEI p(r )

and the following inequalities:

iuEI e2bvI ur i1<iEI e2rbvI i1 , ;r P$1%øR>2, ~23!

Tr~ uAur !<Tr~ uAuq!, ;r>q,iAi<1, ~24!

of which the first one is proven in Ref. 5, Lemma 2.2, while the second is immediate. Usingequations, we derive the following upper bound for the nuclearp-norm of Qb,O :


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. Let

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iuQb,Ouip<expH 2

p (i PI

F S (n51

K1

n D ~ iEfmi~r !e2bv iip

p1iEpmi~r !e2bv iip

p!

1 (n5K11

`1

n~ iEf

mi~r !e2npbv ii11iEpmi~r !e2npbv ii1!G J , ~25!

whereK is the biggest natural number withKp,2.We only need to calculate the nuclearp-norms forp5 2/N. Because of the formula above,

is sufficient to establish upper bounds for the trace norms of the operatorsEfmi(r )e2npbv i and

Epmi(r )e2npbv i. The basic idea is to decompose the above operators intoN Hilbert–Schmidt

operators, whose kernels can easily be estimated. We will describe this briefly forEfmi(r )e2npbv i:

Let xPD(Rs) be a test function, which is identical to 1 onO1 and setx r by x r(x)ªx(r 21x).Denote by the same symbol the corresponding multiplication operator. Let us now definfollowing operators:

hnmi ,r ,b

ªv i1/2~11l2v i

2!s(n21)x r~11l2v i2!2snv i

21/2, ~26!

knmi ,r ,b

ªv i1/2~11l2v i

2!s(n21)x rv i21/2e2bv i. ~27!

Now, setlªmin$r,b%. Using the identityEfmi(r )5Ef

mi(r )v i1/2x rv i

21/2, we obtain the followingdecomposition for everyNPN:

Efmi~r !5Ef

mi~r !h1mi ,r ,b

¯hN21mi ,r ,bkN

mi ,r ,b . ~28!

By an easy, but tedious, calculation, we obtain upper bounds for thep-trace-norms ofEf

mi(r )e2bv i, where the constantC is independent ofr ,b andmi :

iEfmi~r !e2bv iip

p<H CS r

b D s

e2 ~b/4! pmi, b<r ,

CS r

b D @~s21!/2# p

e2 ~b/4! pmi, b.r .

~29!

Inserting this and a similar estimate for the trace norms ofEpmi(r )e2bv i into Eq. ~25!, the claim

follows. j

IV. THE CASE OF AN INFINITE MASS SPECTRUM

In this section we will prove that the scaling limit theories violate the compactness criterithe case of an infinite mass spectrum. For doing this we need lower bounds of thee-content of thephase space maps. Since this investigation of lower bounds does not appear in the literaturour discussion will be more detailed in this part. Here is our final result:

Theorem 4.1:LetA be a generalized free scalar field in d>3 space–time dimensions, havinga discrete infinite mass spectrum, which fulfills the condition( i PNe2bmi,`, for all b.0.

Then none of its scaling limit theories fulfills the compactness criterion.For the proof of this theorem we need some auxiliary results: First, let us fix our notation

xPD(Rs) be a test function with Supp(x),O1 , which is identical to 1 onO1/2. Definex r byx r(x)ªx(r 21x). We use the same notation for the corresponding multiplication operator. Nlet us set

Kf, mi

b ~r !ªv i1/2x rv i

21e22bv ix rv i1/2. ~30!

We can state a first auxiliary result:


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Lemma 4.1: Keeping the definitions made above, the following estimate holds:

lim infl↘0

iKf,lbmi ~lr !i>iKf,b

0 ~r !i . ~31!

Proof: To obtain this, we first present the operatorsKf, lbmi (lr ) as integral kernels in momen

tum space:

Kf,lbmi ~lr !5E dsk

~ upu21mi2!1/4~ uqu21mi

2!1/4

Auku21mi2

e22lbAuku21mi2

3~lr !2sx~~lr !~p2k!!x~~lr !~k2q!!. ~32!

Here the tilde denotes the Fourier transform. Now, letf PD(Rs) be an arbitrary test function withi f i51. Then we definef l by f l(x)ªl2s/2 f (l21x). This implies i f li51. Using the integralkernel presentation above and the self-adjointness ofKf, lb

mi (lr ), we get

iKf,lbmi ~lr !i>^ f l ,Kf,lb

mi ~lr ! f l&5^ f ,Kf,blmi~r ! f &, ~33!

where the last equation follows by substitution. Now, there exists an integrable upper buniform in themi . It is, up to a positive constant, given by

1

11up2ku2s12

1

11uqus11 HAupuAuquuku

1Aupuuku

1Auquuku

11J e22buku. ~34!

Thus, applying Lebesgue’s theorem, we can interchange the integral with the limitl↘0 to obtain

lim infl↘0

iKf,lbmi ~lr !i> lim

l→0^ f ,Kf,b

lmi~r ! f &5^ f ,Kf,b0 ~r ! f &. ~35!

SinceD(Rs) is dense inL2(Rs,C) the lemma follows. j

Before proving the main result of this section, we still need a lemma:Lemma 4.2: There exists a constant C independent of r,b and the mass spectrum, such th

the following estimate holds:

vI 2 1/2x rvI x rvI2 1/2<CEI f~r !. ~36!

Proof: The inequality is obtained as follows:Trivially, we have ivI 21/2x rvI x rvI

21/2i5supi PI iv i21/2x rv ix rv i

21/2i . We presentv i

21/2x rv ix rv i21/2 as an integral kernel in momentum space:

v i2 1/2x rv ix rv i

2 1/2ª~ upu21m2!21/4r 2sE dskx~r ~p2k!!

3~ uku21m2!1/2x~r ~k2q!!~ uqu21m2!21/4.

Now, we can indicate, after a long but straightforward calculation, an upper boundC for its norm,uniform in mi and r . This is achieved by using the criterion of Schur~cf. Ref. 11!, which werephrase for convenience:

Let I 5I (p,q) be an integral kernel, such that the following estimates are finite:


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N1ª suppPRs

E dsquI ~p,q!u,`,

N2ª supqPRs

E dspuI ~p,q!u,`.

Then,I defines a bounded operator andi I i<AN1N2. j

Proof of the theorem:We will prove this theorem using Proposition 2.1, by showing tlim infl↘0 Nlb,lO(e)5`, for somee.0.

By the identity ue2bvI EI f(r )u5 % i PNue2bv i Efmi(r )u we can find orthonormal eigenfunction

Ff, i(l) PLf(Olr), i PN, of ue2lbvI EI f(lr )u corresponding to the eigenvaluesie2lbv i Ef

mi(lr )i .We define local operatorsAf, i

(l) PA(lO), iAf, i(l) i51:

Af,i(l)

ª

1

11e21/2~W~Ff,i(l)!2e2 1/21!. ~37!

With these definitions, the following estimate holds, foriÞ j :

iQlb,lO~Af,i(l)2Af, j

(l) !i25e21

~11e21/2!2 ~exp$ie2lbv iEfmi~lr !i2%1exp$ie2lbv jEf

mj~lr !i2%22!

>e21

~11e21/2!2 ~ ie2lbv iEfmi~lr !i21ie2lbv jEf

mj~lr !i2!. ~38!

Applying Lemma 4.2, we conclude withC independent ofr ,b andmi :

ie2lbv iEfmi~lr !i25ie2lbv iEf

mi~lr !e2lbv ii>CiKf,lbmi ~lr !i . ~39!

Combining the last two equations and Lemma 4.1, we get, foriÞ j ,

lim infl↘0

iQlb,lO~Af,i(l)2Af, j

(l) !i2>CiKf,b0 ~r !i . ~40!

The constantC does not depend onr ,b and i , j . Therefore the theorem follows. j

ACKNOWLEDGMENT

I am grateful to Professor D. Buchholz not only for initiating this line of research but alsomany helpful discussions.

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theory,’’

ersita


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phase space structure and short distance behavior of local quantum field theories

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