Operators on subspaces of hereditarily indecomposable Banach spaces

合集下载

Quantum interference between nuclear excitation by electron capture and radiative recombina

∗ † ‡
This work is part of the doctoral thesis of Adriana Gagyi-P´ alffy, Giessen (D26), 2006. Electronic address: Adriana-Claudia.Gagyi-Palffy@uni-giessen.de
FIG. 1: NEEC and RR recombination mechanisms of a continuum electron into the K shell of a bare ion. The nucleus is schematically represented as undergoing the transition from the ground state (G) to the excited state (E) and again to its ground state.
Quantum interference between nuclear excitation by electron capture and radiative recombination ∗
arXiv:physics/0701207v1 [physics.atom-ph] 18 Jan 2007
interactions are considered explicitly, and the nucleus is described with the help of a geometrical collective model and making use of experimental data. The dynamics of the electron is governed by the Dirac equation. We express the interference term of the cross section using the dimensionless Fano proﬁle parameter for electric and magnetic transitions in Sec. II. The numerical results of the calculation are given in Sec. III, together with an interpretation of the results regarding the possibility of an experimental observation of NEEC. We conclude with a short summary. In this work atomic units are used unless otherwise speciﬁed.

Integrated density of states and Wegner estimates for random Schrodinger Operators

INTEGRATED DENSITY OF STATES AND WEGNER ESTIMATES
99
On the other hand, the electron could be moving in an amorphous medium, in which case there is no large group of symmetries of the Hamiltonian. However, from the physical point of view it is reasonable to assume that the local structure of the medium will be translation invariant on average. This means that we consider the potential which the electron experiences as a particular realisation of a random process and assume stationarity with respect to some group of translations. Moreover, physical intuition suggests to assume that the local properties of the medium in two regions far apart (on the microscopic scale) are approximately independent from each other. Therefore the stochastic process describing the potential should have a correlation function which decays to zero, or — more precisely — should be ergodic. There are interesting models which lie between the two extreme cases of latticeperiodic and amorphous media. They still have an underlying lattice structure which is, however, modiﬁed by disorder. Probably the best studied Hamiltonian with this properties is the alloy type model. We leave its precise deﬁnition for the next paragraph and introduce here a special case on the intuitive level. Consider ﬁrst the potential Vω (x) := uk (ω, x) Each k corresponds to a nucleus sitting on a lattice point. The function uk (ω, ·) describes the atomic or nuclear potential at the site k and depends on the random parameter ω which models the diﬀerent realisations of the conﬁguration of the nuclei. If there is only one type of atom present, which has a spherically symmetric potential, all the uk (ω, ·) are the same, and Vω is periodic. Now assume that there are two kinds a and b of atoms present, which have spherically symmetric atomic potentials of the same shape, but which diﬀer in their nuclear charge numbers. In this case the potential looks like Vω (x) :=

Space-Time Non-Commutativity near Horizon of a Black Hole

a r X i v :g r -q c /0502111v 1 25 F eb 2005Space-Time Non-Commutativity near Horizonof a Black HoleM.Martinis and V.Mikuta-MartinisTheoretical Physics Division,Rudjer Boˇs kovi´c Institute,10001Zagreb,Croatia E-mail:martinis@irb.hr,vmikuta@irb.hr1IntroductionEinstein’s general theory of relativity[1]describes gravity as a manifestation of the curvature of spacetime.A fundamental instability against collapse implies the existence of black holes as stable solutions of Einstein’s equations.A black hole is formed if a massive object(e.g.a star)collapses into an inﬁnitely dense state known as a singularity.In this picture the curvature of spacetime becomes extreme and prevents any particle even light from escaping to inﬁnity.A black hole may have several horizons that fully characterize its structure.The simplest three-dimensional geometry for a black hole is a sphere(known as a Schwarzschild sphere),its surface deﬁnes the event horizon.In the case of a spherical black hole,with Rµν=0,all horizons coincide at the Schwarzschild’s critical radius r s=2GMc−2.The Quantum Field Theory(QFT)in curved spacetime with classical event horizon is,however,troubled by the singularity at the horizon[2].This problem may be solved by treating the black hole as a quantum state which implies that the energy of the black hole and its corresponding time do not commute at the horizon[3].In this picture we study the dynamics of a scalarﬁeld in the near-horizon region described by a static Klein-Gordon(KG)operator which in this case becomes the Hamiltonian of the system.The dynamics of a scalarﬁeld in the near-horizon region,and its associated SO(2,1)conformal symmetry have been studied in many papers[4],[5],[6],[7],[8]in which a complete treatments of conformal quantum mechanics and of near-horizon symmetry were made. In this letter,we present the explicit construction of the time operator in the near-horizon region in terms of the generators of the aﬃne group,and discuss its self-adjointness[9].2Scalar Field in the Near-Horizon RegionThe Schwarzschild geometry of a static spherical black hole is described by the metric(c=¯h=G=1)ds2=−f(r)dt2+[f(r)]−1dr2+r2dΩ2,(1) where rf(r)=r−r s,and dΩ2=dθ2+sin2θdϕ2.Near horizon(r∼r s),f(r) behaves as f(r)∼2κ(r−r s),whereκ=1/2r s denotes the surface gravity.The equation of motion of a free scalarﬁeldΦ(x)in this background metric is −1fω2φ+fφrr+(f′+2f/r)φr−l(l+1)xφ,the KG equation in the near-horizon region,for small x,reduces to a scale invariant Schroedinger equation[d24+Θ2)2(p2+gSinceˆH is x-coordinate dependent,we expect spacetime noncommutativity, [ˆt,x]=0.What isˆt as an operator?Due to Pauli theorem[15]no such self-adjoint operator should exist if the spectrum of the self-adjoint Hamiltonian is semibounded or discrete.In quantum theoryˆH is essentially self-adjoint only for g>3/4in the domainD0={ψ∈L2(R+,dx),ψ(0)=ψ′(0)=0}(8) In this domain it has a continuous spectrum for g≥3/4with E>0but no ground state at E=0[4].For g≤3/4,the Hamiltonian is not essentially self-adjoint[6],[8],but it admits a one-parameter family of self-adjoint extension labeled by a U(1)parameter e iz,where z is a real number,which labels the domains D z of the extended Hamiltonian.The set D z contains all the vectors in D0,and vectors of the formψz=ψ++e izψ−.For g=−1xK02cotzE zandψz exhibits a scale behavior of the typeψz∼√This algebra can be easily extended to the full SO(2,1)conformal algebra by adding to the set(ˆH,ˆD)the conformal generatorˆK=x2/2.In this case the g-dependent constant quadratic Casimir operator is obtainedC2=14−32(ˆDˆH−1+ˆH−1ˆD),(13)which obeys the required commutation relation,[ˆH,ˆt]=i.Although,bothˆH andˆD separately can be made self-adjoint operators in the domain L2(R+,dE) it is not true for aˆt-operator which containsˆH−1[17].It is clear thatˆt is not a self-adjoint operator in the domain L2(R+,dE)whereˆH−→EˆD−→−i(E d2)(14)ˆt−→−i d2=α>0.This allow usto calculate all the matrix elements involving time operator and even to study their classical limit as¯h→0at the horizon.In the limit g−→0,we haveˆH−→H0=p2/2andˆt−→t0whereˆt 0=−1The Aharonov-Bohn time operatorˆt0is not self-adjoint and its eigenfunctions are not orthonormal.4ConclusionIn this paper,we have studied the properties of a scalarﬁeld in the near-horizon region of a massive Schwarzschild black hole.The quantum Hamilto-nian governing the near-horizon dynamics is found to be scale invariant and has the full conformal group as a dynamical symmetry ing only the generators of the aﬃne group,we constructed the time operator near-horizon. The self-adjointness ofˆH andˆt is also discussed.5AcknowledgmentThis work was supported by the Ministry of Science and Technology of the Republic of Croatia under Contract No.0098004.References[1]Nicolson I1981Gravity,Black Holes,and the Universe(David and Charles,London1981)[2]’t Hooft G2004Horizons,gr-qc/0401027.[3]Bai H and Yan Mu-Lin Quantum Horizon,gr-qc/0403064.[4]de Alfaro V et al1976Nuovo Cimento A34569Camblong H E and Ord´o nez C R2003Phys.Rev.D6*******[5]Strominger A1998JHEP9802009Claus P et al1998Phys.Rev.Lett.814553[6]Govindarajam T R et al2000Nucl.Phys.B583291[7]Gibbons G W and Townsand P K1999Phys.Lett.B454187[8]Birmingham D et al2001Phys.Lett.B505191Gupta K S and Sen S2002Phys.Lett.B526121[9]Martinis M and Mikuta V2003Fizika B12285[10]’t Hooft G1996Int.Jour.Mod.Phys.A114623[11]Gupta K S and Sen S2003Hidden Degeneracy in the Brick Wall Model of BlackHoles hep-th/0302183[12]Camblong H E and Ord´o nez C R2004Black Hole Thermodynamics from Near-Horizon Conformal Quantum Mechanics hep-th/0411008[13]’t Hooft G2000Holographic Principle hep-th/0003004[14]Camblong H E et al2000Phys.Rev.Lett.851590Camblong H E et al2001Ann.Phys.(NY)28714and57[15]Pauli W1958”Die allgemeinen Prinzipien der Wellenmechanik”inEncyclopedia of Physics,ed.Fluegge S(Springer-Verlag,Berlin1958)1-168[16]Moretti L and Pinamonti N2002Nucl.Phys.B647131[17]Klauder J R1999Noncanonical Quantization of Gravity I.Foundation of AﬃneQuantum Gravity gr-qc/9906013[18]Aharonov Y and Bohm D1961Phys.Rev.1221649。

Scattering and Inverse Scattering on ACH Manifolds

dt2 + h(t) t2 ∞ ¯ The function ρ := ψ∗ t is where h(t), t ∈ [0, ǫ) is a C 1-parameter family of C ∞ metrics on ∂ X. ¯ ¯ ¯ Note that a boundary deﬁning function in X near ∂ X, which can be extended smoothly to X. n the boundary represents the geometric inﬁnity of X , as does the sphere S for the hyperbolic space Hn+1 the Laplacian of (X, g ) was studied in [27]; it consists a ﬁnite pure point spectrum σpp (∆), which is the set of L2 (X ) eigenvalues, and an absolutely continuous spectrum σac (∆) satisfying The resolvent σac (∆) = n2 /4, ∞ and σpp (∆) ⊂ 0, n2 /4 .
R(λ) = (∆g − λ(n − λ))−1 ,
2000 Mathematics Subject Classiﬁcation. Primary 58J50, Secondary 35P25. Key words and phrases. scattering, pseudoconvex domains, asymptotically complex hyperbolic manifolds. The ﬁrst author was partially supported by French ANR grants no. [JC05-52556] and [JC05-46063], and an Australian National University postdoctoral fellowship. Both authors were supported in part by the National Science Foundation Grant DMS 0500788.

美国赛博空间作战行动Cyberspace _Operations

Challenges to the Joint Force’s Use ofCyberspace..................................................I-11
CHAPTER II
CYBERSPACE OPERATIONS CORE ACTIVITIES
Introduction................................................................................................................II-1
3.应用
a、本出版物中确立的联合原则适用于联合参谋部、作战司令部指挥官、下属统一司令部、联合特遣部队、这些司令部的下属部门、各军种和作战支持机构。
b、本出版物中的指南具有权威性；因此，除非指挥官认为特殊情况另有规定，否则将遵循这一原则。如果本出版物的内容与出版物的内容发生冲突，则以本出版物为准，除非参谋长联席会议通常与其他参谋长联合会成员协调，提供了更为现行和具体的指导。作为多国（联盟或联盟）军事指挥部一部分的部队指挥官应遵循美国批准的多国原则和程序。对于未经美国批准的条令和程序，指挥官应评估并遵循多国司令部的条令与程序，如果适用并符合美国法律、法规和条令。
•联合职能部门和网络空间运作
第三章权限、角色和职责
•简介III-1
•当局III-2
•角色和职责
•法律考虑因素III-11
第四章规划、协调、执行和评估
•联合规划过程和网络空间运营
•网络空间运营规划考虑因素
•对网络空间的情报和操作分析支持
运营计划IV-6
•针对性IV-8
•网络空间部队的指挥与控制

On Bose-Fermi Statistics, Quantum Group Symmetry, and Second Quantization

a rXiv:h ep-th/9611144v119N ov1996July 1996,rev.Nov.1996LMU-TPW 96-17On Bose-Fermi Statistics,Quantum Group Symmetry,and Second Quantization 12Gaetano Fiore Sektion Physik der Ludwig-Maximilians-Universit¨a t M¨u nchen Theoretische Physik —Lehrstuhl Professor Wess Theresienstraße 37,80333M¨u nchen Federal Republic of Germany e-mail:Gaetano.Fiore @physik.uni-muenchen.de Abstract Can one represent quantum group covariant q -commuting “creators,an-nihilators”A +i ,A j as operators acting on standard bosonic/fermionic Fock spaces?We brieﬂy address this general problem and show that the answer is positive (at least)in some simplest cases.1IntroductionIn recent years the idea of Quantum Field Theories(QFT)endowed with Quantum Group[1]symmetries has attracted considerable interest and has been investigated especially in2Dﬁeld theories,in connection with socalled anyonic statistics(when the deformation parameter q is a root of unity).Its application to QFT in higher (e.g.3+1)space-time dimensions relies,among other things,on the condition that Bose and Fermi statistics are compatible with quantum group-symmetry transfor-mations,at other(in particular real)values of q.The latter issue in fact involves two diﬀerent problems,one inﬁrst quantized quantum mechanics and the other in QFT.Theﬁrst problem essentially is whether a Hilbert space can carry both a com-pletely(anti)symmetric representation of the symmetric group S n(so that it can de-scribe the states of n bosons/fermions)and of a quasitriangular non-cocommutative ∗-Hopf algebra H.Contrary to a quite widespread prejudice,we showed in Ref.[2] that this is possible whenever H can be obtained from the universal enveloping Ug of a Lie algebra g by a unitary“Drinfel’d twist”F[3,4].Only,we need to de-scribe the system of n bosons/fermions in an unusual picture,that is related to the standard one[involving(anti)symmetric wavefunctions and symmetric operators] by a unitary transformation F12...n not symmetric under tensor factor permutations; F12...n is derived from F.The relevant point here is that even in this scheme sec-ond quantization naturally leads[5]to creation and annihilation operators A+c i,A j c satisfying the canonical(anti)commutation relations(CCR),and to the standard bosonic/fermionic Fock space representations,exactly as in the standard treatment of second quantization.The second problem,which we brieﬂy address here,is whether nonetheless one can represent,as operators acting on standard bosonic/fermionic spaces,algebraic objects A+i,A j:(1)transforming as conjugate tensors under the action of H;(2) satisfying the quantum,i.e.H-covariant,commutation relations(QCR)[6,7].We report here results(proved in Ref.[8])which allow a positive answer to requirement(1),under the same assumption as above,and a positive answer to requirement(2):a)in the simpler case that H is triangular;b)in the particular case that H=U q su(2)and A+c i,A j c belong to the fundamental representation of su(2)3.We look for a realization of A+i,A j in the form of formal power series in theA+c i,A jc ing F,in sect.3we determine a class of candidates for A+i,A j fulﬁllingrequirement(1).In Sect4we show how to pick out of this class a particular set satisfying requirement(2)under one of the assumptions a),b).These A+i,A j turn out to be well-deﬁned operators on the bosonic/fermion Fock space.In sect.5we brieﬂy comment on the possible application of our results to QFT.2Preliminaries and notation2.1Twisting groups into quantum groupsLet(U g,m,∆c,ε,S c)be the cocommutative Hopf algebra associated with the uni-versal enveloping(UE)algebra U g of a Lie algebra g.m,∆c,ε,S c denote the mul-tiplication comultiplication,counit and antipode respectively;we will often drop the symbol m:m(a⊗b)≡ab.Let F∈U g[[¯h]]⊗U g[[¯h]](we will write F=F(1)⊗F(2),in a Sweedler’s notation with upper indices)be a twist,i.e.an invertible element satisfying the relations(ε⊗id)F=1=(id⊗ε)F(2.1)and F|¯h=0=1⊗1(¯h∈C is the‘deformation parameter’).It is well known[3] that if F also satisﬁes the relation(F⊗1)[(∆c⊗id)(F)]=(1⊗F)[(id⊗∆c)(F),(2.2)then one can construct a triangular non-cocommutative Hopf algebra H=(U g[[¯h]],m,∆,ε,S,R)having the same algebra structure(U g[[¯h]],m),the same counitε,co-multiplication and antipode deﬁned by∆(a)=F∆c(a)F−1,S(a)=γ−1S c(a)γ(2.3)(whereγ−1:=F(1)·S c F(2)),and(triangular)universal R-matrix R:=F21F−1 (F21:=F(2)⊗F(1)).Condition(2.2)ensures that∆is coassociative as∆c.Examples of F’s satisfying conditions(2.2),(2.1)are provided e.g.by the so-called‘Reshetikhin twists’F:=e¯hωij h i∧h j,where{h i}is a basis in the Cartan subalgebra of g andωij∈C.A similar result holds for genuine quantum groups.A well-known theorem by Drinfel’d[4]essentially proves,for any quasitriangular deformation H=(U q g,m,∆,ε,S)[1,9]of U g,with g simple belonging to the classical A,B,C,D series,the ex-istence of an invertible F satisfying condition(2.1)such that H can be obtained from U g through formulae(2.3)as well,after identifying U q g with the isomorphi-cal algebra U g[[¯h]],¯h=ln q.This F does not satisfy condition(2.2),however the(nontrivial)coassociatorφ:=F−112,3F−112F23F1,23∈U g⊗3still commutes with ∆(2)c(U g),thus explaining why∆is coassociative in this case,too.The correspond-ing universal(quasitriangular)R-matrix R is related to F by R:=F21q tSettingσ(X):=ρ(X)ij A+ciA jc(2.7)for all X∈g,oneﬁnds thatσ:g→A±is a Lie algebra homomorphism,so that σcan be extended to all of U g[[¯h]]as an algebra homomorphismσ:U g[[¯h]]→A±[[¯h]];on the unit element we setσ(1U g):=1A.σ(X)commutes with the ‘number of particles’N c:=A+c i A i c.σcan be seen as the generalization of the Jordan-Schwinger realization of su(2),σ(j+)=A+c↑A↓c,σ(j−)=A+c↓A↑c,σ(j0)=1Remark1.If H is a∗-Hopf algebra,ρ,F are unitary5and†is an involution in A±,thenγ′=γ∗and(A i c)†=A+c i⇒σ◦∗=†◦σ,(A i)†=A+i.(3.13)Remark2.Let A±c,inv,A±inv be the subalgebras of A±[[¯h]]invariant underc⊲,⊲(i.e.I∈A±c,inv iﬀx⊲I=ε(x)I,I∈A±inv iﬀx⊲I=ε(x)I).It is notdiﬃcult to prove that A±c,inv=A±inv.An element I∈A±[[¯h]]can be expressed as a function of A i,A+j or of A i c,A+c j,I=f(A i,A+j)=f c(A i c,A+c j).We will prove elsewhere that f=f c in the triangular case,but not in the genuine quasitriangular one.In the latter case,to a polynomial f(resp.f c)there corresponds a highly non-polynomial(tipically a trascendental function)f c(resp.f);so the change of generators A i,A+j↔A i c,A+c j can be used to simplify the functional dependence of I(what might turn useful for practical purposes,e.g.to solve the dynamics associated to some Hamiltonian I).Remark3.A i,A+j are well-deﬁned as operators on the bosonic/fermionic Fock spaces,at least for small¯h[assuming that the tensors T,T′are also of the form 1⊗1+O(¯h)];correspondingly,the transformation A i c,A+c j→A i,A+j is invertible.4Fulﬁlment of the“quantum”commutation re-lationsTheorem1[8]If the noncommutative Hopf algebra H is triangular[i.e.the twist F satisﬁes equation(2.1)],then,setting T≡1⊗1≡T′in eq.(3.11),A i,A+j close the quadratic commutation relationsA i A+j=δi j1A±R ui jv A+u A v,(4.14)A i A j=±R ij vu A u A v(4.15)A+i A+j=±R vu ij A+u A+v(4.16)where R is the(numerical)quantum R-matrix of U g in the representationρ,R ijhk :=[(ρ⊗ρ)(R)]ijhk.(4.17)Theorem2[8]6If g=su(2)andρ≡fundamental representation,it is possibleto determine T,T′such that the elements A i,A+j∈A±[[¯h]](i,j=↑,↓)deﬁned in formulae(3.11)are covariant under U q su(2)and satisfy the U q su(2)-covariant quadratic QCR[6,11,7]A i A+j=1Aδi j±q±1R ui jv A+u A v,(4.18)A i A j=±q∓1R ij vu A u A v(4.19)A+i A+j=±q∓1R vu ij A+u A+v,(4.20)where R is the R-matrix of U q sl(2).Moreover,(A i)†=A+i for the compact section U q su(2)(q∈R).With this choice of T,T′,A+i,A j explicitly read,in the bosonic case,A+↑= N↑c q N↓c A+c↑A+↓= N↓c A+c↓A↑=A↑c N↑c q N↓c A↓=A↓c N↓c,(4.21) and in the‘fermionic’oneA+↑=q−N↓c A+c↑A+↓=A+c↓A↑=A↑c q−N↓c A↓=A↓c,(4.22) where N↑c:=A+c↑A↑c,N↓c:=A+c↓A↓c,(x)q2:=q2x−16In the proof of theorem6we made[8]essential use of the U g[[¯h]]-valued2×2matrix(ρ⊗id)F found in Ref.[12].The operatorsπiα(x)φjα(x)which are obtained from the canonical onesπc iα(x),φjαc(x)through the transformation(3.11),are well-deﬁned(nonlocal)composite op-erators on the Fock space generated byφiαc( x);they act asπc iα(x),φjαc(x)“dressed”in a peculiar way by all theﬁelds considered in the theory.In the case of a trian-gular Hopf algebra H,theorem1implies thatπiα(x),φjα(x)satisfy the quadratic commutation relationsφiα( x)πjβ( x′)=iδ(3)( x− x′)δαβδiαjβ±R lβiαjβmαπlβ( x′)φmα( x)(5.24)φiα( x)φjβ( x′)=±R iαjβmαlβφlβ( x′)φmα( x)(5.25)πiα( x)πjβ( x′)=±R mαlβiαjβπlβ( x′)πmα( x)(5.26)where R iαjβmαlβ=[(ρα⊗ρβ)(R)]iαjβmαlβ.Because of remark2in sect.3,in this casean invariant action S has the same functional dependence onπiα(x),φjα(x)as onπciα(x),φjαc(x).In the quasitriangular case H=U q sl(2)theorem6is not applicable,because in its present form its validity is restricted only to the fundamentalρ(so the opera-torsπiα(x),φjα(x)do not satisfy quadratic commutation relations).Whether some generalization of this theorem to arbitraryρexists and these ideas can be applied to QFT also for quasitriangular H’s,is presently only matter of speculations.AcknowledgmentsI am grateful to Prof.’s V.K.Dobrev and H.-D.Doebner for their invitation to Group21.It is a pleasure to thank J.Wess for his support and for the hospitality at his Institute.References[1]V.G.Drinfeld,Quantum Groups,Proceedings of the International Congressof Mathematicians,Berkeley1986,Vol.1,798.[2]G.Fiore and P.Schupp,Identical Particles and Quantum Symmetries,Nucl.Phys.B470(1996),211.[3]V.G.Drinfeld,Doklady AN SSSR273(1983),531.[4]V.G.Drinfeld,Quasi Hopf Algebras,Leningrad Math.J.1(1990),1419.[5]G.Fiore and P.Schupp,Statistics and Quantum Group Symmetries,Preprint LMU-TPW-96-06ed e-print hep-th/9605133to appear in“Quantum Groups and Quantum Spaces”,Banach Center Publications40(1997),Inst.of Mathematics,Polish Academy of Sciences,Warszawa,P.Budzyski,W.Pusz, S.Zakrweski(editors).[6]W.Pusz,S.L.Woronowicz,Twisted Second Quantization,Reports on Math-ematical Physics27(1989),231.[7]J.Wess and B.Zumino,Covariant Diﬀerential Calculus on the Quantum Hy-perplane Nucl.Phys.Proc.Suppl.18B(1991),302.[8]G.Fiore,Deforming Maps from Group to Quantum Group Covariant Cre-ation&Annihilation Operators,Preprint LMU-TPW96-20,and e-print q-alg/9610005.[9]L.D.Faddeev,N.Y.Reshetikhin and L.A.Takhtajan,Quantization ofLie Groups and Lie Algebras,Algebra i Analysis,1(1989),178;translation: Leningrad Math.J.1(1990),193.[10]B.Jurco,More on Quantum Groups from the Quantization Point of View,Commun.Math.Phys.166(1994),63.[11]W.Pusz,Twisted Canonical Anticommutation Relations,Reports on Mathe-matical Physics27(1989),349.[12]T.L.Curtright,G.I.Ghandour,C.K.Zachos,J.Math.Phys.32(1991),676.。

Approximation in C N Norm Levenberg

Surveys in Approximation Theory Volume 2, 2006. pp. 92–140. Copyright c 2006 Surveys in Approximation Theory. ISSN 1555-578X All rights of reproduction in any form reserved.
arXiv:math/0611249v1 [math.CA] 8 Nov 2006
Approximation in CN
Norm Levenberg
3 November 2006
Abstract. This is a survey article on selected topics in approximation theory. The topics either use techniques from the theory of several complex variables or arise in the study of the subject. The survey is aimed at readers having an acquaintance with standard results in classical approximation theory and complex analysis but no apriori knowledge of several complex variables is assumed. MSC: 32-02, 41-02 1 Introduction and motivation . . . . . . . . . 2 Polynomial hulls and polynomial convexity . . . . 3 Plurisubharmonic functions and the Oka-Weil theorem . . 4 Quantitative approximation theorems in C . . . . 5 The Bernstein-Walsh theorem in CN , N > 1 . . . . 6 Quantitative Runge-type results in multivariate approximation ¯ 7 Mergelyan property and solving ∂ . . . . . . . 8 Approximation on totally real sets . . . . . . 9 Lagrange interpolation and orthogonal polynomials . . . 10 Kergin interpolation . . . . . . . . . . 11 Rational approximation in CN . . . . . . . . 12 Markov inequalities . . . . . . . . . . 13 Appendix on pluripolar sets and extremal psh functions . 14 Appendix on complex Monge-Amp` ere operator . . . 15 A few open problems . . . . . . . . . . References . . . . . . . . . . . . . 92 96 97 103 105 109 111 115 118 121 125 128 130 134 135 136

Splitting an operator Algebraic modularity results for logics with fixpoint semantics

1. INTRODUCTION An important aspect of human reasoning is that it is often incremental in nature. When dealing with a complex domain, we tend to initially restrict ourselves to a small subset of all relevant concepts. Once these “basic” concepts have been ﬁgured out, we then build another, more “advanced”, layer of concepts on this knowledge. A quite illustrative example of this can be found in most textbooks on computer networking. These typically present a seven-layered model of the way in which computers communicate. First, in the so-called physical layer, there is only talk of hardware and concepts such as wires, cables and electronic pulses. Once these low-level issues have been dealt with, the resulting knowledge becomes a ﬁxed base,
Splitting an operator: Algebraic modularity results for logics with ﬁxpoint semantics

A Riemann-Roch-Hirzebruch formula for traces of differential operators

Contents 1. Introduction 2. Hochschild homology of the algebra of diﬀerential operators 3. The third trace 4. The ﬁrst trace is proportional to the third. . . 5. . . . and so is the second 6. Asymptotic topological quantum mechanics Appendix A. Triangulations and signs Appendix B. Heat kernel estimates and asymptotic expansion References 1 3 10 11 14 15 20 20 30
arXiv:math/02461v4 [math.QA] 12 Feb 2008
A RIEMANN–ROCH–HIRZEBRUCH FORMULA FOR TRACES OF DIFFERENTIAL OPERATORS
MARKUS ENGELI AND GIOVANNI FELDER Abstract. Let D be a holomorphic diﬀerential operator acting on sections of a holomorphic vector bundle on an n-dimensional compact complex manifold. We prove a formula, conjectured by Feigin and Shoikhet, giving the Lefschetz number of D as the integral over the manifold of a diﬀerential form. The class of this diﬀerential form is obtained via formal diﬀerential geometry from ∗ ) of the the canonical generator of the Hochschild cohomology HH 2n (Dn , Dn algebra of diﬀerential operators on a formal neighbourhood of a point. If D is the identity, the formula reduces to the Riemann–Roch–Hirzebruch formula. ´sum´ Re e. Soit D un op´ erateur diﬀ´ erentiel holomorphe op´ erant sur les sections d’un ﬁbr´ e vectoriel holomorphe sur une vari´ et´ e complexe de dimension n. Nous d´ emontrons une formule, conjectur´ ee par Feigin et Shoikhet, donnant le nombre de Lefschetz de D comme int´ egrale d’une forme diﬀ´ erentielle sur la vari´ et´ e. La classe de cette forme diﬀ´ erentielle est obtenue, via la g´ eom´ etrie diﬀ´ erentielle formelle, du g´ en´ erateur canonique de la cohomologie de Hoch∗ ) de l’alg` schild HH 2n (Dn , Dn ebre des op´ erateurs diﬀ´ erentiels sur un entourage formel d’un point. Si D est l’identit´ e, la formule se r´ eduit a ` la formule de Riemann–Roch–Hirzebruch.

Space-Time Wireless Systems From Array Processing to MIMO Communications Edited by

† An isotropically distributed unitary matrix has a probability density function that is invariant under left-multiplication by deterministic unitary matrices.
10.1 Introduction In order to achieve high spectral eﬃciency on wireless channels, we need multiple antennas at both transmitter and receiver. Information theoretic results promise considerable capacity gains for wireless communication systems that use multiple transmit and receive antennas for coherent and non coherent reception. Coherent reception means that the receiver knows the channel response but the transmitter not. Non coherent reception means that neither the transmitter nor the receiver know the channel response. In this chapter, we propose to show a non exhaustive presentation of the spacetime codes we can use for the noncoherent case when the communication system uses M transmit antennas and M receive antennas. General assumptions as well as notations are the following. We assume a Rayleigh ﬂat fading channel in order to separate space-time processing and multipath problems. Moreover, this channel is also assumed quasistatic. That means that channel coeﬃcients do not vary during the transmission of a codeword with temporal length T . In that case, the received signal can be expressed as YT ×N = αXT ×M .HM ×N + WT ×N (10.1)

Simplified proof of the Fourier sampling theorem

Simpliﬁed Proof of theFourier Sampling TheoremPeter Høyer∗BRICS†May29,2000AbstractWe give a short and simple proof of Hales and Hallgren’s Fourier Sampling Theorem[“Quantum Fourier Sampling Simpliﬁed”,Pro-ceedings of the Thirty-First Annual ACM Symposium on Theory ofComputing,ACM Press,May1999].The transparency of our proof-technique allows us to generalize and tighten their result.1IntroductionIn the recent years,the theory of quantum computing has been greatly deve-loped and expanded.Two of the most striking results in the area are Grover’s algorithm for searching[3]and Shor’s algorithms for factoring andﬁnding discrete logarithms[6].For an excellent introduction to quantum computing, see for example[2].Any quantum algorithm works on aﬁnite Hilbert space H.Two types of operations are allowed,theﬁrst is unitary operators on H,the second is measurements of the whole or parts of the system.Since we are interested inthe computational complexity of the algorithms,we restrict the operations allowed to only those that can be implemented eﬃciently.One of the primary operators used in the quantum algorithms developed so far,is the quantum Fourier transform.Two of its main uses are to set up a quantum system in an initial state and to perform quantum Fourier sampling.The quantum Fourier transform is actually not a single operator, but a family of operators.One can deﬁne Fourier transforms for anyﬁnite group G.If the group G is Abelian,then there exists exactly one Fourier transform for G,and if G is non-commutative,then there are inﬁnitely many Fourier transforms for G.For every integer n≥1,the quantum Fourier transform over the cyclic group Z n is deﬁned byF n=1nn−1i,j=0ωij n|i j|,(1)whereωn=exp(2π√over Z n,perform quantum Fourier sampling over Z m for some m suﬃciently large compared to n.This idea,however,adds complications to the analysis of the modiﬁed algorithm;for example,one then has to show that the relevant data is still attainable via sampling from the modiﬁed distribution D′.Recently,Hales and Hallgren[4]proposed a general technique for circum-venting such complications.They showed that,for any input state|u ,the original distribution D is contained in the modiﬁed distribution D′by re-striction.This allows us to sample from D via sampling from D′.Weﬁrst explain the notation involved and then we state their theorem.Let1<N<M be integers.For any integer0≤i<N,let i′=⌊iM/N+1/2⌋denote a closest integer to iM/N,and setδi=i′−iM/N. Note that|δi|≤1/2.Given an input state|u = N−1i=0u i|i ,set|v =F N|u and|w =F M|u .Let D v:{0,...,N−1}→[0,1]denote the probability distribution in-duced by measuring|v ,that is,D v(i)=| i|v |2.Deﬁne probability distribu-tion D w:{0,...,M−1}→[0,1]similarly.Let D w′:{0,...,N−1}→[0,1] denote the probability distribution deﬁned by D w′(i)=c·D w(i′),where c= N−1i=0D w(i′) −1is the normalization factor.Thus,we obtain distribu-tion D w′by restricting D w to outcomes j for which j=i′for some0≤i<N, and then relabeling i′by i.Finally,for any two probability distributions D and D′over{0,...,N−1},let|D−D′|= N−1i=0|D(i)−D′(i)|denote their total variation distance.Theorem1(Hales and Hallgren)For any polynomial s(n),there exists a polynomial t(n)such that for all integers N≤2n and M≥t(n)N,and all input states|u = N−1i=0u i|i ,we have|D v−D w′|≤1and secondly,we show that in Theorem1,it suﬃces to pick t(n)to be onthe order of s(n)n.The applications of Hales and Hallgren’s theorem are many.For instance,it allows a simpliﬁed proof of Shor’s theorem for factoring(see Section3of[4]).When applying their theorem,we would use the Fourier transform F M instead of F N.We set up the input state|u ,apply F M and then measurethe system.We repeat this experiment until the measurement produces an outcome j such that j=i′for some0≤i<N.When that happens,we output i and stop.By Theorem2below,the expected number of repetitionsis on the order of MMs|D v−D w′|≤41M) .4sIn the calculations to come,we use many inequalities and bounds.Several of these bounds are not tight as our primary aim is to give a simple and basic proof.Operator A is not necessarily unitary,but it is linear and can be written as a sum A = N −1i,j =0a ij |i j |where a ij ∈C is given bya ij =1N N −1 k =0ωk (i −j +δi N/M )N .(3)Note that |a ij |≤1for all 0≤i,j <N ,that is,every coeﬃcient has absolute value at most 1.The next lemma expresses that every diagonal element a ii is close to 1,whereas every oﬀ-diagonal element a ij (i =j )has small absolute value.The lemma is a variant of Claim 1in [4].Lemma 3For operator A = N −1i,j =0a ij |i j |given by Equation 2,Re(a ii )≥1−5 N |i −j |NN2then,for all 0≤k <N ,we have Re(ωkδi M )≥cos(πN/M )≥1−5 NM 2.Now consider the oﬀ-diagonal element a ij for some 0≤i,j <N withi =j .If δi =0then a ij =0,so suppose ing that the rightmost sum in Equation 3is a geometric series,rewrite a ij =1N 1−ωδi N MM .To lower bound the absolute value of the5denominator,write 1−ωi −j +δi N/M N = sin πi −j M ≥ sin πi −j 212and,by symmetry,that sin(πx )≥2(1−|x |)if1N |i −j |N −πM ,allowing us to conclude that |a ij |≤2M provided M ≥8N .⊓⊔Lemma 3tells us that operator A acts as the identity I = N −1i =0|i i |,modulo some error terms.To analyze the “damage”caused by those error terms,writeA =I +E .Let E = N −1i,j =0e ij |i j |.Then |e ii |2=|Re(a ii )−1|2+|Im(a ii )|2=1+|a ii |2−2Re(a ii )≤10(N10N |i −j |N NM2+log 2(N ) .Proof We prove this lemma by rephrasing it in terms of matrices and vectors,and then introduce a vector norm and its induced matrix norm which we easily can bound.Hence,consider E an N ×N matrix (e ij )N −1i,j =0,and let Norm(·)denote the matrix norm deﬁned byNorm(B )=max B x 2: x =1 where x 2=(x ∗·x )1/2denotes the Euclidean norm of the N ×1column vector x ,and where x ∗denotes the Hermitian adjoint of x .Then,clearly,we have that E |v ≤Norm(E ),and thus it suﬃces to upper bound the matrix norm of E .For this,note that Norm(E )≤Norm(|E |)where |E |denotes the matrix obtained by replacing each entry of E with its absolute value.Observe that,by Lemma 3,we have Norm(|E |)≤Norm(C )where C =(c ij )N −1i,j =0withc ij = 6N|i −j |N NMatrix C is circulant with positive real-valued entries and hence its norm is equal the sum of any row or any column,Norm(C)= N−1j=0c1j= N−1i=0c i1. We upper bound the leftmost sumN−1j=0c1j=N|j|N ≤N j ≤N√2.Then|w′ has unit norm and|v −|w′ ≤52 E|v ≤1b≤1+ E|v .(4)Proof By deﬁnition,b= A|v = (I+E)|v ,so1− E|v ≤b≤1+ E|v .Equation4follows since we assume E|v ≤1/2. Let|y =|v −|w′ .Then|y = I−1b(I+E) |v = 1−1b |v −1b E|v . Hence, |y ≤ 1−1b +1b E|v ≤5Theorem 2follows immediately by composing Lemmata 4,5,and 6.Proof of Proof of Theorem 2Write |w ′ =c RF M |u =1b A |v where c =1b N .By Lemma 4,we have d = E |v ≤3N2N 512,so by Lemma 5, |v −|w ′ ≤5s and |1−1b |≤d ≤1s .⊓⊔3DiscussionOur Theorem 2generalizes Hales and Hallgren’s theorem in two ways.Firstly,if we want to apply Fourier sampling over Z N ,then,by Theorem 2,it suﬃces to be able to implement the Fourier transform F M for some M which is only a log(N )factor larger than N .Thus,for such an M ,we only need log log(N )additional qubits to implement F M instead of implementing F N .Secondly,in Theorem 2,not only are the distributions D v and D w ′close,but so are the states |v and |w ′ just prior the ﬁnal measurement.In the setup studied by Hales and Hallgren,we are given a state |u on which we want to apply a Fourier transform F N which is immediately followed by a measurement of the system.Now,suppose we do not want to measure the state F N |u ,but instead ﬁrst apply some other operations on it and then measure it.In that case,we cannot apply Hales and Hallgren’s theorem,but we can apply Theorem 2.The reason is that Hales and Hallgren’s theorem says that the distributions are close,not that the states themselves are close,as stated in Theorem 2.AcknowledgementsI am very grateful to Sean Hallgren for helpful discussions.I would like to thank an anonymous referee for a careful reading and for many constructive suggestions,and Gilles Brassard and Joan Boyar for comments.References[1]E.Bernstein and U.Vazirani,Quantum complexity theory,SIAM J.Comput.26(1997)1411–1473.8[2]R.Cleve,An introduction to quantum complexity theory.To appearin Collected Papers on Quantum Computation and Quantum Information Theory,World Scientiﬁc,C.Macchiavello,G.M.Palma,and A.Zeilinger, editors,2000.Also available at Los Alamos National Laboratory e-Print archive as</abs/quant-ph/9906111>.[3]L.K.Grover,Quantum mechanics helps in searching for a needle in ahaystack,Phys.Rev.Lett.79(1997)325–328.[4]L.Hales and S.Hallgren,Quantum Fourier sampling simpliﬁed,Proceed-ings of the Thirty-First Annual ACM Symposium on Theory of Comput-ing,ACM Press(1999)330–338.[5]A.Yu.Kitaev,Quantum measurements and the Abelian stabilizer problem(1995).Available at Los Alamos National Laboratory e-Print archive as </abs/quant-ph/9511026>.[6]P.Shor,Polynomial-time algorithms for prime factorization and dis-crete logarithms on a quantum computer,SIAM put.26(1997) 1484–1509.9。

Quanta Without Quantization

a rXiv:h ep-th/975235v129M a y1997Quanta without quantization James T.Wheeler Department of Physics,Utah State University,Logan,UT 84322Abstract The dimensional properties of ﬁelds in classical general relativity lead to a tangent tower structure which gives rise directly to quantum mechanical and quantum ﬁeld theory structures without quantization.We derive all of the fundamental elements of quantum mechanics from the tangent tower structure,including fundamental commuta-tion relations,a Hilbert space of pure and mixed states,measurable expectation values,Schr¨o dinger time evolution,“collapse”of a state and the probability interpretation.The most central elements of string theory also follow,including an operator valued mode expansion like that in string theory as well as the Virasoro algebra with central charges.Introduction The detailed structure of quantum systems follows by fully developing the scaling structure of spacetime.Thus quantum systems -de-spite their Hilbert space of states,operator-valued observables,interferingcomplex quantities,and probabilities -are rendered in terms of classical spacetime variables which simultaneously form a Lie algebra of operators on the space of conformal weights.This remarkable result follows from the tangent tower structure implicit in general relativity.We develop this structure,cite a central theorem,then examine some properties of tensors over the tangent tower.Finally,we apply these ideas to quantum mechanics and string theory,showing how the core elements of both arise classically.1The weight tower and weight maps Consider a spacetime(M,g)with dynamicalﬁelds,{ΦA|A∈A}of various conformal weights and tensorial types.Under a global change of units byλ,letΦA→(λ)w AΦA(1) where w A is called the conformal weight ofΦA.Normally,physicists simply insert these scale factors by hand when needed,without mentioning the im-plicit mathematical structures their use requires,but these structures turn out to be interesting and important.The tower structure begins with the set of conformal weights,W≡{w A| A∈A},which must be closed under addition of any two diﬀerent elements. Possible sets include the reals R,the rationals Q,the integers J,and the ﬁnite set{0,1,−1}.For most physical problems we can choose the unit-weight objects to give W=J.Next,deﬁne the equivalence relationΦA∼=ΦB if w A=w B and∃ηsuch thatΦA=ηΦB(2) which partitions the tangent space T M into a tower of projective Minkowski spaces,P M,one copy for each n∈J=W.This partition enlarges the linear transformation group of the tangent space into the direct product of the Lorentz group and the group of weight maps.While the Lorentz trans-formations have their usual eﬀect,general weight maps act on conformal weight.To see that the group is a direct product,consider the product of an n-weight scalarﬁeld with an m-weight vectorﬁeld.Since the linear trans-formations preserve the Lorentz inner product between arbitrarily weighted vectors,the resulting(n+m)-weight vectorﬁeld remains parallel to the orig-inal m-weight vectorﬁeld under Lorentz transformation.Therefore,Lorentz transformations map diﬀerent weight vectors in the same way.We now investigate weight maps.Just as the only measurable Lorentz objects are scalars,the only measurable tangent tower quantities are zero weight scalars1.Therefore,to readily form zero weight scalars,we classify weight maps and tensors over W by their conformal weights.The generator of global scale changes,D,determines the weights ofﬁelds according to2DΦA=w AΦA(3)while the generators Mαof deﬁnite-weight maps satisfy[D,Mα]=nαMα(4) where nα∈J.Then Mαmaps n-weightﬁelds to(n+nα)-weightﬁelds,making the construction of0-weight quantities straightforward.The following theorem now holds[1]:Theorem.Let V be a maximally non-commuting Lie algebra consisting of exactly one weight map of each conformal weight.Then V is the Virasoro algebra with central charge,[M(m),M(n)]=(m−n)M(m+n)+cm(m2−1)δ0m+n1(5) The lengthy proof relies on explicit construction through a series of induc-tive arguments.It is highly signiﬁcant to note that the real-projective tangent tower produces the same central charge for V as unitarily-projective string theory,despite its classical character.This is theﬁrst concrete evidence that some phenomena widely regarded as“quantum”can be understood from a classical standpoint.Tensors over the weight tower Having understood the algebraic char-acter of deﬁnite-weight weight maps,we next look at the tensors they act on.Since Lorentz transformations decouple from weight maps,the Lorentz and conformal ranks are independent.ThusT a1···a rn1···n s:(P M)r⊗J s→R(6) is a typical rank(r,s)tensor.Weight maps act linearly on each label n i.Particularly relevant to quantum systems are(0,1)tensors.Withηk a k-weight scalar,Dηk=kηk,a general(0,1)tensor is an indeﬁnite-weightlinear combinationΦ=∞k=−∞φkηk(7)We immediately see the need for some convergence criterion.Imposing a norm provides such a criterion,and with a norm these objects form a Hilbert space,H.The simplest norm uses a continuous representation for the(0,1)tensors deﬁned byΦ(x)≡∞n=−∞φn e inx.(8) 3Clearly,Φ(x)exists only under appropriate convergence conditions,provided by including as the(0,1)tensors those vectors satisfying1δ(x−y)+c1(11)∂xfor D andM(k)(x,y)=e ikx(−i∂x−k)δ(x−y)(12) for the k-weight Virasoro operator.In eq.(11),D(x,y)requires the central distribution c to cancel surface terms from the product integral.Such terms are an artifact of the continuous representation.Physical eﬀects of the tangent tower Now we describe physical eﬀects of the tangent tower.We replace the usual(r,0)tensors of quantum mechan-ics and quantumﬁeld theory by(r,1)or(r,2)tensors.In the remaining two sections we do this in detail for quantum mechanics,then brieﬂy for string theory.Conformal weight and quantum mechanics The tangent tower under-pinning of all axiomatic features of quantum mechanics now follows imme-diately:commutation relations of operator-valued position and momentum vectors,a Hilbert space of pure and mixed states,measurable expectation values,Schr¨o dinger time evolution,“collapse”of a state and the probability interpretation.First,consider canonical commutators.Since w x=1and w p/h=−1, canonical coordinates lie in the subalgebra determined by W0={0,1,−1}⊂W.We temporarily consider operators in this subspace,replacing symplectic coordinates Q A=(q i,πj)by weight-map-valued6-vectorsˆQ A=(ˆq i,ˆπj)∈T a nm forming the Lie algebra[ˆQ A,ˆQ B]=c AB1(13)4where projective representation allows arbitrary central charges c AB.The only invariant antisymmetric symplectic tensor is the symplectic2-formΩAB= 0−δi jδj i0 (14)so we set[ˆQ A,ˆQ B]=ΩAB1(15) Furthermore,the natural symplectic metricK AB= 0δi jδj i0 (16)may be diagonalized by a symplectic transformation to new variablesˆR A= (ˆX i,ˆP j)such that˜K AB= δij−δij (17) or equivalentlyˆR′A=(ˆX i,iˆP j)with˜K′AB= δijδij (18)The zero signature of the symplectic metric requires an imaginary unit in relatingˆR′A toˆP j because the physicalˆX i andˆP j have the same metric, +δij.The projective algebra ofˆX i andˆP j is the canonical one[ˆX i,ˆP j]=iδij1(19)[ˆX i,ˆX j]=[ˆP i,ˆP j]=0(20) Thus,“quantum”(ˆX i,ˆP j)commutators follow from classical scaling consid-erations by replacing classical variables with weight tower operators in the {0,1,−1}subalgebra,and using the natural symplectic structure.Other quantum structures follow easily.Thus,deﬁnite weight scalars replace pure quantum states,while indeﬁnite weight objects such asΦ(x)∈H replace mixed states.The construction of expectation values is simply a rule to generate a0-weight scalar.The rule works by matching elements of5H with their complex conjugates but the deﬁnition ofΦ(x)above translates this into the manifestly0-weight sumΦ,Ψ =1dτinﬂat spacetime while uµWµΦgives the action ofa weight map H≡uµWµon the weight superpositionΦ(xν;x).Identifying H as the Hamiltonian operator[2],eq.(22)becomes the Schr¨o dinger equation. The free-particle form for H is the zero-weight quantityH= dˆX jL ,L2φ−2,etc.,with L some standard unitof length.Therefore the rules of quantum mechanics have natural classical interpre-tations in terms of the scale-invariance properties of spacetime.6Conformal weight and quantumﬁeld theory Just asﬁeld theory emerges as the limiting case of multiparticle dynamics,and quantumﬁeld theory emerges as a blend of quantum mechanics and special relativity[3], we can imagine retracing the preceding arguments to derive the principles of quantumﬁeld theory.After all,the quantum state is aﬁeld,even in quantum mechanics.Thus,we may anticipate a classical tangent tower in-terpretation of quantumﬁeld theory.As striking as this conjecture seems, we now demonstrate a more striking claim:the tangent tower contains the essential elements of string theory[4].Minor manipulations of a deﬁnite weight(1,2)tensorαµ(k)(x,y)found bywriting theδ-function in D asδ(x−y)=1References[1]Wheeler,J.T.,String without strings,in preparation.[2]Wheeler,J.T.,Phys.Rev.D44(1990);Wheeler,J.T,Proceedings ofthe Seventh Marcel Grossman Meeting on General Relativity,R.T.Jantzen and G.M.Keiser,editors,World Scientiﬁc,London(1996)pp 457-459.[3]Weinberg,S.,The quantum theory ofﬁelds,Cambridge University Press(1995).[4]Green,M.B.,J.H.Schwarz and E.Witten,Superstring theory,Cam-bridge University Press(1987).[5]Kuchar,K.V.and C.G.Torre,J.Math.Phys.30(8),August1989.8。

Lattice Gauge Fields and Discrete Noncommutative Yang-Mills Theory

† ∗
Contents
1 Introduction and summary 2 Quantum ﬁeld theory on noncommutative spaces 2.1 2.2 2.3 2.4 Sc . . . . . . . . . . . . . . . . . . . . . . . . . . Noncommutative Yang-Mills theory . . . . . . . . . . . . . . . . . . . . . . . Star-gauge invariant observables . . . . . . . . . . . . . . . . . . . . . . . . . The noncommutative torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6 6 9 10 13 14 15 17 23 26 27 31 33 34 38 40 41 44
J. Ambjørn1) ∗ , Y.M. Makeenko1) 2) † , J. Nishimura1) ‡ and R.J. Szabo1) §
1)
The Niels Bohr Institute Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark
2)
Institute of Theoretical and Experimental Physics B. Cheremushkinskaya 25, 117218 Moscow, Russia
2
geometry provides a natural framework to describe nonperturbative aspects of string theory [2, 5]. This belief is further supported by the fact that Matrix Theory [6] and the IIB matrix model [7], which are conjectured to provide nonperturbative deﬁnitions of string theories, give rise to noncommutative Yang-Mills theory on toroidal compactiﬁcations [8]. The particular noncommutative toroidal compactiﬁcation is interpreted as being the result of the presence of a background Neveu-Schwarz two-form ﬁeld, and it can also be understood in the context of open string quantization in D-brane backgrounds [9, 10]. Furthermore, in Ref. [11] it has been shown that the IIB matrix model with D-brane backgrounds is described by noncommutative Yang-Mills theory. The early motivation [12] for studying quantum ﬁeld theory on noncommutative spacetimes was that, because of the spacetime uncertainty relation, the introduction of noncommutativity would provide a natural ultraviolet regularization. However, more recent perturbative calculations [13]–[16] have shown that planar noncommutative Feynman diagrams contain exactly the same ultraviolet divergences that their commutative counterparts do, which implies that the noncommutativity does not serve as an ultraviolet regulator. One therefore needs to introduce some other form of regularization to study the dynamics of noncommutative ﬁeld theories. On the other hand, it has been found that the ultraviolet divergences in non-planar Feynman diagrams [16, 17] exhibit an intriguing mixing of ultraviolet and infrared scales, which can also be described using string-theoretical approaches [18, 19]. Heuristically, this UV/IR mixing can be understood in terms of the induced uncertainty relations among the spacetime coordinates. If one measures a given spacetime coordinate with some high precision, then the remaining spacetime directions will generally extend because of the smearing. Furthermore, noncommutative solitons which do not have counterparts in ordinary ﬁeld theory have been discovered [20] for suﬃciently large values of the noncommutativity parameters, and it has also been shown [19] that noncommutative Yang-Mills theory in four dimensions naturally includes gravity. In order to investigate further the non-trivial dynamics of noncommutative ﬁeld theories, it is important therefore to develop a nonperturbative regularization of these theories. Such a program has been put forward in Refs. [11, 15, 19],[21]–[24] and it is similar to earlier works [25] based on the mapping between large N matrices and spacetime ﬁelds. In particular, in Ref. [22] a uniﬁed framework was presented which naturally interpolates between the two ways that noncommutative Yang-Mills theory has appeared in the context of matrix model formulations of string theory, namely the compactiﬁcation of Matrix theory and the twisted large N reduced model. The model proposed was a ﬁnite N matrix model deﬁned by the twisted Eguchi-Kawai model [26, 27] with a quotient condition analogous to the ones considered in Refs. [8, 28]. It was interpreted as a lattice formulation of noncommutative

Non-associative gauge theory and higher spin interactions

a r X i v :h e p -t h /0412027v 2 10 F eb 2005MCTP-04-64QMUL-PH-04-08hep-th/041202702/12/04Non-associative gauge theory and higher spin interactions Paul de Medeiros 1and Sanjaye Ramgoolam 21Michigan Center for Theoretical Physics,Randall Laboratory,University of Michigan,Ann Arbor,MI 48109-1120,U.S.A.2Department of Physics,Queen Mary University of London,Mile End Road,London E14NS,U.K.pfdm@ ,s.ramgoolam@ Abstract We give a framework to describe gauge theory on a certain class of commutative but non-associative fuzzy spaces.Our description is in terms of an Abelian gauge connection valued inthe algebra of functions on the cotangent bundle of the fuzzy space.The structure of such a gauge theory has many formal similarities with that of Yang-Mills theory.The components of the gauge connection are functions on the fuzzy space which transform in higher spin representations of the Lorentz group.In component form,the gauge theory describes an interacting theory of higher spin ﬁelds,which remains non-trivial in the limit where the fuzzy space becomes associative.In this limit,the theory can be viewed as a projection of an ordinary non-commutative Yang-Mills theory.We describe the embedding of Maxwell theory in this extended framework which follows the standard unfolding procedure for higher spin gauge theories.1IntroductionWe formulate gauge theory on a certain class of commutative but non-associative algebras, developing the constructions initiated in[1].These algebras correspond to so called fuzzy spaces which reduce to ordinary spacetime manifolds in a particular associative limit.We ﬁnd that such gauge theories have a realisation in terms of interacting higher spinﬁeld theories.The non-associative algebra of interest A∗n(M)is a deformation of the algebra of func-tions A(M)on a D-dimensional(pseudo-)Riemannian manifold M.The∗denotes a non-associative product for functions on the fuzzy space whilst n∈Z+provides a quantitative measure of the non-associativity(in particular A∗∞=A).For simplicity,we take M=R D withﬂat metric.Most of our formulas will be independent of the signature of this metric, though we will take it to be Lorentzian in discussions of gauge-ﬁxing etc.Furthermore, although we focus on the deformation for R D,there is a conceptually straightforward gener-alisation for curved manifolds.For example,the deformation A∗n(S2k)has been used in the study of even-dimensional fuzzy spheres in[2].In section2we deﬁne the commutative,non-associative algebra A∗n(R D)which deforms A(R D),and give the derivations of this algebra.In this review,we recall that the associator(A∗B)∗C−A∗(B∗C)of three functions A,B and C on A∗n(R D)can be written as an operator F(A,B)acting on C or as an operator E(A,C)acting on B.These operators have expansions in terms of derivations of the algebra(given in Appendix B)and naturally appear when one attempts to construct covariant derivatives for the gauge theory.Weﬁnd that an inevitable consequence of this structure is that the connection and gauge parameter have to be generalised such that they too have derivative expansions(i.e.they can be understood as functions on the deformed cotangent bundle A∗n(T∗R D)).The inﬁnite number of component functions in these expansions transform as totally symmetric tensors under the Lorentz group.Consequently weﬁnd that this extended gauge theory on A∗n(T∗R D)is related to higher spin gauge theory on A∗n(R D).The local and global structure of this extended gaugetheory is analysed in section3.We observe that the extended gauge theory remains non-trivial even in the limit where the non-associativity parameter goes to zero.In section4we describe certain physical properties in this associative limit.In particular we construct a gauge-invariant action andﬁeld equations for the extended theory using techniques related to the phase space formulation of quantum mechanics initiated by Weyl[20]and Wigner[21].The inﬁnite number higher spin components of the extended gaugeﬁeld become just tensors on R D in the associative limit.We describe various aspects of the extended theory in component form in order to make the connection with higher spin gauge theory more explicit.From this perspective it will be clear that the extended theory(as we have presented it)does not realise all the possible symmetries of the corresponding higher spin theory on R D.We suggest that it could describe a partially broken phase of some fully gauge-invariant theory. We then compare the structure weﬁnd with that of the interacting theory of higher spin ﬁelds discovered by Vasiliev[14].A precise way to embed Maxwell theory in the extended theory is given.The method is identical to the unfolding procedure which has been used by Vasiliev in the context of higher spin gauge theories[16].It can also be understood simply via a change of basis in phase space under a particular symplectic transformation.In section5we describe how the extended theory in the associative limit described in section 4is related to a projection of an ordinary non-commutative Yang-Mills theory.We also describe connections to Matrix theory.We then discuss how one might generalise the results of section4to construct a gauge-invariant action for the non-associative theory.Section6 contains some concluding remarks.2The non-associative deformation A∗nWe begin by deﬁning the non-associative space of interest.Following[1],we consider the commutative,non-associative algebra A∗n(R D)which is a speciﬁc deformation of the commu-tative,associative algebra of functions A(R D)on R D(which is to be thought of as physical spacetime in D dimensions).Another space that will be important in forthcoming discus-sions is the algebra of diﬀerential operators acting on A∗n(R D).This algebra is isomorphic to the deformed algebra A∗n(T∗R D)of functions on the(ﬂat)cotangent bundle T∗R D.This correspondence will be helpful when we come to consider gauge theory on A∗n(R D).The space R D has coordinates xµandﬂat metric.The Euclidean signature metricδµνarises most directly in the Matrix theory considerations motivating[1]but the algebra can be continued to Lorentzian signature by replacing this with Lorentzian metricηµν.The algebraic discussion in this and the next section(and in the appendices)works equally well in either signature,but some additional subtleties related to gauge-ﬁxing discussed in section4are speciﬁc to the Lorentzian case.The deformed algebra A∗n(R D)is spanned by the inﬁnite set of elements{1,xµ,xµ1µ2,...}1,where each xµ1...µs transforms as a totally symmetric tensor of rank s under the Lorentz group.The commutative(but non-associative) product∗for all elements xµ1...µs is deﬁned in[1]and Appendix B(this appendix also deﬁnes a more general set of products with similar properties to∗).The explicit formula is rather complicated but the important point is that xµ1...µs∗xν1...νt equals xµ1...µsν1...νt up to theaddition of lower rank elements with coeﬃcients proportional to inverse powers of n(for example xµ∗xν=xν∗xµ=xµν+11In[1],the elements x were called z and the deformed algebra A∗(R D)was called B∗n(R D).nnot break Lorentz symmetry.One can deﬁne derivations∂µof A∗n(R D)via the rule∂µxµ1...µs=sδ(µ1µxµ2...µs),(1) where brackets denote symmetrisation of indices(with weight1)2.This deﬁnition implies that∂µsatisfy the Leibnitz rule when acting on∗-products of elements of A∗n(R D).This Leibnitz property also holds with respect to the more general commutative,non-associative products described in Appendix B.It is clear that composition of these derivations is a commutative and associative operation.In the associative n→∞limit,∂µjust act as the usual partial derivatives on R D.2.1FunctionsFunctions of the coordinates xµ1...µs are written A(x)∈A∗n(R D).Such functions form a commutative but non-associative algebra themselves with respect to the∗multiplication.A quantitative measure of this non-associativity is given by the associator[A,B,C]:=(A∗B)∗C−A∗(B∗C)(2) for three functions A,B and C.Since A∗n(R D)is commutative then the associator(2)has the antisymmetry[A,B,C]=−[C,B,A].The associator also satisﬁes the cyclic identity [A,B,C]+[B,C,A]+[C,A,B]≡0.An important fact noted in[1]is that such associators can be written as diﬀerential operators involving two functions acting on the third.In particular,one can deﬁne the two operators E(A,B)and F(A,B)via[A,B,C]=:E(A,C)B=:F(A,B)C.(3) The antisymmetry property of the associator implies E(A,B)=−E(B,A)and the cyclic identity implies F(A,B)−F(B,A)=E(A,B).These operators have the following derivativeexpansions(see[1]or Appendix B)E(A,B)=∞s=11s!(Fµ1...µs(A,B))(x)∗∂µ1...∂µs,(4)where the coeﬃcients Eµ1...µs(A,B)and Fµ1...µs(A,B)are both polynomial functions of the algebra transforming as totally symmetric tensors under the Lorentz group3.The properties quoted above follow for each of these coeﬃcients so that Eµ1...µs(A,B)=−Eµ1...µs(B,A)and Fµ1...µs(A,B)−Fµ1...µs(B,A)=Eµ1...µs(A,B).The reason there are no s=0terms in(4)is that the associators[A,1,C]and[A,B,1]are both identically zero.Thus since(4)are valid as operator equations on any function then including such zeroth order terms in(4)would imply their coeﬃcients are identically zero by simply acting on a constant function.Theﬁrst non-vanishing s=1coeﬃcients in(4)can be expressed rather neatly as associators,such that Eµ(A,B)=[A,xµ,B]and Fµ(A,B)=[A,B,xµ].In a similar manner,all subsequent s>1coeﬃcients in(4)can also be expressed in terms of(sums of)associators of A and B with coordinates xµ1...µs(though we do not give explicit expressions as they are unnecessary). An important point to keep in mind is that E(A,B)and F(A,B)vanish in the associative limit as expected.The algebra of the diﬀerential operators in(4)closes under composition and is non-associative (following non-associativity of A∗n(R D))but it is also non-commutative.Since E(A,B)and F(A,B)vanish in the associative limit the algebra of these operators becomes trivially com-mutative when n→∞.As will be seen in the next subsection,more general diﬀerential operators acting on A∗n(R D)also close under composition to form a non-commutative,non-associative algebra.However,this more general algebra remains non-commutative(but as-sociative)when n→∞.For example,the commutator subalgebra of diﬀerential operators acting on R D corresponding to sections of the tangent bundle T R D(i.e.vectorﬁelds over R D)is non-Abelian(even though R D is itself commutative).Indeed this is often how oneconsiders simple non-commutative geometries–as Hamiltonian phase spaces of ordinary commutative position spaces(see e.g.[19]).We will draw on this analogy when we come to construct a gauge theory on A∗n(R D).2.2Diﬀerential operatorsGeneral diﬀerential operators acting on A∗n(R D)are writtenˆA=∞ s=01the algebra of functions is commutative).The deﬁnitions(6)obey the identitiesˆE(ˆA,ˆB)≡−ˆE(ˆB,ˆA)andˆF(ˆA,ˆB)−ˆF(ˆB,ˆA)≡ˆE(ˆA,ˆB)+[ˆA,ˆB](where[ˆA,ˆB]:=ˆAˆB−ˆBˆA is just thecommutator of operators).These reduce to the identities found earlier in terms of functions whenˆA=A andˆB=B.In the associative limit,notice thatˆF(ˆA,ˆB)vanishes identically whilstˆE(ˆA,ˆB)reduces to the commutator[ˆB,ˆA].The explicit derivative expansion forˆE(ˆA,ˆB)is given in Appendix A for later reference (the corresponding expression forˆF(ˆA,ˆB)will not be needed).We should just conclude this review of the relevant algebras associated with A∗n(R D)by noting that,unlike(4),the operator expression forˆE(ˆA,ˆB)includes a non-vanishing zeroth order algebraic term.It is easy to see that this is so by considering C in(6)to be the constant function.In this case all derivative terms inˆE(ˆA,ˆB)on the left hand side vanish whilst the right hand side reduces to the non-vanishing functionˆBA−ˆAB(where A and B are the zeroth order parts ofˆA and ˆB respectively).Thus the zeroth order partˆE(ˆA,ˆB)=ˆBA−ˆAB,which vanishes when(0)ˆA=A andˆB=B as expected.3Non-associative gauge theoryWe begin this section by reviewing the subtleties raised in[1]associated with formulating an Abelian gauge theory on A∗n(R D).We show that a naive formulation is not possible on this non-associative space.Instead it is rather natural to consider an extension of such an Abelian gauge theory on the deformed algebra A∗n(T∗R D)of functions on the cotangent bundle.We describe the local and global gauge structure of this non-associative extended theory.We ﬁnd the structure to be similar to that of a Yang-Mills theory with inﬁnite-dimensional gauge group.We will return to the question of embedding an Abelian gauge theory on A∗n(R D)in this extended structure in later sections.3.1Abelian gauge theory on A∗n(R D)A necessary ingredient in the construction of any gauge theory is the concept of a gauge-covariant derivative.Consider aﬁeldΦwhich is a function of A∗n(R D)and deﬁne it to have the inﬁnitesimal gauge transformation lawδΦ=ǫ∗Φ,(7)whereǫis an arbitrary polynomial function of A∗n(R D).(One reason for the choice of(7)is that it is reminiscent of the inﬁnitesimal gauge transformation for aﬁeld in the fundamental representation of the gauge group in ordinary Yang-Mills theory.)An operator Dµthat is covariant with respect to(7)must therefore obeyδ(DµΦ)=ǫ∗(DµΦ).(8)Clearly the derivation∂µ(1)alone does not obey this covariance requirement sinceδ(∂µΦ)=ǫ∗(∂µΦ)+(∂µǫ)∗Φ.To compensate we must introduce a gauge connection Aµ,which we take to be a function on A∗n(R D)and which transforms such thatδ(Aµ∗Φ)=ǫ∗(Aµ∗Φ)−(∂µǫ)∗Φ.Clearly the existence of such an Aµwould imply thatDµΦ:=∂µΦ+Aµ∗Φ(9) indeed deﬁnes a covariant derivative on functions,satisfying(8).Using(7)then implies that we require Aµto transform such that(δAµ)∗Φ=−(∂µǫ)∗Φ+ǫ∗(Aµ∗Φ)−Aµ∗(ǫ∗Φ).(10) In ordinary gauge theory(10)would allow one to simply read oﬀthe necessary gauge transfor-mation for Aµbut here things are more complicated due to non-associativity.In particular, notice that the last two terms in(10)can be written as the associator[Aµ,Φ,ǫ]and therefore, using(3),we requireδAµ=−(∂µǫ)+E(Aµ,ǫ).(11) This requirement,however,leads to a contradiction since theﬁrst two terms in(11)are algebraic functions on A∗n(R D)whilst(4)tells us that the third term acts only as a diﬀerentialoperator on A ∗n (R D ).Therefore such an A µcan only exist when E (A µ,ǫ)=0,i.e.in theassociative limit where this would simply be an Abelian gauge theory on R D !As indicated in[1],themost conservative way to proceed is therefore to simply generalise thegauge connection A µfrom an algebraic function to a diﬀerential operator ˆAµwith derivative expansionˆA µ=∞ s =01s !ǫα1...αs (x )∗∂α1...∂αs .(13)As noted already,the algebra of such operators is both non-associative and non-commutative.Consequently we must take care when revising the arguments of this subsection in terms of these extended ﬁelds.This revised analysis is described,in the next subsection,within the framework of global gauge transformations for the extended theory.In concluding,it is important to stress that the generalisation we have made is a modiﬁcation of the original theory and therefore the extended theory need not trivially reduce to an Abelian gauge theory on R D in the associative limit.(Notice that the s >0terms in (12)and (13)do not vanish as n →∞.)Indeed we will ﬁnd it does not though we will give a precise way to embed the Abelian theory in its extension on R D .3.2Global structureConsider again aﬁeldΦwhich is a function of A∗n(R D)but now with inﬁnitesimal gauge transformation lawδΦ=ˆǫΦ,(14) whereˆǫis the extended diﬀerential operator(13).Formally this is similar to Yang-Mills theory where one then obtains the global gauge transformation by exponentiating the lo-cal(Lie algebra valued)gauge parameter to obtain a general Lie group element(or more precisely the fundamental representations of these quantities).The main diﬀerence here is that the algebra of local gauge transformations(14)is non-associative.Despite this,given a general diﬀerential operatorˆǫ,there still exists a well-deﬁned exponential exp(ˆǫ)[18].The construction essentially just follows the power series deﬁnition of the exponential map for matrix algebras but here one must choose an ordering for powers ofˆǫ(so as to avoid the potential ambiguities due to non-associativity).We follow[18]and deﬁne powers via a‘left action’rule so thatexp(ˆǫ)Φ:=Φ+ˆǫΦ+13!ˆǫ(ˆǫ(ˆǫΦ))+...,(15)for any functionΦ.It is then clear that the exponentiated operatorˆg:=exp(ˆǫ)is also a diﬀerential operator acting on the algebra(albeit a rather complicated function ofˆǫ)and we deﬁne the‘global’transformation ofΦto beΦ→ˆgΦ.(16)This transformation obviously reduces to(14)in some neighbourhood of the identity where ˆg=1+ˆǫ(the‘identity’here is the unit element of A∗n(R D)).The set of all transformations (16)does not quite form a group under left action composition since it fails to satisfy the associativity axiom(due to non-associativity of the algebra).However,all the other group axioms are satisﬁed4.The derivation∂µis not covariant with respect(16)since this transformation implies∂µΦ→[∂µ,ˆg]Φ+ˆg(∂µΦ).As noted at the end of the previous subsection,we therefore introduce a gauge connectionˆAµwhich must transform such thatˆAµΦ→−[∂µ,ˆg]Φ+ˆg(ˆAµΦ)in order thatˆDΦ:=∂µΦ+ˆAµΦ(17)µtransforms covariantly under(16).This necessary gauge transformation ofˆAµΦunder(16) can be realised provided the gauge transformation ofˆAµis deﬁned such thatˆAΦ→−[∂µ,ˆg](ˆg−1Φ′)+ˆg(ˆAµ(ˆg−1Φ′))(18)µunder the more general function transformationΦ→Φ′.This gives the desired gauge transformation whenΦ′=ˆgΦ.One can obtain the gauge transformation ofˆAµitself by using the operatorˆF(6)to rearrange the brackets in(18).In particular,notice that the right hand side of(18)can be written−[∂µ,ˆg]+ˆgˆAµ−ˆF(ˆg,ˆAµ) (ˆg−1Φ′)(19) = −[∂µ,ˆg]+ˆgˆAµ−ˆF(ˆg,ˆAµ) ˆg−1 Φ′−ˆF −[∂µ,ˆg]+ˆgˆAµ−ˆF(ˆg,ˆAµ) ,ˆg−1 Φ′. ThereforeˆAµmust have the following gauge transformationˆAµ→ −[∂µ,ˆg]+ˆgˆAµ−ˆF(ˆg,ˆAµ) ˆg−1−ˆF −[∂µ,ˆg]+ˆgˆAµ−ˆF(ˆg,ˆAµ) ,ˆg−1 .(20) Settingˆg=1+ˆǫin(20)leads to the inﬁnitesimal form of the gauge transformationδˆAµ=−[∂µ,ˆǫ]+ˆE(ˆAµ,ˆǫ).(21) Of course,at the inﬁnitesimal level,this transformation equivalently follows by the require-ment thatδ(ˆDµΦ)=ˆǫ(ˆDµΦ)under(14).Notice that(20)and(21)do not quite take the form one would expect by naively following the Yang-Mills analogy(that is they diﬀer from what one might expect by associator terms). This is a consequence of the non-associativity of the underlying algebra of functions.In the following section we willﬁnd that the expected Yang-Mills type structure follows exactly in the associative limit.In the discussion above we have only deﬁned covariant derivativesˆDµon functions and not on diﬀerential operators.Although not of the standard Yang-Mills form,(minus)the right hand side of(21)can still be taken as the deﬁnition for the action of the covariant derivative on operatorˆǫ,such thatˆDµ·ˆǫ:=[∂µ,ˆǫ]+ˆE(ˆǫ,ˆAµ).(22) This statement is partially justiﬁed by the fact thatˆDµthen satisﬁes the Leibnitz rule ˆD(ˆǫΦ)=(ˆDµ·ˆǫ)Φ+ˆǫ(ˆDµΦ)(for general operatorˆǫand functionΦ)5.µBased on the transformation law found above,we deﬁne theﬁeld strengthˆFµνasˆF:=ˆE(ˆDν,ˆDµ)=[∂µ,ˆAν]−[∂ν,ˆAµ]+ˆE(ˆAν,ˆAµ).(23)µνIt is clear from this deﬁnition thatˆFµνis indeed a diﬀerential operator which transforms as a two-form under the Lorentz group.In addition,since the gauge transformations above imply thatˆDΦ→ˆg(ˆDµ(ˆg−1Φ′)),(24)µunder(18),then it follows thatˆFµνΦ=ˆDµ(ˆDνΦ)−ˆDν(ˆDµΦ)transforms asˆFΦ→ˆg(ˆFµν(ˆg−1Φ′)),(25)µνand is therefore also gauge-covariant whenΦ′=ˆgΦ.The inﬁnitesimal form of the covariant gauge transformation ofˆFµνisδˆFµν=ˆE(ˆFµν,ˆǫ).(26)From the evidence above,it is clear that there are various subtleties related to the non-associative nature of the theory.Indeed the non-associativity complicates matters even further in the description of more physical aspects of the theory like Lagrangians,ﬁeld equations and the embedding of an Abelian gauge theory in this extended framework.Recall though that this extended theory should have a non-trivial structure,even in the associative limit.We therefore postpone further discussion of the non-associative extended theory to analyse its associative limit in more detail.4Gauge theory on T∗R D and higher spin gauge theory on R DWe begin this section by brieﬂy summarising the results of the previous subsection in the associative limit.We then describe how one can construct a gauge-invariant action and equations of motion for this theory.Writing the extended gaugeﬁeldˆAµin terms of com-weﬁnd that the extended theory describes an interacting theory ponent functions Aα1...αsµinvolving an inﬁnite number of higher spinﬁelds.When written in component form,it will be clear that the extended theory(as we have described it)does not realise all the possible symmetries of the corresponding higher spin gauge theory.We suggest that the extended theory could correspond to a partially broken phase of some fully gauge-invariant higher spin theory.A comparison of the structure weﬁnd with that of the interacting theory of higher spinﬁelds discovered by Vasiliev[14]is then given.We conclude the section by showing how an Abelian gauge theory can be embedded in this extended framework.The embedding is related to the unfolding procedure used by Vasiliev in the context of higher spin gauge theory[16].134.1The associative limitMany expressions found in the previous section retain their schematic form in the associative limit.For example,the gauge transformations for functions are just as in(14),(16)though Φis now simply a function on R D whilst operators likeˆǫin(13)now have the expansionˆǫ=∞ s=01also transforms covariantly.The inﬁnitesimal form of this covariant transformation beingδˆFµν=[ˆǫ,ˆFµν].(32)4.2Action andﬁeld equationsA simple equation of motion to consider for the extended theory in the associative limit is[ˆDµ,ˆFµν]=0.(33)This is theﬁeld equation one would expect from following the Yang-Mills type structure found for the extended theory in the previous subsection.The equation(33)is invariant under the gauge transformation(28).Moreover it is this equation(rather than,say,the also gauge-invariant equationˆDµˆFµν=0)which reduces to the correct Maxwell equation as we will see in section4.5.Following the Yang-Mills analogy further,a natural gauge-invariant action to consider is of the form−1taking the usual gauge-invariant trace(using the Cartan-Killing metric for the gauge group) followed by integrating over spacetime.However,we do not assume a priori that the map(35)can be factorised in thisway6.In the Yang-Mills case the symmetry property of Trsimply follows from the fact that the trace is symmetric.The symmetry of the trace is a rather general property ofﬁnite-dimensional representations–as one considers for Yang-Mills theories with compact gauge groups–since such representations can be expressed in terms ofﬁnite-dimensional square matrices(and for two such matrices X,Y,the trace of XY is just X i j Y j i=Y i j X j i).For the extended theory we are considering thoughﬁelds are valued in the algebra of diﬀerential operators on R D and the situation is very diﬀerent for the case of such inﬁnite-dimensional representations.For example,in quantum mechanics, if the Heisenberg algebra[ˆx,ˆp]=i had any representations ofﬁnite dimension n=0(and hence a symmetric trace)then it would imply the well-known contradiction0=in!The example above is quite pertinent since we will now show thatﬁelds in the extended theory we are considering are related to certain functions in the formulation of quantum mechanics based on the original work of Weyl[20]and Wigner[21]which was later developed by Groenewold[23]and Moyal[24](see[27]for a nice review).Within this framework,there exists a natural concept of the symmetric map Tr.In terms of the abstract canonically conjugate operatorsˆxµandˆpµ,a general operatorˆA of the form(27)is writtenˆA=A(ˆx,ˆp)=∞ s=0i s6As explained in[19],non-commutative gauge theories provide a counter example where such a factori-sation of Tr is not possible.16ordering prescription above7.Given this ordering rule,the Weyl homomorphism[20]says that every operator A(ˆx,ˆp)(37) is naturally associated with an ordinary c-number function˜A on the classical phase space R2D(spanned by coordinates(x,p)),such that1A(ˆx,ˆp)=(2π)2D dy dq dx dp˜A(x,p)yα1...yαs exp(i qµ(ˆxµ−xµ)−i yµpµ).(39) The trace Tr of the operator A(ˆx,ˆp)is deﬁned byTr(ˆA):= dx dp˜A(x,p).(40) This integral is only deﬁned for functions˜A with suitably rapid asymptotic decay properties. We will describe a particular Wigner basis for a class of such integrable functions in the next subsection.The inverse of the relation(38)can then be expressed in terms of this trace,such that˜A(x,p)=1−→∂∂pµ(−i)m ∂∂pµm˜A ∂∂xµm˜B .(43)m!Notice in particular that the m=0term in(43)is just the commutative classical product of functions˜A˜B.The m>0terms are not commutative but are invariant under the combined exchange˜A↔˜B and x↔p.Equation(43)implies that xµ⋆pν=xµpνandpν⋆xµ=xµpν−iδµν,thus conﬁrming that the⋆-product of functions preserves the structureof the Heisenberg algebra.It is also worth noting that partial derivatives(with respect tox or p)act as derivations on the algebra of classical phase space functions with⋆-product since they obey the Leibnitz rule when acting on(43).The deﬁnition(43)implies thatdx dp(˜A⋆˜B)(x,p)= dx dp˜A′(x,p)˜B′(x,p)= dx dp(˜B⋆˜A)(x,p),(44)where the primed phase space functions denote˜A′:=exp i∂xµ∂2∂∂pµ ˜B which are just multiplied with respect to the classical product in(44).Thus the trace(40)of the operator productˆAˆB is indeed symmetric,as required.The precise form of the gauge-invariant action(34)is therefore given by−14 dx dp˜F′µν(x,p)˜F′µν(x,p),(45) where the function˜F′µν:=exp i∂xµ∂4.2.1Wigner basis for integrable functionsWe will now brieﬂy describe a particular basis for a class of classical functions which have ﬁnite integrals over phase space(a more detailed review of this construction is given in[27]). This will show us how to restrict to the class of Weyl-dual operators for which the trace map Tr is well-deﬁned.Of course,this is necessary so that the gauge-invariant action(45)exists.Consider a complete orthonormal basis of eigenfunctions{ψa}for a given Hamiltonian H. To each such eigenfunctionψa(x)on R D,there is an associated Wigner function1f a(x,p)=proportional to exp −i ∂x µ∂4(2π)4D dxdp dydq dy ′dq ′exp −i 4(2π)2D dy dq exp (−i y µq µ)×Tr exp(−i q µˆx µ)ˆF αβexp(−i y µˆp µ) Tr exp(i q µˆx µ)ˆF αβexp(i y µˆp µ)。

The Solution Space of the Unitary Matrix Model String Equation and the

2 2 +1 2 2 +1
1
interpretation of the UMM is not, however, very clear 22]. In view of this it seems worthwhile to explore their structure further. It is well known 24] that the string equation of the (p; q) HMM can be described as an operator equation P; Q] = 1, where P and Q are scalar ordinary di erential operators of order p and q respectively. They are the well de ned scaling limits of the operators of multiplication and di erentiation by the eigenvalues of the HMM on the orthonormal polynomials used to solve the model. The set of solutions to the string equation P; Q] = 1 was analyzed in 25] by means of the Sato Grassmannian Gr. It was proved that every solution of the string equation corresponds to a point in the big cell Gr(0) of Gr satisfying certain conditions. This fact was used to give a derivation of the Virasoro and W -constraints obtained in 26,27] along the lines of 28{31] and to describe the moduli space of solutions to this string equation. The aim of the present paper is to prove similar results for the version of the string equation arising in the UMM. It was shown in 32] that the string equation of the UMM takes the form P ; Q? ] = const., where for the kth multicritical point P and Q? are 2 2 matrices of di erential operators of order 2k and 1 respectively. For every solution of the string equation one can construct, with this result, a pair of points of the Gr(0) obeying certain conditions. These conditions lead directly to the Virasoro constraints for the corresponding -functions and give a description of the moduli space of solutions. We stress that the above results depend solely on the existence of a continuum limit in which the string equation has the form P ; Q? ] = const. and the matrices of di erential operators P and Q? have a particular form to be discussed in detail in subsequent sections. Our results do not depend on other details of the underlying matrix model. The paper is organized as follows. In section 2 we review the double scaling limit of the UMM in the operator formalism 32]. Since the square root of the speci c heat ows according to the mKdV hierarchy we note that its Miura transforms ow according to KdV and thus give rise to two -functions related by the Hirota bilinear equations of the mKdV hierarchy 33{35]. In section 3 we derive a description of the moduli space of the string equation in terms of a pair of points in Gr(0) related by certain conditions. In section 4 we show the correspondence between points in Gr(0) and solutions to the mKdV hierarchy. The Virasoro constraints are derived from invariance conditions on the points of Gr(0) along the lines of 28,29] . This is most conveniently done in the fermionic representation of the -functions of the mKdV hierarchy. Finally in section 5 we determine the moduli space of the string equation. It is found to be isomorphic to the two fold covering of the 2

Fast and correctly rounded logarithms in double-precision

Laboratoire de l’Informatique du ParallélismeÉcole Normale Supérieure de LyonUnitéMixte de Recherche CNRS-INRIA-ENS LYON-UCBL n o5668 Fast and correctly rounded logarithmsin double-precisionFlorent de Dinechin,Christoph Lauter,Septembre2005Jean-Michel MullerResearch Report N o RR2005-37École Normale Supérieure de Lyon46Allée d’Italie,69364Lyon Cedex07,FranceTéléphone:+33(0)4.72.72.80.37Télécopieur:+33(0)4.72.72.80.80Adresseélectronique:lip@ens-lyon.frFast and correctly rounded logarithmsin double-precisionFlorent de Dinechin,Christoph Lauter,Jean-Michel MullerSeptembre2005AbstractThis article is a case study in the implementation of a portable,proven and ef-ﬁcient correctly rounded elementary function in double-precision.We describe the methodology used to achieve these goals in the crlibm library.There are two novel aspects to this approach.Theﬁrst is the proof framework,and in general the techniques used to balance performance and provability.The sec-ond is the introduction of processor-speciﬁc optimizations to get performance equivalent to the best current mathematical libraries,while trying to minimize the proof work.The implementation of the natural logarithm is detailed to illustrate these questions.Keywords:ﬂoating-point,elementary functions,logarithm,correct roundingR´e sum´eCet article montre comment impl´e menter une fonction´e l´e mentaire eﬃcace avec arrondi correct prouv´e en double-pr´e m´e thodologie employ´e e dans ce but par la biblioth`e que crlibm pr´e sente deux aspects novateurs.Le premier concerne la preuve de l’arrondi correct,et plus g´e n´e ralement les techniques employ´e es pour g´e rer les compromis entre performance et facilit´e de preuve. Le second est l’utilisation d’optimisations utilisant des caract´e ristiques les plus avanc´e es des processeurs,ce qui permet d’obtenir une performance´e quivalente aux meilleures impl´e mentations existantes.L’impl´e mentation du logarithme n´e p´e rien est d´e crite en d´e tail`a titre d’illustration.Mots-cl´e s:virguleﬂottante,fonctions´e l´e mentaires,logarithme,arrondi correctFast and correctly rounded logarithms in double-precision1 1Introduction1.1Correct rounding andﬂoating-point elementary functionsFloating-point is the most used machine representation of the real numbers,and is being used inmany applications,from scientiﬁc orﬁnancial computations to games.The basic building blocksofﬂoating-point code are the operators+,−,×,÷and√which are implemented in hardware(or with speciﬁc hardware assistance)on most workstation processors.Embedded processors usuallyrequire lessﬂoating-point performance and have tighter power constraints,and may thereforeprovide only softwareﬂoating point emulation.On top of these basic operators,other buildingblocks are usually provided by the operating system or speciﬁc libraries:elementary functions(exponential and logarithm,trigonometric functions,etc.),operators on complex numbers,linearalgebra,etc.The IEEE-754standard forﬂoating-point arithmetic[2]deﬁnes the usualﬂoating-point formats(single and double precision)and precisely speciﬁes the behavior of the basic operators+,−,×,÷and√.The standard deﬁnes four rounding modes(to the nearest,towards+∞,towards−∞and towards0)and demands that these operators return the correctly rounded result according to the selected rounding mode.Its adoption and widespread use have increased the numerical quality of,and conﬁdence inﬂoating-point code.In particular,it has improved portability of such code and allowed construction of proofs of numerical behavior[17].Directed rounding modes(towards +∞,−∞and0)are also the key to enable eﬃcient interval arithmetic[26,20].However,the IEEE-754standard speciﬁes nothing about elementary functions,which limitsthese advances to code excluding such functions.Currently,several options exist:on one hand,one can use today’s mathematical libraries,which are eﬃcient but without any warranty on theaccuracy of the results.These implementations use combinations of large tables[15,16,29]andpolynomial approximations(see the books by Muller[28]or Markstein[25]).Most modern librariesare accurate-faithful:trying to round to nearest,they return a number that is one of the twoFP numbers surrounding the exact mathematical result,and indeed return the correctly roundedresult most of the time.This behavior is sometimes described using phrases like99%correctrounding or0.501ulp accuracy.However,it is not enough when strict portability is needed,as wasrecently the case for the LHC@Home project:This project distributes a very large computationon a wide network of computers,and requires strictﬂoating-point determinism when checking theconsistency of this distribution,due to the chaotic nature of the phenomenon being simulated.Default libraries on diﬀerent systems would sometimes return slightly diﬀerent results.When such stricter guarantees are needed,some multiple-precision packages like MPFR[27]oﬀer correct rounding in all rounding modes,but are several orders of magnitude slower than theusual mathematical libraries for the same precision.Finally,there are currently three attemptsto develop a correctly-rounded libm.Theﬁrst was IBM’s libultim[24]which is both portableand fast,if bulky,but lacks directed rounding modes needed for interval arithmetic.This projectis no longer supported by IBM,but derivatives of the source code are now part of the GNU Clibrary glibc.The second is crlibm by the Ar´e naire team at ENS-Lyon,ﬁrst distributed in2003.The third is Sun correctly-rounded mathematical library called libmcr,whoseﬁrst beta versionappeared in late2004.Although very diﬀerent,these libraries should return exactly the samevalues for all possible inputs,an improvement on current default situation.This article deals with the implementation of a fast,proven correctly rounded elementary func-tions.The method used to provide eﬃcient correct rounding has been described by Abraham Ziv[31],and is reminded in the sequel.The present article improves Ziv’s work in two important as-pects:First,it proves the correct rounding property.Second,the performance is greatly improved,especially in terms of worst-case execution time and memory consumption.These improvementsare illustrated by a detailed description of the logarithm function.2 F.de Dinechin,uter,J.-M.Muller1.2The Table Maker’s Dilemma and Ziv’s onion peeling strategyWith a few exceptions,the imageˆy of aﬂoating-point number x by a transcendental function f is a transcendental number,and can therefore not be represented exactly in standard number systems. The correctly rounded result(to the nearest,towards+∞or towards−∞)is theﬂoating-point number that is closest toˆy(or immediately above or immediately below respectively).A computer may evaluate an approximation y to the real numberˆy with relative accuracyε. This means that the real valueˆy belongs to the interval[y(1−ε),y(1+ε)].Sometimes however, this information is not enough to decide correct rounding.For example,if[y(1−ε),y(1+ε)] contains the middle of two consecutiveﬂoating-point numbers,it is impossible to decide which of these two numbers is the correctly rounded to the nearest ofˆy.This is known as the Table Maker’s Dilemma(TMD)[28].Ziv’s technique is to improve the accuracyεof the approximation until the correctly rounded value can be decided.Given a function f and an argument x,the value of f(x)isﬁrst evaluated using a quick approximation of accuracyε1.Knowingε1,it is possible to decide if rounding is possible,or if more accuracy is required,in which case the computation is restarted using a slower approximation of accuracyε2greater thanε1,and so on.This approach leads to good average performance,as the slower steps are rarely taken.1.3Improving on Ziv’s approachHowever there was until recently no practical bound on the termination time of Ziv’s iteration:It may be proven to terminate for most transcendental functions,but the actual maximal accuracy required in the worst case is unknown.In libultim,the measured worst-case execution time is indeed three orders of magnitude larger than that of usual libm s(see Table2below).This might prevent using this library in critical applications.A related problem is memory requirement,which is,for the same reason,unbounded in theory,and much higher than usual libm s in practice.Probably for this reason,Ziv’s implementation doesn’t provide a proof of the correct rounding property,and indeed several functions fail to return the correctly rounded result for some input values(although most of these errors have been corrected in the version which is part of the GNU glibc).Sun’s library doesn’t provide a proof,either.Finally,IBM’s library lacks the directed rounding modes(Sun’s library does provide them). These rounding modes might be the most useful:Indeed,in round-to-nearest mode,correct round-ing provides an accuracy improvement over usual libm s of only a fraction of a unit in the last place(ulp),since the values diﬃcult to round were close to the middle of two consecutiveﬂoating-point numbers.This may be felt of little practical signiﬁcance.However,the three other rounding modes are needed to guarantee intervals in interval arithmetic.Without correct rounding in these directed rounding modes,interval arithmetic may loose up to two ulp of precision in each compu-tation.Actually,current interval elementary function libraries are even less accurate than that, because they sacriﬁce accuracy to a very strict proof[18].The goal of the crlibm(Correctly Rounded libm)project is therefore a library which is •correctly rounded in the four IEEE-754rounding modes,•proven,•and suﬃciently eﬃcient in terms of performance(both average and worst-case)and resources (in particular we impose an upper bound of4KB of memory consumed per function[6])to enable the standardization of correct rounding for elementary functions.1.4Organisation of this articleSection2describes the general principles of the crlibm library,from the theoretical aspects to an implementation framework which makes optimal use of current processor technology,and a proof framework which is currently a distinctive feature of this work.Section3is an in-depthFast and correctly rounded logarithms in double-precision3 example of using these frameworks to implement an eﬃcient,proven correctly-rounded natural logarithm.Section4gives measures of performance and memory consumption and shows that this implementation compares favorably to the best available accurate-faithful libm s on most architectures.2The Correctly Rounded Mathematical Library2.1Worst cases for correct roundingLef`e vre and Muller[23,21]computed the worst-caseεrequired for correctly rounding several functions in double-precision over selected intervals in the four IEEE-754rounding modes.For example,they proved that157bits are enough to ensure correct rounding of the exponential function on all of its domain for the four IEEE-754rounding modes,and118bits for the logarithm. Up-to-date information about this quest for worst cases(which functions are covered on which interval)is available in the documentation of crlibm[1].A discussion of the possible strategies in the absence of worst cases is also available in this document.2.2Two steps are enoughThanks to such results,we are able to guarantee correct rounding in two steps only,which we may then optimize separately.Theﬁrst quick step is as fast as current libm,and provides an accuracy between260and280(depending on the function),which is suﬃcient to round correctly to the53 bits of double precision in most cases.The second accurate step is dedicated to challenging cases. It is slower but has a reasonably bounded execution time,being tightly targeted at Lef`e vre/Muller worst cases(contrary to Sun’s and IBM’s library).2.3On portability and performancecrlibm was initially a strictly portable library,relying only on two widespread standards:IEEE-754forﬂoating-point,and C99for the C language.This meant preventing the compiler/processor combination from using advancedﬂoating-point features available in recent mainstream processors, and as a consequence accepting a much lower performance than the default,accurate-faithful libm, typically by a factor2[13,10].Among these advanced features,the most relevant to the implementation of elementary func-tions are:•hardware double-extended(DE)precision,which provides64bits of mantissa instead of the 53bits of the IEEE-754double format,•hardware fused multiply-and-add(FMA),which performs the operation x×y+z in one instruction,with only one rounding.It was suggested that a factor two in performance would be an obstacle to the generalization of correct rounding,therefore our recent research has focussed on exploiting these features.The logarithm is theﬁrst function which has been completed using this approach:In versions of crlibm strictly greater than0.8,there is a compile-time selection between two implementations.•Theﬁrst exploits double-extended precision if available(for ia32and ia64processors),and is referred to as the“DE”version in the following.•The second relies on double-precision only,and is referred to as the“portable”version in the following.Both versions exploit an FMA if available(on Power/PowerPC essentially for the portable version, on Itanium for the DE version).In the absence of an FMA,the portable version is strictly portable in the IEEE-754/C99sense.This choice provides optimized versions(as section4will show)for the overwhelming majority of current mainstream processors.4 F.de Dinechin,uter,J.-M.Muller2.4Fastﬁrst stepThe DE version of theﬁrst step is very simple:as double-extended numbers have a64-bit mantissa, it is easy to design algorithms that compute a function to an accuracy better than2−60using only DE arithmetic[25].For the portable version,we only have double-precision at our disposal.We classically represent a number requiring higher precision(such as y1,the result of theﬁrst step)as the sum of two ﬂoating-point numbers,also called a double-double number.There are well-known algorithms for computing on double-doubles[14].In both versions,we also make heavy use of classical,well proven results like Sterbenz’lemma [17]which gives conditions for aﬂoating-point subtraction to entail no rounding error.2.5Rounding testAt the end of the fast step,a sequence of simple tests on y1either returns a correctly rounded value,or launches the second step.We call such a sequence a rounding test.The property that a rounding test must ensure is the following:a value will be returned only if it can be proven to be the correctly rounded value ofˆy,otherwise(in doubt)the second step will be launched.A rounding test depends on a boundε1onε1,the overall relative error of theﬁrst step.This bound is usually computed statically,although in some case it can be reﬁned at runtime(IBM’s code has such dynamic rounding tests,but for an explained and proven example see crlibm’s tangent[1]).Techniques for computingε1,as well as techniques for proving the validity of a rounding test,will be detailed in Section2.9.The implementation of a rounding tests depends on the rounding mode and the nature of y1(a double-extended for the DE version,or a double-double for the portable version).Besides,in each case,there are several sequences which are acceptable as rounding tests.Some use onlyﬂoating point but require pre-computing onε1[10],othersﬁrst extract the mantissa of y1and perform bit mask operations on the bits after the53rd[1].All these possible tests are cleanly encapsulated in C macros.2.6Accurate second stepFor the second step,correct rounding needs an accuracy of2−120to2−150,depending on the function.We are now using three diﬀerent approaches depending on the processor’s capabilities.•We have designed an ad-hoc multiple-precision library called scslib which is lightweight, very easy to use in the context of crlibm,and more eﬃcient than all other available com-parable libraries[12,7].It allows quick development of the second step,and has been used for the initial implementation of all the functions.It is based on integer arithmetic.•For the DE version of the second step,it has been proven that double-double-extended inter-mediate computations are always enough to ensure correct rounding,even when worst cases have been found requiring more than the128bits of precision oﬀered by this representation[8].Using double-double-extended is not as simple as using scslib,however the algorithmsare those already used and proven for double-double.And it is much more eﬃcient than scslib:we measure a factor10in the worst-case execution time[9].•Finally,we are developping portable second steps based on triple-double arithmetic.This approach is also much more eﬃcient than scslib,but it is also much more diﬃcult to use and to prove.The logarithm presented below is theﬁrst function to be implemented using this technology.The main reason for the performance improvement over scslib is that each computation step can use the required precision,no more.Typically for instance we start a Horner polynomial evaluation in double,continue in double-double,and perform only the last few iterations in triple double.The scslib format doesn’t oﬀer thisﬂexibility.Another advantage is that the accurateFast and correctly rounded logarithms in double-precision5 step can use table-based methods[15,16,29]because triple-double is much less memory-consuming than the scslib format,all the more as these tables can be shared with theﬁrst step,as will be seen in Section3.The main advantage of using scslib is that it leads to very easy error computations.However, being based on integer arithmetic,scslib is also interesting for architectures withoutﬂoating-point hardware.2.7Final roundingThe result of the accurate step will be either a triple-double number,or a double-double-extended number,or a number represented in scslib’s special format.This result mustﬁnally be rounded to a double-precision number in the selected rounding mode.This operation is peculiar to each of the three representations mentioned.•The functions provided by scslib for this purpose are very straightforward,but quite slow.•Processors which support double-extended precision are all able to round the sum of two double-extended numbers to a double,in an atomic operation.Fortunately,this is even easy to express in C[19]asreturn(double)(yh+yl);where yh and yl are double-extended numbers.Note however that more care must be taken for functions whose worst cases may require more than128 bits[8].•The most diﬃcult case is that of the triple-double representation,because it is a redundant representation,and because there is no hardware for doing a ternary addition with only a ﬁnal rounding[11].Again,we have designed sequences of operations for thisﬁnal rounding in the four rounding modes.These sequences involve a dozen ofﬂoating-point operations and nested tests,and their full proof is several pages long[1].2.8Error analysis and the accuracy/performance tradeoﬀThe probability p2of launching the second(slower)step is the probability that the interval[y(1−ε),y(1+ε)]contains the middle of two consecutiveﬂoating-point numbers(or aﬂoating-point number in directed rounding modes).Therefore,it is obviously proportional to the error bound ε1computed for theﬁrst step.This deﬁnes the main performance tradeoﬀone has to manage when designing a correctly-rounded function:The average evaluation time will beT avg=T1+p2T2(1) where T1and T2are the execution time of theﬁrst and second phase respectively(with T2≈100T1 in crlibm using scslib,and T2≈10T1in crlibm using DE or triple-double),and p2is the probability of launching the second phase.Typically we aim at chosing(T1,p2,T2)such that the average cost of the second step is negli-gible:This will mean that the cost of correct rounding is negligible.The second step is built to minimize T2,there is no tradeoﬀthere.Then,as p2is almost proportional toε1,to minimize the average time,we have to•balance T1and p2:this is a performance/precision tradeoﬀ(the more accurate theﬁrst step, the slower)•and compute a tight bound on the overall errorε1.Computing this tight bound is the most time-consuming part in the design of a correctly-rounded elementary function.The proof of the correct rounding property only needs a proven bound,but a loose bound will mean a larger p2than strictly required,which directly impacts6 F.de Dinechin,uter,J.-M.Muller average pare p 2=1/1000and p 2=1/500for T 2=100T 1,for instance.As a consequence,when there are multiple computation paths in the algorithm,it may make sense to precompute diﬀerent values of ε1on these diﬀerent paths [10].2.9Proving correct roundingWith this two-step approach,proving the correct-rounding property resumes to two tasks:•Computing a bound on the overall error of the second step,and checking that this bound is less than Lef`e vre/Muller worst-case accuracy;•Proving that the ﬁrst step returns a value only if this value is correctly rounded,which also requires a proven (and tight)bound on the evaluation of the ﬁrst step.One diﬃculty is that the smallest change in the code (for optimization,and even for a bug ﬁx)will aﬀect the proof.We therefore strive to factorize our code in a way compatible with proof-writing.For example,elementary function are typically implemented using polynomial ap-proximation techniques.The latter can ﬁnally be based on addition and multiplication for the diﬀerent formats.For proof purposes,we want to consider e.g.an addition of two triple-double numbers just like a ﬂoating point addition with its respective “machine epsilon”.The challenge here is the tradeoﬀbetween eﬃciency and provability.Therefore we structure our code and proof as follows.Our basic procedures (including addition and multiplication for various combinations of double,double-double and triple-double arguments and results,but also rounding tests,ﬁnal rounding,etc.)are implemented as C macros.These macros may have diﬀerent processor-speciﬁc implementation,for example to use an FMA when available,so this solution provides both ﬂexibility and eﬃciency.A non-exhaustive list of some properties of these procedures is given by table Operation Property Add12x h +x l =a +b exact (Fast2Sum)Mul12x h +x l =a ·b exact (Dekker)Add22x h +x l ≈(a h +a l )+(b h +b l )+δε≤2−103.5Mul22x h +x l ≈(a h +a l )·(b h +b l )+δε≤2−102Add33x h +x m +x l ≈εdepends on (a h +a m +a l )+(b h +b m +b l )+δoverlap Add233x h +x m +x l ≈εdepends on (a h +a l )+(b h +b m +b l )+δoverlap Mul23x h +x l ≈(a h +a l )·(b h +b l )+δε≤2−149Mul233x h +x l ≈εdepends on (a h +a l )·(b h +b m +b l )+δoverlap Renormalize3x h +x m +x l =a h +a m +a l no overlap in result RoundToNearest3x =◦(x h +x m +x l )(Round-to-nearest)RoundUp3x = (x h +x m +x l )(Round-upwards)TEST_AND_RETURN_RN return ◦(x h +x l )if rounding is safeknowing εcontinue otherwise Table 1:Some basic procedures encapsulated in C-MacrosEach macro has a proof published in [1]and covering all its implementations.What we prove is actually a theorem with hypotheses (validity conditions for the macro to work),and a conclusion (the relative error of the operator is smaller than some epsilon,for example).The proof of a function invoking such a macro will then have to check the hypotheses,and may then use the conclusion.Fast and correctly rounded logarithms in double-precision72.10Using automated error analysis toolsWe strive to automate the error computation to make proofs easier and increase conﬁdence in the result.Aﬁrst approach is to rely on clean Maple scripts to compute the numerical constants, output the C headerﬁles containing these constants,and implement the error computation out of them.Of course,these scripts are part of the crlibm distribution.More recently,we have been making increasing use of Gappa,a tool which manages ranges and errors in numerical code using interval arithmetic[5].This tool takes a code fragment,information on the inputs(typically their ranges and bounds on their approximation errors),and computes and propagates roundoﬀerrors.It is far from being automatic:The user has to provide almost all the knowledge that would go in a paper proof,but does so in an interactive and very safe way,increasing the conﬁdence that all the contribution to the total error are taken properly into account.Besides,this tool relies on a library of theorems which take into account subnormal numbers,exceptional cases,etc,which ensures that theses exceptional cases are considered.This tool outputs a proof in the Coq language[3],and this proof can be machine-checked provided all the support theorems have been proven in Coq.Obviously,our divide-and-conquer approach matches this framework nicely,although we currently don’t have Coq proofs of all the previous theorems.Ultimately,we hope that the“paper”part of the proof will be reduced to an explanation of the algorithms and of the structure of the proof.One of the current weakest point is the evaluation of inﬁnite norms(between an approximation polynomial and the function),which we do in Maple. As we approximate elementary functions on domains where they are regular and well-behaving,we can probably trust Maple here,but a current research project aims at designing a validated inﬁnite norm.Another approach is to rely on Taylor approximations with carefully rounded coeﬃcients, such that mathematical bounds on the approximation error can be computed[18].The main drawback is that it typically leads to polynomials of higher degree for the same approximation error,which results in larger delays,larger memory consumption,and possibly larger rounding errors.3crlibm’s correctly rounded logarithm functionThis section is a detailed example of the previous approach and framework.3.1OverviewThe worst-case accuracy required to compute the natural logarithm correctly rounded in double precision is118bits according to Lef`e vre and Muller[22].Theﬁrst step is accurate to2−60,and the second step to2−120,for all the implementations.For the quick phase we now use a diﬀerent algorithm as the one presented in[10].This choice is motivated by two main reasons:•The algorithm is slightly more complex,but faster.•It can be used for all our diﬀerent implementations(portable or DE).Special cases are handled in all implementations as follows:The natural logarithm is deﬁned over positiveﬂoating point numbers.If x≤0,then log(x)should return NaN.If x=+∞,then log(x)should return+∞.This is true in all rounding modes.Concerning subnormals,the smallest exponent for an non-zero logarithm of a double-precision input number is−53(for the input values log(1+2−52)and log(1−2−52),as log(1+ε)≈εwhen ε→0).As the result is never subnormal,we may safely ignore the accuracy problems entailed by subnormal numbers.The common algorithm is inspired by the hardware based algorithm proposed by Wong and Goto[30]and discussed further in[28].After handling of special cases,consider the argument x written as x=2E ·m,where E is the exponent of x and m its mantissa,1≤m<2.This8 F.de Dinechin,uter,J.-M.Muller decomposition of x into E and m can be done simply by some integer operations.In consequence,one gets log (x )=E ·log(2)+log(m ).Using this decomposition directly would lead to catastrophic cancellation in the case where E =−1and m ≈2.Therefore,if m is greater than approximately √2,we adjust m and E as follows:E = Eif m ≤√2E +1if m >√2y = m if m ≤√2m 2if m >√2All of the operations needed for this adjustment can be performed exactly.We see that y is now bounded by √22≤y ≤√2leading to a symmetric bound for log(y ).The magnitude of y being still too big for polynomial approximation,a further argument reduction is done as follows.The algorithm looks up,using the high magnitude bits of the mantissa of y ,a value r i which approximates relatively well 1y .Setting z =y ·r i −1,we obtain for the reconstructionlog (x )=E ·log (2)+log (1+z )−log (r i )Now z is small enough (typically |z |<2−7)for approximating log (1+z )by a Remez polynomial p (z ).The values for log(2)and log (r i )are tabulated.One crucial point here is the operation sequence for calculating z out of y and r i :z =y ·r i −1.In the DE code,the r i are chosen as ﬂoating-point numbers with at most 10non-zero consecutive bits in their mantissa (they are actually tabulated as single-precision numbers).As y is a double-precision number,the product y ·r i ﬁts in a double-extended number,and is therefore computed exactly in double-extended arithmetic.The subtraction of 1is then also exact thanks to Sterbenz’lemma:y ·r i is very close to 1by construction of r i .Finally the whole range reduction involves no roundoﬀerror,which will of course ease the proof of the algorithm.In the portable version,there is unfortunately no choice of r i that will guarantee that y ·r i −1ﬁts in one double-precision number.Therefore we perform this computation in double-double,which is slower,but still with the guarantee that z as a double-double is exactly z =y ·r i −1.This algorithm allows for sharing the tables between the ﬁrst and the second step:In the portable version,these tables are normalized triple-double,the ﬁrst step using only two of the three values.In the DE version,the tables are double-double-extended,and the ﬁrst step uses only one of the two values.Such sharing brings a huge performance improvement over the previous approach [10]where the two steps were using two distinct algorithms and diﬀerent tables.This improvement has two causes.First,the second step does not need to restart the computation from the beginning.As the argument reduction is exact,the second step doesn’t need to reﬁne it.Second,the table cost of the second step is much reduced,which allows more entries to the table,leading to a smaller polynomial (especially for the second step).Here,tables are composed of 128entries (r i ,log (r i )).3.2A double-extended logarithmThe double-extended version is a straightforward implementation of the previous ideas:Argument reduction and ﬁrst step use only double-extended arithmetic,second step uses a polynomial of degree 14,evaluated in Horner form,with the 8lower degrees implemented in double-double-extended arithmetic.The most novel feature of this implementation is the proof of a tight error bound for the Estrin parallel evaluation [28]of the polynomial of the ﬁrst step,given below.z2=z*z;p67=c6+z*c7;p45=c4+z*c5;p23=c2+z*c3;p01=logirh+z;z4=z2*z2;p47=p45+z2*p67;p03=p01+z2*p23;p07=p03+z4*p47;log=p07+E*log2h;。

1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。

a rX iv:mat h /95227v1[mat h.FA]21Feb1995OPERATORS ON SUBSPACES OF HEREDITARILY INDECOMPOSABLE BANACH SPACES V.Ferenczi February 1,2008Abstract We show that if X is a complex hereditarily indecomposable space,then every operator from a subspace Y of X to X is of the form λI +S ,where I is the inclusion map and S is strictly singular.1Introduction By space (resp.subspace),we shall always mean inﬁnite dimensional space (resp.subspace).A space X is hereditarily indecomposable if no two subspaces of X are in a direct sum.In the whole article,X stands for a hereditarily indecomposable complex Banach space.It was shown in [GM]that every operator from X to X is of the form λI +S ,where I is the identity map and S is strictly singular.We generalize this result by showing that for every subspace Y of X ,every operator from Y to X is of the form λI +S ,where I is the inclusion map and S is strictly singular (this was proved in [GM]in a particular case).In fact,it is a consequence of the following lemma that it is enough to prove this for every subspace Y with a basis with constant 2.2LemmaLemma 1Let Y be H.I.and S an operator from Y to some Banach space.Let Z ⊂Y .Then S is strictly singular if and only if S /Z is strictly singular.ProofConsequence In particular,this lemma can be applied to an operator from Y⊂X to X.Now suppose that the claim of the article is true for every subspace with a basis with constant2.Let Y⊂X,and T:Y→X. We know that Y contains a normalized basic sequence with constant2; let Z be the subspace associated to it.By hypothesis,there existsλsuch that T/Z−λI/Z is strictly singular.By the lemma,T−λI/Y is still strictly singular.3Aﬁlter on block-subspacesA normalized basic sequence(y i)i∈N in X will be denoted by y.We denote by[y]the closed subspace generated by y,by I y the inclusion map from[y]to X,by y k the normalized basic sequence(y i+k)i∈N.For an operator T such that the restriction of T to[y]is deﬁned,T/y stands for this restriction.If(y i)i∈N is a basic sequence such that for every i,y i=0, then|y|denotes the normalized basic sequence(y i/ y i )i∈N.By y′⊂y, we mean that y′is a normalized block basic sequence of y.Let e be a normalized basic sequence in X with constant2.Let J={y/y⊂e}.Deﬁnition1Given y and z in J,we say that they are similar,and write y∼z ifΣ z i−y i <+∞.Properties Similarity is an equivalence relation.Furthermore,for ev-ery y∼z and every y′⊂y,there exists z′⊂z such that y′∼z′.Indeed, for k∈N,let z′k have the same coordinates on(z i)as y′k on(y i):the normalization|z′|of(z′k)k∈N is similar to y′.Deﬁnition2For y and z in J,we say that y≤z if∃z′⊂z/y∼z′.It is a consequence of the properties of similarity that≤is a preorder-ing.It is also aﬁlter.Indeed let y and z in J;using the fact that X is H.I., it is possible toﬁnd y′⊂y and z′⊂z with y′∼z′.We have that y′≤y and y′≤z.Deﬁnition3We deﬁne aﬁlter on the set of block subspaces of[e]by letting Y≤Z if there exists y≤z with Y=[y],Z=[z].Deﬁnition4Let U be an ultraﬁlter on J.For y∈J,let B y be theBanach space of bounded operators from[y]to X,and let .y be theseminorm lim k→+∞ ./y k .Let B be the quotient space of l∞((B y)y∈J)by the kernel of the semi-norm lim U .y.Lemma2Let y∈J and z∈J be similar.The operator p yz from[y] to[z]deﬁned by∀i∈N,p yz(y i)=z i is bounded and p yz−I y y=0.24A morphism from L (Y,X )to BDeﬁnition 5Let T :Y →X with Y =[y ].For z ≤y ,let y ′⊂y such that z ∼y ′,let T z be the element T p zy ′of B z .Let Tbe the element of B associated to (T z )z ≤y (the value of T z for the other values of z has no eﬀect on the value of T,take for example T z =0).This deﬁnition does not depend on the choice of y ′.Indeed,let y ′1∼y ′2be two choices of y ′;let T iz be associated to y ′i ;then T 1z −T 2z =T (p zy ′1−p zy ′2)so that by Lemma 2, T 1z −T 2z z =0.It is easy to check that z → T z z is increasing,so that T is a simple limit.Let B be the set of elements of B of the form T .Remark 1If T :Y →X and U :Y →X ,then T +U = T + U .By Lemma 2,for all y ∼z ,˜p yz =˜I y .5LemmasLemma 3Let T :Y →X with Y =[y ],and let z ≤y .Then ˜Tz = T .ProofProposition 1Let S :Y →X .Then the following properties are equivalent:(a) S =˜0.(b)S is strictly singular.ProofFirst notice that if S :Y →X is such that Σ S (y i ) converges,then S =˜0.Now let T :Y →X .If T =˜0then one can choose T ′=0,Y ′=Y .Suppose now T =˜0.By the ﬁrst consequence of Lemma 3and Proposi-tion 1,passing to a subspace,we may assume that T is an isomorphism.By projections on big enough intervals,selecting block vectors,and using repeatedly the fact that X is H.I.,it is possible to build a normalized block basic sequence y ′⊂y and a (non-normalized)block basic sequence (z i )i ∈N in Z such that the sequence Σ T (y ′i )−z i converges.The operator T ′deﬁned on [y ′]by T ′(y ′i )=z i satisﬁes ˜T′= T by the ﬁrst remark,and takes its values in Z .6A product on BDeﬁnition 6Let T ∈B with T :Z →X .Let U ∈B .Let V ≤Z .By Lemma 4,we can assume that U is associated to U :Y →V .We want to deﬁne T U as T v U .We need to show that T v U does not depend on thechoice of T and U .First,for i =1,2,let V i ≤Z ,and U i :Y i →V i be such that ˜U i = U .Let z ′i ⊂z with v i ∼z ′i .Let y ′≤y 1and y ′≤y 2.ThenT v 2U 2− T v 1U 1= (T v 2U 2)y ′− (T v 1U 1)y ′= T v 2U 2y′−T v 1U 1y ′.Now T v 2U 2y ′−T v 1U 1y ′=T (p v 2z ′2U 2y ′−p v 1z ′1U 1y ′)T v 2U 2y ′−T v 1U 1y ′=T [(p v 2z ′2−I v 2)U 2y ′+(U 2y ′−U 1y ′)+(I v 1−p v 1z ′1)U 1y ′].Using Remark 1,and the fact that the space of strictly singular oper-ators is a two-sided ideal,we get thatT v 2U 2− T v 1U 1=˜0.4We now prove that T v U does not depend on the choice of T .Let for i =1,2T i :Z i →X be such that ˜T i = T .Let V ≤Z 1and V ≤Z 2and U :Y →V be a representative for U .Then (T 2)v U − (T 1)v U = ((T 2)v −(T 1)v )U =˜0.So we can deﬁne T U as T v U without ambiguity.Remark 2If T :Z →X and U :Y →Z ,then T U = T U .Usingappropriate representatives one can then show that B is an algebra;in particular ˜1the common value associated to all I y for y ∈J is neutral for the multiplication.Proposition 2B is a ﬁeld.Proof If λ=˜0,the result is clear.Now assume λ=˜0.For every Y ,if T belongs to L (Y,X )then T ≤ T y y ≤ T ,so that λ ≤inf T =λ T .Furthermore,let T :Z →X satisfy λ= T .For every ǫ>0,there exists z ′⊂z so that T z ′ z′≤(1+ǫ) λ .So there exists k such that T z ′k ≤(1+ǫ)2 λ .As T z ′k = T ,we have inf T =λT ≤ λ .Remark Let T and U be in B .Let ǫ>0.Let T :Y →X be an ǫ-minimal representative for T.Choose an ǫ-minimal representative for U from some Z into Y (it is possible by the previous remark and Lemma 4).Then T U = T U ≤ T U ≤ T U ≤(1+ǫ)2 TU .As ǫis arbitrary,the norm on B is an algebra norm.5Proposition4B is a Banach space.ProofReferences[GM]W.T.GOWERS and B.MAUREY,The unconditional basic se-quence problem,J.Amer.Math.Soc.6(1993),851-874.[LT]J.LINDENSTRAUSS and L.TZAFRIRI,Classical Banach spaces I,Springer-Verlag,New York(1977).Equipe d’Analyse et de Math´e matiques Appliqu´e es,Universit´e de Marne-la-Vall´e e,2,rue de la Butte Verte,93166Noisy le Grand Cedex,France.ferenczi@math.univ-mlv.fr.7。