1+1 dimensional QCD with fundamental bosons and fermions
Fundamentals of Kalman Filtering a Practical Approach
1-6
Numerical Basics
Fundamentals of Kalman Filtering: A Practical Approach
1-7
Numerical Basics Overview
Fundamentals of Kalman Filtering: A Practical Approach
1 - 10
Vector Operations - 2
Vector subtraction
x1 y1 x2 y2 x3 y3 x -y = . - . = . . . . xn yn x1 - y1 x2 - y2 x3 - y3 . . . xn - yn
Fundamentals of Kalman Filtering: A Practical Approach
Paul Zarchan
Fundamentals of Kalman Filtering: A Practical Approach
1-1
Seminar Outline - 1
• Numerical Techniques - Required background - Introduction to source code • Method of Least Squares - How to build batch process least squares filter - Performance and software comparison of different order filters • Recursive Least Squares Filtering - How to make batch process filter recursive - Formulas and properties of various order filters
Off-shell Gauge Fields from BRST Quantization
a rXiv:h ep-th/06589v19M a y26ULB-TH/06-09FIAN-TD/06-06hep-th/0605089Off-shell Gauge Fields from BRST Quantization Maxim Grigoriev a,b a Tamm Theory Department,Lebedev Physics Institute,Leninsky prospect 53,119991Moscow,Russia b Physique Th´e orique et Math´e matique,Universit´e Libre de Bruxelles and International Solvay Institutes,Campus Plaine C.P .231,B-1050Bruxelles,Belgium A BSTRACT .We propose a construction for nonlinear off-shell gauge field theo-ries based on a constrained system quantized in the sense of deformation quanti-zation.The key idea is to consider the star-product BFV–BRST master equation as an equation of motion.The construction is formulated in terms of the BRST extention of the unfolded formalism that can also be understood as an appropri-ate generalization of the AKSZ procedure.As an application,we consider a verysimple constrained system,a quantized scalar particle,and show that it gives rise to an off-shell higher-spin gauge theory that automatically appears in the parent form and properly takes the familiar trace constraint into account.In particular,we derive a geometrically transparent form of the off-shell higher-spin theory on the AdS background.2O FF-SHELL G AUGE F IELDS FROM BRST Q UANTIZATIONC ONTENTS1.Introduction22.Master equation as an equation of motion42.1.Off-shell gauge theories from quantum constrained systems42.2.The basic example72.3.Thefield theory BRST differential83.Off-shell gauge theories in the BRST extended unfolded formulation113.1.BRST extension of the non-linear unfolded formalism113.2.Linearization133.3.Consistent reductions143.4.Putting a quantum constrained system to afiber154.Off-shell higher spinfields174.1.Linearization around Minkowski space204.2.Linearization around arbitrary background225.Off-shell HSfields on AdS in terms of the embedding space245.1.Non-linear off-shell HSfields on AdS245.2.Linearization255.3.Topological HS theory27Acknowledgments30 Appendix A.Proof of the Proposition30 References311.I NTRODUCTIONAn off-shellfield theory is by definition a theory whose equations of motion are equiv-alent to algebraic constraints.This implies that one can in principle solve the equations of motion and eliminate pure gauge degrees of freedom in order to obtain the unconstrained fields and gauge symmetries.Although the off-shell formulation does not actually de-scribe dynamical equations,it can be useful from various standpoints.In particular,an off-shell theory can encode all the information on thefield content and gauge symmetries in the form adapted for introducing consistent interactions.For example,this applies to the higher-spin(HS)theories,where the relevance of the off-shell formulation in constructing nonlinear equations[1](see also[2]for a review)has been recently realized in[3].More recently,a compact and geometrically transparent form ofO FF-SHELL G AUGE F IELDS FROM BRST Q UANTIZATION3the nonlinear off-shell HS theory on Minkowski space was constructed by M.Vasiliev in[4].An off-shell theory can also be regarded as an intermediate step in constructing a La-grangian formulation.Indeed,being algebraic,the off-shell equations of motion can al-ways be made Lagrangian by introducing Lagrange multipliers(we refer the reader to[4] for a more extensive discussion).Moreover,under some regularity assumptions,any off-shell theory can be equivalently formulated in the Lagrangian Batalin–Vilkovisky for-malism[5,6]by introducing the so-called generalized Lagrange multipliers[7].In this paper,we propose the generating procedure for constructing off-shell gauge theories starting from a(quantized)constrained system.The underlying idea is to identify the∗-product version of the quantum Batalin–Fradkin–Vilkovisky(BFV)[8,9]master equation14O FF-SHELL G AUGE F IELDS FROM BRST Q UANTIZATIONThe paper is organized as follows.In Section2,we propose an elementary construc-tion for off-shell gauge theories from quantum constrained systems and discuss their lin-earizations around particular solutions.The simplest example of a system with just one constraint is then explicitly considered.In Section3,we review the BRST extension of the nonlinear unfolded formalism and consistent reductions in this framework,originally described in[15].The linearization of such theories is then briefly discussed.The general construction for off-shell gauge theories starting from quantum constrained systems is given in Section3.4within the BRST-extended unfolded formulation.Section4is devoted to explicit examples leading to the off-shell HS gauge theories on various backgrounds.In Section5,we propose a compact form of the off-shell HS theory on AdS space in terms of the embedding space and discuss its relation to the Vasiliev unfolded formulation.2.M ASTER EQUATION AS AN EQUATION OF MOTION2.1.Off-shell gauge theories from quantum constrained systems.An interesting class of gauge theories can be obtained starting from a quantum phase space of a quantum con-strained system.Suppose we are given with a constrained system quantized in the sense of deformation quantization.This implies the associative∗-product algebra A depend-ing on the quantization parameter .In what follows we restrict ourselves to the formal deformation quantization and therefore allow A to be the algebra of formal power series in with the coefficients being functions on the extended phase space of the system(i.e. functions in phase space coordinates and ghost variables).The associative∗-product on A reduces to the pointwise multiplication and the Poisson bracket in the →0limit:(2.1)f∗g=fg+O( ),f∗g−(−1)|f||g|g∗f= {f,g}+O( 2), where{·,·}denotes the Poisson bracket on the extended phase space and|·|denotes the Grassmann parity.A is also equipped with the ghost number grading denoted by gh(·). For simplicity we suppose that no physical fermions are present so that|φ|=gh(φ) mod2for any homogeneousφ∈A.Let us assume in addition that the extended phase space of the system is a bundle over a manifold X0identified as the space-time manifold in what follows.Algebra A can be then considered as that of sections of the appropriate associated vector bundle H over X0with thefiber being the linear space H of functions on thefiber(here and below we denote by H the vector bundle with thefiber isomorphic to a vector space H).In whatO FF-SHELL G AUGE F IELDS FROM BRST Q UANTIZATION5follows we assume H to be(graded)finite-dimensional,i.e.that one can alwaysfind a suitable degree such that each homogeneous component isfinite-dimensional.1Let e A be a basis in H.Any element of A can be(locally on X0)represented asχ= e AχA(x)where xµare local coordinates on X0.In this representation the∗-product is a bi-linear bidifferential operation on sections of H.In addition we assume that[xµ,xν]∗=0 where xµ,xνare considered as elements of A.Note that relaxing this condition corre-sponds to the interesting possibility to describe noncommutative gaugefield theories.To simplify the exposition we also assume all the bundles to be trivial unless otherwise spec-ified or,which is the same,restrict ourselves to the local analysis.In particular,A can be then identified with the H-valued functions on X0A quantum BRST charge(more precisely,a symbol of the BRST operator)is an ele-mentΦ=e AΦA(x)∈A satisfying the master equation along with the ghost number and the Grassmann parity assignments:(2.2)1[Φ,Λ]∗,where gauge parameterΛis an arbitrary element from A with gh(Λ)=0and|Λ|=0. The physical interpretation of the constructed gaugefield theory has to do with the off-shell description of the backgroundfields.The constraints of the system can be identified with(the generating functions for)the backgroundfields determining e.g.configuration space geometry,background Maxwellfield etc.The master equation ensuring the con-sistency of the constraints imposes the equations(usually equivalent to algebraic i.e.not containing derivatives with respect to xµ)on the backgroundfields and determines their6O FF-SHELL G AUGE F IELDS FROM BRST Q UANTIZATIONgauge symmetries.In Section2.2we consider the scalar-particle system(see also the discussion in[26,12])where this interpretation is transparent.SupposeΦ0be a particular solution to(2.2).The equations of motion and gauge trans-formations expanded aroundΦ0read as(2.4)12 [Φ,Φ]∗=0,δΛΦ=1[Φ,Λ]∗,where by slight abusing notations thefluctuation aroundΦ0is again denoted byΦ.The terms linear inΦdetermines the linearized equations of motion and gauge symmetries.The linearized theory can be naturally interpreted as afield theory associated to a BRST first-quantized system(Ω,Γ(H,X0))with the“space of states”beingΓ(H,X0)∼=A and nilpotent BRST operatorΩdefined byΩφ= −1[Φ0,φ]∗for anyφ∈A.Here and in what follows we use notation(Ω,Γ(H,X0))for thefirst-quantized BRST system specified by the“space of states”Γ(H,X0)(the space of sections of the vector bundle H over X0)and the BRST operatorΩ:Γ(H,X0)→Γ(H,X0).Equations(2.4)can be then rewritten as (2.5)ΩΦ+1[Φ,Λ]∗,so that their linear parts indeed take the familiar formΩΦ=0andδΛΦ=ΩΛ(see[27, 25,28]and references therein for more details onfield theories associated to thefirst-quantized BRST systems).SupposeΩbe an odd nilpotent operatorΩ:A→A not necessarily generated by some Φ0.In this case equations(2.5)still determines a consistent gaugefield theory provided gh(Ω)=1andΩsatisfies Leinbitz ruleΩ(φ∗χ)=(Ωφ)∗χ+(−1)|φ||χ|φ∗(Ωχ).This possibility along with further generalizations of the construction are discussed in more details in Section2.3.Note that in our approach the space of states appears to be the space A of functions on the extended phase space.It does not therefore coincide with the space of quantum states (if it was specified)of the starting point constrained system because the later can be(at least formally)identified with the suitable functions on the Lagrangian submanifold of the phase space(i.e.functions in only one half of the phase space coordinates).In order to describe quantum states in terms of A one needs additional factorization procedure which we do not discuss in this paper.Let us also note that instead of the∗-product description one can use the standard lan-guage of operators.Moreover,all the constructions can be also reformulated in these terms provided one is given with the suitable representation space.In particular,if vari-ables xµare quantized in the coordinate representation the operators can be identified as differential operators on X0with coefficients in linear operators on the“internal”space of states.Such a representation for a constraint operator of a scalar particle has been used in[12]in the related context.O FF-SHELL G AUGE F IELDS FROM BRST Q UANTIZATION7All the considerations of this section remaines true if one takes the classical limit by replacing A with the commutative algebra of the phase space functions equipped with the Poisson bracket(i.e.the classical limit of the∗-product algebra).This can also be un-derstood as replacing the quantum constrained system with the classical one(its classical limit).2.2.The basic example.To illustrate the construction let us consider nearly the sim-plest constrained system,a“scalar particle”on theflat Minkowski space X0,with only one constraint.Let pµbe the momenta conjugated to coordinates xµon X0and F(x,p)the constraint.In order to handle the constraint in the BRST approach we introduce Grass-mann odd ghost variables c,πwith gh(c)=1,gh(π)=−1and[c,π]∗=− .Variables xµ,pνare assumed to carry vanishing ghost degree.The quantum BRST charge is then given by(2.6)Φ=cF(x,p)and automatically satisfies[Φ,Φ]∗=0because F is the only constraint present in the model.According to the general strategy the dynamicalfields are coefficients in the expansion of F with respect to momenta pµ(2.7)F(x,p)=φ0(x)+φµ1(x)pµ+φµν2pµpν+φµνρ3(x)+....and can be identified with the symmetric tensorfields on X0.The gauge symmetries are determined by(2.3)where in this caseΛ=λ(x;p)+cπχ(x;p)(note that terms with nonzero ghost structure do not enter as one can easily see by counting the ghost degree). Explicitly one gets(2.8)δΛF=12{F,χ}∗.where{a,b}∗=a∗b+(−1)|a||b|b∗a denotes the graded symmetric∗-anticommutator.Let us expand the theory around a particular solutionΦ0=c1∂xµλ+18∂∂xµχ+12{F,χ}∗,where thefirst three terms give a linearized gauge transformation.Note that this gauge symmetry has been originally considered in[12]from a slightly different perspective.In order to identify the off-shell gaugefield theory described byΦ=c(1∂pµ∂8O FF-SHELL G AUGE F IELDS FROM BRST Q UANTIZATIONThere are two interpretations of the resulting off-shell theory.The traceless symmetric tensorfields subjected to the gauge transformation above provides the off-shell definition of the conformal HS theory[29,30].It then follows that the off-shell theory determined by(2.9)can be also considered as an off-shell description of the conformal HSfields. We will not discuss conformal HS theory and refer instead to[12],where,in particular, conformal HS theory was constructed in the analogous terms.Although in this case we only reproduced the description from[12]the advantage of our approach is that it can be uniformly extended to more general quantum constrained systems.Another interpretation of the off-shell theory just constructed has to do with the Frons-dal HS gauge theory[31,32].Namely,we show(see Section4.1)that the off-shell theory determined by(2.9)is equivalent through the elimination of generalized auxiliaryfields to the off-shell theory for the Fronsdal HSfields in the parent form[25](see section3.3 for definition of generalized auxiliaryfields).More precisely,in4.1we construct the appropriate extention of the model(2.9),which provides a geometrically transparent for-mulation of the(off-shell)Fronsdal HS theory.Note that the extended model can also serve as an off-shell theory for conformal HSfields.To directly see the relation with the conventional formulation of the Fronsdal theory suppose that the following equations have been in addition imposed on F:(2.10)∂∂xµF(x;p)=0,∂∂pµF(x;p)=0.Together with the condition∂∂pµF=0this coincides with the equations of motionand the partial gaugefixing conditions for Fronsdal HSfields identified in[33].It is important to note,however,that this does not imply that the theory determined by(2.10)along with∂∂pµF=0and the remaining gauge symmetry is equivalent to the on-shell Fronsdal HS theory in the strong sense(i.e.through the elimination/addition of generalized auxiliaryfields).2.3.Thefield theory BRST differential.It is useful to reformulate the procedure in the BRST theory terms.Here,we closely follow the non-Lagrangian BRST formulation from[25](see also[15,7]and references therein).Let e A be a basis in H.We then associate a supermanifold M to the superspace H.To this end we assign a variableψA to each basis element e A and prescribe gh(ψA)=1−gh(e A),|ψA|=1+|e A|mod2.One then defines M to be a supermanifold2with coordinatesψA.In order to define M one alsoO FF-SHELL G AUGE F IELDS FROM BRST Q UANTIZATION9needs tofix the class of functions inψA.Although most of our present considerations do not really depend on this choice,for definiteness we take smooth functions.In what follows we call M the supermanifold associated to H.Consider thefield theory withfieldsψA defined on the space-time manifold X0.The interpretation ofψA depends on its ghost number.In particular,physicalfields are those with vanishing ghost number.If gh(ψA)=0thenψA should be considered as a ghost field or an antifield.The BRST differential determining the theory is given by(2.11)sψA=1[Ψ,Ψ]∗.2It is also useful to expandΨinto components containingfields at given ghost degree:Ψ= kΨ(k)whereΨ(k)=e A kψA k with gh(ψA k)=k.Note that contrary to the conventional stringfield associated with the space of states of thefirst-quantized system,Ψis associated to the algebra of functions on the entire extended phase space.However, as are going to see,Ψcan be naturally interpreted as a conventional stringfield(but associated with the different quantum constrained system)if one considers the linearized theory.The BRST differential determines the equations of motion,the gauge transformation, and the reducibility conditions along with higher order structures of the gauge algebra. In particular,ifψA k denote componentfields enteringΨ(k)(i.e.gh(ψA k)=k)then equations of motion and the gauge transformations have the form(2.13)(sψA−1) ψA k=0,k=0=0,δψA0=(sψA0) ψA k=0,k=0,1,with ghost-number-1fieldsψA1replaced by gauge parametersλA1with|λA1|=|ψA1|+1 mod2.Note that if the theory does not contain physical fermionicfields(as we have assumed) then all ghost-number-zerofieldsψA0are bosonic and can be identified with coefficients10O FF-SHELL G AUGE F IELDS FROM BRST Q UANTIZATIONof the ghost-number-one sub-ΦA0in the expansion ofΦwith respect to the basis e Aspace in H.Analogously,the gauge parameters correspond to ghost-number-onefields ψA1associated to the basis elements of the ghost-number-zero subspace.However,if one wants to consider theories containing physical fermionicfields or build the complete BV-BRST description one needs to replace the coefficientsΦA(x)in the expansion of a generic element from A with thefieldsψA and to prescribe gh(ψA)=1−gh(e A)and |ψA|=1−|e A|mod2,i.e.to replace superspace H with the supermanifold M.The theory expanded around a particular solution to the equations of motion can also be compactly formulated in the BV-BRST ly,letΨ0be a particular solution to dΨ0+(2 )−1[Ψ0,Ψ0]∗=0.The BRST differential of the theory expanded aroundΨ0 is then given by(2.14)sΨ=ΩΨ+1[Ψ,Ψ]∗+12Note that the nilpotency of s is equivalent to the defining relations of L∞-algebra.More-over,L∞-algebra structure is usually defined in terms of an odd nilpotent odd vectorfield on the associated supermanifold.Equation(2.16)generalizes the construction to the case of quantum constrained sys-tems described by the A∞algebra[36].In this case the A∞-structure is determined by the nilpotentΩ:A→A,the homotopy associative∗-product(i.e.associative only in the Ω-cohomology),and the higher order polylinear maps.The L∞-structure is then obtained from the A∞-structure by taking the graded antisymmetrization of the polylinear maps. Quantum constrained systems of this type naturally arise in quantization of some classical constrained systems(see e.g.[37,24]).3.O FF-SHELL GAUGE THEORIES IN THE BRST EXTENDED UNFOLDEDFORMULATION3.1.BRST extension of the non-linear unfolded formalism.In this section we briefly recall the BRST extension[15]of the Vasiliev non-linear unfolded formalism[16,17,1], proposed recently by G.Barnich and the present author.Consider two supermanifolds:a supermanifold X and M playing the roles of(appro-priately extended)space-time and the target space manifolds respectively.Let X be a su-permanifold equipped with a degree gh X(·),an odd nilpotent vectorfield d,gh X(d)=1, and a volume form dµpreserved by d.Let the supermanifold M be equipped with a de-gree gh M,an odd nilpotent vectorfield Q,gh M(Q)=1.In the literature a supermanifold equipped with an odd nilpotent vectorfield is often called Q-manifold while the vector field itself is referred to as Q-structure[38].The basic example for X is the odd tangent bundleΠT X0which has a natural volume form and is equipped with the De Rham differential d.If xµare local coordinates on X0 andθµare associated coordinates on thefibers ofΠT X0then d=θµ∂.δψA(z)Vectorfield s can be considered as a BRST differential of afield theory on X.Indeed,the basic properties s2=0and gh(s)=1hold.In what follows we refer to this system as a quadruple(X,d,M,Q).For the system(X,d,M,Q)it is easy to check using the explicit form(3.1)that (3.2)sψA=dψA+Q A(ψ).This equation can be taken as a definition of the BRST differential in the jet-bundle de-scription of the theory.In this approach componentfields(ψp)Aµ1...µp enteringψA(x,θ)=(ψ0)A(x)+θµ(ψ1)Aµ(x)+...and their derivatives with resoect to xµare treated as inde-pendent coordinates on the jet space.LetψA k denote componentfields with gh(ψA k)=ing the explicit form(3.2)of the BRST differential onefinds the component form of the equations of motion and gauge symmetries(3.3) dψA+Q A(ψ) ψA k=0,k=0=0,and(3.4)δλψA= dψA+Q A(ψ) ψA1=λA1,ψA k=0,k=0,1,whereδλψA denotes variation ofψA under the gauge variation of its physical component fieldsψA0.In particular,if gh(ψA) 0for allfields then the equations of motion deter-mine the so-called free differential algebra[39].If one does not require gh(ψA) 0then the equations of motion can also contain some constraints.In general,instead of(X,d,M,Q)one can similarly consider afibered bundle with X being a base manifold,afiber isomorphic to M,and the transition functions preserving Q-structure.In this case thefield space is the space of sections of the bundle instead of M-valued functions.However,for the sake of simplicity we do not consider here non-trivial bundles unless otherwise specified.Note that the general construction anyway reduces to (X,d,M,Q)locally.In the case where the“target”supermanifold M is in addition equipped with a com-patible Poisson bracket(antibracket){·,·}and Q={S,·}is generated by a“master action”(“BRST charge”)S satisfying the classical master equation{S,S}=0,one can construct afield theory master action S on the space of maps.This procedure was proposed in[18]as an approach for constructing BV-BRST formulations of topological sigma models(see[40,41,42,43,44,27,37,45,46]and references therein for further de-velopments and applications).A generalization that also includes the Hamiltonian BRST formulation has been proposed in[47]and covers the case where S is Grassmann odd and is to be interpreted as a BRST charge of the BFV-BRST formulation of the theory.A simplest but characteristic example is provided by taking M to beΠg with g being a Lie algebra and X=ΠT X0with X0being a space-time manifold.If e i be a basis inthe Lie algebra then c i with |c i |=1are coordinates on M .In addition we prescribe gh(c i )=1and define(3.5)Qc i =1∂ψBψ=ψ0ψB +1∂ψB ∂ψC ψ=ψ0ψC ψB +...,where ...denote terms of higher orders in ψA .In particular,the linearized theory is determined by the following linear differential(3.7)s 0ψA =d ψA +∂R Q A∂ψBψ=ψ0φB ,where φ=e A φA (x,θ)is a general element of the “space of states”which is the space of functions on X with values in the linear space H identified with the tangent space to M .More precisely,e A φA (x,θ)can be considered as a section of the tangent bundle to M pulled back by the map ψ0.From this point of view the BRST differential(3.6)can be naturally understood as that of a non-linear deformation of the linear theory determined by the first-quantized BRST operator Ω.The BRST operator (3.8)has the same structure as that of a parent systems constructed in [25,7].More generally,one can consider a linear gauge field theory on X 0whose BRST differential have the form(3.9)s 0ψA =d ψA +¯ΩA B ψB ,where¯ΩA B=¯ΩA B(x,θ)satisfies“generalized zero curvature”condition(3.10)d¯ΩA B+(−1)|A|+|C|¯ΩA C¯ΩC B=0,needed for nilpotency.The formulation where the BRST differential has the form(3.9)withΩA B satisfy-ing(3.10)can be considered as a BRST extension of the linear unfolded formulation[48, 49].Indeed,if¯Ωis a1-form(i.e.is linear inθµ)and gh(ψA) 0then¯Ωcan be consid-ered a connection1-form and equations of motion determined by s0take the form of a covariant constancy condition(3.11)d(ψp)A+¯ΩA B(ψp)B=0.Hereψp is a ghost-number-zerofield enteringψA=(ψ0)A+θµ(ψ1)Aµ+θµθν(ψ2)Aµν+..., which is identified with a p-form on X0with p=gh(ψA).From this perspective parent theories constructed in[25,7]are particular examples of theories naturally emerging in the BRST extended unfolded form.3.3.Consistent reductions.Two local gaugefield theories formulated within BRST framework are naturally considered equivalent if they are related by elimination/addition of generalized auxiliaryfields.Suppose that after an invertible change of coordinates, possibly involving derivatives,the set offieldsψA splits intoϕα,w a,v a such that equa-tions sw a|w a=0=0(understood as algebraic equations in the space offields and their derivatives)are equivalent to v a=V a[ϕα],i.e.,can be algebraically solved forfields v a. Fields w,v are then generalized auxiliaryfields.Thefield theory described by s is equiva-lent to that described by the reduced differential s acting on the space offieldsϕαand their derivatives and defined by sϕα=sϕα|w a=0,v a=V a[ϕ](see[25]for more details).In the Lagrangian framework,fields w,v are in addition required to be second-class constraints in the antibracket sense.In this context,generalized auxiliaryfields were originally pro-posed in[50].Generalized auxiliaryfields comprise both standard auxiliaryfields and pure gauge degrees of freedom as well as their associated ghosts and antifields.For BRSTfield theory(X,d,M,Q)one easilyfinds generalized auxiliaryfields as originating from contractible pairs for ly,let w a be such that w a,Qw a are inde-pendent constraints on M determining the submanifold M⊂M.The theory(X,d,M,Q) is then equivalent via elimination of generalized auxiliaryfields to(X,d, M, Q)with Q=Q| M.In order to see that Q indeed restricts to M it is enough to observe that (Qw a)| M=0and Q(Qw a)=0.For more details we refer to[15].Analogously,one can consider contractible pairs tαand d tαin the extended space-time manifold ly,suppose that tαand d tαare independent regular constraints deter-mining a submanifold X.One can address the question on the relation of(X,d,M,Q) and( X, d,M,Q)where d=d| X.These theories can not be considered equivalent as localfield theories because they live on different space-time manifolds.However,if thecoordinates transversal to X⊂X are considered as internal degrees of freedom rather than space-time coordinates one can indeed show that respective theories are equivalent. For more details we again refer to[15].In particular,if X=ΠT∗X0with coordinates xµ,θµone can consistently eliminate any pair xν,θν.Note that the auxiliary role of space-time coordinates was observed in[51,52]in the context of HS theories formulated within unfolded framework.If one is given with a particular solutionΨ0satisfying dψA0+Q A(ψ0)=0then the system expanded aroundΨ0can be reduced using the reduction machinery developed in[25,7]for the free theories associated tofist-quantized systems.Indeed,the linearized theory can be identified with the freefield theory associated to thefirst-quantized system described by(3.8).Under the standard assumptions it then follows that the generalized auxiliaryfields for the linearized theory are also generalized auxiliaryfields for its non-linear deformation(see e.g.[7]).3.4.Putting a quantum constrained system to afiber.Let us consider again a con-strained system quantized in the sense of deformation quantization,i.e.the associative ∗-product algebra A of the extended phase space functions depending formally on and equipped with the ghost number grading and the Grassmann parity.Contrary to the con-struction of Section2.1now we are going to achieve a generally covariant(in the sense of X0)description of the theory.To this end we construct an AKSZ-type sigma-model by, roughly speaking,putting the quantum constrained system to the target space.Moreover, we need to change the class of the phase space ly,we replace the space-time coordinates xµwith the formal variables y a so that A consist of formal power series in y a and with coefficients depending on the remaining variables.In addition,we also assume A to be graded-finite dimensional.More technically,wefirst consider a supermanifold associated to A.Let e A be a basis in A andψA coordinates on the associated supermanifold M.The stringfield is then given byΨ=e AψA.Similarly to the considerations in2.1the Q-structure on M is given by(3.12)QψA=¯ΩA BψB+1[Ψ,Ψ]∗.2Note that the construction can be naturally generalized to involve L∞-structure instead of a differential graded Lie algebra.This corresponds to taking odd nilpotent vectorfield Q not necessarily quadratic inψA(see[15]for more details).。
Light-Front QCD
arXiv:hep-ph/0412101v1 7 Dec 2004
Light-Front QCD
∗
Stanley J. Brodsky Stanford Linear Accelerator Center Stanford University, Stanford, California 94309 E-mail: sjbth@
∗
Work supported by Department of Energy contract DE–AC02–76SF00515.
Abstract
In these lectures, I survey a number of applications of light-front methods to hadron and nuclear physics phenomenology and dynamics, including light-front statistical physics. Light-front Fock-state wavefunctions provide a frame-independent representation of hadrons in terms of their fundamental quark and gluon degrees of freedom. Nonperturbative methods for computing LFWFs in QCD are discussed, including string/gauge duality which predicts the power-law fall-off at high momentum transfer of light-front Fock-state hadronic wavefunctions with an arbitrary number of constituents and orbital angular momentum. The AdS/CFT correspondence has important implications for hadron phenomenology in the conformal limit, including an all-orders derivation of counting rules for exclusive processes. One can also compute the hadronic spectrum of near-conformal QCD assuming a truncated AdS/CFT space. Given the LFWFs, one can compute form factors, heavy hadron decay amplitudes, hadron distribution amplitudes, and the generalized parton distributions underlying deeply virtual Compton scattering. The quantum fluctuations represented by the light-front Fock expansion leads to novel QCD phenomena such as color transparency, intrinsic heavy quark distributions, diffractive dissociation, and hidden-color components of nuclear wavefunctions. A new test of hidden color in deuteron photodisintegration is proposed. The origin of leading-twist phenomena such as the diffractive component of deep inelastic scattering, single-spin asymmetries, nuclear shadowing and antishadowing is also discussed; these phenomena cannot be described by lightfront wavefunctions of the target computed in isolation. Part of the anomalous NuTeV results for the weak mixing angle θW could be due to the non-universality of nuclear antishadowing for charged and neutral currents.
Gamma2
u ? d djN i = gA U (p) U (p) ; (1) where U (p) denotes the nucleon spinor. Assuming isospin symmetry, one nds (gA )p = ?(gA )n, and the value of (gA)p = 1:2573 0:0028, extracted from the neutron beta decay, has been quoted in the literature 1]. In nature, the isospin symmetryis broken by the current quark mass di erence as well as the electromagnetic interaction, and thus (gA )p 6= ?(gA )n . Previous studies of the nucleon isovector axial charge in the framework of external eld QCD sum-rule method have been made by various authors 2{4]. However, to our best knowledge, the isospin breaking e ects have been ignored in these studies. The goal of this Letter is to examine the di erence between (gA )p and (gA)n using the external eld QCD sum-rule approach, which has been used in studying various nucleon matrix elements of bilinear quark operators 2{13]. The isospin violation is re ected in mu 6= md and the isospin breaking in the vacuum condensates. Electromagnetic e ects will not be included. Invoking the Goldberger-Treiman relation, we also give an estimate of the isospin breaking in the pion-nucleon coupling constants, which is of interest in the investigation of charge symmetry breaking phenomena 14{17]. Let us start from the correlation function of the nucleon interpolating eld in the presence of a constant external isovector axial vector eld Z
可靠性分析
Reliability assessment of high cycle fatigue under variableamplitude loading:Review andsolutionsAlessandra Altamura ⇑,Daniel StraubEngineering Risk Analysis Group,Technische Universität München,Germanya r t i c l e i n f o Article history:Received 22January 2013Received in revised form 15February 2014Accepted 22February 2014Available online 3April 2014Keywords:Reliability Fatigue crack growth Damage tolerant High cycle fatigue Variable loadsa b s t r a c tIn fatigue reliability assessments,the random load process is commonly represented by itsmarginal distribution (load spectrum)only.However,as shown in this paper,the correla-tion characteristics of the load process can have a strong influence on the fatigue reliabilityand should be accounted for.The paper reviews the modeling of random fatigue crackgrowth under variable amplitude loading for reliability analysis.Solutions for fatigue crackgrowth evaluation at different levels of detailing are described and a fatigue crack growthand failure evaluation algorithm,based on a discretization of the random stress process,ispresented.As an alternative,a mean approximation is described.Finally,effective compu-tational methods for assessing the fatigue reliability under variable amplitude loading areintroduced and applied exemplarily to a case study.The solutions are based on the first-order reliability method FORM and the subset ing a Markov process modelof the loads,the influence of different types of service histories is investigated,by varyingthe correlation length of the stress cycle process.The results show that the correlationlength of the load process has significant influence on the resulting reliability;the resultingprobability of failure can vary up to several orders of magnitude for the same marginalprobability distribution of stress amplitudes.Based on the results of the case study,theinfluences of the stress process correlation and of the adopted failure criteria on the reli-ability are discussed.The mean approximation and the random variable model of therandom load process are demonstrated to be applicable under specific conditions.Ó2014Elsevier Ltd.All rights reserved.1.IntroductionThe reliability of mechanical and structural components subjected to high cycle fatigue can be ensured with a safe-life approach,a damage tolerant design or a fail-safe approach.The safe-life approach requires that the fatigue reliability is suf-ficiently high over the entire service life,which is typically achieved by ensuring that no or only small crack growth occurs[1].Damage tolerant design ensures that the component does not fail between inspection intervals with a sufficiently high reliability [2].The fail-safe approach ensures that damages are limited in case of fatigue failure.In many applications,the fail-safe approach is ruled out,and it is necessary to demonstrate sufficient reliability,with or without inspections.Thereby,the uncertainties in fatigue crack growth must be considered,which are associated with the presence and size of initial flaws or cracks,the mechanical properties of the material,the fatigue crack growth models and its parameters,and the stress /10.1016/j.engfracmech.2014.02.0230013-7944/Ó2014Elsevier Ltd.All rights reserved.⇑Corresponding author.Tel.:+491731758603.E-mail addresses:ale.altamura@tum.de ,aaaltamura@ (A.Altamura).Nomenclaturea crack deptha0initial crack deptha cr critical crack depthb number of cycles in a blockc crack semi-lengthdx i dn general expression of the crack growth rate in direction x ih a¼dadn fatigue crack growth rate in a-directionh c¼dcdn fatigue crack growth rate in c-directionh x fatigue crack growth ratef D r,R is the joint CDF of D r(n)and R(n)g(X),g(X,N)limit state functionk0,k,k min parameters of the toughness distributionn number of fatigue cyclesm,p,q,C,C th,A0Forman–Mettu equation parameters n F number of failuresn s number of samplesp F probability of failurex generic crack lengthz correlation length of the stress range process CDF cumulative density functionE Young’s modulusF cumulative density functionF rmax;i CDF of the maximum stress range in b cyclesFORMfirst order reliability methodG(U)limit state function in the standard normal spaceJ J-integralJ e elastic J-integralJ mat fracture toughness expressed as J-integralK stress intensity factorK mat fracture toughness expressed as K-factorK max maximum stress intensity factorL length of the tubesL r ligament yielding factorL r,max critical value of the ligament yielding factorMCS Monte Carlo simulation methodMA mean approximationN number of cycles(fixed value)N fail number of cycles at failureN target target number of cyclesN stop number of cycles at which the crack growth algorithm stopsNDT non-destructive testsP loadPOD probability of detectionR stress ratio{R(n)}random process of the stress ratioRV random variableRP random processSuS subset simulation methodUTS ultimate tensile strengthWT wall thicknessX vector of random variablesU vector of standard normal uncorrelated random variablesV vector of standard normal correlated random variables{X(n)}general expression for a random processc vector of parameters related to the geometry of the component containing the crackd vector of parameters related to the material propertiesh,f parameters of the POD distributionu standard normal probability density functionv Poisson’s ratioA.Altamura,D.Straub/Engineering Fracture Mechanics121–122(2014)40–6641sequence.Because of these uncertainties,a deterministic fatigue analysis requires large safety factors and leads to a conser-vative design.This motivates the use of reliability analysis for the assessment of fatigue reliability.The reliability of components subjected to high cycle fatigue can be evaluated based on a S–N damage accumulation approach or a fatigue crack growth evaluation approach.The former is based on empirically determined S–N curves and a damage accumulation rule,such as Palmgren–Miner.Its advantages are its simplicity and the fact that material parameters are available in the literature for a range of materials and component designs.Its disadvantages are that the models –due to their empirical nature –cannot be extrapolated beyond the range of experiments,and that the simplified models lead to sig-nificant uncertainty in the predictions,thus necessitating large safety margins in the design.Various effects that are known to influence the fatigue life are not commonly included in the damage accumulation approach,including the influence of the sequence of stress ranges on the fatigue life:fatigue failures are more likely if higher stress ranges occur at the beginning of service life and lower stress ranges towards the end,than if they occur in reverse [3].Such effects can be modeled by a fa-tigue crack growth evaluation approach,which is based on combining linear-elastic fracture mechanics with crack growth models.This approach,however,has the disadvantage of requiring more detailed model inputs and leads to more demand-ing computations.This holds in particular when a reliability assessment is performed,requiring stochastic model inputs and advanced computations.As a consequence,in most studies on fatigue reliability using a crack growth approach,the stress amplitudes are either assumed constant or represented by deterministic block sequences.However,these sequences are not always representative of the random service load history to which structures are subjected.In this paper,effective computational methods for assessing the fatigue reliability under variable amplitude loading are presented.To provide the reader practical guidance for various types of models,we present solutions for different levels of detailing of the crack growth models,e.g.one-and two-dimensional crack growth models,and different failure criteria.The methods are applied to a case study considering a pressurized tube,which demonstrates that the assumptions on the tem-poral variability of the loading can lead to differences in the probability of failure of up to several orders of magnitude.The common assumptions of constant amplitude loading or block sequences are shown to be potentially non-conservative.The crack growth models considered in this paper do not explicitly include retardation and acceleration effects due to load interaction.These effects can be included,but some of the computationally efficient procedures presented in this paper are not applicable,resulting in increased computational efforts for problems where retardation and acceleration effects are relevant.The paper starts out with a detailed introduction to variable amplitude loading.This is followed by a step-by-step intro-duction to the assessment of fatigue crack growth under variable amplitude loading for various degrees of crack growth model complexity (Section 3).Section 4summarizes the probabilistic modeling of crack growth and Section 5presents the failure criteria and proposes efficient modeling strategies for these.Section 6introduces reliability evaluation methods and introduces the algorithms for computing fatigue reliability under variable amplitude loading when fatigue loads are modeled as a Markov random process.Finally,Section 7presents the numerical investigations of the models,which also illustrates the appropriate selection of the presented proposed models and methods under different loading conditions.2.Variable amplitude loading2.1.Historical backgroundMost mechanical and structural components are subjected to variable amplitude loading,also called spectrum loading [4],during their service life.In the 1930s,engineers working in aeronautics studied the variable amplitude characteristics ofr stress r a stress amplitude r y yield strength r y ,cycl cyclic yield strength r min minimum stress value r max maximum stress value r UTS ultimate tensile strength a i FORM sensitivity factorsq geometric parameter in the formulation of L rD K stress intensity factor rangeD K a and D K c stress intensity factor range in directions a and cD K th fatigue thresholdD K th 0fatigue threshold at R =0D r stress rangeD r m membrane component of the stress rangeD r b bending component of the stress range{D r }random process of the stress rangeU standard normal cumulative distribution functionX F failure domain42 A.Altamura,D.Straub /Engineering Fracture Mechanics 121–122(2014)40–66stress cycles in-service.Measurements of service loads were carried out and the first load spectra were published by Kaul [5].In 1939,Gassner introduced the first variable amplitude load sequence for testing aeronautical structures [6].Laboratory experiments require simple load sequences,which however should be representative of the real service conditions.The Eight-Block-Program Test proposed by Gassner is a sequence of loading blocks,now known as ‘‘Gassner sequence’’.Within each block,stress cycles are identical;between blocks,the stress amplitude changes while the mean value remains the same.The lengths of the blocks are defined such that stress amplitudes follow the Lognormal distribution.The Gassner sequence consists of 8varying blocks,whose sequence is fixed and predetermined.After 8blocks,the sequence is repeated.This pro-cedure is the core of the ‘‘Operational fatigue strength’’approach to the design of components under variable amplitude loading [7,8].With the availability of hydraulic testing machines,more realistic load sequences could be applied for testing.Such load sequences can be derived from experimental measurements.For example,the SAE Fatigue and Evaluation Committee se-lected test load sequences from existing strain measurements [9,10].Exhaustive information on fatigue testing under var-iable amplitude loading can be found in [11],while a review of the standard load sequences used for fatigue testing and on the generation of testing load histories from experimental measurements can be found in [12].Lardner [13–15]and Rau [16]proposed to model variable amplitude loading by random processes.In [13,14]an approach for reliability evaluation under random loading is described using the crack propagation law proposed in [17].Rau [16]de-scribes the fatigue crack growth as a random process,since it is the consequence of the application of random process loads.It is suggested that the propagation of the fatigue crack is independent of the order of application of the stress cycles,when the load is a stationary random process and when a high number of stress cycles is applied,so that variations due to the order of application of the stresses average out.At the beginning of the 1970s,Schijve investigated the influence of the load sequence on fatigue life [18].In his study,the effect of the load sequence on crack propagation is investigated by performing experiments applying random loading se-quence with short and long blocks of cycles.It was observed that the random load sequences could lead to fatigue lives that differ from those evaluated using laboratory tests with simplified load sequences,demonstrating the importance of appro-priately representing the randomness of fatigue loads.According to [18],‘‘the predicted life does not depend so much on the sequence,provided that it is random in some way or programmed with a short period’’,which confirms the findings of [16].However,simplified loading sequences consisting of repeated large blocks may lead to unconservative fatigue life predic-tions due to sequence effects.Following these studies,the need to account for the stochasticity of fatigue crack growth under variable loading was recognized.2.2.Fatigue load as a random processWhen describing load sequences from experimental load measurements,procedures for identifying load cycles from the stress-time history are necessary [19].Standardized procedures reported in [4]are:level-crossing counting,peak counting,simple-range counting and rainflow counting.These methods result in a sequence of stress cycles,which are characterized by their stress ranges D r and stress ratios R ,or alternatively by their minimum and maximum stresses r min and r max .Note that the definition of the ordering in which the sequences occur is not necessarily unique if the stress process is a broad band process [20].Statistically,the load sequence can be described by the random processes {D r (n )},{R (n )},i.e.for every stress cycle n there is a random variable pair D r (n )and R (n ).Values of D r (n )and R (n )at different cycles will generally be corre-lated.We limit the discussion to stationary load processes,since the assumption of stationarity is sufficient for most relevant applications.To ease notation,we drop the index n and denote the processes by {D r }and {R }.Under the assumption of a Gaussian copula model (also known as the Nataf distribution model [21]),a stationary random process is fully characterized by its marginal distribution and its autocovariance function [22].These can be determined from observed load sequences.Alternatives are presented by Markov chain models as reported e.g.in [23–26].In general,Markov process models –due to their flexibility –can model the real dependence structure among stress cycles with higher accuracy.For example,switching Markov models [24]can well represent different modes of operations of mechanical systems and structures.However,given the uncertainties associated with determining in-service stresses in real structures,the Gaussian copula model will be sufficiently accurate for many engineering applications.It is pointed out that the marginal distribution of the stress ranges D r and stress ratios R is not affected by these modeling assumptions.A Gaussian copula type model for fatigue load processes,which is simultaneously a Markov process,is described in the following.This model will be applied later for numerical investigations,because it has the advantage that the dependence structure is represented by a single parameter.For simplicity,only the stress range is modeled as a random process {D r },the stress ratio R is assumed to be constant.The marginal distribution of D r (n )is defined through its cumulative distribution function CDF F D r .Let D r (n )be defined through a transformation T 0from a standard Normal variate V (n )as:D r n ðÞ¼T 0V n ðÞðÞ¼F À1D r U V n ðÞðÞ ;ð1Þwhere U is the standard Normal CDF and F À1D r is the inverse CDF of D r (n ).If it is imposed that V (n i )and V (n j )have the jointNormal distribution,then the corresponding pair of stress ranges D r (n i )and D r (n j ),defined through the transformation in Eq.(1),are said to follow the Gaussian copula.A.Altamura,D.Straub /Engineering Fracture Mechanics 121–122(2014)40–6643The autocovariance function of the process {D r }is described through a corresponding autocovariance function K VV of the underlying standard Normal process {V },which is here assumed to be of the exponential type with correlation length z :K VV D n ðÞ¼Cov V n ðÞ;V n þD n ðÞ½ ¼exp ÀD n z :ð2ÞThe process {D r }does not have the same autocorrelation function as the underlying Gaussian process {V },however,the dif-ference between the two is generally small.K D r D r (D n )is obtained from K VV (D n )by means of the Rosenblatt transformation[27]or the Nataf transformation [21].It can be shown that with the exponential autocovariance function (Eq.(2)),the process {V }has the Markovian property[28].Consequently,also the process {D r }is a Markov process:F D r h D r n i ðÞj D r n i À1ðÞ;D r n i À2ðÞ;...;D r n 1ðÞi ¼F D r h D r n i ðÞj D r n i À1ðÞi :ð3ÞWith this model,the dependence among individual stress ranges is characterized solely by the correlation length z of Eq.(2).To illustrate the effect of z ,Fig.1shows three different realizations of stress range processes {D r }with identical marginal distribution but varying correlation length.It is pointed out that by varying the correlation length z ,the load spectrum is not changing,i.e.the marginal distribution shown on the left-hand side of Fig.1of D r is unaltered.However,the correlation length can have an effect on the distribution of the observed realization.In Fig.1,the underlying marginal distribution is clearly visible from one realization of the process with z =1,but this is not the case for Z =500,where all stress ranges in the considered range n ={1,...,500}are highly correlated.In engineering practice,the marginal distribution of D r ,i.e.the load spectrum,is often determined from measured load sequences.Thereby,a correct estimation of this distribution is only possible if the measured load sequence is significantly longer than the correlation length.This effect is clearly visible in Fig.1.In the case of z =1,an empirically determined stress range distribution based on the observed load sequence would be very close to the true underlying marginal distribution shown on the left hand side of Fig.1.In contrast,for z =500,the observed load sequence clearly is not representative of the true underlying distribution of D r .In this case,either a much longer load sequence must be recorded,or several inde-pendent shorter load sequences must be recorded,e.g.load sequences arising from different missions,and combined using the total probability theorem.Alternatively,measurements can be combined with or replaced by fatigue load calculations based on statistical models of the load environment,such as for offshore and marine structures subject to wave loads.2.2.1.Discretization of the fatigue load process into blocksFor practical purposes,it is computationally advantageous to approximate the random load sequence by blocks of cycles with constant amplitude and stress ratio.Such blocks can be defined from the original fatigue load process {D r },{R }by dividing the sequence of cycles into blocks of b cycles.To each block i ,we assign a stress range D r i and a stress ratio R i that are equal to the values of the stress cycle at the mid-point of the block D r i =D r (i À1/2)b .Unlike the blocks of the Gassner sequence,or similar deterministic load sequences,the loading blocks obtained with this method still represent a random process.The resulting stress range process has the same marginal distribution F D r as the original one.The average autocovariance function of the block approximation of D r is:K e D r e D r D n ðÞ¼K D r D r k Áb ðÞÀD n Àk Áb bh K D r D r k Áb ðÞÀK D r D r k þ1ðÞÁb ðÞi ;ð4Þwith k ¼floor D n b:ð5ÞThe error in the covariance is small as long as the correlation length is much larger than the block size,z )b .44 A.Altamura,D.Straub /Engineering Fracture Mechanics 121–122(2014)40–66Fig.2exemplarily illustrates the stress range block sequence corresponding to the realizations of the stress ranges shown in Fig.1,with a block length of25cycles.It can be observed that the approximation becomes better with increasing corre-lation length z.3.Fatigue crack growth evaluation under constant and variable amplitude loading3.1.Models of fatigue crack growthStarting from an initialflaw or notch,cracks will form and grow under cyclic loading.Cracks that grow in two directions usually exhibit a near-elliptical or semi-elliptical shape[29,30].Thereby,the crack front advances in all directions,with coor-dinates x i,x j,x k,...,as depicted in Fig.3.Crack growth in any direction x i is described by a differential equation expressing the crack growth rate d x id n as a function ofthe stress intensity factor range along the crack front in the x i direction,D K xi:d x i d n ¼h xiD K xi;R;dÀÁ;ð6Þwhere R is the stress ratio and d is a set of parameters related to the material properties.For the case of the Paris law we haveh xi ðD K xi;dÞ¼CÁD K mx iwith parameters d¼½C;m .More advanced models for h xiinclude:the bilinear crack growth model adopted in BS7910[31,32],in which the crack growth is described with a Paris modeland two different slopes are used to describe the near-threshold region and the Paris region,respectively:h xi ¼C1ÁD K m1x iand h xi ¼C2ÁD K m2x ifor D K xi>D K th;the Forman–Mettu model[33],which is summarized in Annex A,and which is used in the numerical investigations pre-sented later.When the stress sequence is characterized by overloads,the so-called retardation effect is observed.In the cycles follow-ing the overload,a lower crack propagation rate is observed,due to the plasticity induced closure caused by a larger plastic zone that is the result of the overload[1].Due to this effect,the rate of fatigue crack growth is known to depend on the order in which tensile and compressive overloads are applied[1]and the type of stress sequence has an effect on the fatigue life [34,35].Following the observation of crack closure by Elber[36,37],several models for crack closure were developed to de-scribe the delaying effects of high loads,such as the Wheeler model[38],the Willenborg model[39],the more realistic strip yield model developed by Newman[40]and the partial crack closure model valid in the near-threshold region[41].The evaluation of fatigue crack growth requires knowledge of the stress intensity factor D K xialong the entire crack front.A large body of research has focused on deriving analytical or numerical expressions for D K xi ,including[29,42–45].ForA.Altamura,D.Straub/Engineering Fracture Mechanics121–122(2014)40–6645certain geometries,exact analytical solutions or approximate analytical expressions are available,in other cases FEM anal-ysis is necessary,e.g.[46–49].If the geometry of the crack is approximated by a perfect semi-elliptical or elliptical shape,then it is fully described by the semi-lengths of the two axes,called a and c ,which correspond to the two main growth directions [29],as shown in Fig.4.In the remainder,we will use this approximation.The stress intensity factor ranges in the two directions a or c ,denoted by D K a and D K c ,are a function of the geometry of the component,the crack dimensions a and c ,and the applied stress range D r It is distinguished between the membrane stress range,D r m ,and the bending stress range,D r b ,which varies along the section.In absence of residual stresses,both components can be directly evaluated from a total stress range D r ¼D r m þmax D r b [50].For ease of presentation,we only consider D r .Therefore,we can write the stress intensity factor range in terms of D K a ¼D K a ða ;c ;D r ;c Þand D K c ¼D K c ða ;c ;D r ;c Þand Eq.(6)can be rewritten as follows:d a d n¼h a D K a a ;c ;D r ;c ðÞ;R ;d ðÞ;ð7Þd c d n¼h c D K c a ;c ;D r ;c ðÞ;R ;d ðÞ;ð8Þwhere –d a d n and d c d n are the crack growth rates in directions a and c ;–h a and h c are the functions describing the crack growth rate;–R is the stress ratio;–d is a set of parameters describing material properties;–c is a set of parameters describing the geometry of the component containing the crack.It is reminded that h a and h c can include threshold effects.If retardation is taken into account,a crack closure model with corresponding parameters has to be included in the crack growth rate equations.In these cases,h a and h c are additionally a function of the stresses in previous cycles.3.2.Evaluation of fatigue crack growthIn the following,the evaluation of one-and two-dimensional crack growth under constant and variable amplitude loading is presented.The parameters R ,d and c are assumed constant.Generally they may be modeled as deterministic or random variables.3.2.1.One-dimensional crack growth with constant amplitude loadingFor one-dimensional crack growth,the crack is fully characterized by its depth a ,as shown in Fig.3.Crack growth is thus fully described by Eq.(7).With constant amplitude,the crack growth can be evaluated from the boundary condition on the initial value of crack depth a 0.By reformulating Eq.(7)and integrating on both sides,one obtains:N ¼ZN 0dn ¼Z a a 0da h a ðD K a a ;D r ;c ðÞ;R ;d Þ;ð9Þwhere N is the number of stress cycles to reach a crack depth a .Eq.(9)can be solved numerically.For special cases,analytical solutions exist,e.g.for the simple Paris law,[19].If the interest is in finding the crack depth a as a function of the number of stress cycles n ,a root finding algorithm can be employed;this algorithm requires evaluating the integral in Eq.(9)for different values of a .3.2.2.Two-dimensional crack growth with constant amplitude loadingIn case of two-dimensional crack growth,the crack is described by its depth a and its width c .Crack growth is described by the two coupled differential equations given in Eqs.(7)and (9).An approximate solution of these coupled differential equations is obtained through a step-wise solution.Let D n denote the number of cycles in each step.In the i th step,the crack advances from a i ,c i to a i +1,c i +1.If the ratio of crack depth to width is fixed in each step to a c ÀÁi,then the two differential equations can be integrated separately,as shown in Eqs.(10)and (11).46 A.Altamura,D.Straub /Engineering Fracture Mechanics 121–122(2014)40–66。
弦理论
的个数大约是
，这可以通过计算
Black M2-brane的熵得到：
D4-膜上的耦合常数带有长度量纲：
所以，M5-膜可以通过取D4-膜的强耦合极 限得到。与D2不同的是，D4强耦合对应于 紫外极限。
同样，这样得到的理论应该具有共形不变 性。这个理论更加难以写出，因为这里多 了一个空间维度。
D4-膜的理论中有5个标量场，M5-膜的 理论中也有5个标量场，因为它们有一 样多的横向振动。
以上理论都是基于通常的规范理论，即规 范对称性是通常的规范群，这样规范场在 相应的李代数中取值。
要获得N=8超对称理论，我们可以从N =2理论出发，有了足够多的物质多重态， 也许我们就可以构造出N=8理论。
Schwarz证明，如果多重态是矩阵场，则 不可能构造出N=8理论。
4.2 Bagger-Lambert理论
静态场的运动方程
Verlinde等人说明，以上方程的解的模 参数空间是
更多的进展：
1. Maldacena等人的N=6超共形场论，可
能描述
上的M2-膜
2. 将M2-膜理论用到AdS/CFT
3. 与更多背景场的耦合
4. 与M5膜的关 ……
很多需要考虑的问题： 1. 量子性质，共形不变性的保持 2. 应用M2理论来理解M理论 3. M5-膜理论的构造 4. 李3代数的数学 ……
这个超对称变换不封闭。为了封闭变换，必 须引进用李3代数定义的规范变换：
这样，就必须引进规范场，带两个指标： 修改后的超对称变换为
计算两个超对称变换的交换，我们发现 这个交换可以写成时空平移和规范变换， 但以下的运动方程必须满足
第二个方程正好对应于CS作用量。
超对称变换中的参数也不能任意：
ResearchObjectives
Research ObjectivesThe MILC Collaboration is engaged in a broad research program in Quantum Chromodynamics (QCD).This research addresses fundamental questions in high energy and nuclear physics,and is directly related to major experimental programs in thesefields.It includes studies of the mass spectrum of strongly interacting particles,the weak interactions of these particles,and the behavior of strongly interacting matter under extreme conditions.The Standard Model of High Energy Physics encompasses our current knowledge of the funda-mental interactions of subatomic physics.It consists of two quantumfield theories:the Weinberg-Salaam theory of electromagnetic and weak interactions,and QCD,the theory of the strong interac-tions.The Standard Model has been enormously successful in explaining a wealth of data produced in accelerator and cosmic ray experiments over the past thirty years;however,our knowledge of it is incomplete because it has been difficult to extract many of the most interesting predictions of QCD,those that depend on the strong coupling regime of the theory,and therefore require non-perturbative calculations.At present,the only means of carrying out non-perturbative QCD calculations fromfirst principles and with controlled errors,is through large scale numerical sim-ulations within the framework of lattice gauge theory.These simulations are needed to obtain a quantitative understanding of the physical phenomena controlled by the strong interactions,to de-termine a number of the fundamental parameters of the Standard Model,and to make precise tests of the Standard Model’s range of validity.Despite the many successes of the Standard Model,it is believed by high energy physicists that to understand physics at the shortest distances,a more general theory,which unifies all four of the fundamental forces of nature,will be required.The Standard Model is expected to be a limiting case of this more general theory,just as classical mechanics is a limiting case of the more general quantum mechanics.A central objective of the experimental program in high energy physics,and of lattice QCD simulations,is to determine the range of validity of the Standard Model,and to search for new physics beyond it.Thus,QCD simulations play an important role in efforts to obtain a deeper understanding of the fundamental laws of physics.QCD is formulated in the four-dimensional space-time continuum;however,in order to carry out numerical calculations one must reformulate it on a lattice or grid.It should be emphasized that the lattice formulation of QCD is not merely a numerical approximation to the continuum formu-lation.The lattice regularization of QCD is every bit as valid as continuum regularizations.The lattice spacing a establishes a momentum cutoffπ/a that removes ultraviolet divergences.Stan-dard renormalization methods apply,and in the perturbative regime they allow a straightforward conversion of lattice results to any of the standard continuum regularization schemes.Lattice QCD calculations proceed in two steps.In thefirst,one uses importance sampling tech-niques to generate gauge configurations,which are representative samples from the Feynman path integrals that define QCD.These configurations are saved,and in the second step they are used to calculate a wide variety of physical quantities.It is necessary to generate configurations with a range of lattice spacings,and then perform extrapolations to the zero lattice spacing limit.Fur-thermore,the computational cost of calculations rises as the masses of the quarks,the fundamental constituents of strongly interacting matter,decrease.Until recently,it has been too expensive to carry out calculations with the masses of the two lightest quarks,the up and the down,set to their physical values.Instead,one has performed calculations for a range of up and down quark masses, and extrapolated to their physical values guided by chiral perturbation theory,an effectivefield theory that determines how physical quantities depend on the masses of the lightest quarks.The extrapolations in lattice spacing(continuum extrapolation)and quark mass(chiral extrapolation) are the major sources of systematic errors in QCD calculations,and both must be under control in order to obtain trustworthy results.In our current simulations,we are,for thefirst time,working at or near the physical masses of the up and down quarks.The gauge configurations produced in these simulations greatly reduce,and will eventually eliminate,the systematic errors associatedwith the chiral extrapolation.A number of different formulations of QCD on the lattice are currently in use by lattice gauge theorists,all of which are expected to give the same results in the continuum limit.In recent years, major progress has been made in thefield through the development of improved formulations(im-proved actions)which reducefinite lattice spacing artifacts.Approximately twelve years ago,we developed one such improved action called asqtad[1],which significantly increased the accuracy of our simulations for a given amount of computing resources.We have used the asqtad action to generate an extensive library of gauge configurations with small enough lattice spacings and light enough quark masses to perform controlled calculations of a number of physical quantities. Computational resources provided by the DOE and NSF have enabled us to complete our program of generating asqtad gauge configurations.These configurations are publicly available,and have been used by us and by other groups to study a wide range of physical phenomena of importance in high energy and nuclear physics.Ours was thefirst set of full QCD ensembles that enabled control over both the continuum and chiral extrapolations.We have published a review paper describing the asqtad ensembles and the many calculations that were performed with them up to2009[2]. Over the last decade,a major component of our work has been to use our asqtad gauge config-urations to calculate quantities of importance to experimental programs in high energy physics. Particular emphasis was placed on the study of the weak decays and mixings of strongly interact-ing particles in order to determine some of the least well known parameters of the standard model and to provide precise tests of the standard model.The asqtad ensembles have enabled the calcu-lation of a number of physical quantities to a precision of1%–5%,and will enable many more quantities to be determined to this precision in the coming years.These results are already having an impact on experiments in high energy physics;however,in some important calculations,partic-ularly those related to tests of the standard model,higher precision is needed than can be provided by the existing asqtad ensembles.In order to obtain the required precision,we are now working with the Highly Improved Staggered Quark(HISQ)action developed by the HPQCD Collabora-tion[3].We have performed tests of scaling in the lattice spacing using HISQ valence quarks with gauge configurations generated with HISQ sea quarks[4].We found that lattice artifacts for the HISQ action are reduced by approximately a factor of2.5from those of the asqtad action for the same lattice spacing,and taste splittings in the pion masses are reduced by approximately a factor of three,which is sufficient to enable us to undertake simulations with the mass of the Goldstone pion at or near the physical pion mass.(“Taste”refers to the different ways one can construct the same physical particle in the staggered quark formalism.Although particles with different tastes become identical in the continuum limit,their masses can differ atfinite lattice spacing).More-over,the improvement in the quark dispersion relation enables us to include charm sea quarks in the simulations.The properties of the HISQ ensembles are described in detail in Ref.[5],and the first physics calculations using the physical quark mass ensembles in Refs.[6,7,8].The current status of the HISQ ensemble generation project is described at the link HISQ Lattice Generation and some initial calculations with them at Recent Results.The HISQ action also has major advan-tages for the study of QCD at high temperatures,so we have started to use it in our studies of this subject.Projects using the HISQ action will be a major component of our research for the next several years.Our research is currently focused on three major areas:1)the properties of light pseudoscalar mesons,2)the decays and mixings of heavy-light mesons,3)the properties of strongly interacting matter at high temperatures.We briefly discuss our research in each of these areas at the link Recent Results.References[1]The MILC Collaboration:C.Bernard et al.,Nucl.Phys.(Proc.Suppl.),60A,297(1998);Phys.Rev.D58,014503(1998);G.P.Lepage,Nucl.Phys.(Proc.Suppl.),60A,267(1998);Phys.Rev.D59,074501(1999);Kostas Orginos and Doug Toussaint(MILC),Nucl.Phys.(Proc.Suppl.),73,909(1999);Phys.Rev.D59,014501(1999);Kostas Orginos,Doug Tou-ssaint and R.L.Sugar(MILC),Phys.Rev.D60,054503(1999);The MILC Collaboration:C.Bernard et al.,Phys.Rev.D61,111502(2000).[2]The MILC Collaboration: A.Bazavov et al.,Rev.Mod.Phys.82,1349-1417(2010)[arXiv:0903.3598[hep-lat]].[3]The HPQCD/UKQCD Collaboration: E.Follana et al.,Phys.Rev.D73,054502(2007)[arXiv:hep-lat/0610092].[4]The MILC Collaboration: A.Bazavov al.,Phys.Rev.D82,074501(2010)[arXiv:1004.0342].[5]The MILC Collaboration: A.Bazavov al.,Phys.Rev.D87,054505(2013)[arXiv:1212.4768].[6]The MILC Collaboration: A.Bazavov et al.,Phys.Rev.Lett.110,172003(2013)[arXiv:1301.5855].[7]The Fermilab Lattice and MILC Collaborations:A.Bazavov,et al.,Phys.Rev.Lett.112,112001(2014)[arXiv:1312.1228].[8]The MILC Collaboration:A.Bazavov et al.,Proceedings of Science(Lattice2013)405(2013)[arXiv:1312.0149].。
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一维扩展量子罗盘模型的拓扑序和量子相变
一维扩展量子罗盘模型的拓扑序和量子相变陈西浩;王秀娟【摘要】应用矩阵乘积态表示的无限虚时间演化块算法,研究了扩展的量子罗盘模型.为了深入研究该模型的长程拓扑序和量子相变,基于奇数键和偶数键,引入了奇数弦关联和偶数弦关联,计算了保真度、奇数弦关联、偶数弦关联、奇数弦关联饱和性与序参量.弦关联表现出三种截然不同的行为:衰减为零、单调饱和与振荡饱和.基于弦关联的以上特征,给出了量子罗盘模型的基态序参量相图.在临界区,局域磁化强度和单调奇弦序参量的临界指数β=1/8表明:相变的普适类是Ising类型.此外,保真度探测到的相变点、连续性与非连续性和序参量的结果一致.【期刊名称】《物理学报》【年(卷),期】2018(067)019【总页数】8页(P27-34)【关键词】量子相变;弦关联;拓扑序;临界指数【作者】陈西浩;王秀娟【作者单位】(重庆大学材料科学与工程博士后流动站, 重庆 400030;(重庆大学物理学院, 重庆 400044;(重庆大学, 输配电装备及系统安全与新技术国家重点实验室, 重庆 400044;(重庆大学物理学院, 重庆 400044【正文语种】中文1 引言量子多体系统的量子相与相变一直是凝聚态物理研究的重要内容,尤其是自朗道自发对称破缺相变理论诞生以来,为了获得更多物相本质,人们引入了大量的理论模型.扩展的量子罗盘模型[1](quantum compass model,QCM)就是其中非常重要的一个.该模型可以描述过渡金属氧化物的电子输运特性和量子比特保护机制,特别是量子比特保护机制在量子信息技术中存在潜在应用[2,3].因此该模型的相图本性得到了广泛的研究.人们使用了多种解析和数值模拟方法对存在和不存在横场的QCM 进行了研究[4−16].文献[4—7]的结果表明:模型相变是系统内在引起的且两个无序相间是非连续相变；Sun和Chen[8]的自旋-自旋关联、保真度磁化率和块纠缠熵的有限尺寸标度研究结果支持连续相变的观点；而后Wang和Cho[9]的结果也支持这个观点；文献[10,11]综合了前两种观点,认为连续相边界与非连续相边界交叉处是多临界点.特别地,Su等[11]对该模型进行了全面的研究,得到了全面可信的结果(包含:能量、关联、纠缠、纠缠谱简并度、保真度、序以及临界指数).目前对于该模型的研究,人们把大量的精力都放在了相边界上,却鲜有人关注非局域长程拓扑弦关联序参量(以下简称弦关联序参量).本文在文献[11]的基础上,借助矩阵乘积态表示的无限虚时间演化块算法[17],详细地展示了如何从奇偶数弦关联导出弦关联序参量来研究该模型的量子相与相变的过程.要研究序参量,首先要明确的是序参量的概念.按照朗道自发对称破缺理论的理解,可以表述为:当系统发生自发对称性破缺时,系统就会存在一个局域序,而这个局域序可以用一个局域序参量来刻画[18].该序参量可以将局域相和其他相区分开,对于连续相变还应满足标度定律.然而长程拓扑相是一类新颖的相,超越了朗道连续相变理论,无法使用传统的Landau-Ginzburg-Wilson范式的两个基本概念:自发对称性破缺与局域序参量.但我们可以借助序参量的概念来定义弦关联序参量.局域序参量一般是根据局域磁化强度来确定的.弦关联序参量的定义可以参照以上思路,用弦关联来定义,需满足:1)系统需发生隐性对称破缺；2)该弦关联序参量可以唯一刻画区分对应的长程拓扑相区域；3)对于连续相变,在相变点附近满足标度定律.本文第2部分介绍了模型与相图；第3部分应用单位格点保真度重复确认了相图；第4部分详细介绍了如何导出弦关联序参量的过程,获得了序参量相图并提取了临界指数；第5部分是总结.2 模型扩展的QCM模型的哈密顿量如下:是在格点i上的泡利算符；J1,J2和L是算符间耦合强度.本文中参数L设定为1,作为相图标度.由(1)式知,当J1=0时该模型约化为QCM模型[9].若耦合强度J2=cosθ,L=sinθ,这里的θ是为了研究由J2和L构成的平面引入的幅度角.当θ变化时,系统依次出现x方向的单调奇弦关联Haldane相(−π/4＜θ＜π/4)、z方向的单调偶弦关联Haldane相(π/4＜θ＜3π/4)、x方向的振荡奇弦关联Haldane相(3π/4＜θ＜−3π/4)和z方向的振荡偶弦关联Haldane相(−3π/4＜θ＜−π/4).这些相都是长程拓扑相,没有局域序参量.当J2=0时该模型约化为键轮换的Ising模型[19].若耦合强度L=cosθ,J1=sinθ,当θ变化时,系统依次出现反铁磁相(０＜；θ＜π/2)、奇数反铁磁相(π/2＜θ＜π)、铁磁相(−π ＜θ＜−π/2)、偶数反铁磁相(−π/2＜θ＜0).这些相都是局域相,具有两重简并态.扩展的QCM示意相图已在文献[10,11]中给出.本文先进行了重复性测试(出于简洁考虑,图中仅展示了本文关注区间),如图1.相图被水平连续相边界和竖直非连续相边界分成了四个相区域,区域I和II是长程拓扑相,区域III和IV分别是条纹反铁磁与反铁磁相.连续相边界的中心荷c=1/2,临界指数β=1/8.本文的主要研究在棕色线(1)J2=2(1−J1)参数区间上的弦关联序参量与量子相变.棕色线上的三个样本参数点PII,PI和PIV对应的坐标分别是(J1,J2)=(−0.1,2.2),(0.3,1.4)和(0.6,0.8).此外,以下计算的波函数的截断维数都设定为χ=40.图1 扩展的量子罗盘模型示意相图Fig.1.Schematic phase diagram of the one-dimensional extended quantum compass model.3 保真度保真度是源于量子信息的概念,用于测量两个量子态的相似度；定义为两个量子态波函数的交叠.在矩阵乘积态表述下,保真度可定义为与控制参量J1和对应的波函数|ϕ(J1)⟩和|ϕ(⟩的交叠,即F(J1,=|⟩ϕ(J1)|ϕ(⟩|；其满足：1)归一性F(J1,J1)=1；2)有界性0 6 F(J1,6 1；以及3)对称性F(J1,=F(J,J1).此外,保真度和系统尺度L存在如下标度[20]:d(J1,)是单位格点保真度.这个定义近年来被引入到量子相变的研究,它能准确而方便地给出相变点位置[20].在图2中,我们计算了棕色线(1)J2=2(1−J1)上的参数区间的保真度.保真度平面存在两个挤点(相变点),图2(a)在(J1,=(0,0)发生非连续相变；非连续相变的传统解释是存在能级交叉,一阶导数不连续,则判定系统发生了非连续性相变.从单位格点保真度的角度而言,保真度表面存在着跳跃(不连续)即判定为非连续相变.出现这种跳跃的原因在于波函数内部结构严重的不一致.图2(b)在(J1,=(0.5,0.5)发生连续相变.连续相变的传统解释是能量的一阶导数连续,而二阶及以上导数不连续,则判定系统发生了连续相变.而单位格点保真度则认为只要保真度表面是连续变化的,就认为系统发生了连续相变.发生这种相变的原因在于:系统波函数的内部结构是连续变化的,变化的程度相对远远弱于非连续相变的情况.以上单位格点保真度结果和文献[11]结果一致.图2 单位格点保真度d(J1,随控制参量J1和的变化(a)非连续相变,在(J1,=(0,0)处；(b)连续相变,在(J1,=(0.5,0.5)处Fig.2.Ground-statefidelity per lattice sited(J1,as a function of control parameters J1and(a)discontinuous phasetransition at(J1,=(0,0)；(b)continuous phase transition at(J1,=(0.5,0.5).4 弦关联与量子相变扩展的QCM模型的自旋耦合强度取决于轮换的奇偶数键,奇偶数键上耦合强度的不同会体现在基态波函数的奇偶数键的纠缠的不同,奇偶数键纠缠的不同必然导致所有格点有“二聚化”[21,22]偏向,奇数和偶数格点就会有差异.基于此,我们假设奇偶数弦关联函数可以用来区分长程拓扑相,并采用了如下弦关联函数定义[9,23−28]:这里的α可以是x,y和z；是在格点i处自旋1/2的泡利算符.数值模拟结果表明弦关联存在三种截然不同的行为:衰减为零、单调饱和与振荡饱和.4.1 偶弦关联行为在图3中,我们计算了代表点PII、PI和PIV的偶弦关联的短程行为.在x和y方向,PII,PI和PIV的偶弦关联都衰减为0；在z方向,PII和PI的偶弦关联衰减为0,而PIV的偶弦关联表现为单调负值.图3 偶弦关联函数O,O和O随格点间距|i−j|的变化行为Fig.3. Even string correlations O,Oand Oas a function of the distance of the lattice sites|i−j|.4.2 奇弦关联行为在图4中,类似偶弦关联,我们也计算了代表点PII,PI和PIV的奇弦关联行为.图4(a)点PII奇弦关联O是单调负值,O和O是振荡值.图4(b)点PI的奇弦关联O,O和Osz都是odd单调负值.图4(c)点PIV的奇弦关联O和O非常快地衰减为0,O是单调负值.内插图展示的是奇弦关联的非0长程饱和行为,确保了奇弦关联的非平庸性.自然地,用奇弦关联定义的奇弦关联序参量也就具有了非平庸性.图4 奇弦关联O,O和O随格点间距|i−j|变化行为,内插图展示的是饱和行为Fig.4. Behaviors of the odd string correlations with the distance of the lattice sites|i−j|.The insets show the saturation behaviors.4.3 弦关联序参量相图为了更深刻地理解系统发生的量子相变,需要找到系统的序参量(当系统处于长程拓扑相时应找到弦关联序参量).4.1和4.2节的结果表明:1)偶弦关联,在x和y方向,不管是在长程拓扑相区域还是局域相区域,偶弦关联都衰减为0,这是平庸的,无法用作弦关联序参量的定义;在z方向,在局域相区域里,偶弦关联是单调负值,这和该区域存在明显的局域序相违,所以偶弦关联无法用作弦关联序参量的定义；2)对于奇弦关联,1在x方向,在长程拓扑相区域中都表现出单调负值,在局域相区域,衰减为0；在两个长程拓扑相区域中,偶弦关联行为一致；更为特别的是相图1中的相变点(J1/L,J2/L)=(0,2),和文献[9]中的深度处于长程拓扑序中的点(Jy,Jz)=(8,4)等价,也就是说相变点(J1/L,J2/L)=(0,2)深度处于单调O长程序中,故x方向的奇弦关联在整个长程拓扑区域(含相变点)都具有O长程拓扑序,不可用于弦关联序参量的定义；2在y方向,在长程拓扑相区域II中,奇弦关联表现为振荡行为,即+−+−......+−+−；而在区域I中表现为单调负值,即−−−−......−−−−；在局域相区域IV中衰减为0；3在z方向,奇弦关联一直存在,不可用作弦关联序参量的定义.综上,弦关联序参量仅可采用y方向奇弦关联来定义.结合示意相图特性,可考虑如下两种类型奇弦关联序参量定义:但需要注意弦关联序参量是按照格点间距|i−j|来定义的,理论上来说该间距可做到无穷大,这是有限的时间和计算资源所不允许的,通常人们会设定一个误差范围(本文设定的误差范围是ε＜1e−8),当达到这个误差范围时就认为该弦关联值就是弦关联序参量的值.从相邻相的区分意义上,Wang和Cho[9]成功地找到了一组弦序参量；而弦关联的线性组合弦序参量：适用于全局参数区间.图5给出了序参量相图,包含y方向的振荡奇弦关联序相(J1＜0)、y方向的单调奇弦关联序相(＜；J1＜0.5)、和反铁磁相(J1＞0.5).单调(振荡)奇(偶)弦关联序参量是Hida等[23]在研究轮换交替作用的海森伯模型时,为了描述偶弦关联Haldane相和二聚相共存的情况而引入.而后Wang和Cho[9]为了刻画区分四种不同的Haldane相而引入到QCM模型中.扩展的QCM模型中虽有轮换交替耦合,却不存在二聚相,但轮换交替耦合确实会导致基态波函数存在“二聚化”偏向.故在引入单调(振荡)奇(偶)弦关联时是假设它们可以用于长程拓扑相区域刻画.经过数值证实,由“二聚化”偏向引入的单调(振荡)奇弦关联序参量可以很好地刻画区分长程拓扑相区域.y方向振荡奇弦关联序参量和单调奇弦关联序参量值非0表明在这两个子区间发生了隐性的Z2×Z2对称破缺.随着控制参量J1增大到0.5后,它打破了原来的长程序,系统发生自发对称破缺,进入反铁磁序.磁化强度表现为符号相反、大小相等.图5 以J1为控制参量的基态序参量相图Fig.5. The ground-state phase diagram of order parameters as a function of control parameter J1.4.4 临界指数图6 (a)βO；(b) β；图标是数据,实线为拟合曲线Fig.6. (a)β；(b) β.The icons are data.The solid line is fitting line.临界指数是研究系统临界性质的参数.提取临界指数能从更深的层次上揭示相变的性质.图6展示了渐近临界点J=0.5过程中,y方向的单调奇弦关联序参量和局域磁化强度随着|J1−J|的幂次律变化关系.它们的幂次律拟合方程分别为O=|J1 −|2β 和|⟩|=|J1 − J|β.拟合临界指数βO=0.1236 和β|⟩σ⟩|=0.1246都趋近于准确值β=1/8表明:该处发生相变的普适类为Ising类型.5 结论本文应用矩阵乘积态表述的无限虚时间演化块算法,研究了扩展的QCM模型的长程拓扑相和量子相变.为了深入研究该模型的长程拓扑相和量子相变,按照奇数和偶数键,本文引入了奇数弦关联和偶数弦关联.弦关联表现出三种截然不同的行为:衰减为零、单调饱和与振荡饱和.基于弦关联的以上特征导出的y方向的振荡 (单调)奇弦关联序参量可描述该模型的长程拓扑相和量子相变.y方向的振荡(单调)奇弦关联序参量的非零值表明扩展的QCM发生了隐性的Z2×Z2对称破缺.在临界点J=0.5附近,局域磁化强度和单调奇弦关联序参量的临界指数β=1/8表明相变的普适类是Ising类型.此外,保真度探测到的相变点、连续性与非连续性和序参量结果一致.以上方法对长程拓扑相和量子相变的研究具有指导意义.参考文献【相关文献】[1]Kugel K I,Khomskii D I 1973 Zh.Eksp.Teor.Fiz.64 1429[2]Doucot B,Feigelán M V,IoffeL B,Ioselevich A S 2005 Phys.Rev.B 71 024505[3]Milman P,Maineult W,Guibal S,Guidoni L,Douot B,Ioffe L,Coudreau T 2007Phys.Rev.Lett.99 020503[4]Brzezicki W,Dziarmaga J,Olés A M 2007 Phys.Rev.B 75 134418[5]You W L,Tian G S 2008 Phys.Rev.B 78 184406[6]Brzezicki W,Olés A M 2009 Acta Phys.Pol.A 115 162[7]Sun K W,Zhang Y Y,Chen Q H 2009 Phys.Rev.B 79 104429[8]Sun K W,Chen Q H 2009 Phys.Rev.B 80 174417[9]Wang H T,Cho S Y 2015 J.Phys.:Condens.Matter 27 015603[10]Eriksson E,Johannesson H 2009 Phys.Rev.B 79 224424[11]Liu G H,Li W,You W L,Tian G S,Su G 2012 Phys.Rev.B 85 184422[12]Wang L C,Yi X X 2010 Eur.Phys.J.D 77 281[13]Jafari R 2011 Phys.Rev.B 84 035112[14]Motamedifar M,Mahdavifar S,Shayesteh S F 2011 Eur.Phys.J.B 83 181[15]You W L 2012 Eur.Phys.J.B 85 83[16]Liu G H,Li W,You W L 2012 Eur.Phys.J.B 85 168[17]Vidal G 2007 Phys.Rev.Lett.98 070201[18]Zhou H Q 2008 arXiv:0803.0585v1[cond-mat.stat-mech][19]Wang H T,Cho S Y,BatchelorM T 2015 arXiv:1508.01316[quant-ph][20]Zhou H,Barjaktarevi J P 2008 J.Phys.A:Math.Theor.41 412001[21]Su Y H,Chen A M,Wang H L,Xiang C H 2017 Acta Phys.Sin.66 120301(in Chinese)[苏耀恒,陈爱民,王洪雷,相春环2017物理学报66 120301][22]Kennedy T,Tasaki H 1992 Phys.Rev.B 45 304[23]Hida K 1992 Phys.Rev.B 45 2207[24]Chen X H,Cho S Y,Zhou H Q 2016 J.Korean Phys.Soc.68 1114[25]Wang H T,Li B,Cho S Y 2013 Phys.Rev.B 87 054402[26]Hatsugai Y 2007 J.Phys.:Condens.Matter 19 145209[27]Pollmann F,Berg E,Turner A,Oshikawa M 2012 Phys.Rev.B 85 075125[28]Su Y H,Cho S Y,Li B,Wang H L,Zhou H Q 2012 J.Phys.Soc.Jpn.81 074003。
科技英语答案分析解析
I. Mark the following statements with T (true) or F (false) according to the passage
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( F ) 1. Today’s industries cannot survive worldwide competition unless they introduce new products with better QCD. ( T ) 2. The manufacturing process starts from the design specifications and ends with shipping of the actual products. ( F ) 3. Most of the information generated and handled in the synthesis subprocess is quantitative. ( T ) 4. Once a design has been completed, after optimization or some tradeoff decision, the design evaluation phase begins. ( T ) 5. The outcome of process planning is a production plan, a materials order, and machine programming.
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2. The product cycle is composed of two main processes：the design process and the manufacturing process．
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Tomoyasu Ichihara2 Department of Physics Tokyo Institute of Technology Oh-okayama, Meguro-ku Tokyo 152, JAPAN
Abstract We analyze the properties of mesons in 1+1 dimensional QCD with bosonic and fermionic “quarks” in the large Nc limit. We study the spectrum in detail and show that it is impossible to obtain massless mesons including boson constituents in this model. We quantitatively show how the QCD mass inequality is realized in two dimensional QCD. We find that the mass inequality is close to being an equality even when the quarks are light. Methods for obtaining the properties of “mesons” formed from boson and/or fermion constituents are formulated in an explicit manner convenient for further study. We also analyze how the physical properties of the mesons such as confinement and asymptotic freedom are real43;1 dimensional QCD with fundamental bosons and fermions
arXiv:hep-th/9506058v1 9 Jun 1995
Kenichiro Aoki1 Department of Physics, Hiyoshi Campus Keio University Hiyoshi, Kouhoku-ku Yokohama 223, JAPAN
will study the case when the quarks are heavy analytically. For mesons involving light quarks, we establish a number of results analytically and further analyze the problem numerically. The numerical approach will be formulated explicitly in all the three ff, bf and bb cases which should be useful for further study. We use these results to see how the QCD mass inequality[11] [12] applies to two dimensional QCD in the large Nc limit. Given two different types of quarks, the QCD mass inequality states that the meson made from the same quarks is on average lighter than the meson made from different quarks. This non–trivial inequality, however, does not tell us by how much these meson masses differ, a question which we are able to answer analytically in some cases and numerically for all quark masses. Also, while there is no reason to doubt this important inequality, as was pointed out in the original articles themselves, the inequality is not completely rigorous. We find it satisfying to be able to study how the inequality is realized in a simplified version of QCD. To our knowledge, the QCD mass inequality has not been previously shown to be satisfied in such a concrete manner. Here, the inequality is applied to ff, bf and bb cases. Except in the bf case, the inequality is necessarily an equality when the constituent quark masses are the same. In the bf case, such needs not be the case and indeed we find that it is always a strict inequality in the bf case. The paper is organized as follows. In section 2, we briefly review the integral equation satisfied by the wave function of the meson in the large Nc limit, partly to fix the notation. In section 3, we analytically study the static properties of mesons when the quark masses are large. We further formulate the bound state equation as a linear eigenvalue problem for general values of the quark masses and analyze the problem numerically. In sect 4, we analyze some matrix elements and see how confinement and asymptotic freedom is realized in the bf case. We conclude with a brief discussion.
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email: a00500@cc.hc.keio.ac.jp email: tomo@th.phys.titech.ac.jp
1. Introduction In confining gauge theories, physically observable particles at low energies have no gauge charge and are bound states of the charged matter that appear in the gauge theory Lagrangian. A prototype of such a theory is QCD wherein the gauge group is SU(3) and the charged matter fields are vector fermions in the fundamental representation. Several natural generalizations of QCD exist; we may use gauge groups other than SU(3), we may use chiral fermions or we may choose representations more complicated than the fundamental representation for the matter fields. In this case, we cannot put in too many or too large representations if we want to preserve asymptotic freedom. Another generalization is to consider boson constituents as well as fermion constituents. It is this last, as well as the first generalization in two dimensions in the large Nc limit that we shall investigate in this work. The possibility of boson constituents arise necessarily in several contexts, such as technicolor with multiple stages of symmetry breaking[1], QCD or technicolor with supersymmetry[2], so called bosonic technicolor[3] as well as the Standard Model[4]. When both boson and fermion constituents exist in the theory, “meson” states of both Bose and Fermi statistics may arise. In general, it is difficult to derive the properties of the bound states from first principles. By using the large Nc limit in two dimensions, we may analyze the properties of these meson states concretely. Two dimensional QCD in the large Nc limit has greatly contributed to our current understanding of the gauge theory dynamics by providing us with a model where the properties are explicitly calculable analytically. Also, two dimensional QCD has been used to test the validity of various approaches and approximation schemes applied to QCD. The model was first solved by ’t Hooft [5] and some further physical properties such as some current matrix elements, the asymptotic freedom of mesons were studied in some subsequent works [6][7]. The formulation was extended to include boson–boson bound states [8][9] and boson–fermion bound sates[10]. Mesons made only from fermionic quarks or the bosonic quarks obey Bose statistics but the boson–fermion bound state obeys Fermi statistics. (Hereafter, often referred to as ff, bb and bf cases.) In this work, we shall extend the investigation of the physical properties of the mesons made from fermions and generalize the results to the mesons made from bosons only and bosons and fermions. The results will enable us to compare the three cases and see the differences and similarities that arise between mesons made from constituents of various statistics. The spectrum of mesons is investigated both analytically and numerically. We 1