Generalized Gauged Thirring Model on Curved Space-Times

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T.F. Cootes, Active Appearance Models

T.F. Cootes, Active Appearance Models

tracking a single deforming object we match a model which can fit a whole class of objects.
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BACKGROUND
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1 INTRODUCTION
THE ªinterpretation through synthesisº approach has received
AbstractÐWe describe a new method of matching statistical models of appearance to images. A set of model parameters control modes of shape and gray-level variation learned from a training set. We construct an efficient iterative matching algorithm by learning the relationship between perturbations in the model parameters and the induced image errors. Index TermsÐAppearance models, deformable templates, model matching.
Several authors have described methods for matching deformable models of shape and appearance to novel images. Nastar et al. [12] describe a model of shape and intensity variations using a 3D deformable model of the intensity landscape. They use a closest point surface matching algorithm to perform the fitting, which tends to be sensitive to the initialization. Lanitis et al. [11] and Edwards et al. [3] use a boundary finding algorithm (an ªActive Shape Modelº) to find the best shape, then use this to match a model of image texture. Poggio et al. [6], [9] use an optical flow algorithm to match shape and texture models iteratively, and Vetter [14] uses a general purpose optimization method to match photorealistic human face models to images. The last two approaches are slow because of the high dimensionality of the problem and the expense of testing the quality of fit of the model against the image. Fast model matching algorithms have been developed in the tracking community. Gleicher [7] describes a method of tracking objects by allowing a single template to deform under a variety of transformations (affine, projective, etc.). He chooses the parameters to minimize a sum of squares measure and essentially precomputes derivatives of the difference vector with respect to the parameters of the transformation. Hager and Belhumeur [8] describe a similar approach, but include robust kernels and models of illumination variation. Sclaroff and Isidoro [13] extend the approach to track objects which deform, modeling deformation using the low energy modes of a finite element model of the target. The approach has been used to track heads [10] using a rigid cylindrical model of the head. The Active Appearance Models described below are an extension of this approach [4], [1]. Rather than tracking a particular object, our models of appearance can match to any of a class of deformable objects (e.g., any face with any expression, rather than one persons face with a particular expression). To match rapidly, we precompute the derivatives of the residual of the match between the model and the target image aபைடு நூலகம்d use them to compute the update steps in an iterative matching algorithm. In order to make the algorithm insensitive to changes in overall intensity, the residuals are computed in a normalized reference frame.

Abstract Integrating objective and subjective hazard risk in decision-aiding system design

Abstract Integrating objective and subjective hazard risk in decision-aiding system design
Keywords: Alerting; Probabilistic modeling; Hazards; Decision-aids; Decision theory; Risk
1. Introduction Real-time decision aiding and alerting systems are often used to assist human operators in controlling processes ef®ciently and in preventing undesirable incidents from occurring (such as a collision in a vehicle control application, or exceeding temperature limits in process control). There are many types of real-time decision-aids, ranging from process status displays, to planning tools, to safety- and time-critical warning systems. To date, warning systems have generally been restricted to cases in which there is a clear de®nition of hazardous states. For example, traf®c collision risk can be de®ned in concrete, objective terms (e.g. no closer than 100 m separation between aircraft), which then is translated into algorithms and decision thresholds. This can be classi®ed as a case of objective assessment of hazard risk. Due to sensor and prediction errors, there may still be uncertainty in whether a decision to change the process' trajectory is needed. These uncertainties, however, can also be objectively estimated and used when de®ning decision thresholds to balance false alarms and missed detections, and optimize system performance from the human operator's perspective. In cases in which the distinction between hazard and nonhazard is less distinct (i.e. the hazard risk is subjective), decision-aids typically display the state information but leave the decision-making to the human operator. Aviation

Baryogenesis and phase transition in the standard model

Baryogenesis and phase transition in the standard model

a rXiv:h ep-th/96837v17A ug1996Baryogenesis and phase transition in the standard model D.Karczewska 1R.Ma´n ka 2Department of Astrophysics and Cosmology,University of Silesia,Uniwersytecka 4,40-007Katowice,Poland ABSTRACT The sphaleron type solution in the electroweak theory,generalized to include the dilaton field,is examined.The solutions describe both the variations of Higgs and gauge fields inside the sphaleron and the shape of the dilaton cloud surrounding the sphaleron.Such a cloud is large and extends far out-side.These phenomena may play an important role during the baryogenesis which probably took place in the Early Universe.1.INTRODUCTION In this paper the electroweak theory will be extended by the inclusion of dilatonic field.The Glashow-Weinberg-Salam dilatonic model with SU L (2)×U Y (1)symmetry is described by the lagrangian L =−14e 2ϕ(x )/f B µνB µν+12igW a µσa −12W a µσa arerespectively local gauge fields associated with U Y (1)and SU L (2)symmetrygroups.Y denotes the hypercharge.The gauge group is a simple product of U Y (1)and SU L (2)hence we have two gauge couplings g and g ′.The generators of gauge groups are:a unit matrix for U Y (1)and Pauli matrices for SU L (2).In the simplest version of the standard model a doublet of Higgsfields is introduced H = H +H 0 = 012v ,with the Higgs potentialU (H +,H,ϕ)=λ H +H −1√rdescribe the hedgehog structure.Thisproduces a nontrivial topological charge of the sphaleron.The topological charge is equal to the Chern-Simons number.Such a hedgehog structure determines the asymptotic shape of the sphaleron with gauge fields different from zero W a i =ǫaij n j 1−s (r )2f 2 d ′(r )2v 20h ′(r )2−14r 2v 20(3−s (r ))2h (r )2−1r 2(3−4s (r )+s (r )2)2+2s ′(r )2 −12 (5)1Then we switch to dimensionless variables x=M W r=r/r W,where M2W= 1M W∼10−18cm.The resulting Euler-Lagrange equa-tions are following:s(x)function,which describes the gaugefield in theelectroweak theory and satisfies the equations′′(x)+2d(x)′d(x)2(3−s(x))+1x +1M2W(d(x)2−h(x)2)h(x)−8C2x2(s(x)−3)2h(x)=0,(7) where M2H=2λv20determines the Higgs mass.The d(x)function describing the dependence of a dilatonfield on x in extended electroweak theory obeys the equation:d′′(x)+2d(x)+M2Hx4(s(x)−3)2(s(x)−1)2d(x)3+2λv40(d(x)2−h(x)2)d(x)3+4gf )2∼10−9.Thispractically means that the dilatonfield is a freefield.The simplest solutions h(x)=1(shown in Fig.1),s(x)=3(Fig.2),d(x)=1(Fig.3)are global ones corresponding to the vacuum with broken symmetry in the standard model.It is obvious that far from the center of the sphaleron our solutions should describe the normal broken phase which is very well known from the standard model.Knowing the asymptotic solutions we are able to construct a two-parameter family of solutions(for details see[5]):s(x)=1+2tanh2(tx)(9)h(x)=tanh(ux)(10)2d(x)=a+(1−a)tanh2(kx)(11) where t,u,a,k are parameters to be determined by the variational procedure. The relevant values of the parameters are those which minimize the energy. For example,with the standard values of M W=80.6GeV,M Z=91.16GeV, M H=350GeV we found the numeric solutions t,u,k,as functions depend-ing on the initial conditions of the dilatonfield d(0)=a in the center of the sphaleron.Our solutions describe both the behavior of Higgsfield and gauge field inside the sphaleron and the shape of the dilaton cloud surrounding the sphaleron.Such a cloud is large and extends far outside the sphaleron. The sphalerons are created during thefirst order phase transition in the ex-panding universe as inhomogeneous solutions of the motion equations.These phase transition bubbles,which probably took place in the early universe, break the CP and C symmetry on their walls and can cause the breaking of baryonic symmetry.Detailed consideration of this problem will be the subject of a separate paper.3.CONCLUSIONSNumerical solutions suggest that sphaleron possess an‘onionlike’structure. In the small inner core the scalarfield is decreasing with global gauge symme-try restoration SU(2)×U(1).In the middle layer the gaugefield undergoes sudden change.The sphaleron coupled to dilatonfield has also an outer shell, where dilatonfield changes drastically.The spherically symmetric dilaton so-lutions coupled to the gaugefield or gravity are interesting in their own rights and may further influence the monopole catalysis of baryogenesis induced by sphaleron.This paper is sponsored by the Grant KBN2P30402206. References[1]A.Riotto,Phys.Rev.D49,730-738,(1994),Singlet Majoron model withhidden scale invariance.[2]R.Manka,D.Karczewska,Z.Phys.C57,417-420,(1993),The neutrinoball in the standard model.[3]R.Manka I.Bednarek D.Karczewska,Phys.Scr.52,36-40,(1995),On theneutrino ball model.3[4]B.Kleihaus,J.Kunz,Y.Brihaye,Phys.Lett.B273,(1991)100.[5]R.Manka D.Karczewska,Baryogenesis in the dilatonic electroweak the-ory,preprint U´SL-TH-96-2.FIGURE CAPTIONSFigure1.The dependence of the Higgsfield h(x)on x.Figure2.The dependence of the gaugefield s(x)on x.Figure3.The dependence of the dilatonfield d(x)on x.4xxx。

Abelian projection and studies of gauge-variant quantities in lattice QCD without gauge fix

Abelian projection and studies of gauge-variant quantities in lattice QCD without gauge fix

a rXiv:h e p-la t/961129v127Nov1996Abelian projection and studies of gauge-variant quantities in the lattice QCD without gauge fixing Sergei V.SHABANOV 1Institute for Theoretical Physics,Free University of Berlin,Arnimallee 14,WE 2,D-14195,Berlin,Germany Abstract We suggest a new (dynamical)Abelian projection of the lattice QCD.It contains no gauge condition imposed on gauge fields so that Gribov copying is avoided.Configurations of gauge fields that turn into monopoles in the Abelian projection can be classified in a gauge invariant way.In the continuum limit,the theory respects the Lorentz invariance.A similar dynamical reduction of the gauge symmetry is proposed for studies of gauge-variant correlators (like a gluon propagator)in the lattice QCD.Though the procedure is harder for numerical simulations,it is free of gauge-fixing artifacts,like the Gribov horizon and copies.1.One of the important features of the QCD confinement is the existence of a stable chromoelectrical field tube connecting two color sources (quark and antiquark).Numerical studies of the gluon field energy density between two color sources leave no doubt that such a tube exists.However,a mechanism which could explain its stability is still unknown.It is believed that some specific configurations (or excitations)of gauge fields are re-sponsible for the QCD confinement,meaning that they give a main contributions to the QCD string tension.Numerical simulations of the lattice QCD shows that Abelian (com-mutative)configurations of gauge potentials completely determine the string tension in the full non-Abelian gauge theory [1].This phenomenon is known as the Abelian dominance.Therefore one way of constructing effective dynamics of the configurations relevant to the QCD confinement is the Abelian projection [2]when the full non-Abelian gauge group SU(3)is restricted to its maximal Abelian subgroup (the Cartan subgroup)U(1)×U(1)by a gauge fixing.Though dynamics of the above gauge field configuration cannot be gauge dependent,a right choice of a guage condition may simplify its description.There is a good reason,supported by numerical simulations [3],[4],to believe that the sought configurations turn into magnetic monopoles in the effective Abelian theory,and the confinement can be due to the dual mechanism [5]:The Coulomb field of electric charges is squized into a tube,provided monopole-antimonopole pair form a condensate like the Cooper pairs in superconductor.It is important to realize that the existence of monopoles in the effective Abelian theory is essentially due to the gauge fixing,in fact,monopoles are singularities of thegaugefixing.Note that monopoles cannot exist as stable excitations in pure gauge the-ory with simply connected group like SU(3).Since the homotopy groups of SU(3)and of U(1)×U(1)are different(the one of SU(3)is trivial),a gauge condition restricting SU(3)to U(1)×U(1)should have singularities which can be identified as monopoles[2].A dynamical question is to verify whether all configurations of non-Abelian gaugefields relevant to the confinement(in the aforementioned sense)are”mapped”on monopoles of the Abelian theory(the monopole domimance[4]).It appears that monopole dynam-ics may depend on the projection recipe[6].There are indications that some Abelian projections exhibit topological singularities other than magnetic monopoles[7].Though the lattice QCD is,up to now,the only relible tool for studying monopole dynamics,the true theory must be continuous and respect the Lorentz invariance.In this regard,Abelian projections based on Lotentz invariant gauge conditions play a dis-tinguished role.For example,the gauge can be chosen as follows D HµA offµ=0where D Hµ=∂µ+igA Hµ,A Hµare Cartan(diagonal)components of guage potentials Aµ,while A offµare its non-Cartan(off-diagonal)components.This gauge restricts the gauge sym-metry to the maximal Abelian(Cartan)subrgoup and is manifestly Lorentz invariant. The lattice version of the corresponding Abelian theory is known as the maximal Abelian projection.The above homotopy arguments can be implemented to this gauge to show that it has topological singularities and Gribov’s copying[9](in the continuum theory, zero boundary conditions at infinity have to be imposed[10]).The Gribov copying makes additional difficulties for describing monopole dynamics(even in the lattice gluodynamics [11]).In this letter,a new(dynamical)Abelian projection is proposed.It involves no gauge condition to be imposed on gaugefields.The effective Abelian theory appears to be non-local,though it can be made local at the price of having some additional(ghost)fields.All configurations of gaugefields that turn into magnetic monopoles in the effective Abelian theory are classified in a gauge invariant way.The effective Abelian theory fully respects the Lorentz symmetry and the Gribov problem is avoided.Another important aspect of the QCD confinement is the absence of propagating color charges,meaning that a nonperturbative propagator of colored particles,gluons or quarks, has no usual poles in the momentum space.It has been argued that such a behavior of a gluon propagator in the Coulomb gauge could be due to an influence of the so called Gribov horizon on long-wavefluctuations of gaugefields[9],[12].The result obviously depends on the gauge chosen,which makes it not very reliable.The situation looks more controversial if one recalls that a similar qualitative behavior of the gluon propagator has been found in the study of Schwinger-Dyson equations[13]. In this approach,the Gribov ambiguities have not been accounted for.So,the specific pole structure of the gluon propagator occurred through a strong self-interaction of gauge fields.In this letter,we would also like to propose a method for how to study gauge-variant quantities,like a gluon propagator,in the lattice QCD,avoiding any explicit gaugefixing. The method is,hence,free of all the aforementioned gaugefixing artifacts.It gives a hope that dynamical contributions(self-interaction of gaugefields)to the pole structure of the gluon propagator can be separated from the kinematical(gauge-fixing)ones.2.To single out monopoles in non-Abelian gauge theory,onefixes partially a gauge so that the gauge-fixed theory possesses an Abelian gauge group being a maximal Abelian subgroup of the initial gauge group.The lattice formulation of the Abelian projection has been given in[8].The idea is to choose a function R(n)of link variables Uµ(n),n runs over lattice sites, such thatR(n)→g(n)R(n)g−1(n)(1) under gauge transformations of the link variablesUµ(n)→g(n)Uµ(n)g−1(n+ˆµ),(2) where g(n)∈G,G is a compact gauge group,andˆµis a unit vector in theµ-direction.A gauge is chosen so that R becomes an element of the Cartan subalgebra H,a maximal Abelian subalgebra of a Lie algebra X of the group G.In a matrix representation,the gauge condition means that off-diagonal elements of R are set to be zero.Clearly,the gaugefixing is not complete.A maximal Abelian subgroup G H of G remains as a gauge group because the adjoint action(1)of G H leaves elements R∈H untouched.A configuration Uµ(n)contains monopoles if the corresponding matrix R(n)has two coinciding eigenvalues.So,by construction,dynamics of monopoles appears to be gauge-dependent,or projection-dependent.It varies from gauge to gauge,from one choice of R to another[6].Yet,the monopole singularities are not the only ones in some Abelian projections[7].In addition,Abelian projections may suffer offthe Gribov ambiguities [11].To restrict the full gauge symmetry to its maximal Abelian part and,at the same time, to avoid imposing a gauge condition on link variables,we shall use a procedure similar to the one discussed in[14]in the framework of continuumfield theory.A naive continuum limit of our procedure poses some difficulties.To resolve them,a corresponding operator formalism has to be developed.It has been done in[15]for a sufficiently large class of gauge theories.Consider a complex Grassmannfieldψ(n)(a fermion ghost)that realizes the adjoint representation of the gauge group:ψ(n)→g(n)ψ(n)g−1(n),(3)ψ∗(n)→g(n)ψ∗(n)g−1(n).(4) Let the fermion ghost be coupled to gaugefields according to the actionS f= n,µtrDµψ∗(n)Dµψ(n),(5)where Dµψ(n)=ψ(n+ˆµ)−U−1µ(n)ψ(n)Uµ(n)is the lattice covariant derivative in the adjoint representation.We assume thatψ(n)=ψi(n)λi,whereλi is a matrix represen-tation of a basis in X normalized as trλiλj=δij,andψi(n)are complex Grassmann variables.The partition function of the fermion ghostfield readsZ f(β)= n(dψ∗(n)dψ(n))e−βS f=detβD TµDµ,(6)where the integration over Grassmann variables is understood,and D Tµdenotesa trans-position of Dµwith respect to a scalar product induced by n,µtr in(5).Note that the action(5)can be written in the form S f= ψ∗D TµDµψ.Consider a pair of real Lie-algebra-valued scalarfieldsϕ(n)andφ(n)(boson ghosts) with an actionS b=1(2π)dim G e−βS b=(detβD TµDµ)−1.(10) We have the identityZ b(β)Z f(β)=1.(11) By making use of this identity,the partition function of gaugefields can be transformed to the formZ Y M(β)=v−L G µ,n dUµ(n)e−βS W Z b(β)Z f(β)=(12)=v−L G D UµDψ∗DψDϕDφe−β(S W+S b+S f),(13)where S W is the Wilson action of gaugefields,v G a volume of the group manifold G,L a number of lattice sites,and D denotes a product of correspondingfield differentials over lattice sites.The effective actionS eff=S W+S b+S f(14) is invariant under gauge transformations(2)–(4)and(8),(9).The factor v−L G is included to cancel the gauge group volume factorizing upon the integration overfield configurations in(13).Now we may take the advantage of having scalarfields in the adjoint representation and restrict the gauge symmetry to the Cartan subgroup without imposing gauge conditions on the link variables.We make a change of the integration variables in(13)φ(n)=˜g(n)h(n)˜g(n)−1,(15) where˜g(n)belongs to the coset space G/G H,dim G/G H=dim G−dim G H,and h(n)∈H.Other newfields denoted˜Uµ(n),˜ϕand˜ψ∗,˜ψare defined as the corresponding gauge transformations of the initialfields with g(n)=˜g−1(n).No restriction on their values is imposed.Relation(15)determines a one-to-one correspondence between old and new variables if and only if˜g(n)∈G/G H and h(n)∈K+,where K+is the Weyl chamber in H.An element h of the Cartan subalgebra H belongs to the Weyl chamber K+⊂H if for any simple rootω,(h,ω)>0;(,)stands for an invariant scalar product in X.In a matrix representation of X,it is proportional to tr(see[16],pp.187-190).With the help of the adjoint transformation,any element of a Lie algebra can be brought to the Cartan subalgebra.Since the Cartan subalgebra is invariant under the adjoint action of the Cartan subgroup,˜g(n)must be restricted to the coset G/G H.There are discrete transformations in G/G H which form the Weyl group W[16].Any element of W is a composition of reflections in hyperplanes orthogonal to simple roots in H.Its action maps H onto H itself.The Weyl group is a maximal isomorphism group of H[16].Therefore, a one-to-one correspondence in(15)is achieved if h(n)∈H/W≡K+.Due to the gauge invariance of both the measure and exponential in(13),the integral over group variables˜g(n)is factorized and yields a numerical vector that,being divided by v L G,results in(2π)−Lr,r=dim H=rank G.This factor is nothing but a volume of the Cartan gauge group G H.The integration over h(n)inquires a nontrivial measure,and the integration domain must be restricted to the Weyl chamber K+.So,in(13)we havev−1G dφ(n)=(2π)−r K+dh(n)µ(n).(16) The measure has the form[17]µ(n)= α>0(h(n),α)2,(17)whereαranges all positive roots of the Lie algebra X.The Cartan subalgebra is isomor-phic to an r-dimensional Euclidean space.The invariant scalar product can be thought as an ordinary vector scalar product in it.Relative orientations and norms of the Lie algebra roots are determined by the Cartan matrix[16].The integration measure for the otherfields remains unchanged.For example,G=SU(2),then r=1,µ=h2(n)where h(n)is a real number because H SU(2)is isomorphic to a real axis.The Weyl chamber is formed by positive h(n).The su(3)algebra has two simple rootsω1,2(r=2).Their relative orientation is determined by the Cartan matrix,(ω1,ω2)=−1/2,|ω1,2|=1.The Weyl chamber is a sector on a plane(being isomorphic to H SU(3))with the angleπ/3.The algebra has three positive rootsω1,2andω1+ω2.So,the measure(17)is a polynom of the sixth order.Its explicit form is given by(28).Thefield h(n)is invariant under Abelian gauge transformationsg H(n)h(n)g−1H(n)=h(n),g H(n)∈G H.(18) Therefore,after integrating out the coset variables˜g(n)in accordance with(16),we represent the partition function of Yang-Mills theory as a partition function of the effective Abelian gauge theoryZ Y M(β)=(2π)−Lr D˜Uµe−βS W F(˜U),(19)whereF(˜U)=(detβD TµDµ)1/2 K+ n(dh(n)µ(n))e−βS H,(20)S H=1/2 n,µtr h(n+ˆµ)−˜U−1µ(n)h(n)˜Uµ(n) 2.(21)To obtain(19),we have done the integral over both the Grassmann variables and the boson ghostfield˜ϕ(n),which yields(detβD TµDµ)1/2.The function F(˜U)is invariant only with respect to Abelian gauge transformations,˜Uµ(n)→g H(n)˜Uµ(n)g−1H(n+ˆµ).It provides a dynamical reduction of the full gauge group to its maximal Abelian subgroup.Since no explicit gauge condition is imposed on the link variables˜Uµ(n),the theory do not have usual gaugefixing deceases,like the Gribov copies or horizon.We shall call the Abelian projection thus constructed a dynamical Abelian projection.3.Making a coset decomposition of the link variables[8]˜Uµ(n)=U Hµ(n)U chµ(n),(22) where U Hµ(n)=exp u Hµ(n),u Hµ(n)∈H and U chµ(n)=exp u chµ(n),u chµ(n)∈X⊖H,we conclude that lattice Yang-Mills theory is equivalent to an Abelian gauge theory with the actionS A=S W−β−1ln F.(23) The link variables U chµ(n)play the role of chargedfields,while U Hµ(n)represents”electro-magnetic”fields.In the naive continuum limit,U Hµbecome Abelian potentialsU Hµ(n)→exp n+ˆµndxµA Hµ,A Hµ∈H.(24)Note that thefield h(n)carries no Abelian charge and does not interact with U Hµas easily seen from(22)and(21)because(U Hµ)−1(n)h(n)U Hµ(n)=h(n).Bearing in mind results on simulations of the Polyakov loop dynamics on the lattice, one should expect that the Coulombfield of charges in the effective Abelian theory is squeezed into stable tubes connecting opposite charges.A mechanism of the squeezing has to be found from a study of dynamics generated by(23).First,one should verify if the dual mechanism can occur in the effective Abelian theory.In our approach,configurations U Hµ(n)containing monopoles can exist.Kinematical arguments for this conjecture are rather simple.Let G be SU(N).In a matrix represen-tation,the change of variables(15)becomes singular at lattice sites where thefieldφ(n) has two coinciding eigenvalues.This condition implies three independent conditions on components ofφ(n)which can be thought as equations for the singular sites.At each moment of lattice time,these three equations determine a set of spatial lattice vertices (locations of monopoles).Therefore on a four-dimensional lattice,the singular sites form world-lines which are identified with world-lines of monopoles[2].The new link variables˜Uµ(n)=˜g(n)Uµ(n)˜g−1(n+ˆµ)(25)inquires monopole singularities via˜g(n).Their density can be determined along the lines given in[8].So,monopole dynamics is the dynamics of configurationsφ(n)with two equal eigenval-ues in the full theory(13).If such configurations are dynamically preferable,then one can expect that in the dynamical Abelian projection,effective monopoles and antimonopoles form a condensate.All monopole-creating configurations of the scalarfieldφ(n)can easily be classified in a gauge invariant way.First of all we observe that the change of variables(15)is singular if its Jacobian vanishes nµ(n)=0.(26) We have to classify all configurationsφ(n)which lead toµ(n)=0.The polynom(17)is invariant with respect to the Weyl group.According to a theorem of Chevalley[16],any polynom in H invariant with respect to W is a polynom of basis(elementary)invariant polynoms tr h l(n)with l=l1,l2,...,l r being the orders of independent Casimir operators of G[16].Therefore,µ(n)=P(tr h l1(n),tr h l2(n),...,tr h l r(n))==P(trφl1(n),trφl2(n),...,trφl r(n))=0.(27) Solutions of this algebraic equation determine all configurationsφ(n)which will create monopoles in the dynamical Abelian projection(19).For G=SU(3),we have r=2,l1= 2,l2=3and[18]1µsu(3)(n)=As follows from(21)and(22),the Abelianfield U Hµ(n)and the Cartanfield h(n) are decoupled because[U Hµ(n),h(n)]=0.So,in the full theory,we define Abelian link variables by the relation[Uφµ(n),φ(n)]=0.(29) The coset decomposition assumes the formUµ(n)=Uφµ(n)U chµ(n).(30) One can regard it as a definition of chargedfields U chµ(n)for given Uµ(n)andφ(n).Consider a vector potential corresponding to Uφµ(n)as determined by(24).It has theformAφµ(n)=rα=1Bαµ(n)eφα(n),(31)where Bαµ(n)are real numbers,and Lie algebra elements eφα(n)form a basis in the Cartan subalgebra constructed in the following wayeφα=λi trλiφlα−1.(32) It is not hard to be convinced that[18][eφα,eφβ]=0.(33) Since for any group G one of the numbers lαis equal to2,one of the elements(32) coincides withφitself.The elements(32)are linearly independent in X becausedet Pαβ≡det tr eφαeφβ=const·P.(34) So,a generic elementφof X has a stationary group Gφ⊂G with respect to the adjoint action of G in X,gφφg−1φ=φ,gφ∈Gφ.This stationary group is isomorphic to the Cartan subgroup G H.All linear combinations of the elements(32)form a Lie algebra of Gφ∼G H.In fact,the basis(32)can be constructed without an explicit matrix representation ofλi.We recall that for each compact simple group G and its Lie algebra X,there existr=rank G=dim H symmetrical irreducible tensors of ranks lα,d i1,i2,...,i lα,invariant withrespect to the adjoint action of G in X.Clearly,(eφα)i=d ij1...j lα−1φj1···φjlα−1.Now it is easy to see that the Abelian potentials Bφµ(n)are singular at lattice sites whereφ(n)satisfies(27).Indeed,from(31)we getBαµ(n)=Pαβ(n)tr eφβ(n)Aφµ(n),(35) where PαβPβγ=δαγ.The determinant of the matrix Pαβ(n)vanishes at the sites where µ(n)=P(n)=0.At these sites,the inverse matrix Pαβ(n)does not exist,and thefields Bαµ(n)are singular.For unitary groups SU(N),lα=2,3,...,N,the singular sites form lines in the four-dimensional lattice[2],[8].These lines are world-lines of monopoles.5.The above procedure of avoiding explicit gaugefixing can be implemented to re-move the gauge arbitrariness completely and,therefore to study gauge-variant correlators,like the gluon propagator,or some other quantities requiring gaugefixing on the lattice [20].The advantage of dynamical gaugefixing is that it is free of all usual gaugefixing dynamical artifacts,Gribov’s ambiguities and horizon[14].It is also Lorentz covariant.Recent numerical studies of the gluon propagator in the Coulomb gauge[19]show that it can befit to a continuum formula proposed by Gribov[9].The same predictions were also obtained in the study of Schwinger-Dyson equations where no effects of the Gribov horizon have been accounted for[13].The numerical result does not exclude also a simple massive boson propagator for gluons[19].So,the problem requires a further investigation.Gaugefixing singularities(the Gribov horizon)occur when one parametrizes the topo-logically nontrivial gauge orbit space by Cartesian coordinates.So,these singularities are pure kinematical and depend on the parametrization(or gauge)choice.They may,how-ever,have a dynamical evidence in a gauge-fixed theory[21].For example,a mass scale determining a nonperturbative pole structure of the gluon propagator in the infrared region(gluon confinement)arises from the Gribov horizon[9],[12]if the Lorentz(or Coulomb)gauge is used.From the other hand,no physical quantity can depend on a gauge chosen.There is no gauge-invariant interpretation(or it has not been found yet) of the above mass scale.That is what makes the gluon confinement model based on the Gribov horizon looking unsatisfactory.Here we suggest a complete dynamical reduction of the gauge symmetry in lattice QCD,which involves no gauge condition imposed on gaugefields and,hence,is free of the corresponding kinematical artifacts.For the sake of simplicity,we discussfirst the gauge group SU(2).Consider two auxiliary(ghost)complexfieldsψandφ,Grassmann and boson ones,respectively.Let they realize the fundamental representation of SU(2),i.e.they are isotopic spinors.The identity(11)assumes the formZ b(β)Z f(β)= Dφ+DφDψ+Dψe−β(S b+S f)=1,(36) where S f= n(∇µψ)+∇µψand S b=1/2 n(∇µφ)+∇µφ,and the lattice covariant deriva-tive in the fundamental representation is defined by∇µφ(n)=φ(n+ˆµ)−U−1µ(n)φ(n). Inserting the identity(36)into the integral representation of the Yang-Mills partition func-tion(12),we obtain an effective gauge invariant action.The ghostfields are transformed asφ(n)→g(n)φ(n)andψ(n)→g(n)ψ(n).In the integral(13),we go over to new variables to integrate out the gauge group volumedφ+(n)dφ(n)=v su(2)∞dρ(n)ρ3(n),(37)whereφ(n)=˜g(n)χρ(n),χ+=(10),ρ(n)is a real scalarfield,and˜g(n)is a generic element of SU(2).A new fermion ghostfield and link variables˜Uµare related to the old ones via a gauge transformation with g(n)=˜g−1(n).Since the effective action is gaugeinvariant,the integral over˜g(n)yields the gauge group volume v Lsu(2).We end up withthe effective theoryZ Y M(β)= D˜Uµe−βS W F(˜U),(38)F(˜U)=(detβ∇+µ∇µ)1/2∞0 ndρ(n)ρ3(n) e−βS(ρ),(39)S(ρ)=1/2 n,µ ρ(n+ˆµ)−χ+˜U−1µ(n)χρ(n) 2.(40)The function(39)is not gauge invariant and provides the dynamical reduction of the SU(2) gauge symmetry.A formal continuum theory corresponding to(38)has been proposed and discussed in[14].Expectation values of a gauge-variant quantity G(U)are determined byG(U) ≡ F(U)G(U) W= D Uµe−βS W F(U)G(U).(41)For example,for the gluon two-point correlator one sets G(U)=Aµ(n)Aµ′(n′)where the gluon vector potential on the lattice reads2iaAµ(n)=Uµ(n)−U+µ(n)−1Though the integration domain is restricted in the sliced path integral(20),this re-striction will disappear in the continuum limit because of contributions of trajectories reflected from the boundary∂K+[17],[18].It is rather typical for gauge theories that a scalar product for physical states involves an integration over a domain with boundaries which is embedded into an appropriate Euclidean space.The domain can even be com-pact as,for example,in two-dimensional QCD[22].In the path integral formulation,this feature of the operator formalism is accounted for by appropriate boundary conditions for the transition amplitude(or the transfer matrix)rather than by restricting the integration domain in the corresponding path integral[22],[23].In turn,the boundary conditions are to be found from the operator formulation of quantum gauge theory[18],[22],[23].So,a study of the continuum limit requires an operator formulation of the dynamical reduction of a gauge symmetry,which has been done in[15].The dynamical Abelian projection can be fulfilled in the continuum operator formal-ism.The whole discussion of monopole-like singular excitations given in sections3and4 can be extended to the continuum theory.So,it determines Lorentz covariant dynamics of monopoles free of gaugefixing artifacts.To study monopole dynamics in the continuum Abelian gauge theory,one has to introduce monopole-carrying gaugefields[24].AcknowledgementI express my gratitude to F.Scholtz for valuable discussions on dynamical gauge fixing,to A.Billoire,A.Morel and V.K.Mitrjushkin for providing useful insights about lattice simulations,and D.Zwanziger and M.Schaden for a fruitful discussion on the Gribov problem.I would like to thank J.Zinn-Justin for useful comments on a dynamical evidence of configuration space topology in quantumfield theory.I am very grateful to H.Kleinert for a stimulating discussion on monopole dynamics.References[1]T.Suzuki and I.Yotsuyanagi,Phys.Rev.D42(1990)4257.[2]G.’t Hooft,Nucl.Phys.B190[FS3](1981)455.[3]H.Shiba,T.Suzuki,Phys.Lett.B333(1994)461.[4]J.D.Stack,S.D.Neiman and R.J.Wensley,Phys.Rev.D50(1994)3399.[5]S.Mandelstam,Phys.Rep.23(1976)245;’t Hooft,in:High Energy Physics,ed.M.Zichichi(Editrice Compositori,Bologna, 1976).[6]see,for example,L.Del Debbio,A.Di Giacomo,G.Pafutti and P.Pier,Phys.Lett.B 355(1995)255.[7]M.N.Chernodub,M.I.Polikarpov and V.I.Veselov,Phys.Lett.B342(1995)303.[8]A.S.Kronfeld,G.Schierholz and U.-J.Wiese,Nucl.Phys.B293(1987)461.[9]V.N.Gribov,Nucl.Phys.B139(1978)1.[10]I.M.Singer,Commun.Math.Phys.60(1978)7.[11]S.Hioki,S.Kitahara,Y.Matsubara,O.Miyamura,S.Ohno and T.Suzuki, Phys.Lett.B,271(1991)201.[12]D.Zwanziger,Nucl.Phys.B378(1992)525.[13]M.Stingl,Phys.Rev.D34(1986)3863.[14]F.G.Scholtz and G.B.Tupper,Phys.Rev.D48(1993)1792.[15]F.G.Scholtz and S.V.Shabanov,Supersymmetric quantization of gauge theories, FU-Berlin preprint,FUB-HEP/95-12,1995.[16]S.Helgason,Differential Geometry,Lie Groups,and Symmetric Spaces(Academic Press,NY,1978).[17]L.V.Prokhorov and S.V.Shabanov,Phys.Lett.B216(1989)341;pekhi34(1991)108.[18]S.V.Shabanov,Theor.Math.Phys.78(1989)411.[19]C.Bernard,C.Parrinello and A.Soni,Phys.Rev.D49(1994)1585.[20]Ph.de Forcrand and K.-F.Liu,Nucl.Phys.B(Proc.Suppl.)30(1993)521.[21]V.G.Bornyakov,V.K.Mitrjushkin,M.M¨u ller-Preussker and F.Pahl,Phys.Lett.B 317(1993)596.[22]S.V.Shabanov,Phys.Lett.B318(1993)323.[23]S.V.Shabanov,Phys.Lett.B255(1991)398;Mod.Phys.Lett.A6(1991)909.[24]H.Kleinert,Phys.Lett.B293(1992)168.。

Generalized Measures in Gauge Theory

Generalized Measures in Gauge Theory

a r X i v :h e p -t h /9310201v 1 31 O c t 1993Generalized Measures in Gauge TheoryJohn C.BaezDepartment of MathematicsUniversity of CaliforniaRiverside CA 92521October 30,1993Abstract Let P →M be a principal G -bundle.We construct well-defined substi-tutes for “Lebesgue measure”on the space A of connections on P and for “Haar measure”on the group G of gauge transformations.More precisely,we define algebras of “cylinder functions”on the spaces A ,G ,and A /G ,and de-fine generalized measures on these spaces as continuous linear functionals on the corresponding algebras.Borrowing some ideas from lattice gauge theory,we characterize generalized measures on A ,G ,and A /G in terms of graphs embedded in M .We use this characterization to construct generalized mea-sures on A and G ,respectively.The “uniform”generalized measure on A is invariant under the group of automorphisms of P .It projects down to the gen-eralized measure on A /G considered by Ashtekar and Lewandowski in the case G =SU (n ).The “generalized Haar measure”on G is right-and left-invariant as well as Aut(P )-invariant.We show that averaging any generalized measure on A against generalized Haar measure gives a G -invariant generalized measure on A .1Introduction The space A of connections on a principal bundle is an infinite-dimensional affinespace,and the while the notion of the “Lebesgue measure”D A on A has been very fruitful,it is mathematically ill-defined.Some of the infinities in quantum field theory calculations can be avoided by projecting down D A to the space A /G of connections modulo gauge transformations,but certainly not all.While the theory of cylinder measures on infinite-dimensional vector spaces [6]provides a rigorous framework for interpreting the Gaussian “measures”appearing in the physics of the free boson field[5],it is usually quite difficult to apply this theory to the study of gauge fields,except in the case of 2d Yang-Mills theory.It is thus desirable to generalize the concept of measure in a manner more suited to the needs of gauge theory.Recently Ashtekar and Isham [1]have proposed an approach based on the idea of a Wilson loop,that is,the trace of the holonomy of a connection around a loop in the base manifold M .Wilson loops are very natural observables in gauge theory,and in the “loop representation”of gauge theories they play a primary role [7].In the caseof the group G=SU(n),Ashtekar and Lewandowski[2]used this approach to define and construct a very natural“generalized measure”µAL on A/G,which is invariant under diffeomorphisms of M.The author[3]extended this approach to construct a rich variety of diffeomorphism-invariant generalized measures on A/G when G is compact.These generalized measures give rise to invariants of multiloops(collections of loops)in the base manifold,and their classification involves a combination of singularity theory and knot theory.In this paper we show that one can define generalized measures on A.All of these project down to generalized measures on A/G,but even when one is interested in gauge-invariant quantities,it is sometimes easier to work“upstairs”on A.In particular,when G is compact,there is a“uniform”generalized measureµu on A that projects down toµAL under the map A→A/G.This generalized measureµu is in some respects a rigorous substitute for the ill-defined“Lebesgue measure”on A,but it is actually built using Haar measure on G.We also define generalized measures on G,and when G is compact we construct a natural exampleµH that is a rigorous substitute for Haar measure on G.As an application of this“generalized Haar measure”we show that any generalized measure on A can be averaged against µH to give a G-invariant generalized measure on A.We emphasize that these constructions are no panacea;in particular,they are unlikely to be of much use in4-dimensional Yang-Mills theory,where one expects that only“smeared”Wilson loops will serve as physical observables[4].With some modification,these constructions might allow the construction of the Chern-Simons path integral as a generalized measure,as discussed in[3].They may also be suited for rigorous work on the loop representation of quantum gravity[7,8].2Generalized MeasuresLet M be a manifold,possibly with boundary,let G be a Lie group,and letπ:P→M be a principal G-bundle.Let A be the space of connections on P and let G be the group of gauge transformations.Let Diff(M)denote the group of diffeomorphisms of M restricting to diffeomorphisms of∂M.Everything in this section applies equally to the following three cases:1.The C∞case:M and P are smooth,πis smooth,and A,G,and Diff(M)consist of smooth connections,gauge transformations,and diffeomorphisms, respectively.2.The Cωcase:M and P are real-analytic,πis real-analytic,and A,G,andDiff(M)consist of real-analytic connections,gauge transformations,and diffeo-morphisms,respectively.3.The hybrid case:M is real-analytic and P is smooth,πis smooth,A andG consist of smooth connections and gauge transformations,respectively,andDiff(M)consists of real-analytic diffeomorphisms.We will define sub-C*-algebras of the bounded continuous complex functions on A,G,and A/G(with their C∞topologies).By a“generalized measure”on one of these spaces we will mean simply a continuous linear functional on the corresponding C*-algebra.Everyfinite regular Borel measure on one of these spaces defines such a generalized measure,but the most interesting generalized measures are not of this form.It is easiest to construct generalized measures in case3above,so in the next section we will restrict attention to that case,even though case1is in some ways the most natural.Given a pathγ:[0,1]→M,let Aγbe the space of all maps from thefiber Pγ(0)to thefiber Pγ(1)that can be obtained as holonomies alongγof some connection A∈A. Note that there is a natural mappγ:A→AγA→Aγassigning to each connection A∈A its holonomy Aγalongγ.One should think of pγ:A→Aγas picking out a small piece of information about the connection A; one can reconstruct A from all the pieces{Aγ}.Fixing trivializations of P at the endpointsγ(0)andγ(1),Aγcan be identified with an open and closed subspace of the group G.We give Aγthe subspace topology(which is independent of the choice of trivialization).This makes the map pγcontinuous.We say f is a cylinder function on A if it is of the formf(A)=F(Aγ1,...,Aγn),where{γi}is afinite set of paths in M(required to be real-analytic in the Cωand hybrid cases)andF: i Aγi→Cis a bounded continuous function.Let Fun0(A)denote the space of cylinder func-tions on A,which is a∗-subalgebra of the bounded continuous functions on A.The completion of Fun0(A)with respect to the sup norm,which we denote by Fun(A),is thus a C*-subalgebra of the bounded continuous functions on A.Note that G acts as∗-automorphisms of Fun0(A)bygf(A)=f(g−1A).The G-invariant functions in Fun0(A)may be regarded as functions on A/G,and we denote the algebra of all such functions as Fun0(A/G).We call these cylinder functions on A/G.We denote the completion of Fun0(A/G)with respect to the sup norm by Fun(A/G).This may be regarded either as a C*-subalgebra of the bounded continuousfunctions on A/G(with its quotient topology),or of the bounded continuous G-invariant functions on A.It can be seen that as special cases of this algebra one obtains the“holonomy C*-algebra”defined in the smooth case by Ashtekar and Isham [1]for G=SU(2)and the“analytic holonomy C*-algebra”defined in the hybrid case by Ashtekar and Lewandowski[2]for G=SU(n).We may also define cylinder functions on the group G of gauge transformations. Given a point x∈M,let G x be thefiber at x of the bundle P×Ad G,with its subspace topology.An element g∈G is a section of P×Ad G,so there is a natural mapp x:G→G xg→g xOne should think of p x:g→g x as picking out a small piece of information about the gauge transformation g;one can reconstruct g from the pieces{g x}.Note that G x is naturally a group,and that p x is a homomorphism.We say f is a cylinder function on G if it is of the formf(g)=F(g x1,...,g xn),where{x i}is afinite set of points in M andF: i Gγi→Cis bounded and continuous.The completion of the algebra Fun0(G)of cylinder func-tions on G is a C*-subalgebra of the bounded continuous functions on G,which we denote by Fun(G).By a generalized measure on A,G,or A/G we mean a continuous linear functional on Fun(A),Fun(G),or Fun(A/G),respectively.Note that every generalized measure on A“projects down”to a generalized measure on A/G;this operation of“projecting down”is really just restriction of a continuous linear functional on Fun(A)to the subalgebra Fun(A/G).Let Aut(P)denote the group of bundle automorphisms g such that for some h∈Diff(M),π(g(p))=h(π(p))for all p∈P.(In cases2and3,recall that h must be real-analytic.)Then we have an exact sequence1→G→Aut(P)→Diff(M)→1.The group Aut(P)acts as∗-automorphisms of Fun(A)bygf(A)=f(g−1A).so we obtain an action of Diff(M)as∗-automorphisms of Fun(A/G).By duality, Aut(P)acts on the generalized measures on A,and Diff(M)acts on the generalized measures on A/G.Any Aut(P)-invariant generalized measure on A projects down to a Diff(M)-invariant generalized measure on A/G.Note also that Aut(P)acts on the generalized measures on G,as do left and right translation.3Characterizing Generalized MeasuresIn this section and the next we restrict our attention to the“hybrid case,”case2of the previous section.Ashtekar and Lewandowski recognized the importance of this case when they used it to construct a very natural sort of Diff(M)-invariant generalized measure on A/G for G=SU(2).Subsequently they,and independently the author, were able to generalize this construction to more general compact Lie groups,and also to give a rather concrete characterization of all generalized measures on A/G. In addition,the author has given a recipe for constructing many Diff(M)-invariant examples of such generalized measures.Here we give concrete characterizations of generalized measures on A,G,and A/G when G is any Lie group.First,we need a notion of an embedded graph in M,a slight variant of that in[3].We define an embedded graphφin M to be afinite collection of real-analytic pathsφj:[0,1]→M such that:1.for all j,φj is one-to-one,2.for all j,φj|(0,1)is an embedding,3.for all j and k,φj[0,1]∩φk[0,1]⊆{φj(0),φj(1)}.The pathsφj are called the edges ofφ,and the pointsφj(0),φj(1)are called the vertices ofφ.Somewhat redundantly,we write E(φ)for the set of edges ofφand V(φ)for the set of(distinct)vertices.Note that the set|φ|= jφj[0,1]⊆Mequipped with the subspace topology indeed has the topology of afinite graph.The following lemma proved by Ashtekar and Lewandowsi plays a key technical role.Lemma1.[2]Let{γi}be afinite collection of real-analytic paths in M.Then there exists an embedded graphφsuch that for eachγi there exist paths inφsuch that γi is equivalent to a product of these paths and their inverses,up to a continuous orientation-preserving reparametrization.Given an analytic graphφin M,letAφ= γ∈E(φ)Aγ,the Cartesian product over all edgesγofφof the spaces Aγ,equipped with the product topology.Similarly,letGφ= x∈V(φ)G xequipped with the product topology.We may write any element of Gφas a tuple (g x)x∈V(φ)where g x∈G x.Similarly,we may write any element of Aφas a tuple (Aγ)γ∈E(φ),where Aγ:Pγ(0)→Pγ(1).There are natural maps pφ:A→Aφand pφ:G→Gφ,given bypφ(A)=(Aγ)γ∈E(φ),pφ(g)=(g x)x∈V(φ).Though we denote both of these maps by pφ,the meaning should be clear from context.Both these maps are onto,since we can alwaysfind a connection having any specified holonomies in the sets Aφ,and we can alwaysfind a gauge transformation having any specified values at the vertices ofφ.Given A∈A and g∈G,we will sometimes write Aφfor pφ(A)and gφfor pφ(g).In the above we are borrowing an idea from lattice gauge theory,in which“connec-tions”assign group elements to the edges of a lattice,while“gauge transformations”assign group elements to vertices.We can make this analogy very precise if the group G is connected.In this cases,we can trivialize P over|φ|for any embedded graph φ.Fixing a trivialization gives an identification of Aγ,for any edgeγofφ,with the group G,henceAφ∼=G E(φ).Similarly,fixing a trivialization gives an identification of G x,for any vertex x ofφ, with G,soGφ∼=G V(φ).The group Gφacts on the space Aφas follows:)γ∈E(φ).(g x)x∈V(φ)(Aγ)γ∈E(φ)=(gγ(1)Aγg−1γ(0)This action is compatible with the action of G on A,as follows:pφ(g)pφ(A)=pφ(gA)for any g∈G,A∈A.Let Fun(Aφ)denote the algebra of bounded continuous functions on Aφ.We will identify F∈Fun(Aφ)with the function f on A given byf(A)=F(Aφ),allowing us to writeFun(Aφ)⊆Fun0(A).Since a generalized measureµon A is just a continuous linear functional on Fun(A), we can restrictµto a continuous linear functionalµφon Fun(Aφ).(When G is compact,Aφis compact,so the Riesz-Markov theorem allows us to identify continuous linear functionals on Fun(Aφ)withfinite regular Borel measures on Aφ.)A set ofcontinuous linear functionalsµφ∈Fun(Aφ)∗,one for each embedded graphφ,will be called a family.The following theorem gives necessary and sufficient conditions for a family{µφ}to come from a generalized measure on A.As in[3],these conditions can be used to construct concrete examples of generalized measures.Given embedded graphsφ,ψ,we say thatφis included inψ,which we write asφ֒→ψ,if every edge ofφis,up to orientation-preserving reparametrization,a product of edges ofψand their inverses.Note thatφ֒→ψimplies that every vertex ofφis a vertex ofψ,and that|φ|⊆|ψ|.It also implies that Fun(Aφ)⊆Fun(Aψ). We say that the family{µφ}is consistent ifφ֒→ψimplies that the restriction of µψto Fun(Aφ)isµφ.We say that the family{µφ}is uniformly bounded if there is a constant C>0such that µφ <C for allφ.Theorem1.Supposeµis a generalized measure on A,that is,a continuous linear functional on Fun(A).For any embedded graphφin M,letµφdenote the restriction of µto Fun(Aφ).Then{µφ}is a consistent and uniformly bounded family.Conversely, if{µφ}is a consistent and uniformly bounded family,there is a unique generalized measureµon A whose restriction to Fun(Aφ)isµφ.Proof-Ifµis a generalized measure on A the family{µφ}obtained by restriction is consistent,and µφ ≤ µ ,so it is uniformly bounded.Conversely,suppose we are given a consistent and uniformly bounded family{µφ}.Wefirst define a linear functionalµon Fun0(A)as follows.Any element f∈Fun0(A)is of the formf(A)=F(Aγ1,...,Aγn),where{γi}are paths in M.In this situation we say that f can be expressed in terms of the paths{γi}.Construct an embedded graphφfrom the pathsγi as in Lemma 1.Then f∈Fun(Aφ).Defineµ(f)=µφ(f).We need to check thatµis well-defined,linear,and extends to a continuous linear functional on Fun(A).If the extension exists,it is unique,since Fun0(A)is dense in Fun(A).For well-definedness,suppose that f can be expressed in two ways,in terms of paths{γi}or in terms of paths{γ′j}.Using Lemma1,construct embedded graphs φfrom the paths{γi},φ′from the paths{γ′j},andψfrom the paths{γi,γ′j}.Note thatφ֒→ψandφ′֒→ψ.Thusµφ(f)=µψ(f)=µφ′(f).For linearity,suppose f,g∈Fun0(A).Then f+g∈Fun0(A)and there exist paths{γi}in terms of which f,g,and f+g can all be ing Lemma1, construct an embedded graphφfrom the paths{γi}.Thenµ(f+g)=µφ(f+g)=µφ(f)+µφ(g)=µ(f)+µ(g).Clearlyµ(λf)=λµ(f)for allλ∈C.Finally,to show thatµextends to a continuous linear functional on Fun(A)it suffices to note that there exists C>0with µ(f) ≤C f for any f∈Fun0(A), by the uniform boundedness of the family{µφ}.⊓⊔Completely analogous results holds for generalized measures on G and A/G.Let Fun(Aφ/Gφ)denote the subalgebra of Fun(Aφ)consisting of functions invariant un-der the action of Gφ.Alternatively,Fun(Aφ/Gφ)may be regarded as the algebra of bounded continuous functions on Aφ/Gφ,equipped with its quotient topology.A gen-eralized measureµon A/G restricts to a family of elementsµφ∈Fun(Aφ/Gφ)∗,one for each embedded graphφ.(When G is compact these are the same asfinite regular Borel measures on Aφ/Gφ.)We define consistency and uniform boundedness of such families as before,and obtain:Theorem2.Supposeµis a generalized measure on A/G.For any embedded graph φin M,letµφdenote the restriction ofµto Fun(Aφ/Gφ).Then{µφ}is a consistent and uniformly bounded family.Conversely,if{µφ}is a consistent and uniformly bounded family,there is a unique generalized measureµon A/G whose restriction to Fun(Aφ/Gφ)isµφ.Proof-The proof follows that of Theorem1.⊓⊔Let Fun(Gφ)denote the algebra of bounded continuous functions on Gφ.We will identify F∈Fun(Gφ)with the function f on G given by f(g)=F(gφ),allowing us to writeFun(Gφ)⊆Fun0(G).Thus a generalized measureµon G restricts to a family{µφ}of elements of Fun(Gφ)∗. (When G is compact,elements of Fun(Gφ)∗are the same asfinite regular Borel mea-sures on Gφ.)We define consistency and uniform boundedness of such families as in the case of A.Theorem3.Supposeµis a generalized measure on G.For any embedded graphφin M,letµφdenote the restriction ofµto Fun(Gφ).Then{µφ}is a consistent and uniformly bounded family.Conversely,if{µφ}is a consistent and uniformly bounded family,there is a unique generalized measureµon G whose restriction to Fun(Gφ)is µφ.Proof-The proof follows that of Theorem1.⊓⊔We conclude with an alternate description of Fun(A/G).Recall that functions in Fun(A/G)may be regarded as limits of G-invariant cylinder functions on A.At least when G is amenable(for example,compact,abelian,or an extension of an amenable group by an amenable group),these are precisely the same as G-invariant functions on A that are limits of cylinder functions:Theorem4.Suppose that G is amenable.Then Fun(A/G)is equal to the subalgebra of G-invariant functions in Fun(A).Proof-It is immediate that elements of Fun(A/G)are G-invariant and lie in Fun(A).To prove the opposite inclusion,suppose f∈Fun(A)is G-invariant.Then there exists a sequence f i∈Fun0(A)with f i→f.To show f∈Fun(A/G)it suffices to show the existence of a sequence of G-invariant elements of Fun0(A)converging tof.We may suppose f i∈Fun(Aφi ).The group Gφi,being isomorphic to a product ofcopies of G,is amenable.Let M i:Fun(A)→Fun(A)denote the result of averagingover the action of Gφi with respect to an invariant mean.Noting that M i:Fun(Aφi)→Fun(Aφi ),that M i is a contraction,and that M i f=f,we conclude that M i f i is asequence of G-invariant elements of Fun0(A)converging to f.⊓⊔It is not clear whether the hypothesis of amenability is necessary.4ExamplesNow we consider the case where G is compact.In this case we construct a generalized measure on A that we call the uniform generalized measure.This generalized measure is invariant under all of Aut(P).It thus projects down to a generalized measure on A/G that is Diff(M)-invariant,as described at the end of Section2.The result is the generalized measure on A/G constructed for G=SU(n)by Ashtekar and Lewandowski[2].We also construct a generalized measure on G called generalized Haar measure,which is both left-and right-invariant as well as Aut(P)-invariant. We also show how to average any generalized measure on A against generalized Haar measure on G to obtain a G-invariant generalized measure on A.As in the previous section,we assume P→M is a G-principal bundle and work in the“hybrid case,”case2of Section2.Let m denote normalized Haar measure on G,which is assumed compact.Supposeγis any path in M.Fixing a trivialization of P atγ(0)andγ(1)we obtain an identification of Aγwith a closed and open subset X⊆G.Define the measureµγon Aγto be the restriction of m to Aγ.A change of trivialization of P atγ(0)changes X⊆G by a left translation,while a change of trivialization atγ(1)changes X by a right translation.Since m is left-and right-invariant,it follows thatµγis independent of the choice of trivializations atγ(0)and γ(1).Recall that for any embedded graphφAφ= γ∈E(φ)Aγ.Defineµφto be thefinite regular Borel measure on Aφgiven by the product of the measuresµγ.Note thatµφ ≤1for allφ.One can show that{µφ}is a consistent family in the sense of Theorem1 (for details,see[2,3]).Theorem1thus implies the existence of a unique generalized measureµon A such that for allφ,µφis the restriction ofµto Fun(Aφ).We call this generalized measureµu the uniform generalized measure on A.It is easily seen from the natural way in which it was constructed thatµu is Aut(P)-invariant,and thus projects down to a Diff(M)-invariant generalized measureµAL on A/G.We call µAL the Ashtekar-Lewandowski generalized measure on A/G.Similarly,given any point x∈M,a trivialization of P at x gives an identification of G x with ing this identification the Haar measure m on G gives rise to a measureµx on G x.Since m is left-and right-invariant,µx independent of the choice of trivialization of P at x.Define thefinite regular Borel measureµφonGφ= x∈V(φ)G xto be the product of the measuresµx.The family{µφ}is consistent and uniformly bounded,so by Theorem3there is a unique generalized measureµH on G such that for allφ,µφis the restriction ofµH to Fun(Gφ).We call this generalized measure generalized Haar measure on G.By naturality,µH is invariant under the action of Aut(P)on G.By the invariance properties of Haar measure,µH is also left-and right-invariant.That is,the left and right actions of G on itself give rise to actions of G on Fun(G),hence dually on Fun(G)∗,andµH is preserved by these actions.We can convolve generalized measures on G as follows.For each embedded graph φone can convolvefinite regular Borel measures on the compact Lie group Gφ:∗:Fun(Gφ)∗×Fun(Gφ)∗→Fun(Gφ)∗by the usual formula(µ∗ν)(f)= Gφ×Gφf(gh)dµ(g)dν(h),and one has the boundµ∗ν ≤ µ ν .Ifφ֒→ψthere is a natural group homomorphism Gψ→Gφ,hence a homomorphism of convolution algebras Fun(Gψ)∗→Fun(Gφ)∗.This allows us to give Fun0(G)∗,which is the inverse limit of the spaces Fun(Gψ)∗,the structure of an algebra in a unique way such that all the mapsFun0(G)∗→Fun(Gφ)∗are algebra homomorphisms.We write the product in Fun0(G)as∗.Since this product satisfies the boundµ∗ν ≤ µ ν ,it extends uniquely by continuity to a product on Fun(G)∗,again written∗and called the convolution of generalized measures on G.Moreover,we can average(or convolve)generalized measures on A against gen-eralized measures on G as follows.For embedded graphφthe convolution algebra Fun(Gφ)∗acts on Fun(Aφ)∗,∗:Fun(Gφ)∗×Fun(Aφ)∗→Fun(Aφ)∗by the usual formula(µ∗ν)(f)= Gφ×Aφf(gA)dµ(g)dν(A),and one has the boundµ∗ν ≤ µ ν .Using these facts,an inverse limit argument like the one above shows that the convolu-tion algebra Fun(G)∗acts on Fun(A)∗.Below,we apply this to construct G-invariant generalized measures on A from generalized measures on A by convolution against Haar generalized measure on G:Theorem5.Letνbe a generalized measure on A and letµH denote Haar generalized measure on G.ThenµH∗νis a G-invariant generalized measure on A.Proof-Suppose f∈Fun0(A).Then f∈Fun(Aφ)for some embedded graphφ,so f depends on A only through Aφ;we write f(A)=F(Aφ).Letνφbe the restriction ofνto Fun(Aφ).Writingµφfor Haar measure on Gφ,we have,for any g∈G,(µH∗ν)(gf)=(µφ∗νφ)(gφF)= Gφ×AφF(g−1φhAφ)dµφ(h)dνφ(Aφ)= Gφ×AφF(hAφ)dµφ(h)dνφ(Aφ)=(µφ∗νφ)(F)=(µH∗ν)(f).Since Fun0(A)is dense in Fun(A),we conclude that(µH∗ν)(gf)=(µH∗ν)(f)for all f∈Fun(A),soµH∗νis G-invariant.⊓⊔5ConclusionsThere is much more one can do to generalize the theory of Lie groups and homo-geneous spaces to groups of gauge transformations and spaces of connections,using the framework introduced here.For example,there is a Hilbert space completionL2(A,µu)of Fun(A),on which Aut(P)has a unitary representation.Similarly,there is a Hilbert space L2(A/G,µAL)on which Diff(M)has a unitary representation,and a Hilbert space L2(G,µH)which is a unitary representation of G acting by left(or right)translation,as well as a unitary representation of Aut(P).It is still unclear how useful these structures will be in physics,but they have many of the properties one would naively expect of nonrigorous constructions using“Lebesgue measure”on A and“Haar measure”on G.References[1]A.Ashtekar and C.J.Isham,Representations of the holonomy algebra of gravityand non-abelian gauge theories,Jour.Class.and Quant.Grav.9(1992),1069-1100.[2]A.Ashtekar and J.Lewandowski,Representation theory of analytic holonomyC*-Algebras,to appear in Knots and Quantum Gravity,ed.J.Baez,Oxford U.Press.[3]J.Baez,Diffeomorphism-invariant generalized measures on the space of connec-tions modulo gauge transformations,to appear in the proceedings of the Confer-ence on Quantum Topology,eds.L.Crane and D.Yetter,hep-th/9305045. [4]J.Baez,Link invariants,holonomy algebras and functional integration,U.C.Riverside preprint(1993),hep-th/9301063.[5]J.Baez,I.Segal and Z.Zhou,An Introduction to Algebraic and ConstructiveQuantum Field Theory,Princeton U.Press,Princeton,1992.[6]A.N.Kolmogorov,Foundations of the Theory of Probability,Chelsea,New York,1956.[7]R.Loll,Chromodynamics and gravity as theories on loop space,Penn State Uni-versity preprint(1993)CGPG-93/9-1,hep-th9309056.[8]C.Rovelli and L.Smolin,Loop representation for quantum general relativity,Nucl.Phys.B331(1990),80-152.。

Interacting Modified Variable Chaplygin Gas in Non-flat Universe

Interacting Modified Variable Chaplygin Gas in Non-flat Universe

a r X i v :0802.1146v 2 [a s t r o -p h ] 3 M a y 2008Interacting Modified Variable Chaplygin Gas in aNon-flat UniverseMubasher Jamil ∗and Muneer Ahmad Rashid †Center for Advanced Mathematics and Physics National University of Sciences and Technology Peshawar Road,Rawalpindi,46000,PakistanMay 3,2008AbstractWe have presented a more generalized model of interacting Chaplygin gas.The effective equation of states corresponding to matter and dark energy have been derived in this generalized model.Moreover,the case of phantom energy arises by putting some constraints on the parameters involved.Keywords :Interaction;Chaplygin Gas;Dark Matter.1IntroductionHosts of results coming from the observations of WMAP [1,2]have convincingly shown the validity of standard Big Bang model with cosmological constant Λ,so called ΛCDM model.Surprisingly the energy density corresponding to Λis two third of the critical density or ΩΛ≈0.7,the remaining is due to baryonic and non-baryonic dark matter.The genesis of this energy,commonly called the dark energy,and its subsequent evolution poses serious problems to investigate.The dark energy is commonly represented by a barotropic equation of state (EoS)p =−ρ(or ω=−1).Numerous models have been proposed and developed in literature to explain the accelerated expansion of the universe which include those based on homogeneous and time dependent scalar fields termed as quintessence [3],phantom energy with EoS ω<−1[4],Chaplygin gas (CG)with the EoS p =−A/ρ[5],dissipative or viscous cosmology [6]and modified gravity models including Dvali-Gabadadze-Porrati (DGP)model [7]and higher order f (R )gravity theories [8].Among all of these alternative theories,CG EoS effectively explains the evolution of the universe right from the earlier matter dominated era (decelerating phase)to the late dark energy dominated era(accelerating phase).This EoS wasfirst introduced in aerodynamical context[9]and has been hugely investigated in recent years due to its adequate efficiency in interpolating the astrophysical data.The energy conservation equation˙ρ+3H(ρ+p)=0yieldsρ= a6,(1) where A is a positive constant and B is the constant of integration.Above,p andρare the pressure and energy densities of the rge values of a(t)mimics cosmological constant.The CG model favors a spatiallyflat universe with95.4%confidence level which agrees with the observational data of Sloan Digital Sky Survey(SDSS)and Supernova Legacy Survey(SNLS)[10].CG also possesses a property of giving accelerated expansion even if it gets coupled with other scalarfields like quintessence or dissipative matterfields[11].But at the same time CG also has a drawback as it produces oscillations or exponential blow up of dark matter power spectrum which is inconsistent with observations[12].Similar results have been obtained in later generalizations of CG[13].Various generalizations of CG have been proposed in the literature(see e.g.[14,15,16,17,18,19,20,21]).All these models based on the inhomogeneous Chaplygin gas offer unified picture of dark matter and dark energy[22].Earlier,it had been suggested that interacting models of CG with dark matter can effectively ameliorate the coincidence problem which is not possible in pure CG models[23,24,25].Moreover,this problem has also been investigated in the context of dissipative CG and successfully explains the cosmic conundrum problem[11].The interacting model effectively explains the phantom divideω=−1i.e.the transition fromω>−1toω<−1[26].It also yields stable stationary attractor solution of the Friedmann-Robertson-Walker(FRW)equations at late times.This situation motivates us to investigate the more generalized form of interacting CG and tofind the effective EoS of matter and the dark energy in the respective model.The formalism adopted here is from[19].2Interacting modified variable Chaplygin gasWe assume the background spacetime to be homogeneous and isotropic represented by the FRW metricds2=−dt2+a2(t)[dr2a2=1respectively.Eqs.(5)and(6)show that the energy conservation for dark energy and matter would not hold independently if there is an interaction between them but would hold globally for the whole interacting system as is manifested in Eq.(4).Further,we further define the density ratio r m as r m≡ρmdt =ρmρm−˙ρΛr mΓ3H,ωeffm=−13H,(9)which involve the contribution from the interaction between matter and dark ing Eq.(9)in Eq.(5)and(6),we get˙ρΛ+3H(1+ωeffΛ)ρΛ=0,(10) and˙ρm+3H(1+ωeffm)ρm=0.(11) Further from the standard FRW model,the density parameters corresponding to matter,dark energy and curvature are defined asΩm=ρm3H2M2p,(12)ΩΛ=ρΛ3H2M2p,(13)Ωk=kΩΛ.(16) In our generalized model,we choose the modified variable Chaplygin gas(MVG)is defined aspΛ=AρΛ−B(a)where A is a positive constant and0≤α≤1with B(a)varies as B o a−n,where n and B o are positive constants.Notice that for B=0,Eq.(17)represents the EoS of perfectfluid.Also if B(a)=B o,it gives EoS of modified CG.Again if A=0andα=1,it represents the usual CG.Recently it is deduced using the latest supernovae data that models withα>1are also possible[30].The modified form of the CG is also phenomenologically motivated and can explain theflat rotational curves of galaxies[31].The galactic rotational velocity V c is related with the MVG parameter A as A2=V c/√1+α,(18) where s≡3(1+α)(1+A),∆≡3(1+α)B o1+α[(∆a−n+Ca−s)−α3(1+α)[(∆a−n+Ca−s)−αΩΛ−1.(21)Using Eq.(16)and(21),we haveΓ=3b2H1+Ωk3(1+α)[(∆a−n+Ca−s)−1(∆na−n+Csa−s)]−1.(23)Notice that the cosmological models with phantom energy arise whenωeffΛ<−1which is possible if thequantity in the square brackets in Eq.(23)is less then zero.Therefore,we have two possible cases: Case(1).If(∆a−n+Ca−s)>0and(∆na−n+Csa−s)<0,then we have a constraint on a(t)−C∆n.(24) This is possible when−C∆n<a s−n<−C∆>0and s<n.Thus the phantom dark energy case is possible if scale factor a(t)is constrained as given in Eq.(24) and(25).Also,Eq.(23)represents usual dark energy if the quantity in the square brackets vanishes identically.4Also we note that the quantity in Eq.(18)must be positive so that a(t)>(−C s−n.Therefore,the minimum value of a(t)lies ata min= −C s−n.(26) The universe can bounce before reaching the singularity when a>a min which corresponds to the bouncing universe model.This bounce is possible due to our assumption of positive curvature.Moreover,the effective EoS of matter is determined from Eqs.(9),(16)and(20)asωeff m =−b2(1+ΩΛ[13]Carturan D and Finelli F2003Phys.Rev.D6*******[14]Bento M C et al2002Phys.Rev.D6*******[15]Cruz N et al2007Phys.Lett.B646177[16]Benaoum H B,hep-th/0205140[17]Setare M R2007Phys.Lett.B648329[18]Zhang H and Zhu Z,arXiv:0704.3121v2[19]Setare M R2007Eur.Phys.J.52689;arXiv:0711.0524[20]Guo Z and Zhang Y2007Phys.Lett.B645326;astro-ph/0506091v3[21]Debnath U2007Ap.Sp.Sci.312295;arxiv:0710.1708v1[22]Bili´c et al2002Phys.Lett.B53517[23]Zhang H and Zhu Z2006Phys.Rev.D7*******[24]Wu P and Yu H2007Class.Quantum Grav.244661[25]Campo S et al2006Phys.Rev.D7*******[26]Sadjadi H M and Alimohammadi M2003Phys.Rev.D7*******[27]Kim H et al2006Phys.Lett.B632605[28]Waterhouse T P and Zipin J P,astro-ph/0804.1771v1[29]Paris C M and Visser M1999Phys.Lett.B45590[30]Bertolami O et al2004Mon.Not.Roy.Ast.Soc.353329;astro-ph/0402387v2[31]Tekola A G,gr-qc/0706.0804v1[32]Wang B et al2005Phys.Lett.B6241416。

Temperature- and Curvature Dependence of the Chiral Symmetry Breaking in 2D Gauge Theories

Temperature- and Curvature Dependence of the Chiral Symmetry Breaking in 2D Gauge Theories

a rXiv:h ep-th/93185v114O ct1993ETH-93-38September 1993Temperature-and Curvature Dependence of the Chiral Symmetry Breaking in 2D Gauge Theories I.Sachs 1and A.Wipf 2Institute for Theoretical Physics Eidgen¨o ssische Technische Hochschule,H¨o nggerberg CH-8093Z¨u rich,Switzerland Abstract The partition function and the order parameter for the chiral sym-metry breaking are computed for a family of 2-dimensional interacting theories containing the gauged Thirring model.In particular we derive non-perturbative expressions for the dependence of the chiral conden-sate on the temperature and the curvature.Both,high temperature and high curvature supress the condensate exponentially and we can associate an effective temperature to the curvature.1IntroductionDespite of the considerable amount of work devoted to the subject of chiral symmetry breaking in gauge theories and in particular QCD,the understand-ing of this non-perturbative phenomenon is still unsatisfactory[1].Also the behaviour of quantum systems in a hot and dense enviroment(eg.in neu-tron stars or in the early universe)are still under active investigation[1].On another front there has been much effort on the apparently different problem of quantizing self-interacting theories in a background gravitationalfield[2].Rather than seeking new partial results for realistic4-dimensional theo-ries we analyse a family of interacting theories of charged fermions,scalars, pseudo-scalars and photons propagating in2-dimensional curved spacetime in detail.These models are defined by the actionS= √FµνFµν ,(1)4where Fµνis the electromagneticfield strength and Dµ=∇µ−ieAµthe generally-and gauge covariant derivative.This family contains in particular the Schwinger model(g i=0,i=1,..,3)[3]and the gauged Thirring model (g21=−g22=g2,g3=0)[4,5]in curved spacetimeS T h= √4jµjµ−1.Lβis the inverse temperature and L is the infrared cut-offwhich will be removed after the correlation have been calculated.Furthermore,finite tem-perature boundary conditions are imposed on the quantumfields[6].On the torus a general gauge potential with non-vanishingflux can be decomposed asAµ=A kµ+tµ+∂µα−ηµν∂νϕ,where the last3terms are recognized as Hodge decomposition of the single valued part of A and A k is an instanton potential giving rise to a quantized flux e F=2πk.As a consequence the corresponding Dirac operator has |k|zero modes[6]of chirality sign{k}.These zero modes are responsiblefor a non vanishing chiral condensate ¯ψψ as can be seen by inspecting the fermionic generating functional[6]in the externalfields2A,φ,λ,h and sources η,¯η|k| p=1(¯η,ψ0p)(ψ†0p,η)det′(i/D)e− √Z F[A k,φ,λ,hµ,η,¯η]=gφ△φ+λ△λ−hµhµ−12the harmonicfield h is needed for a consistent quantization on the torus,analogous to the harmonic part tµof the gaugefield.On the torus the action(1)is changed to S→S+ √After a covariant gaugefixing (3)is promoted to a well defined quantity.From (2)it is clear that only the trivial topological sector contributes to the partition function.Then Z F [A,φ,λ,h ]equals the determinant of the Dirac operator /D which is related by conformal-and chiral transformations toi ˆ/D ≡ˆγµ(∂µ−2πiL a µ=et µ+g 2h µ−τµδ0,µ.Hatted quantities refer to flat metric and constant gauge potentials.The chemical potential is contained in the last term in a µ.Integrating the chiral-and conformal anomalies [8]we finddet(i /D )=det(i ˆ/D )exp [12π √△G ],(4)whereS L =1△R is the Liouville action and G =g 2ϕ+eφ.One must be careful in computingthe hatted determinant since the gauge potential is complex.This has been done in [9]with the resultdet(i ˆ/D )=12πVe β|η(τ)|412(−△+m 2γ)exp (1g and m 2γ=e 22π+g 22is the dynamically generated ”photon”mass.This result already indicates that in the trivial topological sector the theory (1)should be equivalent to a free,massive,neutral,boson even in curved space-time.Note that the mass depends on g 2.In particular (6)shows that only the transversal part of the current-current interaction contributes to the mass renormalization in the3Thirring model.Note also that the chemical potential does not appear in thefinal result for the partition function.This may not come as a surprise, because of the equivalence to a uncharged boson.Also∂µZ[µ]=0is the only result consistent with Gauss’s law.We consider this consistency as a confirmation of our definition of the fermionic determinant which differs from previous ones in the literature[10].The non-minimal coupling to gravity(for g3=0)contributes to the gravitational anomaly and therefore affects the intensity of the Hawking radiation3Chiral CondensateThe chiral condensate ¯ψψ is the order parameter for the chiral symmetry breaking,responsible for the mass term in(6).Here we evaluate the depen-dence of the order parameter on temperature and curvature.Recalling(2) we see that only configurations within the topological sectors k=±1can contribute to this expectation value.More precicely¯ψ(x)P+ψ(x) =12(1+γ5)is the projector on states with positive chirality.Z0has been computed in the previous section(6).The generalization of(4)to non-zero k readsdet′(i/D)=det Nψ24πS L)·exp(1gG1V √ˆV2σˆψp0+(x)andχ(x)satisfies the differential equation√g2πˆg2πAll information about the harmonics and the chemical potential is contained in the zero modes.However,d2t det′(iˆ/D)ψ†01ψ01=12βLand hence ψ†P+ψ = ˆV|η(τ)|2e−2π2/e2V+2π/ˆV √g 1π△)ϕ−e2πφ△ϕ .A formal calculation of the resulting Gaussian integrals yieldsψ†P+ψ = ˆV|η(τ)|2e−2π2/e2V+2π/ˆV √e2K(x,x)]exp[2πg22△2−m2γ△|y =12πlog[2π|η(τ)|2For σ=0the Green’s function K has been computed in[6].Substitution of this Green’s function leads,after removing the infrared cut-off,to the following exact formula for the chiral condensate on flat spaceψ†P +ψ β=−Tm γ2π+g 22exp −π2m γ2π+g 22F ,(12)whereF (β)= n>0 14π+π4π2g 22/(2π+g 22)exp 2π2πT g 22e 2T for T →∞.(15)Hence the chiral condensate decays exponentially for high temperatures ap-proaching zero assymptotically.The coupling to the pseudoscalars φweakens the effect of the temperature while the scalar field λhas no effect.For the gauged Thirring model this result implies that only the transversal part of the current-current coupling affects the chiral condensate.Finally note that,as the partition function,the chiral condensate does not depend on the chemical potential.How does the gravitational field affect the chiral condensate?To answer this question we need to know the massive Green’s function,entering in (11),6for non-trivial gravitationalfields(for simplicity we assume T=0).Let us first consider a space with constant positive curvature.Then G mγhas been computed explicitely[13].Here we only need the short distance expansion, given byG mγ(x,y)=−18)+ψ(12−α)+O(s2)},(16)whereα2=1R andψ(z)is the Digamma function.Substituting(16)into(11)we end up with the exact formula for the chiral condensate for constant curvatureψ†P+ψ R= ψ†P+ψ R=0·exp π2m2γ)+ψ(12−α)} .(17)The assymptotic expansions for large-and small curvatures are easily worked out inserting the corresponding expansions for the Digamma function[14]. Wefindψ†P+ψ R= ψ†P+ψ R=0·exp −πmγ→0(18) andψ†P+ψ R= ψ†P+ψ R=0·(R2π+g22exp −π4e2γ for R4πmγ.(20) In passing we note that if we compare the prefactors,rather than the expo-nentials,we would writeT eff=R14π√The latteridentification actually coincides(up to factor of2)with the Hawk-ing temperature of free scalars in de Sitter space[15].The correct iden-tification involves the(dynamical)mass of the gaugefield and is therefore not universal.From this observation we learn that the temperature associ-ated with curvature depends on the matter content.Notefinally that the non-minimal coupling(g3)has no effect on the chiral condensate.In Fig. 2we have plotted the chiral condensate for arbitrary constant values of the curvature.For gravitational backgrounds with non-constant curvature we have to refer to perturbative methods for the calculation of the massive Green’s func-tion.Again we only need the short distance expansion of G mγ.For geodesic distances s small compared to m−1γthe massive Green’s function may be approximated by the Seeley DeWitt expansion[16]G m(x,y)∼1∂m2)j H(2)0(ms),(22)where H(2)0is the Hankel function of the second kind and order zero.InparticularH(2)0(z)→22+γ]for z→0.Inserting(22)into(11)we end up with the following expansion for the chiral condensate in an arbitrary backgroundψ†P+ψ R= ψ†P+ψ R=0·exp −πe)2∞ 1a j(x)(j−1)!6R and reproduces the assymptotic behaviour(18).Higher order contributions must be taken into account to uncover the effect of variablecurvature.For this one has to substitute is the corresponding Seeley DeWitt coefficients a j into(23).These have been computed up to j=5[17].4SummaryWe have computed the partition function and the order parameter of the chiral symmetry breaking for a Thirring-like gauge theory.In particular we8find that both,high temperature and high curvature supress the condensate paring the two results,we defined a curvature induced ef-fective temperature which,unlike the Hawking temperature,depends on the matter content and is therefore not universal.Furthermore we have shown that a non-minimal coupling to gravity affects the Hawking radiation while it has no effect on the chiral symmetry breaking.The non-minimal coupling choosen in our model is however not unique and this result is therefore not general.Finally we obtain that a chemical potential for the electric charge does affect neither the partition function,nor the chiral condensate,in con-sitency with Gauss’s law.5AcknowledgementsThis work has been supported by the Swiss National Science Foundation. We would like to thank A.Dettki and H.M¨u ller for discussions. References[1]E.V.Shuryak,The QCD vacuum,hadrons and superdense matter,World Scientific,Singapore,1988.[2]N.D.Birrell and P.C.W.Davies,Phys.Rev.D18(1978)4408.[3]J.Schwinger,Phys.Rev.128(1962)2425.[4]K.Johnson,Nuovo Cim.20(1964)773.[5]A.J.da Silva,M.Gomes and R.K¨o berle,Phys.Rev.D34(1986)504;M.Gomes and A.J.da Silva,Phys.Rev.D34(1986)3916.[6]I.Sachs and A.Wipf,Helv.Phys.Acta65(1992)653.[7]A.Actor,Fortschritte der Phys.35(1987)793.[8]S.Blau,M.Visser and A.Wipf,Int.J.Mod.Phys.A4(1989)1467.[9]I.Sachs,A.Wipf and A.Dettki,preprint ETH-TH/93-10.9[10]D.Z.Freedman and K.Pilch,Phys.Lett.213B(1988)331;D.Z.Freed-man and K.Pilch,Ann.Phys.192(1989)331;S.Wu,Comm.Math.Phys.124(1989)133.[11]S.Coleman,Phys.Rev.11(1975)2088.[12]K.Gawedzky,Conformalfield theory,to appear in Birkh¨a user.[13]T.S.Bunch and P.C.W.Davies,Proc.R.Soc.Lond.A.360(1978)117.[14]M.Abramowitz and I.S.Stegun,Handbook of Mathematical Func-tions,Dover Publications Inc.NY(1972).[15]N.D.Birrell and P.C.W.Davies,Quantumfields in curved space,Cam-bridge Univ.Press,1982.[16]S.M.Christensen,Phys.Rev.D14(1976)2490;M.L¨u scher,Ann.Phys.142(1982)359.[17]P.Gilkey,Invariance theory,the heat equation and Athiyah SingerIndex theorem,Publish or Perish,1984;A.E.M.van deVen,Nucl.Phys.B250(1985)593.10。

Generalization of Lambert's Reflectance Model

Generalization of Lambert's Reflectance Model
gure 1: (a) Real image of a cylindrical clay vase. (b) Image of the vase rendered using the Lambertian re ectance model. In both cases, illumination is from the viewing direction. of the cylindrical vase will decrease as we approach the occluding boundaries on both sides. However, the real vase is very at in appearance with image brightness remaining almost constant over the entire surface. The vase is clearly not Lambertian 1 . This deviation from Lambertian behavior can be signi cant for a variety of real-world materials, such as, concrete, sand, and cloth. An accurate model that describes body re ection from such commonplace surfaces is imperative for realistic image rendering.
which is due to subsurface scattering. Most of the previous work on physically-based rendering has focused on accurate modeling of surface re ectance. They predict ideal specular re ection from smooth surfaces as well as wide directional lobes from rougher surfaces 13]. In contrast, the body component has most often been assumed to be Lambertian. A Lambertian surface appears equally bright from all directions. This model was advanced by Lambert 18] more than 200 years ago and remains one of the most widely used models in computer graphics. For several real-world objects, however, the Lambertian model can prove to be a poor and inadequate approximation to body re ection. Figure 1(a) shows a real image of a clay vase obtained using a CCD camera. The vase is illuminated by a single distant light source in the same direction as the sensor. Figure 1(b) shows a rendered image of a vase with the same shape as the one shown in Figure 1(a). This image is rendered using Lambert's model, and the same illumination direction as in the case of the real vase. As expected, Lambert's model predicts that the brightness

DRG-SC系列信号条件器模块说明说明书

DRG-SC系列信号条件器模块说明说明书

H-23DRG-SC Series Signal ConditionersߜModels Available for Thermocouples,RTDs, DC Voltage and Current, Frequency,Strain Gage Bridge,AC Voltage and CurrentߜField Configurable Input and Output Ranges ߜFive FieldConfigurable Output Ranges: 0-5 V, 0-10 V,0-1 mA, 0-20 mA and 4-20 mAߜSlim Housing Mounts on DIN Rail for High Density Installations ߜ1800 Volts Isolation Between Input, Output and Power SupplyThe DRG Series signal conditioner modules accept a wide variety of input signals such asthermocouples, RTDs, strain gages,DC voltages/currents,AC voltages/currents, frequency and potentiometers and produce a proportional conditioned process output. The inputs and outputs are both field configurable and offer flexible wide ranging capability. The slim housing mounts on a DIN rail and is ideal for high density installation. All modules provide 1800 Vdc isolation between the input, output and power supply.Field ConfigurableOne advantage of the DRG series is the field configurable input andoutput ranges. Each module can be set to a number of ranges by dip switch selection. Wide ranging precision zero and spanpotentiometers provide even further adjustment. The signal conditioners may be set for an almost limitless number of ranges. Rangeadjustment requires the use of a calibrator or reference source.DRG-SC Series$245Basic unitSpecificationsDRG-SC-ACRange (voltage mode):100 mV to 200 VacImpedance (voltage mode):>100KΩOverload (voltage mode):300Vac, max.Range (current mode):10 mA to 100 mAACImpedance (current mode):20 Ω, typicalOvercurrent (current mode):200mAACOvervoltage (current mode):60V rmsFrequency Range:40 to 400 Hz, factory calibrated at 60 Hz Accuracy (including linearity, hysteresis):±0.1% of span, typical;±0.5% of span, maximum. Response Time:(10-90%)250mS., typicalPower:9-30 Vdc, 1.5W typical, 2.5W max.DRG-SC-BGRange:10 mV to ±200 mV Impedance:>1 MΩOvervoltage:400 VRMS max. (intermittent); 264 VRMS, max. (continuous)Accuracy (including linearity, hysteresis):±0.1% typical, ±0.2% max. of range @25°CBridge Excitation:1 to 10 Vdc, 120mA max.Response Time:(10-90%)<200mS., typicalPower:18-30 Vdc, 1.5W typical, 2.5W max.(one 350Ωbridge), 4W max.(four 350Ωbridges)DRG-SC-DCRange (voltage mode): 10 mV to 100 VImpedance (voltage mode):>100KΩOverload (voltage mode):400VRMS, max.Range (current mode):1 mA to 100 mAImpedance (current mode):20 Ω, typicalOvercurrent (current mode):170mA RMS max.Overvoltage (current mode):60Vdc Accuracy (including linearity,hysteresis):<2 mA/20 mV:±0.35%fs, typical; 0.5% max.;>2mA/20 mV:±0.1% fs typical,0.2% max.Response Time: (10-90%)200mS., typicalPower:9-30 Vdc, 1.5 W typical,2.5W maxDRG-SC-FRFrequency Range:2Hz to10,000HzAmplitude Range:50 mV to150VRMSAccuracy (including linearity,hysteresis):±0.1% of selectedrangeImpedance:>10KΩOver-Voltage:180V rms, max.Over-Range:20Khz, max.Response Time:(10-90%):500mSec., or 100 times the periodof the full scale frequency.Power:9-30Vdc, 1.5W typical,2.5W maxDRG-SC-PTResistance (End to End): 100 Ωup to 100 KΩAccuracy (including linearity,hysteresis):±0.1% maximum@25°CInput Impedance: >1 MΩInput Excitation:500 mV, 5 mAmaximum driveResponse Time:(10-90%)<200mS., typicalDRG-SC-RTD:Sensor Types:RTD, Pt100, Pt500,Pt1000 (a = 0.00385 or 0.00392);Cu10, Cu25, Cu100Sensor Connection:3 wireRange: See Range TableAccuracy (including linearity,hysteresis):±0.1% typical, ±0.2%max. the maximum inputtemperature range @ 25°C, 0 Ωlead resistance.Excitation Current:<2 mA forPt100, Pt500, Pt1000; <5 mA forCu100; <10 mA for Cu10, Cu25Leadwire Resistance:40% of basesensor resistance or 100 Ω(whichever is less), max. per lead.Leadwire Effect:Less than 1% ofthe maximum input temperaturespan.Response Time:(10-90%)200mS., typicalPower:9-30 Vdc(DRG-SC-BG: 18-30 Vdc),1.5 W typical,2.5W maxDRG-SC-TCSensor Types:J, K, T, R, S, E, BRanges:See Range TableAccuracy:J±2°C (-200 to 750°C)K±5°C (-200 to -140°C)±2°C (-140 to 1250°C)±4°C (1250 to 1370°C)E±2.5°C (-150 to 1000°C)T±3°C (-150 to 400°C)R & S±6°C (50 to 1760°C)B±5°C (500 to 1820°C)Bias Current (burnout detection):<1.5 microampImpedance:>1 MΩOvervoltage: ±10 V differentialResponse Time (10 to 90%):500mSec. Typical.Power:9-30 Vdc, 1.5 W typical,2.5W maxSPECIFICATIONS COMMON TOALL MODULESOutput*Voltage Output:Output:0-5 V, 0-10 VImpedance:<10 ΩDrive:10 mA max.Current OutputOutput:0-1 mA, 0-20 mA,4-20 mACompliance:0-1 mA; 7.5 V, max.(7.5 KΩ)0-20 mA: 12 V, max.(600 Ω)4-20 mA: 12 V, max.(600 Ω)Isolation:1800 Vdc between inputoutput and power.Mounting:Standard 32 mm or35mm DIN railESD Susceptibility:MeetsIEC801-2, level 2 (4 KV)Humidity (Non-Condensing):Operating:15 to 95% (@45°C),Soak:90% for 24 hours (@65°C)Temperature Range:Operating: 0to 55°C (32 to 131°F),Storage: -25 to 70°C (-13 to 158°F)*DRG-SC-DC-B has a ±5 V and ±10 Voutput onlyH-24H-25DRG-SC-TC Thermocouple Input Signal ConditionerThe DRG-SC-TC is a DIN rail mount thermocouple input signal conditioner. It can be field configured for over 60 different thermocouple temperature ranges.The output is linear to temperature and can be set for either 0-5 V, 0-10V, 0-1 mA, 0-20 mA or 4-20 mA. Zero and span pots allow 50% adjustability of offset and span turn down within each of the ranges. For example the 500-1000°C range could be offset and turned down to provide a 4-20 mA signal representing 750-1000°C.Ordering Example:DRG-SC-TC Thermocouple input signal conditioner $285.$285Lifetime WarrantyH-26DRG-SC-DCDC Input Signal ConditionerDRG-SC-BG Bridge/Strain Gage Input Signal ConditionerThe DRG-SC-BG is a DIN rail mount bridge or strain gage input signal conditioning module. The field configurable input and output offers flexible, wideranging capability for bridge or strain gage applications from 0.5 mV/V to over 50 mv/V. Wide ranging,precision zero and span pots allow 50% adjustability of offset and gain within each of the 11 switch selectable input ranges. The output can be set for either 0-5 V, 0-10 V, 0-1 mA, 0-20 mA or 4-20 mA. This flexibility,combined with an adjustable (1 to 10 Vdc) bridge excitation source, provides the user a reliable,accurate instrument to isolate and condition virtually any bridge or strain gage input.INPUT RANGES0-10 mV, 0-20 mV, 0-50 mV, 0-100 mV, 0-200 mV,±5mV, ±10 mV, ±20 mV, ±50 mV, ±100 mV, ±200 mVOrdering Example:DRG-SC-BG bridge input signal conditioner $285.The DRG-SC-DC is a DIN rail mount DC voltage and current input signal conditioning module. The input can be field configured for any one 12 voltage ranges from 10 mV to 100 V or 6 current ranges from 1mA to 100 mA. The output is linear to the input and can be set to either 0-5 V, 0-10 V, 0-1 mA, 0-20mA or 4-20mA for the DRG-SC-DC-U (unipolar outputs) and -5 V to +5 V or -10 V to +10 V for the DRG-SC-DC-B (bipolar outputs). Zero and span pots allow 50%adjustability of offset and span turn down within each of the ranges. For example the 0-2 mA input range could be turned down to 0-1 mA and provide a full scale output signal (e.g. 4-20 mA).INPUT RANGES (UNIPOLAR AND BIPOLAR)Voltages:20 mV, 50 mV, 100 mV, 200 mV 500 mV, 1 V, 2 V, 5 V, 10V, 25 V, 50 V, 100 VCurrent:2 mA, 5 mA, 10 mA, 20mA, 50 mA, 100 mAComes with operator’s manual.Ordering Example: DRG-SC-DC-U DC voltage/current input signal conditioner $245.$245$285Lifetime WarrantyLifetime WarrantyHH-27Comes with operator’s manual.Ordering Example:DRG-SC-RTD RTD input signal conditioner $285.DRG-SC-RTD RTD Input Signal Conditioner$285Lifetime WarrantyInput Ranges (°C):PT100, PT500 & PT1000:-18 to 500, -18 to 600Cu10, Cu25 & Cu100:-200 to -100, -100 to 260, -100 to 100, -50 to 50, -18 to 50, -18 to 100, -18 to 260DRG-SC-FR Frequency Input Signal ConditionerThe DRG-SC-FR is a DIN rail mount frequency input signal conditioning module. The field configurable input and output offers flexible, wide ranging capability for a variable frequency drives, magnetic pickups, turbine meters and other pulse or frequency outputtransducers. The output can be set for either 0-5 V,0-10 V, 0-1 mA, 0-20 mA or 4-20 mA. The DRG-SC-FR can be configured for virtually any frequency input to DC signal output within the ranges specified.Calibration utilizes technology where the user simply applies minimum and maximum input frequencies,touching a recessed button to configure the minimum and maximum output range.Input Range:2 Hz to 10,000 Hz, 50 mVp to 150 VrmsOrdering Example:DRG-SC-FR frequency input signal conditioner $285.H-28DRG-SC-PT Potentiometer Input Signal ConditionerOrdering Example:DRG-SC-AC Potentiometer input signal conditioner $285.The DRG-SC-PT is a DIN rail mount potentiometerinput signal conditioning module.The input provides a constant voltage and is designedto accept any three-wire potentiometer from 100 Ωto100 KΩ. The field configurable output can be set foreither 0-5 V, 0-10 V, 0-1 mA, 0-20 mA or 4-20 mA.Wide ranging, precision zero and span pots, used inconjunction with DIP switches, allow 80% adjustabilityof offset and gain to transmit a full scale output fromany 20% portion of the potentiometer input.INPUT RANGE100 Ωto 100 KΩ$285LifetimeWarrantyH。

基于交通时空特征的车辆全局路径规划算法

基于交通时空特征的车辆全局路径规划算法

ISSN 1674-8484汽车安全与节能学报, 第12卷第1期, 2021年J Automotive Safety and Energy, Vol. 12 No. 1, 2021基于交通时空特征的车辆全局路径规划算法杜 茂,杨 林*,金 悦,涂家毓(上海交通大学机械与动力工程学院,上海 200240,中国)摘要:为降低混合动力汽车(HEV)的出行时间和出行能耗,提出了一种基于时空动态交通信息的路径规划算法。

分析了影响车辆通行时间和全程最低能耗的因素。

一种基于广义回归网络(GRNN)模型,拟合计算了道路通行时间以及整体路径的全程能耗。

构建了基于并行A*算法的车辆路径规划算法,为确定起终点位置后的车辆,规划了一条耗时更短、更加节能的路径。

进行了仿真对比试验。

结果表明:相比于依据平均车速与道路功率的计算方法,该算法能够获得更优的出行路径,可降低车辆能耗11%以上,缩短行车时间13%以上。

因而,该算法可为车辆规划更优的路径。

关键词:混合动力汽车(HEV);城市交通;路径规划;时空搜索中图分类号: U 469.7 文献标识码: A DOI: 10.3969/j.issn.1674-8484.2021.01.005 Vehicle global path planning algorithm based on spatio-temporal characteristics of trafficDU Mao, YANG Lin*, JIN Yue, TU Jiayu(School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China) Abstract: A path planning algorithm based on spatiotemporal dynamic traffic information was proposed toreduce the travel time and energy consumption of hybrid electric vehicles (HEV). The factors that affect thevehicle travel time and the minimum energy consumption in the whole path were analyzed. The travel timeand the path energy consumption are calculated based on the generalized regression network (GRNN) model.A vehicle path planning algorithm based on parallel A * algorithm was constructed to plan a shorter time-consuming or more energy-saving path for vehicles after determining the starting and ending positions. Thevirtual simulation test was implemented. The results show that the proposed algorithm can obtain a better travel path and reduce vehicle energy consumption by more than 11% or driving time by more than 13% comparedwith the calculation methods through average speed or power parameters. Therefore, the proposed algorithm can plan a better route for vehicles.Key words: hybrid electric vehicle (HEV); urban traffic; path planning; space-time searching收稿日期 / Received :2020-11-11。

生成对抗网络人脸生成及其检测技术研究

生成对抗网络人脸生成及其检测技术研究

1 研究背景近年来,AIGC(AI Generated Content)技术已经成为人工智能技术新的增长点。

2023年,AIGC开启了人机共生的时代,特别是ChatGPT的成功,使得社会对AIGC的应用前景充满了期待。

但AIGC在使用中也存在着诸多的风险隐患,主要表现在:利用AI生成不存在的虚假内容或篡改现有的真实内容已经达到了以假乱真的效果,这降低了人们对于虚假信息的判断力。

例如,2020年,MIT利用深度伪造技术制作并发布了一段美国总统宣布登月计划失败的视频,视频中语音和面部表情都高度还原了尼克松的真实特征,成功地实现了对非专业人士的欺骗。

人有判断力,但AI没有,AI 生成的内容完全取决于使用者对它的引导。

如果使用者在使用这项技术的过程中做了恶意诱导,那么AI所生成的如暴力、极端仇恨等这样有风险的内容会给我们带来很大的隐患。

因此,对相关生成技术及其检测技术的研究成为信息安全领域新的研究内容。

本文以A I G C在图片生成方面的生成技术为目标,分析现有的以生成对抗网络(G e n e r a t i v e Adversarial Network,GAN)为技术基础的人脸生成技术。

在理解GAN的基本原理的同时,致力于对现有的人像生成技术体系和主要技术方法进行阐述。

对于当前人脸伪造检测的主流技术进行综述,并根据实验的结果分析检测技术存在的问题和研究的方向。

2 GAN的基本原理GAN由Goodfellow等人[1]于2014年首次提出。

生成对抗网络人脸生成及其检测技术研究吴春生,佟 晖,范晓明(北京警察学院,北京 102202)摘要:随着AIGC的突破性进展,内容生成技术成为社会关注的热点。

文章重点分析基于GAN的人脸生成技术及其检测方法。

首先介绍GAN的原理和基本架构,然后阐述GAN在人脸生成方面的技术模式。

重点对基于GAN在人脸语义生成方面的技术框架进行了综述,包括人脸语义生成发展、人脸语义生成的GAN实现。

Generalizedadditivemixedmodels

Generalizedadditivemixedmodels

Generalized Additive Mixed ModelsInitial data-exploratory analysis using scatter plots indicated a non linear dependence of the response on predictor variables. To overcome these difficulties, Hastie and Tibshirani (1990) proposed generalized additive models (GAMs). GAMs are extensions of generalized linear models (GLMs) in which a link function describing the total explained variance is modeled as a sum of the covariates. The terms of the model can in this case be local smoothers or simple transformations with fixed degrees of freedom (e.g. Maunder and Punt 2004). In general the model has a structure of:Where and has an exponential family distribution. is a response variable, isa row for the model matrix for any strictly parametric model component, is the correspondingparameter vector, and the are smooth functions of the covariates, .In regression studies, the coefficients tend to be considered fixed. However, there are cases in which it makes sense to assume some random coefficients. These cases typically occur in situations where the main interest is to make inferences on the entire population, from which some levels are randomly sampled. Consequently, a model with both fixed and random effects (so called mixed effects models) would be more appropriate. In the present study, observations were collected from the same individuals over time. It is reasonable to assume that correlations exist among the observations from the same individual, so we utilized generalized additive mixed models (GAMM) to investigate the effects of covariates on movement probabilities. All the models had the probability of inter-island movement obtained from the BBMM as the dependent term, various covariates (SST, Month, Chlorophyll concentration, maturity stage, and wave energy) as fixed effects, and individual tagged sharks as the random effect. The GAMM used in this study had Gaussian error, identity link function and is given as:Where k = 1, …q is an unknown centered smooth function of the k th covariate andis a vector of random effects following All models were implemented using the mgcv (GAM) and the nlme (GAMM) packages in R (Wood 2006, R Development Core Team 2011).Spatially dependent or environmental data may be auto-correlated and using models that ignore this dependence can lead to inaccurate parameter estimates and inadequate quantification of uncertainty (Latimer et al., 2006). In the present GAMM models, we examined spatial autocorrelation among the chosen predictors by regressing the consecutive residuals against each other and testing for a significant slope. If there was auto-correlation, then there should be a linear relationship between consecutive residuals. The results of these regressions showed no auto-correlation among the predictors.Predictor terms used in GAMMsPredictor Type Description Values Sea surface Continuous Monthly aver. SST on each of the grid cells 20.7° - 27.5°C Chlorophyll a Continuous Monthly aver. Chlo each of grid cells 0.01 – 0.18 mg m-3 Wave energy Continuous Monthly aver. W. energy on each of grid cells 0.01 – 1051.2 kW m-1Month Categorical Month the Utilization Distributionwas generated January to December (1-12)Maturity stage Categorical Maturity stage of shark Mature male TL> 290cmMature female TL > 330cmDistribution of residual and model diagnosticsThe process of statistical modeling involves three distinct stages: formulating a model, fitting the model to data, and checking the model. The relative effect of each x j variable over the dependent variable of interest was assessed using the distribution of partial residuals. The relative influence of each factor was then assessed based on the values normalized with respect to the standard deviation of the partial residuals. The partial residual plots also contain 95% confidence intervals. In the present study we used the distribution of residuals and the quantile-quantile (Q-Q) plots, to assess the model fits. The residual distributions from the GAMM analyses appeared normal for both males and females.MalesResiduals distribution ResidualsF r e q u e n c y-202402004006008001000120-4-2024-2024Q-Q plotTheorethical quantilesS a m p l e q u a n t i l e sFemalesHastie, T.J., and R.J. Tibshirani. 1990. Generalized Additive Models. CRC press, Boca Raton,FL. Latimer, A. M., Wu, S., Gelfand, A. E., and Silander, J. A. 2006. Building statistical models toanalyze species distributions. Ecological Applications, 16: 33–50. Maunder, M.N., and A.E. Punt. 2004. Standardizing catch and effort: a review of recentapproaches. Fisheries Research 70: 141-159. Wood, S.N. 2006. Generalized Additive Models: an introduction with R. Boca Raton, CRCPress.。

generalized maxwell model

generalized maxwell model

Generalized Maxwell Model1. IntroductionThe Maxwell model is a linear viscoelastic model used to describe the rheological behavior of viscoelastic materials. It consists of a spring and a dashpot in parallel, and ismonly used to model the behavior of polymers, gels, and otherplex fluids. In this article, we will explore the generalized Maxwell model, which is an extension of the original Maxwell model and provides a more accurate representation of the viscoelastic properties of materials.2. The Maxwell modelThe Maxwell model, first proposed by James Clerk Maxwell in the 19th century, consists of a spring and a dashpot in parallel. The spring represents the elastic behavior of the material, while the dashpot represents the viscous behavior. The constitutive equation of the Maxwell model is given by:σ(t) = Eε(t) + ηdε(t)/dtWhere σ(t) is the stress, ε(t) is the strain, E is the elastic modulus, η is the viscosity, and dε(t)/dt is the rate of strain. The Maxwellmodel is simple and easy to understand, but it fails to capture the nonlinear viscoelastic behavior of many materials.3. The generalized Maxwell modelTo ovee the limitations of the original Maxwell model, the generalized Maxwell model introduces multiple springs and dashpots in parallel, each with its own elastic modulus and viscosity. This allows for a more accurate representation of theplex viscoelastic behavior of materials. The constitutive equation of the generalized Maxwell model is given by:σ(t) = ∑(Eiε(t) + ηidε(t)/dt)Where the summation is taken over all the springs and dashpots in the model, and Ei and ηi are the elastic moduli and viscosities of the individual elements. By including multiple elements with different relaxation times, the generalized Maxwell model can accurately describe the behavior of materials with nonlinear viscoelastic properties.4. Applications of the generalized Maxwell modelThe generalized Maxwell model has found wide applications in various fields, including polymer science, biomedicalengineering, and materials science. It has been used to study the viscoelastic behavior of polymers, gels, and foams, and to design materials with specific viscoelastic properties. In biomedical engineering, the model has been used to study the mechanical behavior of soft tissues and to develop new biomaterials for tissue engineering. In materials science, the model has been used to characterize the viscoelastic properties ofposites and to optimize their performance.5. Comparison with other viscoelastic modelsThe generalized Maxwell model is just one of many viscoelastic models used to describe the rheological behavior of materials. Other popular models include the Kelvin-Voigt model, the Burgers model, and the Zener model. Each of these models has its own advantages and limitations, and the choice of model depends on the specific material and the behavior of interest. The generalized Maxwell model is particularly useful for materials withplex viscoelastic behavior, as it allows for a more detailed description of the relaxation processes.6. ConclusionIn conclusion, the generalized Maxwell model is a powerful tool for describing the viscoelastic behavior of materials. Byextending the original Maxwell model to include multiple springs and dashpots, the generalized Maxwell model provides a more accurate representation of the nonlinear viscoelastic properties of materials. It has found wide applications in various fields and has contributed to our understanding of the mechanical behavior ofplex fluids and solids. As our knowledge of viscoelastic materials continues to grow, the generalized Maxwell model will undoubtedly remain an important tool for researchers and engineers alike.。

Stata 数据分析软件 - 选择模型示例说明书

Stata 数据分析软件 - 选择模型示例说明书

Title Example45g—Heckman selection modelDescription Remarks and examples References Also seeDescriptionTo demonstrate selection models,we will use the following data:.use https:///data/r18/gsem_womenwk(Fictional data on women and work).summarizeVariable Obs Mean Std.dev.Min Maxage2,00036.2088.286562059educ2,00013.084 3.0459121020married2,000.6705.470149201children2,000 1.6445 1.39896305wage1,34323.69217 6.305374 5.8849745.80979.notes_dta:1.Fictional data on2,000women,1,343of whom work.2.age.......age in yearsc......years of schooling4.married...1if married spouse present5.children..#of children under12years6.wage......hourly wage(missing if not working)See Structural models8:Dependencies between response variables and Structural models9: Unobserved inputs,outputs,or both in[SEM]Intro5for background.Remarks and examples Remarks are presented under the following headings:The Heckman selection model as an SEMFitting the Heckman selection model as an SEMTransforming results and obtaining rhoFitting the model with the BuilderThe Heckman selection model as an SEMWe demonstrate below how gsem can be used tofit the Heckman selection model(Gronau1974;Lewis1974;Heckman1976)and produce results comparable to those of Stata’s dedicated heckman command;see[R]heckman.Our purpose is not to promote gsem as an alternative to heckman.We have two other purposes.One is to show that gsem can be used to generalize the Heckman selection model to response functions other than linear and,in addition or separately,to include multilevel effects when such effects are present.The other is to show how Heckman selection models can be included in more complicated SEM s.12Example 45g —Heckman selection modelFor those unfamiliar with this model,it deals with a continuous outcome that is observed only when another equation determines that the observation is selected,and the errors of the two equations are allowed to be correlated.Subjects often choose to participate in an event or medical trial or even the labor market,and thus the outcome of interest might be correlated with the decision to participate.Heckman won a Nobel Prize for this work.The model is sometimes cast in terms of female labor supply,but it obviously has broader application.Nevertheless,we will consider a female labor-supply example.Women are offered employment at a wage of w ,w i =X i β+ iNot all women choose to work,and w is observed only for those women who do work.Women choose to work ifZ i γ+ξi >0wherei ∼N (0,σ2)ξi ∼N (0,1)corr ( ,ξ)=ρMore generally,we can think of this model as applying to any continuously measured outcome w i ,which is observed only if Z i γ+ξi >0.The important feature of the model is that the errors ξi of the selection equation and the errors i of the observed-data equation are allowed to be correlated.The Heckman selection model can be recast as a two-equation SEM —one linear regression (for the continuous outcome)and the other censored regression (for selection)—and with a latent variable L i added to both equations.The latent variable is constrained to have variance 1and to have coefficient 1in the selection equation,leaving only the coefficient in the continuous-outcome equation to be estimated.For identification,the variance from the censored regression will be constrained to be equal to that of the linear regression.The results of doing this are the following:tent variable L i becomes the vehicle for carrying the correlation between the two equations.2.All the parameters given above,namely,β,γ,σ2,and ρ,can be recovered from the SEM estimates.3.If we call the estimated parameters in the SEM formulation β∗,γ∗,and σ2∗,and let κdenote the coefficient on L i in the continuous-outcome equation,thenβ=β∗γ=γ∗/σ2∗+1σ2=σ2∗+κ2ρ=κ/(σ2∗+κ2)(σ2∗+1)This parameterization places no restriction on the range or sign of ρ.See Skrondal and Rabe-Hesketh (2004,107–108).Example45g—Heckman selection model3 Fitting the Heckman selection model as an SEMWe wish tofitWhat makes this a Heckman selection model is1.the inclusion of latent variable L in both the continuous-outcome(wage)equation and thecensored-outcome selection equation;2.constraining the selected<-L path coefficient to be1;3.constraining the variance of L to be1;and4.constraining the error variances to be equal.Before we canfit this model,we need to create new variables selected and notselected.selected will equal0if the woman works(wage is not missing)and missing otherwise.notselected is the complement of selected:it equals0if the woman does not work(wage is missing)and missing otherwise.selected and notselected will be used as the dependent variables in the censored regression,providing the equivalent of a scaled probit regression..generate selected=0if wage<.(657missing values generated).generate notselected=0if wage>=.(1,343missing values generated).tabulate selected notselected,missingnotselectedselected0.Total001,3431,343.6570657Total6571,3432,000Old-time Stata users may be worried that because wage is missing in so many observations,namely, all those corresponding to nonworking women,there must be something special we need to do so that gsem uses all the data.There is nothing special we need to do.gsem counts missing values on an equation-by-equation basis,so it will use all the data for the censored regression part of the model while simultaneously using only the working-woman subsample for the continuous-outcome(wage) part of the model.We use all the data for the censored regression because gsem understands the meaning of missing values in the censored dependent variables so long as one of them is nonmissing.4Example45g—Heckman selection modelTofit this model in command syntax,we type.gsem(wage<-educ age L)>(selected<-married children educ age L@1,>family(gaussian,udepvar(notselected))),var(L@1e.wage@a e.selected@a)Fitting fixed-effects model:Iteration0:Log likelihood=-5752.6506Iteration1:Log likelihood=-5260.9961Iteration2:Log likelihood=-5209.2571Iteration3:Log likelihood=-5208.9039Iteration4:Log likelihood=-5208.9038Refining starting values:Grid node0:Log likelihood=-5208.7006Fitting full model:Iteration0:Log likelihood=-5208.5322(not concave)Iteration1:Log likelihood=-5208.0269Iteration2:Log likelihood=-5202.872(not concave)Iteration3:Log likelihood=-5202.0258Iteration4:Log likelihood=-5198.6178Iteration5:Log likelihood=-5193.0576Iteration6:Log likelihood=-5191.8655Iteration7:Log likelihood=-5178.5058Iteration8:Log likelihood=-5178.3095Iteration9:Log likelihood=-5178.3046Iteration10:Log likelihood=-5178.3046Generalized structural equation model Number of obs=2,000 Response:wage Number of obs=1,343 Family:GaussianLink:IdentityLower response:selected Number of obs=2,000 Upper response:notselected Uncensored=0 Family:Gaussian Left-censored=657 Link:Identity Right-censored=1,343Interval-cens.=0 Log likelihood=-5178.3046(1)[selected]L=1(2)-[/]var(e.wage)+[/]var(e.selected)=0(3)[/]var(L)=1Coefficient Std.err.z P>|z|[95%conf.interval] wageeduc.9899512.053255218.590.000.8855729 1.094329age.2131282.02060210.350.000.172749.2535074L 5.923736.184681832.080.000 5.561767 6.285706 _cons.4859114 1.0768650.450.652-1.624705 2.596528 selectedmarried.6242746.1054319 5.920.000.4176319.8309173children.6152095.06520029.440.000.4874196.7429995 educ.0781542.0162868 4.800.000.0462327.1100757age.0511983.0066377.710.000.0381901.0642066L1(constrained)_cons-3.493217.3730379-9.360.000-4.224357-2.762076var(L)1(constrained)var(e.wage).9664635.2689653.560141 1.66753 var(e.sele~d).9664635.2689653.560141 1.66753Example 45g —Heckman selection model 5Notes:1.Some of the estimated coefficients and parameters above will match those reported by the heckman command and others will not.The above parameters are in the transformed structural equation modeling metric.That metric can be transformed back to the Heckman metric and results will match.The relationship to the Heckman metric isβ=β∗γ=γ∗/σ2∗+1σ2=σ2∗+κ2ρ=κ/(σ2∗+κ2)(σ2∗+1)2.βrefers to the coefficients on the continuous-outcome (wage)equation.We can read those coefficients directly,without transformation except that we ignore the wage <-L path:wage =0.9900educ +0.2131age +0.48593.γrefers to the selection equation,and because γ=γ∗/√σ2∗+1,we must divide the reportedcoefficients by the square root of σ2∗+1.What has happened here is that the scaled probit hasvariance σ2∗+1,and we are merely transforming back to the standard probit model,which has variance 1.The results arePr (selected =0)=Φ(0.4452married +0.4387children +0.0557educ +0.0365age −2.4910)4.To calculate ρ,we first calculate σ2=σ2∗+κ2and then calculate ρ=κ/ σ2(σ2∗+1):σ2=0.9664+5.92372=36.0571ρ=5.9237/σ2(.9664+1)=0.70355.These transformed results match the results that would have been reported had we typed.heckman wage educ age,select(married children educ age)(output omitted )6.There is an easier way to obtain the transformed results than by hand,and the easier way provides standard errors.That is the subject of the next section.Transforming results and obtaining rhoWe can use Stata’s nlcom command to perform the transformations we made by hand above,and we can obtain standard errors.Let’s start by obtaining σ2and ρ.To remind you,the formulas areσ2=σ2∗+κ2ρ=κ/σ2(σ2∗+1)We must describe these two formulas in a way that nlcom can understand.The Stata notation forparameters σ2∗and κfit by gsem isσ2∗:b[/var(e.wage)]κ:b[wage:L]6Example45g—Heckman selection modelWe cannot remember that notation;however,we can type gsem,coeflegend to be reminded.We now have all that we need to obtain the estimates ofσ2andρ.Because heckman reportsσrather thanσ2,we will tell nlcom to report the sqrt(σ2):.nlcom(sigma:sqrt(_b[/var(e.wage)]+_b[wage:L]^2))>(rho:_b[wage:L]/(sqrt((_b[/var(e.wage)]+1)*(_b[/var(e.wage)]>+_b[wage:L]^2))))sigma:sqrt(_b[/var(e.wage)]+_b[wage:L]^2)rho:_b[wage:L]/(sqrt((_b[/var(e.wage)]+1)*(_b[/var(e.wage)]>+_b[wage:L]^2)))Coefficient Std.err.z P>|z|[95%conf.interval]sigma 6.004758.165647136.250.000 5.680095 6.32942rho.703489.051186113.740.000.603166.8038119 The output above nearly matches what heckman reports.heckman does not report the test statistics and p-values for these two parameters.In addition,the confidence interval that heckman reports for ρwill differ slightly from the above and is better.heckman uses a method that will not allowρto be outside of−1and1,whereas nlcom is simply producing a confidence interval for the calculation we requested and in absence of the knowledge that the calculation corresponds to a correlation coefficient.The same applies to the confidence interval forσ,where the bounds are0and infinity.To obtain the coefficients and standard errors for the selection equation,we type.nlcom(married:_b[selected:married]/sqrt(_b[/var(e.wage)]+1))>(children:_b[selected:children]/sqrt(_b[/var(e.wage)]+1))>(educ:_b[selected:educ]/sqrt(_b[/var(e.wage)]+1))>(age:_b[selected:age]/sqrt(_b[/var(e.wage)]+1))married:_b[selected:married]/sqrt(_b[/var(e.wage)]+1)children:_b[selected:children]/sqrt(_b[/var(e.wage)]+1)educ:_b[selected:educ]/sqrt(_b[/var(e.wage)]+1)age:_b[selected:age]/sqrt(_b[/var(e.wage)]+1)Coefficient Std.err.z P>|z|[95%conf.interval]married.445177.0673953 6.610.000.3130847.5772693children.4387126.027778815.790.000.3842671.4931581educ.0557326.0107348 5.190.000.0346927.0767725age.0365101.00415348.790.000.0283696.0446505 The above output matches what heckman reports.Fitting the model with the BuilderUse the diagram in Fitting the Heckman selection model as an SEM above for reference.1.Open the dataset and create the selection variable.In the Command window,type.use https:///data/r18/gsem_womenwk.generate selected=0if wage<..generate notselected=0if wage>=.2.Open a new Builder diagram.Select menu item Statistics>SEM(structural equation modeling)>Model building and estimation.Example45g—Heckman selection model73.Put the Builder in gsem mode by clicking on the button.4.Create the independent variables.Select the Add observed variables set tool,,and then click in the diagram about one-fourth of the way in from the left and one-fourth of the way up from the bottom.In the resulting dialog box,a.select the Select variables radio button(it may already be selected);e the Variables control to select the variables married,children,educ,and age in thisorder;c.select Vertical in the Orientation control;d.click on OK.If you wish,move the set of variables by clicking on any variable and dragging it.5.Create the generalized response for selection.a.Select the Add generalized response variable tool,.b.Click about one-third of the way in from the right side of the diagram,to the right of themarried rectangle.c.In the Contextual Toolbar,select Gaussian,Identity in the Family/Link control(it mayalready be selected).d.In the Contextual Toolbar,select selected in the Variable control.e.In the Contextual Toolbar,click on the Properties...button.f.In the resulting Variable properties dialog box,click on the Censoring...button in theVariable tab.g.In the resulting Censoring dialog box,select the Interval-measured,depvar is lower boundaryradio button.In the resulting Interval-measured box below,use the Upper bound control to select the variable notselected.h.Click on OK in the Censoring dialog box,and then click on OK in the Variable propertiesdialog box.The Details pane will now show selected as the lower bound and notselected as the upper bound of our interval measure.6.Create the endogenous wage variable.a.Select the Add observed variable tool,,and then click about one-third of the way in fromthe right side of the diagram,to the right of the age rectangle.b.In the Contextual Toolbar,select wage with the Variable control.7.Create paths from the independent variables to the dependent variables.a.Select the Add path tool,.b.Click in the right side of the married rectangle(it will highlight when you hover over it),and drag a path to the left side of the selected rectangle(it will highlight when you can release to connect the path).c.Continuing with the tool,create the following paths by clickingfirst in the right side ofthe rectangle for the independent variable and dragging it to the left side of the rectangle for the dependent variable:8Example45g—Heckman selection modelchildren->selectededuc->selectedage->selectededuc->wageage->wage8.Clean up the direction of the error terms.We want the error for selected to be above the rectangle and the error for wage to be below the rectangle,but it is likely they have been created in other directions.a.Choose the Select tool,.b.Click in the selected rectangle.c.Click on one of the Error rotation buttons,,in the Contextual Toolbar until the erroris above the rectangle.d.Click in the wage rectangle.e.Click on one of the Error rotation buttons,,in the Contextual Toolbar until the erroris below the rectangle.9.Create the latent variable.a.Select the Add latent variable tool,,and then click at the far right of the diagram andvertically centered between the selected and wage variables.b.In the Contextual Toolbar,type L in the Name control and press Enter.10.Draw paths from the latent variable to each endogenous variable.a.Select the Add path tool,.b.Click in the upper left quadrant of the L oval,and drag a path to the right side of theselected rectangle.c.Continuing with the tool,create another path by clickingfirst in the lower-left quadrantof the L oval and dragging a path to the right side of the wage rectangle.11.Place constraints on the variances and on the path from L to selected.a.Choose the Select tool,.b.Click on the L oval.In the Contextual Toolbar,type1in the box and press Enter.c.Click on the error oval attached to the wage rectangle.In the Contextual Toolbar,type a inthe box and press Enter.d.Click on the error oval attached to the selected rectangle.In the Contextual Toolbar,typea in the box and press Enter.e.Click on the path from L to selected.In the Contextual Toolbar,type1in the boxand press Enter.12.Clean up the location of the paths.If you do not like where a path has been connected to its variables,use the Select tool,, to click on the path,and then simply click on where it connects to a rectangle and drag the endpoint.Example45g—Heckman selection model9 13.Estimate.Click on the Estimatebutton,,in the Standard Toolbar,and then click on OK in the resultingGSEM estimation options dialog box.You can open a completed diagram in the Builder by typing.webgetsem gsem_selectReferencesGronau,R.1974.Wage comparisons:A selectivity bias.Journal of Political Economy82:1119–1143.https:///10.1086/260267.Heckman,J.J.1976.The common structure of statistical models of truncation,sample selection and limited dependent variables and a simple estimator for such models.Annals of Economic and Social Measurement5:475–492.Lewis,ments on selectivity biases in wage comparisons.Journal of Political Economy82:1145–1155.Skrondal, A.,and S.Rabe-Hesketh.2004.Generalized Latent Variable Modeling:Multilevel,Longitudinal,and Structural Equation Models.Boca Raton,FL:Chapman and Hall/CRC.Also see[SEM]Example34g—Combined models(generalized responses)[SEM]Example46g—Endogenous treatment-effects model[SEM]Intro5—Tour of models[SEM]gsem—Generalized structural equation model estimation commandStata,Stata Press,and Mata are registered trademarks of StataCorp LLC.Stata andStata Press are registered trademarks with the World Intellectual Property Organizationof the United Nations.Other brand and product names are registered trademarks ortrademarks of their respective companies.Copyright c 1985–2023StataCorp LLC,College Station,TX,USA.All rights reserved.®。

Generalization of the Dick Model

Generalization of the Dick Model
∗ †
mslus@.pl wereszcz@.pl
1
where δ > 0, Λ is a dimensional constant, φ is massless scalar field coupled to a the gauge field Aa µ and Fµν is defined in the standard manner. The action we consider emerges from Dick model [2] with more general coupling between the scalar and the gauge fields. In the contradistinction to the Dick model our scalar field is massless and there are not simple connections between solutions of these models. One can add a potential or a mass term to the action but as far as we do not know how to choose the ground state for the scalar field it seems better and more general to consider the model without potential. Because of the fact that we are interested in the long range behaviour of the fields i.e. in the small energy limit, we neglect the denominator which is present in the original coupling. However, it is possible to consider the full model with the denominator. Then the solutions will contain a well-know part corresponding to the standard Yang-Mills equations. The field equations for (1) take the form: Dµ φ Λ

standardizingmea...

standardizingmea...

CommentaryStandardizing Measures for Early Psychosis: What Are Our Goals?Sophia VinogradovThe former Director of the National Institute for Mental Health, Tom Insel,used to tell this anecdote.He had justfinished presenting new advances in psychiatric neuroscience to par-ents of young people with serious mental illness,when a father raised his hand.“Those may be great discoveries,”the father said.“But we are telling you that the house is onfire and you are telling us the chemical structure of the paint.”If you work clinically with young individuals who are expe-riencing afirst(or second,or third)episode of psychosis,you know that they and their family members want to know only one thing:How do we put out thisfire(i.e.,what treatment will work)?In this issue of Biological Psychiatry:Cognitive Neurosci-ence and Neuroimaging,Öngür et al.(1)note that early psy-chosis research“has seen an explosion of interest and activity in the past decade.accompanied by a growing diversity of tools and approaches.”But how do we turn this activity into true“fire-fighting efforts”that help patients and families? Ideally,we would create data-driven predictive and prescrip-tive analytics in early psychosis that could generalize across samples.This requires ourfield to acquire large,well-harmonized longitudinal data sets of reliable measures from a wide range of individuals who are followed over adequate time periods across different treatments and settings.Many programs around the world have pursued this goal locally, studying diverse measures and treatment models across co-horts from Asia and Australia to Europe and the Americas.Despite this large body of work,high-level evidence sup-porting structured decision making in early psychosis is still limited.We know some of the factors that predict worse“fire danger”but we do not have well-validated algorithms that predict optimal treatment sequences for different individual presentations.In the meantime,we might borrow from oncology and develop objective consensus methods for treatment recommendations(2).In this approach,one obtains treatment decision trees from experts across different sites, using the same terminology and rules across sites(such as specific symptom cut-off scores measured on a reliable in-strument).For example,Figure1presents the decision tree when an early psychosis patient with trauma symptoms pre-sents to the University of California Davis Sacramento Early Diagnosis and Preventative Treatment clinic(3).Combining decision tree information from multiple expert sources,one can assess treatment recommendations for every possible combination of parameters;in this example,relevant parame-ters would likely include gender,age,specific severity of psychosis and trauma symptoms,and level of distress.The ensemble of recommendations from the multiple experts serves as a basis for developing an objective consensus.But even this approach requires that the objective decision-making criteria(such as symptom ratings,distress,functional impair-ment,and biomarkers)are measured in the same way.Thus,in order to achieve the next stage of clinically mean-ingful progress in early psychosis—be it data-driven or expert consensus–driven—we must engage in data harmonization across large numbers of individuals in clinical and research programs(1).To meet this goal,the National Institute of Mental Health has added early psychosis research and clinical care to its Specialty Collections in the PhenX Toolkit.The widespread use of standard Early Psychosis Specialty Collection(EPSC) measures will support measurement-based care and more precise communication among clinical practitioners,along with quality improvement.It can also support the rapid development of objective consensus clinical decision trees,which would be a boon to thefield.As the aggregated data sets grow,they can be mined to discover new insights regarding pathophysiologic and clinical heterogeneity in early psychosis and to provide new leads for treatment development.Öngür et al.(1)present a focused overview of the processes and rationale that went into thefinal selection of measures in the EPSC.The Early Psychosis Working Group was convened in early2016to develop clinical measures and translational research measures.The authors provide a summary table of the EPSC measures,key features,and relevant literature.Note that this table lists4sets of cognitive measures(requiring2hours to obtain),brain anatomical imaging and task functional magnetic resonance imaging,9assessment instruments requiring a trained interviewer or clinician administration(each requiring several minutes to 1.5hours to obtain),and13self-administered measures(each requiring5–20minutes to com-plete).What is not provided is a“hierarchy”of the relative importance of these measures,given a collective goal of creating widespread opportunities for data harmonization. Perhaps thefield is not yet mature enough to agree upon critical common data elements that should be obtained in every sample.Perhaps this will be the goal of future workgroups.Several fruitful translational domains known to play a key role in long-term functional outcomes are not included in the EPSC measures.For example,the assessment of reward responsivity/ effort expenditure/motivated behavior(via self-report or laboratory-based measures)is lacking,despite research indi-cating that deficits in this domain are associated with poorer functioning in early psychosis(4).Social cognition measures are embedded in the Computerized Neurocognitive Battery but are not called out despite their strong association with both moti-vation and functional outcome in psychosis(5).Measures of104ª2019Society of Biological Psychiatry.Published by Elsevier Inc.All rights reserved.https:///10.1016/j.bpsc.2019.11.006 Biological Psychiatry:Cognitive Neuroscience and Neuroimaging January2020;5:4–/BPCNNI ISSN:2451-9022insight are not included,even though clinically they may be among the most important of prognostic indicators.Meta-cognitive capacities in general are likely to also play a key role in outcome and are not captured at present in the EPSC (6).An important question remains:What does the young per-son with a lived experience of psychosis want?What do family members want?It is easier to say what the young person experiencing psychosis does not want.They do not want to have an illness identity.They do not want to come to clinic;they do not want to take medications that make them gain weight;they do not want to endlessly answer questions from treatment providers.On any given clinic day,most of my pa-tients are not able or willing to participate in a lengthy battery of cognitive and self-report measures.If our ultimate goal is to acquire meaningful longitudinal data from a representative sample (and not just from an atypical handful of individuals who have retained a high degree of insight and who are cooperative),then either a streamlined version of EPSC mea-sures or a less burdensome approach to data acquisition will need to be developed.As for family members,they are generally in a state of shock,grief,and denial.While many are deeply interested in understanding more about the illness and how to help their loved one,they are often experiencing information and emotion over-load.I believe it behooves us to share the results of our measures with the people we treat and to create a feedback loop where data are communicated and interpreted with the individual and family,as appropriate.However,this must be done in a manner that brings clarity and does not add to their sense of being overwhelmed.This,too,is an important area for further development.Finally,we need to remember the treatment team.They do not want any process that adds to their already overburdened and complex work flow.Some of them may be interested in measurement-based care,but there are still large barriers to its implementation (7).Most frontline clinicians genuinely wish to understand the illness and make optimal treatment recom-mendations,but this means that the measurement-based in-formation provided to them must be actionable,and we are not quite there yet.Perhaps what lies ahead is a series of incremental steps,of “satis ficing ”strategies.At some point we may be able to home in on a smaller number of reliable standardized measures that are not burdensome to the individual,that do not requirespecializedFigure 1.Example of a decision tree guiding treatment recommendations for a patient presenting with early psychosis symptoms and with traumabining decision tree information from multiple expert sources allows one to develop objective consensus methods according to Putora et al.(2).This requires that the objective decision-making criteria are measured in the same way across all samples.APS,attenuated positive symptoms;CBTp,cognitive behavioral therapy for psychosis;PTSD,posttraumatic stress disorder;SCID,Structured Clinical Interview for DSM;TI-CBTp,trauma-integrated cognitive behavioral therapy for psychosis.[Reproduced with permission from Folk et al.(3)].CommentaryBiological Psychiatry:Cognitive Neuroscience and Neuroimaging January 2020;5:4–/BPCNNI 5staff to acquire,and that show concurrent validity with con-structs of interest and/or have high criterion validity as predictors of outcome,rather than relying on2hours of cognitive assess-ments and6hours of interviews and patient-reported outcome measures.These may well include passive sensor measures, other forms of digital phenotyping,and/or brief gamified ecological momentary assessment sampling and symptom self-reports on smartphones(8,9).It is also highly likely that trial-by-trial data obtained from digital measures will provide computational parameters that are more deeply reflective of clinically relevant brain information processing variations than data obtained from traditional clinical and cognitive measures (10).Such approaches may help identify latent variables or latent classes that are more closely aligned with underlying patho-physiology or neurobehavioral resilience than our current clinical observations.In an ideal future,these kinds of metrics will be easily merged with data from neuroimaging to provide individu-alized decision-making support,and perhaps even to prevent the“fires”from starting in thefirst place.Acknowledgments and DisclosuresSV is on the Scientific Advisory Boards of Alkermes,PsyberGuide,and Mindstrong,and has been a site investigator on a National Institute of Mental Health Small Business Innovation Research grant to Positscience, Inc.,a company with commercial interests in cognitive training software. Article InformationFrom the Department of Psychiatry,University of Minnesota,Minneapolis, Minnesota.Address correspondence to Sophia Vinogradov,M.D.,Department of Psychiatry,University of Minnesota,F282/2A West Building,8393A2450 Riverside Ave.,Minneapolis,MN55454;E-mail:****************.Received Nov14,2019;accepted Nov14,2019.References1.Öngür D,Carter CS,Gur RE,Perkins D,Sawa A,Seidman LJ,et al.(2020):Common data elements for National Institute of Mental Health–funded translational early psychosis research.Biol Psychiatry Cogn Neurosci Neuroimaging5:10–22.2.Putora PM,Panje CM,Papachristofilou A,Dal Pra A,Hundsberger T,Plasswilm L(2014):Objective consensus from decision trees.Radiat Oncol9:270.3.Folk JB,Tully LM,Blacker DM,Liles BD,Bolden KA,Tryon V,et al.(2019):Uncharted waters:Treating trauma symptoms in the context of early psychosis.J Clin Med8:E1456.4.Chang WC,Chu AOK,Treadway MT,Strauss GP,Chan SKW,Lee EHM,et al.(2019):Effort-based decision-making impairment in patients with clinically-stabilizedfirst-episode psychosis and its rela-tionship with amotivation and psychosocial functioning.Eur Neuro-psychopharmacol29:629–642.5.Halverson TF,Orleans-Pobee M,Merritt C,Sheeran P,Fett AK,Penn DL(2019):Pathways to functional outcomes in schizophrenia spectrum disorders:Meta-analysis of social cognitive and neuro-cognitive predictors.Neurosci Biobehav Rev105:212–219.6.Davies G,Greenwood K(2018):A meta-analytic review of the rela-tionship between neurocognition,metacognition and functional outcome in schizophrenia[published online ahead of print Oct31].J Ment Health.7.Lewis CC,Boyd M,Puspitasari A,Navarro E,Howard J,Kassab H,et al.(2019):Implementing measurement-based care in behavioral health:A review.JAMA Psychiatry76:324–335.8.Kumar D,Tully LM,Iosif AM,Zakskorn LN,Nye KE,Zia A,Niendam TA(2018):A mobile health platform for clinical monitoring in early psy-chosis:Implementation in community-based outpatient early psy-chosis care.JMIR Mental Health5:e15.9.Kidd SA,Feldcamp L,Adler A,Kaleis L,Wang W,Vichnevetski K,et al.(2019):Feasibility and outcomes of a multi-function mobile health approach for the schizophrenia spectrum:App4Independence(A4i).PLoS One14:e0219491.10.Cooper JA,Barch DM,Reddy LF,Horan WP,Green MF,Treadway MT(2019):Effortful goal-directed behavior in schizophrenia:Computa-tional subtypes and associations with cognition.J Abnorm Psychol 128:710–722.Commentary6Biological Psychiatry:Cognitive Neuroscience and Neuroimaging January2020;5:4–/BPCNNI。

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a r X i v :h e p -t h /9308067v 1 13 A u g 1993ETH-TH/93-14Generalized Gauged Thirring Model onCurved Space-TimesA.Dettki 1,I.Sachs 2and A.Wipf 21Max-Planck Institut f¨u r Physik,Werner-Heisenberg Institut f¨u r Physik,P.O.Box 401212,Munich,Germany 2Institute for Theoretical Physics,Eidgen¨o ssische Technische Hochschule,H¨o nggerberg,CH-8093Z¨u rich,Switzerland Abstract:We analyse the interacting theory of charged fermions,scalars,pseuso-scalars and photons propagating in 2-dimensional curved spacetime in detail.For certain values of the coupling constants the theory reduces to the gauged Thirring model and for others the Schwinger model incurved spacetime.It is shown that the interaction of the fermions with the pseudo-scalars shields the electromagnetic interaction,and that the non-minimal coupling of the scalars to the gravitational field amplifies the Hawking radiation.We solve the finite temperature and density model by using functional techniques and in particular derive the exact equation of state.The explicit temperature and curvature dependence of the chiral condensateis found.When the electromagnetic field is switched offthe model reduces to a conformal field theory.We determine the physically relevant expectation values and conformal weights of the fundamental fields in the theory.1.IntroductionThe response of physical systems to a change of external conditions is of em-minent importance in physics.In particular the dependence of expectation values on temperature,the particle density,the space region,the imposed boundary conditions or external fields has been widely studied [1].Despite2 A.Dettki,I.Sachs and A.Wipfall these efforts we are still unable to understand,for example,the mecha-nism leading to the spontaneous symmetry breaking of the SU A(N)in low temperature QCD[2].Clearly such subtle effects require a better under-standing of the nonperturbative effects and in particular nonperturbative vacuum sector of gauge theories.From our experience with2-dimensional gauge theories[3]which we suppose to mimic one-flavour QCD[4],we are lead to believe that gaugefields with windings are responsible for the non-vanishing chiral condensate and in particular its temperature dependence.A related problem is how quantum systems behave in a hot and dense en-vironment as it exists or existed in heavy ion collision,neutron stars or the early epochs of the universe[2].On another front there has been much effort to quantize selfinteract-ingfield theories in a background gravitationalfield[5].For example,one is interested whether a black hole still emits thermal radiation when self-interaction is included.Due to general arguments by Gibbons and Perry[6] this question is intimately connected with universality of the second law of thermodynamics.Rather than seeking new partial results for more general and realistis-tic4-dimensional systems we have chosen an idealized2-dimensional model with self-interaction to investigate the questions mentioned and others.It is a theory containing photons1,charged massless fermions,scalars and pseu-doscalars in interaction with themselves and a gravitational background field.The model has the actionS= √4FµνFµν+i¯ψγµ(∇µ−ig1∂µλ+ig2ηνµ∂νφ)ψ+gµν(∂µφ∂νφ+∂µλ∂νλ)−g3Rλ ,(1.1) where Fµνis the electromagneticfield strength,the gamma-matrices in curved space are related to theflat ones asγµ=eµaˆγa,∇µ=∂µ+iωµ−ieAµGeneralized Gauged Thirring Model on Curved Space-Times3 is the generally and gauge covariant derivative containing the U(1)gauge potential and spin connection,ηµν=√g22)via the Schinger mechanism.For2finite volumes the theory possesses instantons which mimimize the euclidean action.These instantons lead to chirality violating vacuum expectation val-ues.For example,a non-zero chiral condensate develops which only for high temperature and large curvature vanish exponentially.For e=0all coupling constants are dimensionless and the theory be-comes conformally invariant.In this limit the vacuum structure becomes trivial.Despite its complexity the general model(1.1)is solvable for arbi-trary classical backgrounds gµνand allows for an analytical treatment.This in turn enables the entire stress tensor in any curved space,the induced currents,their correlators and the equation of state to be constructed.The physical role of the coupling constants is the following:The coupling ofφto the transversal current decreases the effective electromagnetic inter-action between fermions.For example,the electric charge becomes renormal-=e/ 2.ized to eR4 A.Dettki,I.Sachs and A.WipfThe mass in the bosonised theory depends on g2.In the ungauged sector the Kac-Moody central extension,conformal weights and U(1)charges de-pendend on g2.The coupling constant g3amplifies the Hawking radiation which remains thermal for the interacting model.It is(3+24πg23)times as strong as that of a free massless scalarfield.The central charge and con-formal weights depend on g3.Actually,the weights of the fermionicfields become complex for g3=0.However,g3does not enter in thefinite sizeeffects.The coupling constant g1to the longitudinal current weakens the long range gauge invariant electron-electron correlators in the one-instanton sector(see6.27).In the ungauged sector it enters in expectation values of local operators and in particular in the short distance expansions of the fermionicfields and energy momentum tensor.It does not influence the thermodynamics of the model.Since for particular choices of the coupling constants the model reduces to wellknown and wellstudied exactly soluble models there are many earlier works which are related to ours.Some of them concentrated more on the gauge sector and investigated the renormalization of the electric charge in the gauged Thirring model by the four-fermi interaction[8]or the non-trivial vacuum structure in the Schwinger model[3,9].Others concentrated on the ungauged conformal sector.Freedman and Pilch calculated the partition function of the ungauged Thirring model on arbitrary Riemann surfaces[10]. We do not agree with their result and in particular show that there is no holomorphic factorization for general fermionic boundary conditions.Also we deviate from Destri and deVega[11]which investigated the ungauged model on the cylinder with twisted boundary conditions.We shall comment on the discrepancies in sections3and7.Other papers which are relevant and are dealing with different aspects of certain limiting cases of(1.1)are [12],where the thermodynamics of the Thirring model has been studied or [5]in which the Hawking radiation has been derived.Generalized Gauged Thirring Model on Curved Space-Times5 The paper is organized as follows:In section2we analyse the classical model to prepare the ground for the quantization.In particular we derive the general solution of thefield equations,discuss the conservation laws and investigate the limiting theories.By employing the graded structure we derive the classical Poissson(anti)commutators of the fundamentalfields with the energy momentum tensor.In the following section we quantize the finite temperature model.To avoid infrared problems we assume space to befinite.Together with thefinite temperature boundary conditions we are lead to considering the theory on the2-dimensional euclidean torus.Due to the twists in the fermionic boundary conditions,the non-trivial vacuum structure and the associated instantons and fermionic zero-modes the quan-tization is rather subtle.Actually we show that some of the results in the literature are incorrect.In section4the general results are applied to derive the partition function of the gauged model.Its dependence on the spatial size,temperature and gravitationalfield is explicitly found.In section5we show that for equal couplings the gauged model on curved spacetime can be bosonized.It turns out that only the non-constant parts of the currents can be bosonized and that for this part the wellknownflat spacetime rules need just be covariatized.In the following section the chiral symmetry breaking is studied.The exact form of the chiral condensate is found.On theflat torus the formula simplifies to(6.13).Various limits,e.g.L→∞,T→0, T→∞or g2→∞are investigated.By comparing the temperature andcurvature dependence of the condensate we derive an effective curvature induced temperature.In section7the thermodynamics of the ungauged model is studied.We derive the ground state energy and its dependence on the coupling constants,size of the system and boundary conditions.We compute the equation of state and our result does not agree with[12].In the last section we investigate the conformal sector of(1.1),that is the ungauged model inflat spacetime.Besides the Virasoro algebra the model contains an U(1)Kac-Moody algebra.We calculate the important commu-6 A.Dettki,I.Sachs and A.Wipftators and in particular determine the conformal weights and U(1)-charges of the fundamentalfields fromfirst principles.Also we show that thefinite size effects are in general not proportional to the central charge as has been conjectured by Cardy[13].The appendix A contains our conventions and scaling formulae for the various geometrical objects.In appendix B we col-lected some useful variational formulae which we have used in this work.In appendix C we derive the partition function within the canonical approach.2.Classical theory2.1Equations of MotionThefield equations of the model(1.1)areiγµ(∇µ−ig1∂µλ+ig2ηµν∂νφ)ψ≡iγµDµ=02∇2λ=−g3R−g1∇µjµ(2.1)2∇2φ=−g2∇µj5µ∇νFµν=e jµ,which are the Dirac equation for massless charged fermions propagating in a curved space-time and interacting with the scalar and pseudoscalar-fields, Klein Gordon type of equations and Maxwell equation.Here j5µis the axial vector current which is defined byj5µ=¯ψγµγ5ψ=ηµνjν.(2.2) When one decomposes the gaugefield as√Aµ=∂µα−ηµρ∂ρϕso that F01=σ/∂e−iF−iγ5G+12Generalized Gauged Thirring Model on Curved Space-Times7 Hence,ifψ0(x)solves the free Dirac equation inflat Minkowski space time, thenψ(x)≡e iF+iγ5G−1√−g∇µjµ=∂µ√8 A.Dettki,I.Sachs and A.Wipfand depend on4chiral functions which arefixed by the initial data on some spacelike hypersurface.The solutions of the free Dirac equations depend on 2chiral functions asψ0= ψ−(x−)ψ+(x+) .(2.11) In these coordinate system the Maxwell equations(2.1)can easily be inte-grated and onefinds∂+∂−φ=F01=2e2σ x− ψ†−(ξ)ψ−(ξ)dξ−x+ ψ†+(ξ)ψ+(ξ)dξ .(2.12) To go further we mustfix the gauge.Conveniently one chooses the Lorentz gauge such thatα=0in(2.3)and thusφin(2.12)determines Aµ.We see that in isothermal coordinates and this gauge the general solution of(2.1) is given by(2.10),(2.12)and(2.5),that is in terms of6chiral functions.Besides the currents the symmetric energy-momentum tensor of the mat-terfieldsTµν≡−2gδSgµνFσρFσρ−FσνFµσ+i4(2.14)jµ(g1∇νλ−g2ηνα∇αφ)+(µ↔ν)2+g2gµνjαηαβ∇βφ−2g2jαηα(µ∇ν)φ,where we have introduced the symmetrization A(µBν)=1Generalized Gauged Thirring Model on Curved Space-Times9 always present when one couples scalars non-minimally to a background cur-vature.The remaining terms reflect the interaction between the fermionic and auxiliaryfields.On shell Tµνis conserved as required by general ing the field equations forψandλits trace reads1Tµµ=g23R−S p where S p= √∇2R(2.16)4the variation of which isδS p= 4[gµνR−∇µ∇ν1∇2R ∇ν 1∇α 1−gδgµν(2.17)∇2RThe trace of the modified energy momentum tensor is now zero,and for gµν→ηµνthe Lagrangian corresponds to a conformalfield theory in Minkowski spacetime.Choosing the coupling constants appropriately,the model reduces to various well known exactly solvable models:-For the special choice g1=g2=e=0and for vanishing gaugefield theλ-dependent part of(1.1)is just the Lagrangian of scalarfields coupled to a background charge and and for imaginary g3describes the minimal models of conformalfield theory[17].10 A.Dettki,I.Sachs and A.Wipf-For g3=0and g21=−g22=g2the fermionic sector reduces to the gaugedversion of the Thirring model[18]in curved space time.To see that we solve the Klein Gordon equations in(2.1)for the U(1)current which yieldsjµ=−2g2ηνµ∂νφ.(2.18)Inserting this into the Dirac equation wefindiγµ∇µψ−g24jµjµ−12σ/∂e−iF+γ5G+1In this subsection we investigate the Hamiltonian structure of the model (1.1)in the conformal limit,i.e.inflat Minkowski space and for vanishing gaugefield.In the presence of both fermions and bosons it is convenient to exploit the graded Poisson structure[19].We recall,that the equal time Poisson brackets are{A(x),B(y)}≡ O dz1 A(x)←−δδO(z)∓A(x)←−δδπO(z) x0=y0.(2.23a) The sum is over all fundamentalfields O(x)in the theory.The sign is minus if one or both of thefields A and B are bosonic(even)and it is plus if both are fermionic(odd).The momentum densitiesπO(x)conjugate to the O-fields are given by functional left-derivatives−→δSπO(x)=(πλ−ig1πψψ)2+14x±=x0±x1so that∂±=12(1±γ5)ψ.Then T−−in(2.14)simplifies toT−−=−12∂−f{T f,ψ+}=f∂−ψ++12(1+ig1g3)ψ†+∂−f.(2.29)Whereasφandψ+are primaryfields,λis not.Actually,the non-primary character ofλis very much linked with the g3-dependent term in the transformation of the Diracfield.To see that more clearly we note that under an infinitesimal left conformal transformation generated by¯T f= dx+f(x+)T++the scalar and fermifield transform as{¯T f,λ}=f∂−λ−g3appearing in the action is only conformally invariant becauseλtransforms inhomogenously like a spin connection.It may be surprising that the sym-metry transformations depend on the coupling constant g3which is not present in theflat space time Lagrangean.However,the same happens for example in4dimensions if one couples a scalarfield conformally,that is non-minimally,to gravity.Although the Lagrangeans for the minimally and conformally coupled particles are the same on Minkowski spacetime,their energy momentum tensors are not.The same happens for the conformally invariant nonabelian Toda theories wich admit several energy momentum tensors and hence several conformal structures[20].The current transforms as{T f,j−}=f∂−j−+j−∂−f(2.31) and the energy momentum tensor as{T f,T−−}=f∂−T−−+2T−−∂−f−g23∂3−f.(2.32) Recalling that a primaryfield O with weight h transforms as{T f,O}=f∂−O+hO∂−fand comparing with the above results we have found the following structure:-The pseudoscalarfieldφis primary with hφ=0.The scalarfieldλis only primary for g3=0in which case hλ=0.=1-The Diracfieldψ+is primary with hψ+3.Quantization of the generalized gauged ThirringmodelIn this section we quantize the general model(1.1)in curved space-times. The results are then applied in the following sections,where we calculate the partition function,ground state energy,equation of state and certain correlators of interest and their dependence on the chemical potential,vol-ume of space,temperature and background metric.To do that we couple the conserved U(1)-charge to a chemical potentialµ.We enclose the system in a box with length L to avoid infrared divergences.To investigate the temperature dependence the time is taken to be purely imaginary in the functional approach[21].The imaginary time x0varies then from zero to the inverse temperatureβand we must impose periodic-and antiperiodic boundary conditions for the bosonic-and fermionicfields,respectively.Thus to study thefinite temperature model we must assume that space-time is an euclidean torus[0,β]×[0,L].To see how the partition function and correlators depend on the gravi-tationalfield we assume that the torus is equipped with an arbitrary met-ric with euclidean signature or equivalently with a2-bein eµa.The curved gamma matrices areγµ=eµaˆγa and in particularγ5=−ig.We allow for the general twisted boundary conditions for the fermionsψ(x0+L,x1)=−e2πi(α0+β0γ5)ψ(x0,x1)(3.2)ψ(x0,x1+L)=−e2πi(α1+β1γ5)ψ(x0,x1).The parametersαi andβi represent vectorial and chiral twists,respectively. We could allow for twisted boundary conditions for the(pseudo)scalars as well,e.g.φ(x0+nL,x1+mL)=φ(x1,x0)+2π(m+n).However,to recover the Thirring model for certain values of the couplings we assume that these fields are periodic.Forσ=0,τ=iβ/L andα0=β0=0the partition function has then the usual thermodynamical interpretation.Its logarithm is proportional to the free energy at temperature T=1/β.3.1Fermionic path integralTwisted boundary conditions as in(3.2)require some care in the fermionic path integral.Indeed the fermionic determinant is not uniquely defined when one allows for such twists.The ambiguities are not related to the unavoidable ultra-violett divergences but to the transition from Minkowski-to Euclidean space-time.To see that more clearly let S±denote the set of fermionicfields in Minkowski space-time with chirality±1.Since both the commutation relations and the action do not connect S+and S−we can consistently impose different boundary conditions on S+and S−.On the other hand,in the euclidean pathintegral for the generating functionalZ F[η,¯η]= Dψ†Dψe √g ¯ηψ+ψ†η ,(3.3) the Dirac operator/D= 0D−D+0exchanges the two chiral components ofψ,i.e./D:S±→S∓.Thus,in con-trast to the situation in Minkowski space the two chiral sectors are related in the action.Of course,the eigenvalue problem for i/D is then not well defined.This is the origin of the ambiguity in the definition of the deter-minant.It is related to the ambiguities one encounters when one quantizes chiral fermions[22].To solve this problem we shall analytically continue the well-defined determinants in the untwisted sectorβ=0toβ=0.Theresulting determinants do not factorize into(anti-)holomorphic pieces and differ from previous ones in the literature[10].In appendix C we give fur-ther arguments in favour of our result by calculating the determinants in a different way.Let us now study the generating functional for fermions in an external gravitational and gaugefield and coupled to the auxiliaryfields.For that we observe that on the torus the decompositon(2.2)of the gauge potential generalises toAµ=A Iµ+2πgΦ/V.As instanton potential we chooseeA Iµ=eˆA Iµ−Φηνµ∂νχ,where eˆA I=−√ˆV x1,0 (3.4b)is the instanton potential on theflat torus with the sameflux butfield strength√g ΦˆgΦg△χ.(3.4c)The solution of this equation is given byχ(x)=−1△e−2σ (x)=1g(y)G0(x,y)e−2σ(y),(3.4d)whereG0(x,y)= x|1λn(3.6a)is the Greenfunction belonging to−△.In deriving(3.4d)we have used that 1V of△is missing. Note that2-dimensional gauge theories are not scale or Weyl invariant as 4-dimensional ones are.For that reason the instantons on conformallyflat spacetimes are not just the’flat’instantons.To be more explicit we relate G0to the GreenfunctionˆG0on theflat torus with the hatted metric[26]ˆG 0(x,y)=−1η(τ) 1L1L (0,τ)|2,whereξ=x−y.(3.6b)For that we note that due to the missing zero-mode in(3.6a)the usualflat spacetime equations for the Greenfunctions are modified to−△x G0(x,y)=δ(x−y)g−1√ˆV.(3.7a) Furthermore one sees at once that both Green functions annihilate the cor-responding constant zeromodesd2y ˆgˆG0(x,y)=0.(3.7b) From these two equations one concludes that Greenfunction on the curved torus is related to theflat one(3.6b)asG0(x,y)=ˆG0(x,y)+1g(u)g(v)ˆG0(u,v)−1g(u)d2u−1g(u)ˆG0(u,y)(3.8)and this replaces the infinite space relations G0=ˆG0[27].Our choice for the instanton potential(3.4)corresponds to a particular trivialization of the U(1)-bundle over the torus[3].In other words,the gauge potentials and fermionfields at(x0,x1)and(x0,x1+L)are necessarily related by a nontrivial gauge transformation with windingsAµ(x0,x1+L)−Aµ(x0,x1)=∂µα(x)ψ(x0,x1+L)=−e ieα(x)e2πi(α1+β1γ5)ψ(x0,x1).(3.9a) For the choice(3.4b)wefindeα(x)=−ΦLg0hµ+g1∂µλ−g2ηµν∂νφ(3.10) appears in the Dirac operator in(1.1)on the torus.λandφcouple to the divergence of the vector and axial vector currents.The harmonicfields hµcouple to the harmonic part of the current and are needed to recover the Thirring model in the limit g20=g21=g22.Also,we shall see that tµand hµare essential to obtain the correct answer for the thermodynamic potential.Note that Bµcontains no instanton part since it couples to the gauge invariant fermionic current.Finally we introduce a chemical potential for the conserved U(1)charge. In the euclidean functional approach this is equivalent to coupling the fermions to a constant imaginary gauge potential A0[29].Inserting the above decompositions and the chemical potential into the Dirac operatorfinally yields in isothermal coordinates/D=γνDν=e iF+γ5(G+Φχ)−32σ,where/ˆD=γµ ∂µ+iˆωµ−ieˆA Iµ−2πi2πµδµ0.(3.11)Hereˆωis the spin connection belonging toˆeµa.It vanishes for our choice of the reference zweibein.ˆA I is the instanton potential(3.4b)on theflat torus. The scalar and pseudoscalar functions F,G andχhave been introduced in(2.4)and(3.4d).In the chosen coordinates t and h and hence H are all constant.In[3]it has been shown that/D possesses|k|zero-modes of definite chirality and their chirality is given by the sign of k.They are crucial in any correct quantization.For example,if one would leave out instanton sectors in which i/D has zero-modes then the cluster property would be violated.In afirst step we quantize the fermions in theflat instanton and har-monic background and reference metricˆgµν,that is we assume/D→/ˆD in (3.3).The dependence on the remainingfields F,G,χandσ,that is the relation between Z F andˆZ F,is then found by integrating the chiral and trace anomalies[30]and exploiting the relation(3.11)between/D and/ˆD. We expand the fermionicfield in a orthonormal basis of the Hilbert spaceψ(x)= n a nψn+(x)+ n b nψn−(x)ψ†(x)= n¯a nχ†n+(x)+ n¯b nχ†n−(x),(3.12) where a n,b n,¯a n,¯b n are independent Grassmann variables.Topologically trivial sectorFor k=0or vanishing instanton potential we can immediately write down a basisψn±(x)=1Ve i(p±n,x)e±,where(p±n)i=2π2+αi±βi+n i),(3.13)and e±are the eigenvectors ofγ5.Theψn+andψn−must obey the S+ and S−boundary conditions,respectively.These boundary conditionsfix the admissable momenta p±n in(3.13).Since the Dirac operator maps S±into S∓theχn±must then obey the same boundary conditions as theψn∓. Thusχn±(x)is obtained fromψn±(x)by exchanging p+n and p−n.It follows then thati/ˆDψn±=λ±nχn∓(3.14a) withλ+n=2π2+a1+β1+n1)−(1τ0L[τ(12+a0−β0+n0)].(3.14b)Here we have introduced aµ≡αµ−Hµ−µµ.Substituting(3.12,14)into the generating functional(3.3)and applying the standart Grassmann integra-tion rules we arrive atˆZF[η,¯η]=det i/ˆD e− ¯η(x)ˆS(x,y)η(y),det i/ˆD= nλ+nλ−n,ˆS(x,y)= n ψn+(x)χ†n−(y)λ−n .(3.15)ˆS is the fermionic Green function in the0-instanton sector.Note that both the’eigenvalues’and the Green function depend on the Teichmueller pa-rameter,harmonic potentials,twists and chemical potential.We proceed to calculate the infinte product or generalized determinant in(3.15).This is one of the central points of our article and for non-zero chiral twists and chemical potential our result deviates from previous ones [10].Actually the twists and chemical potential are related as one can see from(3.14).One may be tempted so identifydet(D+D−)∼ λ+nλ−n and det D+det D−∼ λ+n λ−m(3.16)and thus conclude that the determinant is a product,f (τ)¯f(τ),that is factorizes into holomorphic and anti-holomorphic pieces (the overall factor ∼1/τ0L in the eigenvalues (3.14b)drops in the infinite product,since the torus has vanishing Euler number).However,the infinite product in (3.15)must be regularized and the two expressions in (3.16)may differ.In conformal field theory [26]one is naturally lead to consider the individual chiral sectors and thus finds holomorphic factorization.For Dirac fermions one uses /D 2to regularize the product and this leads to the determinant of the product D +D −.To continue we recast the infinite product in the form∞ λ+n λ−n = n ∈Z 22π2+c µ+n µ)(1τ0τ1−|τ|21−τ1 .(3.17b )The point is that for real c µ,that is for vanishing chiral twists βµand chemical potential (see the definitions of a µbelow (3.14b)and µµin (3.11))the zeta function defined byζ(s )= n λ+n λ−n −s (3.17c )has a well defined analytic continuation to s<1via a Poisson resummation.An explicit calculation yields [3,31,38]det(i /ˆD )≡ n λ+n λ−n reg =e −ζ′(s )|s =0,where ζ′(s ) s =0=−log 1√det(i/ˆD)=e2π(|η(τ)|2Θ −a1+β1a0−β0 (0,τ)¯Θ −¯a1−β1¯a0+β0 (0,τ).(3.19)It can be shown that this determinant is gauge invariant,i.e.invariant un-derαµ→αµ+1,but not invariant under chiral transformatins,βµ→βµ+1, as expected.Furthermore it transforms covariantly under modular transfor-mationsτ→τ+1andτ→−1/τ.In other words,det i/ˆD is invariant under modular transformations if at the same time the boundary conditions are transformed accordingly.The exponential prefactor is needed for modular covariance and is not present in the literature[10].It correlates the two chiral sectors and will have important consequences.In the appendix C we confirm(3.19)with operator methods.Topologically nontrivial sectorsFor definiteness we assume k>0.Then i/ˆD possesses k zero-modesˆψp0+,p= 1,...,k with positive chirality and S+boundary conditions.Together with the excited modesψn+they form a basis of S+.Thus we must add the zero-mode contribution c nˆψp0+toψin(3.12).Similarly we must add¯c nˆχp†0+to ψ†in(3.12).The zero modesˆχp0+inψ†must obey S−boundary conditions (see the discussion below(3.13)).Thus the zero-modes in the expansions of ψandψ†have the same chirality but obey different boundary conditions. This is required for the zero-and excited modes to form a complete basis and is consistent since i/D does not relate the zero-mode sector of S+with S−.The Grassmann integral over the variables belonging to the excited modes is performed as in the trivial sector.Also,the integration over the c n and¯c n can easily be done and one obtainsˆZ F [η,¯η]=|k| p=1(¯η,ˆψp0+)(ˆχp0+,η)det′i/ˆD e− ¯η(x)ˆS e(x,y)η(y),det′i/ˆD= λ±n=0λ+nλ−n,ˆS e(x,y)= λn=0 ψn+(x)χ†n−(y)√ggµνDν−12ηµνFµνγ5(3.21) simplifies in the instanton backgroundˆA I and on theflat torus to−/ˆD2=−ˆgµνˆDµˆDν−ΦWith the explicit spectrum at hand we can compute the zero-mode trun-cated determinant with zeta-function methods andfind[3]det′(i/ˆD)= πˆV2+α0+β0),(3.25a) where we have assumed k>0.The choice of c p is dictated by the time-like boundary conditions in(3.2).Inserting this ansatz into the zero mode equation/˜D2˜χp=0yields|τ|2d2L4y2−2iτ1Φdy−iτΦk(c p−H0−µ0).This is just the differential equation for the ground state of a generalized harmonic oscillator to which it reduces forτ=iτ0.The solution is given by ξp=exp −Φk(c p−H0−µ0) 2 .These functions do not obey the boundary condition(3.9),but the cor-rect eigenmodes can be constructed as superpositions of them.For that we observe that˜χp(x0,x1+L)=e−iΦx0/βe2iπH1˜χp+k(x0,x1)so that the sumsˆψp 0+=(2kτ0)1kτ0e2πi(H0−α0−12+α1+β1−H1)˜χp+nk e+,(3.25b)where p=1,...,k,obey the boundary conditions and thus are the k required zero-modes.We have chosen the phase such that the accompanying zero-modes in(3.20)are just。

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