The Gauged Vector Model in Four-Dimensions Resolution of an Old Problem

a rXiv:g r-qc/96213v18Feb1996On the Gauge Aspects of Gravity †Frank Gronwald and Friedrich W.Hehl Institute for Theoretical Physics,University of Cologne D-50923K¨o ln,Germany E-mail:fg@thp.uni-koeln.de,hehl@thp.uni-koeln.de ABSTRACT We give a short outline,in Sec.2,of the historical development of the gauge idea as applied to internal (U (1),SU (2),...)and external (R 4,SO (1,3),...)symmetries and stress the fundamental importance of the corresponding con-served currents.In Sec.3,experimental results with neutron interferometers in the gravitational field of the earth,as interpreted by means of the equivalence principle,can be predicted by means of the Dirac equation in an accelerated and rotating reference ing the Dirac equation in such a non-inertial frame,we describe how in a gauge-theoretical approach (see Table 1)the Einstein-Cartan theory,residing in a Riemann-Cartan spacetime encompassing torsion and curvature,arises as the simplest gravitational theory.This is set in con-trast to the Einsteinian approach yielding general relativity in a Riemannian spacetime.In Secs.4and 5we consider the conserved energy-momentum cur-rent of matter and gauge the associated translation subgroup.The Einsteinian teleparallelism theory which emerges is shown to be equivalent,for spinless mat-ter and for electromagnetism,to general relativity.Having successfully gauged the translations,it is straightforward to gauge the four-dimensional affine group R 4⊃×GL (4,R )or its Poincar´e subgroup R 4⊃×SO (1,3).We briefly report on these results in Sec.6(metric-affine geometry)and in Sec.7(metric-affine field equations (111,112,113)).Finally,in Sec.8,we collect some models,cur-rently under discussion,which bring life into the metric-affine gauge framework developed.Contents1.Introduction2.Remarks on the history of the gauge idea2.1.General relativity and Weyl’s U(1)-gauge theory2.2.Yang-Mills and the structure of a gauge theory2.3.Gravity and the Utiyama-Sciama-Kibble approach2.4.E.Cartan’s analysis of general relativity and its consequences3.Einstein’s and the gauge approach to gravity3.1.Neutron matter waves in the gravitationalfield3.2.Accelerated and rotating reference frame3.3.Dirac matter waves in a non-inertial frame of reference3.4.‘Deriving’a theory of gravity:Einstein’s method as opposed to thegauge procedure4.Conserved momentum current,the heuristics of the translation gauge4.1.Motivation4.2.Active and passive translations4.3.Heuristic scheme of translational gauging5.Theory of the translation gauge:From Einsteinian teleparallelism to GR5.1.Translation gauge potentialgrangian5.3.Transition to GR6.Gauging of the affine group R4⊃×GL(4,R)7.Field equations of metric-affine gauge theory(MAG)8.Model building:Einstein-Cartan theory and beyond8.1.Einstein-Cartan theory EC8.2.Poincar´e gauge theory PG,the quadratic version8.3.Coupling to a scalarfield8.4.Metric-affine gauge theory MAG9.Acknowledgments10.ReferencesFrom a letter of A.Einstein to F.Klein of1917March4(translation)70:“...Newton’s theory...represents the gravitationalfield in a seeminglycomplete way by means of the potentialΦ.This description proves to bewanting;the functions gµνtake its place.But I do not doubt that the daywill come when that description,too,will have to yield to another one,for reasons which at present we do not yet surmise.I believe that thisprocess of deepening the theory has no limits...”1.Introduction•What can we learn if we look at gravity and,more specifically,at general relativity theory(GR)from the point of view of classical gaugefield theory?This is the question underlying our present considerations.The answer•leads to a better understanding of the interrelationship between the metric and affine properties of spacetime and of the group structure related to gravity.Furthermore,it •suggests certain classicalfield-theoretical generalizations of Einstein’s theory,such as Einstein–Cartan theory,Einsteinian teleparallelism theory,Poincar´e gauge theory, Metric-Affine Gravity,that is,it leads to a deepening of the insight won by GR.We recently published a fairly technical review article on our results29.These lectures can be regarded as a down-to-earth introduction into that subject.We refrain from citing too many articles since we gave an overview a of the existing literature in ref.(29).2.Remarks on the history of the gauge idea2.1.General relativity and Weyl’s U(1)-gauge theorySoon after Einstein in1915/16had proposed his gravitational theory,namely general relativity(GR),Weyl extended it in1918in order to include–besides grav-itation–electromagnetism in a unified way.Weyl’s theoretical concept was that of recalibration or gauge invariance of length.In Weyl’s opinion,the integrability of length in GR is a remnant of an era dominated by action-at-a-distance theories which should be abandoned.In other words,if in GR we displace a meter stick from one point of spacetime to another one,it keeps its length,i.e.,it can be used as a standardof length throughout spacetime;an analogous argument is valid for a clock.In con-trast,Weyl’s unified theory of gravitation and electromagnetism of1918is set up in such a way that the unified Lagrangian is invariant under recalibration or re-gauging.For that purpose,Weyl extended the geometry of spacetime from the(pseudo-) Riemannian geometry with its Levi-Civita connectionΓ{}αβto a Weyl space with an additional(Weyl)covectorfield Q=Qαϑα,whereϑαdenotes thefield of coframes of the underlying four-dimensional differentiable manifold.The Weyl connection one-form reads1ΓWαβ=Γ{}αβ+ψ,D ψA)mat L (DJ=0A theorem local gauge symmetry coupling A Noether’s J <dJ=0of Lagrangian(d ψ),L mat ψgauge potentialsymmetry rigid ConservedJA(connection)current Fig.1.The structure of a gauge theory `a la Yang-Mills is depicted in this diagram,which is adapted from Mills 53.Let us quote some of his statements on gauge theories:‘The gauge principle,which might also be described as a principle of local symmetry ,is a statement about the invariance properties of physical laws.It requires that every continuous symmetry be a local symmetry ...’‘The idea at the core of gauge theory...is the local symmetry principle:Every continuous symmetry of nature is a local symmetry.’The history of gauge theory has been traced back to its beginnings by O’Raifeartaigh 69,who also gave a compact review of its formalism 68.the electromagnetic potential is an appendage to the Dirac field and not related to length recalibration as Weyl originally thought.2.2.Yang-Mills and the structure of a gauge theoryYang and Mills,in 1954,generalized the Abelian U (1)-gauge invariance to non-Abelian SU (2)-gauge invariance,taking the (approximately)conserved isotopic spin current as their starting point,and,in 1956,Utiyama set up a formalism for the gauging of any semi-simple Lie group,including the Lorentz group SO (1,3).The latter group he considered as essential in GR.We will come back to this topic below.In any case,the gauge principle historically originated from GR as a concept for removing as many action-at-a-distance concept as possible –as long as the group under consideration is linked to a conserved current.This existence of a conserved current of some matter field Ψis absolutely vital for the setting-up of a gauge theory.In Fig.1we sketched the structure underlying a gauge theory:A rigid symmetry ofa Lagrangian induces,via Noether’s theorem,a conserved current J ,dJ =0.It can happen,however,as it did in the electromagnetic and the SU (2)-case,that a conserved current is discovered first and then the symmetry deduced by a kind of a reciprocal Noether theorem (which is not strictly valid).Generalizing from the gauge approach to the Dirac-Maxwell theory,we continue with the following gauge procedure:Extending the rigid symmetry to a soft symmetry amounts to turn the constant group parameters εof the symmetry transformation on the fields Ψto functions of spacetime,ε→ε(x ).This affects the transformation behavior of the matter La-grangian which usually contains derivatives d Ψof the field Ψ:The soft symmetry transformations on d Ψgenerate terms containing derivatives dε(x )of the spacetime-dependent group parameters which spoil the former rigid invariance.In order to coun-terbalance these terms,one is forced to introduce a compensating field A =A i a τa dx i (a =Lie-algebra index,τa =generators of the symmetry group)–nowadays called gauge potential –into the theory.The one-form A turns out to have the mathematical mean-ing of a Lie-algebra valued connection .It acts on the components of the fields Ψwith respect to some reference frame,indicating that it can be properly represented as the connection of a frame bundle which is associated to the symmetry group.Thereby it is possible to replace in the matter Lagrangian the exterior derivative of the matter field by a gauge-covariant exterior derivative,d −→A D :=d +A ,L mat (Ψ,d Ψ)−→L mat (Ψ,A D Ψ).(4)This is called minimal coupling of the matter field to the new gauge interaction.The connection A is made to a true dynamical variable by adding a corresponding kinematic term V to the minimally coupled matter Lagrangian.This supplementary term has to be gauge invariant such that the gauge invariance of the action is kept.Gauge invariance of V is obtained by constructing it from the field strength F =A DA ,V =V (F ).Hence the gauge Lagrangian V ,as in Maxwell’s theory,is assumed to depend only on F =dA ,not,however,on its derivatives dF,d ∗d F,...Therefore the field equation will be of second order in the gauge potential A .In order to make it quasilinear,that is,linear in the second derivatives of A ,the gauge Lagrangian must depend on F no more than quadratically.Accordingly,with the general ansatz V =F ∧H ,where the field momentum or “excitation”H is implicitly defined by H =−∂V /∂F ,the H has to be linear in F under those circumstances.By construction,the gauge potential in the Lagrangians couples to the conserved current one started with –and the original conservation law,in case of a non-Abelian symmetry,gets modified and is only gauge covariantly conserved,dJ =0−→A DJ =0,J =∂L mat /∂A.(5)The physical reason for this modification is that the gauge potential itself contributes a piece to the current,that is,the gauge field (in the non-Abelian case)is charged.For instance,the Yang-Mills gauge potential B a carries isotopic spin,since the SU(2)-group is non-Abelian,whereas the electromagnetic potential,being U(1)-valued and Abelian,is electrically uncharged.2.3.Gravity and the Utiyama-Sciama-Kibble approachLet us come back to Utiyama(1956).He gauged the Lorentz group SO(1,3), inter ing some ad hoc assumptions,like the postulate of the symmetry of the connection,he was able to recover GR.This procedure is not completely satisfactory, as is also obvious from the fact that the conserved current,linked to the Lorentz group,is the angular momentum current.And this current alone cannot represent the source of gravity.Accordingly,it was soon pointed out by Sciama and Kibble (1961)that it is really the Poincar´e group R4⊃×SO(1,3),the semi-direct product of the translation and the Lorentz group,which underlies gravity.They found a slight generalization of GR,the so-called Einstein-Cartan theory(EC),which relates–in a Einsteinian manner–the mass-energy of matter to the curvature and–in a novel way –the material spin to the torsion of spacetime.In contrast to the Weyl connection (1),the spacetime in EC is still metric compatible,erned by a Riemann-Cartan b (RC)geometry.Torsion is admitted according to1ΓRCαβ=Γ{}αβ−b The terminology is not quite uniform.Borzeskowski and Treder9,in their critical evaluation of different gravitational variational principles,call such a geometry a Weyl-Cartan gemetry.secondary importance in some sense that some particularΓfield can be deduced from a Riemannian metric...”In this vein,we introduce a linear connectionΓαβ=Γiαβdx i,(7) with values in the Lie-algebra of the linear group GL(4,R).These64components Γiαβ(x)of the‘displacement’field enable us,as pointed out in the quotation by Einstein,to get rid of the rigid spacetime structure of special relativity(SR).In order to be able to recover SR in some limit,the primary structure of a con-nection of spacetime has to be enriched by the secondary structure of a metricg=gαβϑα⊗ϑβ,(8) with its10componentfields gαβ(x).At least at the present stage of our knowledge, this additional postulate of the existence of a metric seems to lead to the only prac-ticable way to set up a theory of gravity.In some future time one may be able to ‘deduce’the metric from the connection and some extremal property of the action function–and some people have tried to develop such type of models,but without success so far.2.4.E.Cartan’s analysis of general relativity and its consequencesBesides the gauge theoretical line of development which,with respect to gravity, culminated in the Sciame-Kibble approach,there was a second line dominated by E.Cartan’s(1923)geometrical analysis of GR.The concept of a linear connection as an independent and primary structure of spacetime,see(7),developed gradually around1920from the work of Hessenberg,Levi-Civita,Weyl,Schouten,Eddington, and others.In its full generality it can be found in Cartan’s work.In particular, he introduced the notion of a so-called torsion–in holonomic coordinates this is the antisymmetric and therefore tensorial part of the components of the connection–and discussed Weyl’s unifiedfield theory from a geometrical point of view.For this purpose,let us tentatively callgαβ,ϑα,Γαβ (9)the potentials in a gauge approach to gravity andQαβ,Tα,Rαβ (10)the correspondingfield ter,in Sec.6,inter alia,we will see why this choice of language is appropriate.Here we definednonmetricity Qαβ:=−ΓD gαβ,(11) torsion Tα:=ΓDϑα=dϑα+Γβα∧ϑβ,(12)curvature Rαβ:=′′ΓDΓαβ′′=dΓαβ−Γαγ∧Γγβ.(13)Then symbolically we haveQαβ,Tα,Rαβ ∼ΓD gαβ,ϑα,Γαβ .(14)By means of thefield strengths it is straightforward of how to classify the space-time manifolds of the different theories discussed so far:GR(1915):Qαβ=0,Tα=0,Rαβ=0.(15)Weyl(1918):Qγγ=0,Tα=0,Rαβ=0.(16)EC(1923/61):Qαβ=0,Tα=0,Rαβ=0.(17) Note that Weyl’s theory of1918requires only a nonvanishing trace of the nonmetric-ity,the Weyl covector Q:=Qγγ/4.For later use we amend this table with the Einsteinian teleparallelism(GR||),which was discussed between Einstein and Car-tan in considerable detail(see Debever12)and with metric-affine gravity29(MAG), which presupposes the existence of a connection and a(symmetric)metric that are completely independent from each other(as long as thefield equations are not solved): GR||(1928):Qαβ=0,Tα=0,Rαβ=0.(18)MAG(1976):Qαβ=0,Tα=0,Rαβ=0.(19) Both theories,GR||and MAG,were originally devised as unifiedfield theories with no sources on the right hand sides of theirfield equations.Today,however,we understand them10,29as gauge type theories with well-defined sources.Cartan gave a beautiful geometrical interpretation of the notions of torsion and curvature.Consider a vector at some point of a manifold,that is equipped with a connection,and displace it around an infinitesimal(closed)loop by means of the connection such that the(flat)tangent space,where the vector‘lives’in,rolls without gliding around the loop.At the end of the journey29the loop,mapped into the tangent space,has a small closure failure,i.e.a translational misfit.Moreover,in the case of vanishing nonmetricity Qαβ=0,the vector underwent a small rotation or–if no metric exists–a small linear transformation.The torsion of the underlying manifold is a measure for the emerging translation and the curvature for the rotation(or linear transformation):translation−→torsion Tα(20) rotation(lin.transf.)−→curvature Rαβ.(21) Hence,if your friend tells you that he discovered that torsion is closely related to electromagnetism or to some other nongravitationalfield–and there are many such ‘friends’around,as we can tell you as referees–then you say:‘No,torsion is related to translations,as had been already found by Cartan in1923.’And translations–weFig.2.The neutron interferometer of the COW-experiment11,18:A neutron beam is split into two beams which travel in different gravitational potentials.Eventually the two beams are reunited and their relative phase shift is measured.hope that we don’t tell you a secret–are,via Noether’s theorem,related to energy-momentum c,i.e.to the source of gravity,and to nothing else.We will come back to this discussion in Sec.4.For the rest of these lectures,unless stated otherwise,we will choose the frame eα,and hence also the coframeϑβ,to be orthonormal,that is,g(eα,eβ)∗=oαβ:=diag(−+++).(22) Then,in a Riemann-Cartan space,we have the convenient antisymmetriesΓRCαβ∗=−ΓRCβαand R RCαβ∗=−R RCβα.(23) 3.Einstein’s and the gauge approach to gravity3.1.Neutron matter waves in the gravitationalfieldTwenty years ago a new epoch began in gravity:C olella-O verhauser-W erner measured by interferometric methods a phase shift of the wave function of a neutron caused by the gravitationalfield of the earth,see Fig.2.The effect could be predicted by studying the Schr¨o dinger equation of the neutron wave function in an external Newtonian potential–and this had been verified by experiment.In this sense noth-ing really earth-shaking happened.However,for thefirst time a gravitational effect had been measured the numerical value of which depends on the Planck constant¯h. Quantum mechanics was indispensable in deriving this phase shiftm2gθgrav=gpath 1path 2zx~ 2 cm~ 6 cmA Fig.3.COW experiment schematically.the neutron beam itself is bent into a parabolic path with 4×10−7cm loss in altitude.This yields,however,no significant influence on the phase.In the COW experiment,the single-crystal interferometer is at rest with respect to the laboratory,whereas the neutrons are subject to the gravitational potential.In order to compare this with the effect of acceleration relative to the laboratory frame,B onse and W roblewski 8let the interferometer oscillate horizontally by driving it via a pair of standard loudspeaker magnets.Thus these experiments of BW and COW test the effect of local acceleration and local gravity on matter waves and prove its equivalence up to an accuracy of about 4%.3.2.Accelerated and rotating reference frameIn order to be able to describe the interferometer in an accelerated frame,we first have to construct a non-inertial frame of reference.If we consider only mass points ,then a non-inertial frame in the Minkowski space of SR is represented by a curvilinear coordinate system,as recognized by Einstein 13.Einstein even uses the names ‘curvilinear co-ordinate system’and ‘non-inertial system’interchangeably.According to the standard gauge model of electro-weak and strong interactions,a neutron is not a fundamental particle,but consists of one up and two down quarks which are kept together via the virtual exchange of gluons,the vector bosons of quantum chromodynamics,in a permanent ‘confinement phase’.For studying the properties of the neutron in a non-inertial frame and in low-energy gravity,we may disregard its extension of about 0.7fm ,its form factors,etc.In fact,for our purpose,it is sufficient to treat it as a Dirac particle which carries spin 1/2but is structureless otherwise .Table 1.Einstein’s approach to GR as compared to the gauge approach:Used are a mass point m or a Dirac matter field Ψ(referred to a local frame),respectively.IF means inertial frame,NIF non-inertial frame.The table refers to special relativity up to the second boldface horizontal line.Below,gravity will be switched on.Note that for the Dirac spinor already the force-free motion in an inertial frame does depend on the mass parameter m .gauge approach (→COW)elementary object in SRDirac spinor Ψ(x )Cartesian coord.system x ids 2∗=o ij dx i dx jforce-freemotion in IF (iγi ∂i −m )Ψ∗=0arbitrary curvilinear coord.system x i′force-free motion in NIF iγαe i α(∂i +Γi )−m Ψ=0Γi :=1non-inertial objects ϑα,Γαβ=−Γβα16+24˜R(∂{},{})=020global IF e i α,Γi αβ ∗=(δαi ,0)switch on gravity T =0,R =0Riemann −Cartang ij |P ∗=o ij , i jk |P ∗=0field equations 2tr (˜Ric )∼mass GR2tr (Ric )∼massT or +2tr (T or )∼spinECA Dirac particle has to be described by means of a four-component Dirac spinor. And this spinor is a half-integer representation of the(covering group SL(2,C)of the)Lorentz group SO(1,3).Therefore at any one point of spacetime we need an orthonormal reference frame in order to be able to describe the spinor.Thus,as soon as matterfields are to be represented in spacetime,the notion of a reference system has to be generalized from Einstein’s curvilinear coordinate frame∂i to an arbitrary, in general anholonomic,orthonormal frame eα,with eα·eβ=oαβ.It is possible,of course,to introduce in the Riemannian spacetime of GR arbi-trary orthonormal frames,too.However,in the heuristic process of setting up the fundamental structure of GR,Einstein and his followers(for a recent example,see the excellent text of d’Inverno36,Secs.9and10)restricted themselves to the discussion of mass points and holonomic(natural)frames.Matter waves and arbitrary frames are taboo in this discussion.In Table1,in the middle column,we displayed the Ein-steinian method.Conventionally,after the Riemannian spacetime has been found and the dust settled,then electrons and neutron and what not,and their corresponding wave equations,are allowed to enter the scene.But before,they are ignored.This goes so far that the well-documented experiments of COW(1975)and BL(1983)–in contrast to the folkloric Galileo experiments from the leaning tower–seemingly are not even mentioned in d’Inverno36(1992).Prugoveˇc ki79,one of the lecturers here in Erice at our school,in his discussion of the classical equivalence principle,recognizes the decisive importance of orthonormal frames(see his page52).However,in the end,within his‘quantum general relativity’framework,the good old Levi-Civita connection is singled out again(see his page 125).This is perhaps not surprising,since he considers only zero spin states in this context.We hope that you are convinced by now that we should introduce arbitrary or-thonormal frames in SR in order to represent non-inertial reference systems for mat-ter waves–and that this is important for the setting up of a gravitational gauge theory2,42.The introduction of accelerated observers and thus of non-inertial frames is somewhat standard,even if during the Erice school one of the lecturers argued that those frames are inadmissible.Take the text of Misner,Thorne,and Wheeler57.In their Sec.6,you willfind an appropriate discussion.Together with Ni30and in our Honnef lectures27we tailored it for our needs.Suppose in SR a non-inertial observer locally measures,by means of the instru-ments available to him,a three-acceleration a and a three-angular velocityω.If the laboratory coordinates of the observer are denoted by x x as the correspond-ing three-radius vector,then the non-inertial frame can be written in the succinct form30,27eˆ0=1x/c2 ∂c×B∂A.(25)Here ‘naked’capital Latin letters,A,...=ˆ1,ˆ2,ˆ3,denote spatial anholonomic com-ponents.For completeness we also display the coframe,that is,the one-form basis,which one finds by inverting the frame (25):ϑˆ0= 1+a ·c 2 dx 0,ϑA =dx c ×A dx A +N 0.(26)In the (3+1)-decomposition of spacetime,N and Ni βαdx0ˆ0A =−Γc 2,Γ0BA =ǫABCωC i α,with e α=e i ,into an anholonomic one,then we find the totallyanholonomic connection coefficients as follows:Γˆ0ˆ0A =−Γˆ0A ˆ0=a A x /c 2 ,Γˆ0AB =−Γˆ0BA =ǫABC ωC x /c 2 .(28)These connection coefficients (28)will enter the Dirac equation referred to a non-inertial frame.In order to assure ourselves that we didn’t make mistakes in computing the ‘non-inertial’connection (27,28)by hand,we used for checking its correctness the EXCALC package on exterior differential forms of the computer algebra system REDUCE,see Puntigam et al.80and the literature given there.3.3.Dirac matter waves in a non-inertial frame of referenceThe phase shift (24)can be derived from the Schr¨o dinger equation with a Hamilton operator for a point particle in an external Newton potential.For setting up a grav-itational theory,however,one better starts more generally in the special relativistic domain.Thus we have to begin with the Dirac equation in an external gravitational field or,if we expect the equivalence principle to be valid,with the Dirac equation in an accelerated and rotating,that is,in a non-inertial frame of reference.Take the Minkowski spacetime of SR.Specify Cartesian coordinates.Then the field equation for a massive fermion of spin1/2is represented by the Dirac equationi¯hγi∂iψ∗=mcψ,(29) where the Dirac matricesγi fulfill the relationγiγj+γjγi=2o ij.(30) For the conventions and the representation of theγ’s,we essentially follow Bjorken-Drell7.Now we straightforwardly transform this equation from an inertial to an accel-erated and rotating frame.By analogy with the equation of motion in an arbitrary frame as well as from gauge theory,we can infer the result of this transformation:In the non-inertial frame,the partial derivative in the Dirac equation is simply replaced by the covariant derivativei∂i⇒Dα:=∂α+i previously;we drop the bar for convenience).The anholonomic Dirac matrices are defined byγα:=e iαγi⇒γαγβ+γβγα=2oαβ.(32) The six matricesσβγare the infinitesimal generators of the Lorentz group and fulfill the commutation relation[γα,σβγ]=2i(oαβγγ−oαγγβ).(33) For Dirac spinors,the Lorentz generators can be represented byσβγ:=(i/2)(γβγγ−γγγβ),(34) furthermore,α:=γˆ0γwithγ={γΞ}.(35) Then,the Dirac equation,formulated in the orthonormal frame of the accelerated and rotating observer,readsi¯hγαDαψ=mcψ.(36) Although there appears now a‘minimal coupling’to the connection,which is caused by the change of frame,there is no new physical concept involved in this equation. Only for the measuring devices in the non-inertial frame we have to assume hypotheses similar to the clock hypothesis.This proviso can always be met by a suitable con-struction and selection of the devices.Since we are still in SR,torsion and curvatureof spacetime both remain zero.Thus(36)is just a reformulation of the‘Cartesian’Dirac equation(29).The rewriting in terms of the covariant derivative provides us with a rather ele-gant way of explicitly calculating the Dirac equation in the non-inertial frame of an accelerated,rotating observer:Using the anholonomic connection components of(28) as well asα=−i{σˆ0Ξ},wefind for the covariant derivative:Dˆ0=12c2a·α−ii∂2¯hσ=x×p+1∂t=Hψwith H=βmc2+O+E.(39)After substituting the covariant derivatives,the operators O and E,which are odd and even with respect toβ,read,respectively30:O:=cα·p+12m p2−β2m p·a·x4mc2σ·a×p+O(1Table2.Inertial effects for a massive fermion of spin1/2in non-relativistic approximation.Redshift(Bonse-Wroblewski→COW)Sagnac type effect(Heer-Werner et al.)Spin-rotation effect(Mashhoon)Redshift effect of kinetic energyNew inertial spin-orbit couplingd These considerations can be generalized to a Riemannian spacetime,see Huang34and the literature quoted there.。

高斯引理英语Gauss's theorem, also known as the divergence theorem,is an important mathematical concept in the field of vector calculus. This theorem is widely used in physics, engineering, and other related fields to describe the relationship between the flux of a vector field and its source.The theorem is named after German mathematician Johann Carl Friedrich Gauss, who first introduced this concept inthe 19th century. Gauss's theorem states that the flux of a vector field through a closed surface is equal to theintegral of the divergence of the vector field over thevolume enclosed by the surface.To understand this theorem in more detail, it isimportant to first understand the concept of a vector field.A vector field is a function that assigns a vector to every point in space. For example, the wind velocity at every point in the atmosphere can be represented as a vector field. Similarly, the magnetic field around a magnet can also be represented as a vector field.The flux of a vector field through a closed surface is defined as the amount of the vector field that passes through the surface. This concept is closely related to the idea of flow rate, and is an important quantity in fluid dynamics and other similar fields.The divergence of a vector field is a measure of how much the vector field "spreads out" or "converges" at a particular point in space. Mathematically, it is defined asthe dot product of the vector field with the del operator.Using Gauss's theorem, we can relate the flux of a vector field through a closed surface to the divergence of the vector field over the volume enclosed by the surface. This is a powerful mathematical tool that has many important applications in physics and engineering. For example, it can be used to calculate the electric field around a charged object, or the flow rate of a fluid through a pipe.In conclusion, Gauss's theorem is an important mathematical concept that has wide-ranging applications in physics, engineering, and other related fields. It is a powerful tool that enables us to relate the flux of a vector field to its source, and is an essential building block in the development of more complex mathematical models.。

a r X i v :0804.0252v 1 [h e p -t h ] 1 A p r 2008A Matrix Model for 2D Quantum Gravity defined by Causal Dynamical Triangulations J.Ambjørn a,b ,R.Loll b ,Y.Watabikic ,W.Westrad and S.Zohren eaThe Niels Bohr Institute,Copenhagen UniversityBlegdamsvej 17,DK-2100Copenhagen Ø,Denmark.email:ambjorn@nbi.dkb Institute for Theoretical Physics,Utrecht University,Leuvenlaan 4,NL-3584CE Utrecht,The Netherlands.email:loll@phys.uu.nlc Tokyo Institute of Technology,Dept.of Physics,High Energy Theory Group,2-12-1Oh-okayama,Meguro-ku,Tokyo 152-8551,Japan email:watabiki@th.phys.titech.ac.jpd Department of Physics,University of Iceland,Dunhaga 3,107Reykjavik,Iceland email:wwestra@raunvis.hi.ise Blackett Laboratory,Imperial College,London SW72AZ,UK,and email:stefan.zohren@ Abstract A novel continuum theory of two-dimensional quantum gravity,based on a ver-sion of Causal Dynamical Triangulations which incorporates topology change,has recently been formulated as a genuine string field theory in zero-dimensionaltarget space (arXiv:0802.0719).Here we show that the Dyson-Schwinger equa-tions of this string field theory are reproduced by a cubic matrix model.This matrix model also appears in the so-called Dijkgraaf-Vafa correspondence if the superpotential there is required to be renormalizable.In the spirit of this model,as well as the original large-N expansion by ’t Hooft,we need no special double-scaling limit involving a fine tuning of coupling constants to obtain the continuum quantum-gravitational theory.Our result also implies a matrix model represen-tation of the original,strictly causal quantum gravity model.11IntroductionDynamical triangulations(DT)were introduced as a regularization of the Polyakov bosonic string and of two-dimensional quantum gravity[1,2,3].Using this regu-larization,one could show that a tachyon-free version of Polyakov’s bosonic string theory does not exist in target space dimensions d>1[4].However,when viewed as a theory of2d quantum gravity coupled to matter with central charge c≤1, the theory(non-critical string theory)did make ing matrix-model tech-niques and other combinatorial methods,it was sometimes even advantageous to use the regularized theory for analytic calculations.Related attempts to use DT as a regularization of higher-dimensional quantum gravity[5]were less successful [6].This triggered the introduction of Causal Dynamical Triangulations(CDT), which use causal,Lorentzian instead of Euclidean curved spacetimes as a fun-damental input.Evidence has been accumulating that they provide us with a non-trivial theory of quantum gravity in four dimensions[7,8].While the higher-dimensional DT and CDT theories of quantum gravity at this point rely strongly on numerical simulations,the2d CDT theory of quantum gravity can be solved analytically[9],like its2d Euclidean DT counterpart.This is described in detail in two recent papers,where we have also developed a com-plete stringfield theory in a zero-dimensional target space for the CDT version of2d quantum gravity[10,11].1This stringfield theory or third quantization of2d quantum gravity uses the formalism already developed by Ishibashi,Kawai and collaborators for the DT version of2d quantum gravity in the context of non-critical string theory[12,13].For non-critical string theory,it is known from [12]that the stringfield theory reproduces the results of the double-scaling limit of the matrix models whenever the results can be compared.Given the formal similarity between the CDT stringfield theory and the non-critical stringfield theory,it is natural to ask whether there also exists a matrix model which reproduces the results of the former.Below we will show that the answer is in the affirmative.However,since the scaling found in the CDT model is different from the conventional double-scaling limit of matrix models,a different limit needs to be taken.We will show that the limit is simply the conventional limit used in the context of the Dijkgraaf-Vafa duality to U(N)supersymmetric gauge theories[14].2CDT stringfield theoryWe have recently developed a stringfield theory for Causal Dynamical Triangu-lations[11].The starting point of the CDT quantization of gravity is the as-sumption that in a gravitational path integral over spacetimes with a Lorentzian signature only causal geometries should be included,an idea dating back at least to[15].How this can be done explicitly in a regularized theory,how one can rotate to Euclidean signature to perform explicit calculations,and eventually take the cut-off(or lattice spacing)to zero is described in detail in[9]for two and in[16]for three and four spacetime dimensions.We demonstrated in[10] how one can still solve the2d model analytically when the original formulation is extended to allow the light-cone structure to become degenerate in isolated points.In[11]we generalized these results to a genuine stringfield theory,which enabled us in principle to calculate the amplitudes of certain spatial correlators, for two-dimensional worldsheets of any topology2.Let us briefly define the CDT stringfield theory,while referring to[11]for details.We will work in a Euclidean notation,which means that we started out with a Lorentzian signature,regularized the theory,rotated it to Euclidean signature as described in[9]and then took the lattice cut-offa to zero.In particular,this implies that all quantities discussed below are already continuum quantities.We have a“free”Hamiltonian H0which describes the causal propagation of a spatial universe with respect to proper time t.Let a spatial universe with the topology of a circle of length l2(the“exit”loop)be separated a geodesic distance t from another spatial loop of length l1(the“entrance”loop),and denote the(l1,l2;t).It is represented by the path integral corresponding amplitude by G(0)λG(0)(l1,l2;t)= D[gµν]e−S[gµν],(1)λwith the(Euclidean)gravity actionS[gµν]=λ d2ξ2For earlier results in this direction we refer to[17].3topology of a cylinder,S1×[0,1].In a Hilbert space language one has[9](l1,l2;t)= l2|e−t H0(l)|l1 ,H0(l)=−l d2G(0)λΨ†(l)H0(l)Ψ(l)−g dl1 dl2Ψ†(l1)Ψ†(l2)Ψ(l1+l2)(4)l−αg dl1 dl2Ψ†(l1+l2)Ψ(l2)Ψ(l1)− dlFigure 1:A typical geometry in the string field theory contributing to the amplitude w (l 1,...,l n )of eq.(6).Proper time progresses upwards.The dots mark singular points of the causal structure.evolve and eventually vanish into the vacuum,while forming a connected two-dimensional geometry (c.f.Fig.1).The amplitudes w (l 1,...,l n )are determined from the string field theory “partition function”Z (J )=lim t →∞ 0|e −t ˆH e R d lJ (l )Ψ†(l )|0 (7)through the prescriptionw (l 1,...,l n )=δn F (J )δJ (l )−δ(l )−gl l 0dl ′δ2F (J )δJ (l ′)δF (J )δJ (l +l ′).(9)One obtains the Dyson-Schwinger equations for the amplitudes w (l 1,...,l n )by differentiating (9)n times with respect to J (l )and then setting J (l )=0.The general equation at order n can be written down easily,but is involved.We will5give only thefirst three equations explicitly,from which the general structure should be clear.The Dyson-Schwinger equations are most conveniently formu-lated in terms of the Laplace-transformed amplitudesw(x1,...,x n)≡1g (λ−x2),V(x)=13x3 ,(11)we obtain from(9)(see[11]for details)the equations0=∂x −V′(x)w(x)+w2(x)+αw(x,x) −1x−y ,0=∂x [−V′(x)+2w(x)]w(x,y,z)+αw(x,x,y,z) +∂y [−V′(y)+2w(y)]w(x,y,z)+αw(x,y,y,z) +∂z [−V′(z)+2w(z)]w(x,y,z)+αw(x,y,z,z) +(14) 2∂x[w(x,y)w(x,z)]+2∂y[w(x,y)w(y,z)]+2∂z[w(x,z)w(y,z)]+2 ∂x∂y w(x,z)−w(y,z)x−z+∂y∂z w(x,y)−w(x,z)3Note that both w and w h are still g-dependent,although we do not write the dependence explicitly here.6equations at orderα0allow us to determine w0(x),w0(x,y),...,and similarly the equations at general orderαh determine w h(x),w h(x,y),etc.For example,onefindsw0(x)=1g(x−c)212(c−+c+)(x+y)+c−c+(x−c−)(x−c+)2g/c.(18)Writing the amplitudes in this fashion leads one to the surprising realization that w0(x)and w0(x,y)coincide with the large-N limit of the resolvent and the planar loop-loop correlator[18,19,20]of the Hermitian matrix model with potentialV(M)=λ3gM3!(19)This is a potentially exciting result,because so far no standard formulation in terms of matrix models has been found for a CDT model,in contrast to the“old”Euclidean DT models.We will in the following section prove a more general result, which will identify the Dyson-Schwinger equations derived above with the loop equations of a Hermitian matrix model with the cubic potential(19).3Matrix loop equationsLet M denote an N×N Hermitian matrix,V(M)a potential of the formV(M)=−∞k=1g kx1−M)···(tr1d M e−N tr V(M).(22)7It is well known that the matrix integrals corresponding to(21)possess a large-N expansion.Assume we have the so-called one-cut solution related to this expansion.The invariance of the matrix integral under a change in variables leads to the loop equation[18,19,20]C d z x−z W(z)=W2(x)+1W(x1),(24)d V(x2)···d V(x n)where the insertion operator is given byddx k+1W(x,x) −1N2W(x,x,y) +N2+∂x∂y W(x)−W(y)W(x,x,y,z) +N22∂x W(x,z)W(x,y) +(28)∂x∂y W(x,z)−W(y,z)x−z .Using that W(x1,...,x n)is a symmetric function of its arguments,we see that eqs.(26)-(28)lead to exactly the same coupled equations for W as do(12)-(14)8for w if we identify1α=W h(x1,...,x n)(30)N2hof the multi-loop correlators(see,for instance,[21]or the more recent papers [22,23]).The iterative solution of these so-called loop equations is uniquely determined by W0(x)(and the assumption that W(x1,...,x n)is analytic in those x i that do not belong to the cut of the matrix model),and we have already seen that W0(x)=w0(x).4Discussion and OutlookLet us consider the matrix model corresponding to the potential(19).We can√perform a simple change of variables M→−M−g 13M3 ,m=2√the fact that the link length of the triangles(the lattice spacing of the dynamical lattice)was taken to zero in the continuum limit.The situation here is different. Although CDT can be constructively defined as the continuum limit of a dynam-ical lattice,we have in the present work been dealing only with the associated continuum theory.Thus in our case the matrix model with the potential(19)(or (31))already describes a continuum theory of2d quantum gravity.Its coupling constants can be viewed as continuum coupling constants and the role of N is exactly as in the original context of QCD,namely,to reorganize the expansion in the coupling constant g.’t Hooft’s large-N expansion of QCD is a reorganization of the perturbative series in the Yang-Mills coupling g YM,with1/N taking the role of a new expansion parameter.In this framework,after the coefficient of the term1/N2h of some observable has been calculated as function of the’t Hooft coupling g2H=g2YM N,one must take N=3for SU(3),say.The situation in CDT stringfield theory is entirely analogous:starting from a perturbative expansion in the“string coupling constant”g(in fact,in the dimensionless coupling constant g/λ3/2,as described in[10,11]),we can reorganize it as a topological expansion in the genus of the worldsheet by introducing the expansion parameterα.For the multi-loop correlators this expansion is exactly the large-N expansion of the matrix model(19)and the coefficients,the functions W h(x1,...,x n),are exactly the multi-loop correlators for genus-h worldsheets of the CDT stringfield theory withα=1.As a“bonus”for our treatment of generalized(and therefore slightly causality-violating)geometries,we also obtain a matrix formulation of the original two-dimensional CDT model proposed in[9],where the spatial universe was not al-lowed to split.Working out the limit as g→0of the various expressions derived above,we see that this model corresponds to the large-N limit of the matrix model where the coupling constants go to infinity,but at the same time the cut shrinks to a point in such a way that the resolvent(or disk amplitude)survives,that is,w0(x)→1λ=w CDT(x).(33)The existence of a matrix model describing the algebraic structure of the Dyson-Schwinger equations leads automatically to the existence of Virasoro-like opera-tors L n,n≥−1[18,19],which can be related to redefinitions of the time variable t in the stringfield theory.This line of reasoning has already been pursued by Ishibashi,Kawai and collaborators in the context of non-critical stringfield the-ory.It would be interesting to perform the same analysis in the CDT model and show that reparametrization under the change of time-variable will reappear in a natural way in the model via the operators L n.The results should be simpler and more transparent than the corresponding results in non-critical stringfield theory since we have a non-trivial free Hamiltonian H0in the CDT model.10AcknowledgmentJA,RL,WW and SZ acknowledge support by ENRAGE(European Network on Random Geometry),a Marie Curie Research Training Network in the European Community’s Sixth Framework Programme,network contract MRTN-CT-2004-005616.RL acknowledges support by the Netherlands Organisation for Scientific Research(NWO)under their VICI program.References[1]J.Ambjørn,B.Durhuus,J.Fr¨o hlich:Diseases of triangulated random sur-face models,and possible cures,Nucl.Phys.B257(1985)433-449;J.Ambjørn,B.Durhuus,J.Fr¨o hlich and P.Orland:The appearance of critical dimensions in regulated string theories,Nucl.Phys.B270(1986) 457-482.[2]F.David:A model of random surfaces with nontrivial critical behavior,Nucl.Phys.B257(1985)543-576;A.Billoire and F.David:Scaling properties of randomly triangulated planarrandom surfaces:a numerical study,Nucl.Phys.B275(1986)617-640. [3]V.A.Kazakov,A.A.Migdal and I.K.Kostov:Critical properties of ran-domly triangulated planar random surfaces,Phys.Lett.B157(1985)295-300.[4]J.Ambjørn and B.Durhuus:Regularized bosonic strings need extrinsic cur-vature,Phys.Lett.B188(1987)253-257.[5]J.Ambjørn and J.Jurkiewicz:Four-dimensional simplicial quantum gravity,Phys.Lett.B278(1992)42-50.[6]P.Bialas,Z.Burda, A.Krzywicki and B.Petersson:Focusing on thefixed point of4d simplicial gravity,Nucl.Phys.B472(1996)293-308[hep-lat/9601024].[7]J.Ambjørn, A.G¨o rlich,J.Jurkiewicz and R.Loll:Planckian birthof the quantum de Sitter universe,Phys.Rev.Lett.100(2008)091304 [arXiv:0712.2485,hep-th].[8]J.Ambjørn,J.Jurkiewicz and R.Loll:Reconstructing the universe,Phys.Rev.D72(2005)064014[hep-th/0505154];Spectral dimension of the universe,Phys.Rev.Lett.95(2005)171301[hep-th/0505113];Semiclassical universe fromfirst principles,Phys.Lett.B607(2005)205-21311[hep-th/0411152];Emergence of a4D world from causal quantum gravity,Phys.Rev.Lett.93 (2004)131301[hep-th/0404156].[9]J.Ambjørn and R.Loll:Non-perturbative Lorentzian quantum gravity,causality and topology change,Nucl.Phys.B536(1998)407-434[hep-th/9805108].[10]J.Ambjørn,R.Loll,W.Westra and S.Zohren:Putting a cap on causalityviolations in CDT,JHEP0712(2007)017[arXiv:0709.2784,gr-qc]. [11]J.Ambjørn,R.Loll,Y.Watabiki,W.Westra and S.Zohren:A stringfieldtheory based on causal dynamical triangulations,[arXiv:0802.0719,hep-th];Topology change in causal quantum gravity[arXiv:0802.0896,hep-th]. [12]N.Ishibashi and H.Kawai:Stringfield theory of noncritical strings,Phys.Lett.B314(1993)190-196[hep-th/9307045];Stringfield theory of c≤1noncritical strings,Phys.Lett.B322(1994) 67-78[hep-th/9312047];A background independent formulation of noncritical string theory,Phys.Lett.B352(1995)75-82[hep-th/9503134].[13]H.Kawai,N.Kawamoto,T.Mogami and Y.Watabiki:Transfer matrix for-malism for two-dimensional quantum gravity and fractal structures of space-time,Phys.Lett.B306(1993)19-26[hep-th/9302133];M.Ikehara,N.Ishibashi,H.Kawai,T.Mogami,R.Nakayama and N.Sasakura:Stringfield theory in the temporal gauge,Phys.Rev.D50 (1994)7467-7478[hep-th/9406207];A note on stringfield theory in the temporal gauge,Prog.Theor.Phys.Suppl.118(1995)241-258[hep-th/9409101];Y.Watabiki:Construction of noncritical stringfield theory by transfer ma-trix formalism in dynamical triangulation,Nucl.Phys.B441(1995)119-166 [hep-th/9401096];H.Aoki,H.Kawai,J.Nishimura and A.Tsuchiya:Operator product expan-sion in two-dimensional quantum gravity,Nucl.Phys.B474(1996)512-528 [hep-th/9511117];J.Ambjørn and Y.Watabiki:Non-critical stringfield theory for2d quan-tum gravity coupled to(p,q)–conformalfields,Int.J.Mod.Phys.A12(1997) 4257-4289[hep-th/9604067].[14]R.Dijkgraaf and C.Vafa:Matrix models,topological strings,and supersym-metric gauge theories,Nucl.Phys.B644(2002)3-20[hep-th/0206255];A perturbative window into non-perturbative physics[hep-th/0208048].12[15]C.Teitelboim:Causality versus gauge invariance in quantum gravity andsupergravity,Phys.Rev.Lett.50(1983)705-708;The proper time gauge in quantum theory of gravitation,Phys.Rev.D28 (1983)297-309.[16]J.Ambjørn,J.Jurkiewicz and R.Loll:Reconstructing the universe,Phys.Rev.D72(2005)064014[hep-th/0505154];Dynamically triangulating Lorentzian quantum gravity,Nucl.Phys.B610 (2001)347-382[hep-th/0105267].[17]R.Loll and W.Westra:Space-time foam in2d and the sum over topologies,Acta Phys.Polon.B34(2003)4997-5008[hep-th/0309012];Sum over topologies and double-scaling limit in2D Lorentzian quantum grav-ity,Class.Quant.Grav.23(2006)465-472[hep-th/0306183];R.Loll,W.Westra and S.Zohren:Taming the cosmological constant in 2D causal quantum gravity with topology change,Nucl.Phys.B751(2006) 419-435[hep-th/0507012].[18]F.David:Loop equations and nonperturbative effects in two-dimensionalquantum gravity,Mod.Phys.Lett.A5(1990)1019-1030.[19]J.Ambjørn,J.Jurkiewicz and Yu.M.Makeenko:Multiloop correlators fortwo-dimensional quantum gravity,Phys.Lett.B251(1990)517-524. [20]J.Ambjørn and Yu.M.Makeenko:Properties of loop equations for the Her-mitean matrix model and for two-dimensional quantum gravity,Mod.Phys.Lett.A5(1990)1753-1764.[21]J.Ambjørn,L.Chekhov,C.F.Kristjansen and Yu.Makeenko:Matrix modelcalculations beyond the spherical limit,Nucl.Phys.B404(1993)127-172;Erratum-ibid.B449(1995)681[hep-th/9302014].[22]B.Eynard:Topological expansion for the1-hermitian matrix model correla-tion functions,JHEP0411(2004)031[hep-th/0407261].[23]L.Chekhov and B.Eynard:Hermitean matrix model free energy:Feynmangraph technique for all genera,JHEP0603(2006)014[hep-th/0504116].13。

a r X i v :0708.2250v 1 [h e p -t h ] 16 A u g 2007Lorentz Invariance Violation from String Theory ∗Speaker.†Thiswork has been supported in part by the European Science Foundation network programme Quantum Geom-etry and Quantum Gravity and by the European Union through the FP6Marie Curie Research and Training Network UniverseNet (MRTN-CT-2006-035863).√√2iDµ(x0)aµ−4+...δ3(x0)b−1c−3+...}|0 (2.1) where x0denotes a generic space time point and aµn are the appropriate creation operators,which upon acting on the(first quantized)open string vacuum state|0 ,create the various excitation modes:scalars(tachyons)with amplitude T(x0),which are characteristic of the open bosonic(in general broken supersymmetric)string,vectors,with amplitudes Aµ,...,tensors with amplitudes Bµν,...etc..Ghostfields,with creation operators b n,c n are also included,which arise fromfixing the gauge invariances of the string state.The open stringfield theory action is cubic in thefieldΨ.Upon expanding about a squeezed state backgroundΨB,Ψ=ΨB+∆,as appropriate for our discussion on Lorentz violating vacua in strings[2],one obtains for the stringfield theory action I(Ψ):I(Ψ)=13ΨB⋆ΨB⋆ΨB+13∆⋆∆⋆∆.(2.2) In the above equation,⋆denotes the appropriate gauge invariant inner product of open stringfield theory,α′is the Regge slope,and the suffix B denotes background,whilst g is the ghost number. The quantity Q is the nilpotent(Q2=0)BRST operator,and Q B is defined by its action on a generic string stateΦin the backgroundΨB:Q BΦ=QΦ+gα′[ΨB⋆Φ−(−1)g(Φ)Φ⋆ΨB].As becomes evident from(2.2),in such models,there are cubic terms in an effective low-energy(target-space)Lagrangian involving the tachyonic scalarfield T,that characterizes thebosonic string vacuum,and invariant products of higher-tensorfields Bµ1...µn that appear in themode expansion of a stringfield:T Bµ1...µnBµ1...µn.(2.3) The negative mass squared tachyonfield,then,acts as a Higgsfield in such theories,acquiring a vacuum expectation value,which,in turn,implies non-zero vacuum expectation values for the tensorfields B,leading in this way to energetically preferable configurations that are Lorentz Invariance Violating(LIV).From the point of view of string theory landscape these are perfectly acceptable vacua[2],given that they respect world-sheet conformal invariance of thefirst quantized string theory.An effective target-spacefield theory framework to discuss the phenomenology of such LIV theories is the so-called Standard Model Extension(SME)[10].For our purposes in this section, it suffices to simply give an example of SME effects on phenomenology of particle physics,by considering the SME Modified Dirac Equation for spinorfieldsψ,representing leptons and quarksHµνσµν+icµνγµDν+idµνγ5γµDν ψ=0,(2.4)2where Dµ=∂µ−A aµT a−qAµis an appropriate gauge-covariant derivative.The non-conventional terms proportional to the coefficients aµ,bµ,cµν,dµν,Hµν,...,stem from the corresponding local operators of the effective Lagrangian which are phenomenological at this stage.The set of terms pertaining to aµ,bµentail CPT and Lorentz Violation,while the terms proportional to cµν,dµν,Hµνexhibit Lorentz Violation only.It should be stressed that,within the SME frame-work,as is also the case with the decoherence approach to Quantum Gravity(QG)[9],CPT viola-tion does not necessarily imply mass differences between particles and antiparticles.Some remarks are now in order,regarding the form and order-of-magnitude estimates of the Lorentz and/or CPT violating effects.In the approach of[10]the SME coefficients have been taken to be constants.Unfortunately there is not yet a detailed microscopic model available,which would allow for concrete predictions of their order of magnitude.Theoretically,the(dimensionful,with dimensions of energy)SME parameters can be bounded by applying renormalization group and naturalness assumptions to the effective local SME Hamiltonian,which leads to bounds on bµof order10−17GeV.At present all SME parameters should be considered as phenomenological and to be constrained by experiment.In general,however,the constancy of the SME coefficients may not be true.In fact,in certain string-inspired or stochastic models of space-time foam that violate Lorentz symmetry[9,13],the coefficients aµ,bµ...are probe-energy(E)dependent,as a result of back-reaction effects of matter onto thefluctuating space-time.Specifically,in stochastic models of space-time foam,one mayfind[13]that on average there is no Lorentz and/or CPT violation, i.e.,the respective statistical v.e.v.s(over stochastic space-timefluctuations) aµ,bµ =0,but this is not true for higher order correlators of these quantities(fluctuations),i.e., aµaν =0, bµaν = 0, bµbν =0,....In such a case the SME effects will be much more suppressed,since by dimensional arguments suchfluctuations are expected to be at most of order E4/M2P,and hence much harder to detect.3.Non-Critical String theory as an alternative to Landscape and LIVCritical strings,in afirst quantized form[14],endow world-sheet conformal invariance.As a result,from a target space time view point the strings propagate in classical space time backgrounds that obey equations of motion,derived from an effective action,and thus they describe by construc-tion equilibrium situations infield theory.On the other hand,non-critical(Liouville)strings[15] correspond toσ-models deformed by world-sheet vertex operators that are non conformal.As a result,the corresponding target-space backgrounds,which the string propagates on,are off-shell, that is they do not obey equations of motion,although the corresponding world-sheet beta func-tions are proportional to off-shell variations of the string effective action.This latter result stems from a generic property of stringyσ-models,according to which the world-sheet renormalization group beta function,βi(g),expressing the running of the“renormalized”background g i(µ)with the world-sheet renormalization scaleµ,is always proportional to the variations with respect to g i of an off-shell scalar function S in the space of backgrounds{g i},which thus plays the rôle of a(3.1)δg jwhere G i j is the inverse of Zamolodchikov metric[17]in the theory space of strings,{g i},which is related[17,16]to world-sheet short distance divergencies of the two point function of the vertex operators for the backgrounds g i:G i j=Lim z→0z2z)V j(0,0) .In this sense,non-critical strings represent non equilibrium situations in string theory,and in fact have been used[18,19]to discuss an approach to equilibrium in cosmology,in an attempt to explain the smallness of the observed current-epoch cosmological constant(or,more accurately, dark energy),by viewing it as a relaxation phenomenon(seefig.1).The framework may be viewed as providing alternatives to landscape scenarios in string theory[3,19].Crucial to the above in-terpretation was the identification[3]of time with the Liouville mode of non-critical strings[15], which is allowed in certain supercritical string models[20],described byσ-models whose central charge exceeds the critical value.We note that recently,supercritical strings,especially from the cosmological point of view,attracted some attention,either from a theoretical point of view,in an attempt to discuss the initial value problem of our Universe and the associated cosmological in-stabilities[21],or from a purely phenomenological view point,as modifying astroparticle physics constraints on interesting particle physics models,such as supersymmetric[22].In this presentation we shall consider non-critical strings[3]as providing[23]situations in which Lorentz symmetry of the string vacuum may be broken,in the sense of leading to a non-trivial vacuum refractive index,i.e.modified dispersion relations,for photons.In fact,to our knowledge,this was thefirst instance where modified dispersions have been proposed in the context of concrete Lorentz violating approaches to quantum gravity.Subsequently,many other proposals have been made,which entailed non-standard dispersion relations for a variety of reasons that I will not discuss here.For the purposes of this talk,I will single out the so-called Deformed Special8π Σd2ξ γg i(φ)V i(X),(3.3)where γis afiducial world-sheet metric,and the plus(minus)sign in front of the kinetic term of the Liouville mode pertains to subcritical(supercritical)strings.The dressed couplings g i(φ)are obtained by the following procedure:d2z g i V i(X)→ d2z g i(φ)eαiφV i(X),(3.4)whereαi is the“gravitational”anomalous dimension.If the original non-conformal vertex operator has anomalous scaling dimension∆i−2(for closed strings,to which we restrict ourselves for definiteness),where∆i is the conformal dimension,and the central charge surplus of the theory is Q2=c m−c∗+ 4+2−∆i.(3.6)2The gravitational dressing is trivial for marginal couplings,∆i=2,as it should be.This dressing applies also to higher orders in the perturbative g i expansion.For instance,at the next order,where the deviation from marginality in the deformations of the undressedσmodel is due to the operator product expansion coefficients c iin theβi function,the Liouville-dressing procedure implies thejkreplacement[25]:g i→g i eαiφ+πφd2z√γg i V i≡1Here the concepts of Fourier transforms and plane waves should be understood as being appropriately generalized to curved target spaces,with the appropriate geodesic distances taken into account.For our purposes below,the details of this will not be relevant.We shall work with macroscopically-flat space-times,where the quantum-gravity structure appears through quantumfluctuations of the vacuum,leading simply to non-criticality of the string,in the sense of non-vanishingβi functions.(3.14)M Pwithηa dimensionless quantity,which parametrizes our present ignorance of the quantum struc-ture of space time.For the purposes of the present work,we restrict ourselves to non-critical strings onfixed-genus world sheets,in which caseηin(3.14)is real.This should be viewed as describ-ing only part of the quantum-gravity entanglement,namely that associated with the presence of global string modes[3].The full string problem involves a summation over genera,which in turn implies complexη’s,arising from the appearance of imaginary parts in Liouville-string correlation functions on re-summed world sheets,as a result of instabilities pertaining to microscopic black-hole decay in the quantum-gravity space-time foam[3].Such imaginary parts inηwill produce frequency-dependent attenuation effects in the amplitudes of quantum-mechanical waves for low-energy string modes.This type of attenuation effect leads to decoherence of the type appearing in the density-matrix approach to measurement theory[3].For our purpose of deriving bounds on the possible accuracy of distance measurements,however,such attenuation effects need not be taken into account,and the simple entanglement formula(3.14),with realη,will be sufficient.The value ofηdepends in general on the type of masslessfield considered.In particular,in the case of photonsηis further constrained by target-space gauge invariance,which restricts the structure of the relevantσ-modelβfunction.With this caveat in mind,(3.14)represents a maximal estimate of∆αi,compatible with the generic structure of perturbations of theσ-modelβfunctions for closed strings.The equations(3.7),(3.13)and(3.14)indicate that the dressedσ-model deformation(3.4) corresponds to waves of the form2k|2e i| k|t+i k. X+iη|2Note that eq.(3.15)is consistent with target-space gauge invariance,since it may describe one of the polarization states of the photon,and the transversality condition on the photon polarization tensor,kµAµ=0,can be maintained.3The reader’s attention is called here upon the necessary presence of a D-brane world when D-particles are em-bedded in the bulk space,given that isolated D-particles cannot exist[29]due to the requirement of conservation of the U(1)gaugefluxes that characterize D-branes[28].In general,in string/brane theory higher-rank Dp-branes(i.e.with p longitudinal directions)provide endpoints for lower-p branes,and the latter govern the dynamics of the former.i +∞−∞dωe iωX0ε,which describes the duration of the scattering,not to beconfused with the life time of the intermediate composite string state offig.2,which is much smaller,of the order of the string lengthℓs[32]4.It turns out that,for reasons pertaining to the closure of the logarithmic algebra[31]the limitε→0+cannot be taken independently of the world sheet renormalization scale lnΛ,whereΛis the world-sheet area.In fact,one must haveε−2∼lnΛ(4.3)Hence,only close to an infraredfixed point situation,whereΛ→∞one hasε→0+,and the duration of the scattering is much larger than all other scales in the problem.In general,1/εmust be keptfinite,and this expresses the life time of the induced space time deformation,which we now proceed to discuss.To this end,we take into account[31]that the anomalous dimension of the operators(4.1),(4.2) is−ε2/2and hence these deformations are relevant from a world-sheet renormalization group viewpoint.The corresponding open stringσ-model is thereby non conformal,in need of Liouville2.By rewrit-ing the boundary dressed operator(4.4)as a bulk one,using Stokes theorem,and performing the relevant partial integrations on the world sheet,using also the(world-sheet)string equations of motion,we may arrive at the following bulk world-sheetΣ-deformation[4]:V bulk∼−εu iΣeεϕ/√∂X i∼−εu iΣeε(ϕ/√∂X i(4.5)where we approximatedΘε(X0)∼e−εX0,for X0>0where our formalism applies.We now observe that for times of order X0∼∆t∼1/ε,i.e.within the duration of the scattering,the bulk operator (4.5)implies an off-diagonal deformation of the space-time metric[4],withϕ−i componentsg iϕ∼eε(ϕ/√√√5A more complicated situation,leading to different proportionality constants between Liouville mode and target time has been considered in[32],where a semi-microscopic model to describe D-particle recoil induced by strings propagating in brane worlds has been considered.The induced four-dimensional metric in such a situation is more complicated than(4.7),as is obtained by averaging appropriately over statistical populations of D-particles(c.f.some discussion below).The results of both analyses,however,as far as D-particle-recoil-induced modified dispersion re-lations for matter probes are concerned,are qualitatively similar and,hence,for our purposes in this article we shall consider(4.9)from now on.6In the original works[4]we have considered times of order1/εso we ignored the decay,and concentrated only on the metric(4.12).Here we keep the X0-dependence explicit,in order to make manifest the relaxation nature of the induced deformation,implying an asymptotic Minkowski metric.(4.14)M swhere the parameterξ>0depends on the details of the D-particle foam,such as the D-particle density[32].Such details cannot be estimated by theoretical considerations at present,as we are lacking a fundamental microscopic model.However,progress towards this direction is made by constructing explicitly superstring/super-brane world models in higher dimensional bulk space times[19].The big issue pending of course,towards phenomenologically realistic four dimen-sional models,is the appropriate compactification procedure,which unfortunately entails all sorts of complications regarding supersymmetry breaking etc.The main conclusion from this section so far,therefore,is that in the non-critical stringy mod-els of space-time foam,involving D-particles,there is an induced modification to the space time metric,(4.12),which depends linearly on the incident momentum,and eventually leads to sublu-minal modified dispersion relations(4.13)on average.This subluminality feature will distinguish the approach from others in LIV Quantum Gravity,such as certain loop quantum gravity models and other proposals[36],where both subluminal and superluminal propagation may be allowed, thereby implying experimentally detectable(in principle)birefringence effects.8Moreover,since neutrinos avoid clustering,they have also been conjectured in[40]to contribute to the dark energy of the Universe in a non-trivial way,which I do not have the time or space to explain here.2θµν∂xµ∂xν f(x)g(y)|x=y),can be made physically equivalent to a subset of a general Lorentz violating extension of the standard model(SME),of the type considered by Kostelecky and collaborators[10]and mentioned in section2of the present article.It is interesting to notice[44]that,since many non commutativefield theories satisfy CPT invariance, the resulting SME effective theories involve non renormalizable Lorentz violating terms,which however respect CPT symmetry.For example,regarding SME quantum electrodynamics,which was the subject of our discussion in section2,the following subset of terms may be physically equivalent to a CPT conserving non-commutativefield theory to leading order inθµν[44]:L=1ψγµ↔Dµψ−m4FµνFµν−1ψγµ↔Dµψ+1ψγµ↔Dβψ+1ψψ−18qθαβFαβFµνFµν.(5.2)with q being the electric charge and Dµdenoting the gauge covariant derivative as usual.The reader should compare(5.2)with the corresponding expression for SME,leading to(2.4):all the CPT-violating terms are absent,since the microscopic non commutativefield theory is CPT invariant.For our purposes in this work,we mention that in general,non-commutativefield theories appear to have problems with either unitarity and/or causality[45].Indeed,the authors of[45] considered the elastic scattering of two wavepackets into two outgoing ones,in a generic non-commutativefield theory(5.1),and demonstrated the existence of acausal(“advanced”)out-going wave packets,with negative delays∆t,i.e.occurring before even the scattering of the incident waves took place.Also their calculation showed that rigid rods grow instead of Lorentz contracting at high energies.This unphysical situation,which seems to be generic to non commutativefield theories,is eliminated if one considers stringy effects.Indeed,the authors of[45]repeated the scattering calculation by considering wavepackets in open(super)string theory.The result shows no advanced waves,and more interestingly,the outgoing wave packet splits into a series of packets,one located at the origin,x=0,and the others at x=16πnα′p0(where p0is the energy of the wave packet,α′=ℓ2s is the Regge slope,andℓs is the string length scale),with increasing spread and decreasing amplitude as the order n of the packet increases.There is an intermediate stretched string state formed,which oscillates,thereby producing the series of the outgoing packets.The string has total energy p0,and its length begins to grow up to order L∼α′p0storing the energy as potential energy. However,the scattering is causal,and the positive time delay,obtained in the calculation,increases linearly with the energy p0:∆t=α′p0,(5.3).We mention at this point that Genus re-summation effects are still not G+1fully understood in string theory(due to issues regarding re-summability of the genus expansion), and hence truly non perturbative expressions for the above time delays are not known at present.In tests of photon dispersion relations,such as those in[39]studying the arrival times of photons from distant gamma-ray bursts,uncertainties due to critical-string interactions,as in(5.3), should be considered as source effects,describing for instance interactions among photons-viewed as open string states-at the source regions.Such uncertainties would result in non simultaneous emissions of photons,and should be taken in principle into account in studies like[39],in order to correct the analyses in searches for quantum-gravity propagation effects.However,even for large string length scales,ℓs≫ℓPlanck,the number of total photon-photon scatterings at the source are such that these effects are negligible,when compared to the currently available time resolutions in these measurements.The time delays/uncertainties(5.3),that increase linearly with the energy,bear some formal similarities with our D-particle/open-string-state interactions.There again,we have had the for-mation of an intermediate composite state(seefig.2),but the difference from theflat-space-time case of open-string scattering,was the induced Finsler-like metric(4.12).Nevertheless,precisely due to such metrics,there will be time delays for highly energetic photons,as compared to the less energetic ones,which will grow linearly with the energy,as follows from the induced refractive index effects(4.14).We note that,in the D-particle case,the total time delay of a matter probe in the D-particle foam,depends on the details of the defects distribution.If the latter is non uniform, as in the model of[40],where the D-particle concentration is high near massive celestial bodies, due to the D-particle mass M s/g s,and very low in the“empty”space outside the body,then the arrival time delay effects due to(4.14)are suppressed in GRB photon tests[39],compared to the case of uniform D-particle distribution over the D-brane world.This is a consequence of the fact that,in such a case,the effects are largest near the massive source,whilst photon propagation be-tween the source and the detector is virtually free.However,it seems to me that,to obtain such source-D-foam induced delays that could be measurable by the current technology,would require an unphysically large concentration of D-particles near the source of the GRB.∂2F22gµν(x,u)uµuν(6.2)A particle,of mass m,moving in a Finsler geometry is characterized by an actionI=m f i F(x,˙x)dτ(6.3) where the overdot denotes derivative with respect to time t,and F is the Finsler norm defined above (6.1),(6.2).The norm can depend on several physically important parameters,such as the mass of the particle,or,as is the case with the D-particle recoil metric,the string scale and coupling,and in general the Planck scale for QG-induced modified dispersion relations.The canonical momenta in this formalism are defined as[7]:(6.4)pµ=m∂FFand the modified dispersion relations are obtained as a result of the mass-shell condition with respect to the canonical momentahµν(x,p)pµpν=−m2(6.5) where hµν(x,p)is the inverse of the velocity-dependent metric,hµν(x,p)=gµν(x,˙x).m(1−b) 2b/(1+b)(6.11) implying modified dispersion relations,as generically expected in a Finsler geometry.Bounds on b can be found by looking for anisotropy limits in mass kinetic terms19I note here that an attempt to discuss deformed Lorentz symmetries of the type appearing in DSR models[5], leading to modified dispersion relations,in world-sheetσ-models,was also made in[51].22。

高二英语数学建模方法单选题20题1. In the process of mathematical modeling, "parameter" means _____.A. a fixed valueB. a variable valueC. a constant valueD. a random value答案：A。

解析：“parameter”常见释义为“参数”，通常指固定的值，选项 A 符合；选项B“variable value”意为“变量值”；选项C“constant value”指“常数值”；选项D“random value”是“随机值”，在数学建模中“parameter”通常指固定的值。

2. When building a mathematical model, "function" is often used to describe _____.A. a relationship between inputs and outputsB. a set of random numbersC. a single valueD. a group of constants答案：A。

解析：“function”在数学建模中常被用来描述输入和输出之间的关系，选项 A 正确；选项B“a set of random numbers”表示“一组随机数”；选项C“a single value”是“单个值”；选项D“a group of constants”指“一组常数”。

3. In the context of mathematical modeling, "optimization" refers to _____.A. finding the best solutionB. creating a new modelC. changing the parameters randomlyD. ignoring the constraints答案：A。