Extra Gauge Invariance from an Extra Dimension

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On the Gauge Aspects of Gravity

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.。

Currents and Superpotentials in classical gauge invariant theories I. Local results with ap

a rXiv:g r-qc/98429v 228May1998gr-qc/9804029LPTENS 98/06Currents and Superpotentials in classical gauge invariant theories I.Local results with applications to Perfect Fluids and General Relativity.B.Julia and S.Silva Laboratoire de Physique Th´e orique CNRS-ENS 24rue Lhomond,F-75231Paris Cedex 05,FranceABSTRACTE.Noether’s general analysis of conservation laws has to be completed in a Lagrangian theory with local gauge invariance.Bulk charges are replaced byﬂuxes at a suitable singularity(in general at inﬁnity)of so-called superpotentials,namely local functions of the gaugeﬁelds(or more generally of the gauge forms).Some gauge invariant bulk charges and current densities may subsist when distinguished one-dimensional subgroups are present.We shall study mostly local consequences of gauge invari-ance.Quite generally there exist local superpotentials analogous to those of Freud or Bergmann for General Relativity.They are parametrized by inﬁnitesimal gauge transformations but are aﬄicted by topological ambiguities which one must handle case by case.The choice of variational principle:variables,surface terms and bound-ary conditions is crucial.As aﬁrst illustration we propose a new Aﬃne action that reduces to General Rela-tivity upon gaugeﬁxing the dilatation(Weyl1918like)part of the connection and elimination of auxiliaryﬁelds.We can also reduce it by similar considerations either to the Palatini action or to the Cartan-Weyl moving frame action and compare the associated superpotentials.This illustrates the concept of Noether identities.We formulate a vanishing theorem for the superpotential and the current when there is a(Killing)global isometry or its generalisation.We distinguish between,asymptotic symmetries and symmetries deﬁned in the bulk.A second and independent application is a geometrical reinterpretation of the convec-tion of vorticity in barotropic nonviscousﬂuidsﬁrst established by Helmholtz-Kelvin, Eckart and Ertel.In the homentropic case it can be seen to follow by a general theorem from the vanishing of the superpotential corresponding to the time indepen-dent relabelling symmetry.The special diﬀeomorphism symmetry is,in the absence of dynamical gaugeﬁeld and spin,associated to a vanishing internal transverse mo-mentumﬂux density.We consider also the nonhomentropic case.We identify the one-dimensional subgroups responsible for the bulk charges and thus propose an im-pulsive forcing for creating or destroying selectively helicity resp.enstrophies in odd resp.even dimensions.This is an example of a new and general Forcing Rule.21Introduction1.1GeneralitiesEighty years ago E.Noether[1]assembled together in a series of theorems some consequences of continuous symmetries of classical actions.Any rigid (Lie)symmetry gives rise to a current with the general formula∂LJ:=Σ−δφ∧1950[2,3,4],namely there are local superpotentials U((D−2)-forms)such that on shell,assuming onlyﬁelds and theirﬁrst derivatives contribute to the action:Jξ:=ξ.J+dξ.∧U(2) is conserved for allξ’s and hence J=dU.Bergmann,see[3]introduced the term strong conservation laws.In fact local invariance is still widely and wrongly believed to actually prevent the existence of any invariant conserved charge,see however the recent[5].Note that the locality of U does not follow from that of J even when spacetime is contractible.If one restricts attention to inﬁnitesimal gauge transformations along aﬁxed generator,one may still multiply the latter by a scalar coeﬃcient depending arbitrarily on spacetime coordinates and apply the Hilbert-Noether-Bergmann construction to that subalgebra,one obtains thenJξ≈dUξ:=d(ξ.U)(3) Independently of these results it was shown in1981[6]that a p-form gauge invariance corresponding to a(p+1)-form potential leads to a(D−p−2)-form J that is closed on shell.In other words dJ≈0modulo the equations of motion,generalizing the p=0and p=−1cases.If one views Yang-Mills invariance as a mixture of p=0and p=−1invariances one recovers the analog of Bergmann’s analysis.We recognize one half of Maxwell’s equations in the strong conservation equationJ≈dU=d∗F(4) In the nonabelian case we may still pick a direction of gauge transformations with arbitrary(scalar and x dependent)magnitudeξ(x)then for this par-ticular abelian subgroup of gauge transformations we have the same formula 3.This is the origin of the’t Hooft abelian charges of the dyons,see for instance[7].The discussion of higher conservation laws has been recently carefully extended to a generalised Noether theorem relating symmetries of various types with generalised charges[8]in a cohomological framework.Now in the case of rigid symmetry,J is already aﬄicted by ambigui-ties,it is well known that they permit the constructions of the symmetrized or improved energy-momentum tensors.This arbitrariness becomes much more serious in the case of gauge invariance as the ambiguity of the super-potential U seems to be total.In fact the litterature on General Relativity4is littered with a host of superpotentials without clear status of respectabil-ity.We shall concentrate in this paper on the local aspects of the theory,in particular on the formulas that generalize1,their dependence on the order of diﬀerentiation of theﬁelds and on possible surface terms.But let us recall that the physical measurement of the force leads to the value of a gravitational mass far from its source by actually assuming asymptoticﬂatness all around it.One could also expand around an arbitrary background near inﬁnity and deﬁne a mass parameter there,this has been carried out in particular for anti de Sitter asymptotics.In this paper we shall focus on the asymptoticallyﬂat case.One puts the laboratory at inﬁnite distance from the source(s)in some direction in the sense that the metricbecomesﬂat up to order1/r corrections then the limit of r2(g ii−1)in asymptotic rest frame coordinates is the physical mass deviating test particles.The use of arbitrary coordinates requires a geometrical deﬁnition of asymptotics,in other words of the boundary at inﬁnity(we shall consider spatial inﬁnity in thisﬁrst paper).We must choose a model manifold for the neighbourhood of inﬁnity but not its coordinates.Note that this manifold does not have to be close to ours except there.A side remark is that local but not global asymptoticﬂatness would force us to distinguish between a local deﬁnition of mass from formulas involving totalﬂuxes.Let us take the example of electromagnetism and consider an orbifold ALM space obtained by quotienting I R4by Z2(the sphere at inﬁnity is replaced by I R P2).Clearly if the electricﬁeld is e4ǫijk g j0r k again if one assumes global trivial topology at inﬁnity.We leavethis global issue for subsequent work.To the above physical and local deﬁnition of charge one can compare mathematical formulas,for instance the charge may take the form of aﬂux at inﬁnity,this is the case for the celebrated ADM expression[9]for the total mass of a curved spacetime or the generalisation by Regge and Teitel-boim[10]in a Hamiltonian description.We shall follow here a Lagrangian approach and invert the conventional order of the constructions:we shall look for a bulk density such that its integral is equal to the physical mass given by such aﬂux at inﬁnity.It has not been widely recognized that when there is a singularity,even if it is hidden behind an horizon and contrary to the abelian case,bulk integrals may not make physical sense.The Nester-Witten form[11,12]can be used outside horizons and will be5discussed in the next paper of this series hereafter called paper II.The proof of the positivity of total ADM mass for a general solution with black holes uses the existence of a supersymmetric extension of the theory it localises all the energy outside the horizons,and the“energy density”is positive there.The special case of a global Killing vector is essentially bringing us into an abelian framework as Kaluza-Klein inspired ideas may suggest.In a gen-eral gauge theory we shall call Killing symmetry a Lie algebra generator preserving the value of the gaugeﬁeld,for instance isometries of a metric, isotropy gauge transformations in Yang-Mills theory,Killing spinors for su-persymmetry etc...The existence of a global(bulk)Killing symmetry leads in general to the vanishing of the gauge part of the current density as a generalisation of the vanishing charge of the photons and of all the Fourier zero modes of the Kaluza-Klein dimensional reduction.In the case of diﬀeo-morphisms the gauge current may fail to vanish because of a surface term but it does vanish for spatial Killing directions and in the vacuum as we shall see.We shall discuss the general formalism in section2but mostly focus on the rich case of diﬀeomorphisms.1.2PerfectﬂuidsThe reader interested in the conservation laws ofﬂuids will at this stage be able to skip the middle sections(3-5)on our new formulation of General Rel-ativity and should go directly to section6,if he so wishes.Relabelling sym-metries allow a physically suggestive interpretation of the conserved quan-tities of perfectﬂuids.Theseﬂuids obey a variational principle involving independent Lagrangian coordinates(the labels),theﬁelds are simply the Eulerian,or laboratory,coordinates,they admit time independent space relabeling gauge invariance without any gaugeﬁeld.In the homentropic (possibly compressible)case the relabelings are arbitrary volume preserving diﬀeomorphisms.The corresponding spatial Noether current is purely lon-gitudinal because there is no propagating gaugeﬁeld in this gauge invariant theory,this is the local vorticity conservation in comoving cordinates[13]. Noether’s theorem has been invoked before but without superpotentials(see the nice review[14])and when it was precisely formulated it was the global theorem that was used as in Taub’s description ofﬂows(see the review [15])where the roles of Lagrangian and Eulerian variables are exchanged. It turns out that global(bulk)conservation laws do exist even in the ab-sence of boundaries.This may seem surprising to aﬁeld theorist;we shall6explain this phenomenon and identify the rigid symmetries responsible for these charges.We hope to return to the eﬀect of boundaries in the future.We shall also identify a simple mechanism of creation of these charges by forcing with an optimum scheme that could be implemented numerically and almost experimentally.The problem with experimental implementation is not serious,in most(=slow speed)situations the incompressible approxi-mation is valid and thus one may identify at any chosen instant Lagrangian and Eulerian coordinates so the forcing mechanism can be formulated either theoretically in Lagrangian coordinates or practically of course in Eulerian ones.We shall return to the incompressible case in the next paper.This forcing,although impulsive,is reminiscent of the generation of the electric charge of electromagnetic dyons by uniform rotation in internal space [16]and of geodesic motion of quasistatic solutions of the variational problem of magnetic monopole theory[17].We shall also explain the relation between homentropic and non homentropic situations,in fact a partial breaking from (D+1)to D-dimensional relabeling symmetry by some marker like the value of the entropy changes dramatically the number of local invariants and exchanges the properties of even and odd numbers of space-dimensions.1.3General RelativityThe organisation of the rest of the paper is as follows.First the notions of cascade and abelian cascade of currents and superpotentials:J or T,U,V... are introduced in subsection2.1and illustrated on simple examples includ-ing electromagnetism,Yang-Mills theory,p-forms gaugeﬁelds,in the rest of section2.The identiﬁcation of Noether currents for selected generators re-duces the problem ofﬁnding the invariant charges to the selection of abelian subgroups of gauge symmetries.In section3,Hilbert’s action in second order form is analysed and the need for longer cascades appears.The spin term of Belinfante’s symmetrized energy-momentum tensor[18]for matter is derived from the matter contri-bution to the superpotential.The mechanism is that tensorﬁelds with spin do transform under diﬀeomorphisms with derivative terms as gaugeﬁelds do.In the fourth section a newﬁrst order aﬃne gauge theory of gravitation is deﬁned.Its symmetries include diﬀeomorphisms,local linear frame trans-formations and a new gauge symmetry without gaugeﬁeld,let us call it the7Einstein-Weyl symmetry,we shall see why momentarily.The latter gauge symmetry does not have any propagating gaugeﬁelds,as a consequence the associated superpotential and currents vanish.This is a special case of the so-called Noether identities which is formulated as a general vanishing theorem in subsection4.2.The various superpotentials are easily analysed. In subsection4.3other Noether identities are discussed and the Sparling-Dubois-Violette-Madore rewriting of Einstein’s equations as a closure,or conservation,condition[19,20]is adapted to our aﬃne theory.Supergrav-ity practitioners should not be surprised by such a result,see for instance [21].What happens here is that the conservation laws encode all the equa-tions of motion and not only some combinations of them.Gauge invariance far from being a nuisance has the power to determine the dynamics.The aﬃne theory leads,see subsection4.4,either toﬁrst order Poincar´e (Cartan-Weyl)theory by going to orthonormal frames and using the metric-ity of the connection or to the Palatini formalism by going to a coordinate frame and eliminating the torsion.In both cases one eliminates part of the linear connection by its equation of motion and byﬁxing a residual1-form gauge invariance(without gauge2-form):the arbitrariness of the scaling part of the linear connection.In other words the invariance of the action under the shift ofΓρµνby a scaling(Weyl)component,Aµ(x)δρν,is a gauge symmetry that generalizes the so-called Einstein symmetry[22].In sum-mary,modulo this“Einstein-Weyl”arbitrariness which is due to the form of the scalar curvature,the vanishing of torsion and nonmetricity follow from the variations of suitable components of the connectionﬁeld.The name of H.Weyl is associated with the invention of(scaling)gauge invariance and is appropriate despite diﬀerences in the implementation.For this new gauge invariance one explains again the vanishing of the superpotential and hence of the current by the absence of propagating gaugeﬁeld.Recall the examples of kappa-symmetry or(string theory)Weyl currents...The local invariance of our aﬃne action with respect to those gl(D,I R) generators that are not in the Lorentz subalgebra deﬁned by the metric (and consequently do not propagate)leads also to the vanishing of the as-sociated U(ab)superpotentials.Frauendiener[23]also considered the full frame bundle to investigate energy-momentum pseudotensors.Finally the Hilbert action follows from our action by going to second order formalism via the Cartan-Weyl action for instance.In Subsection4.5we compare the superpotentials associated to these four actions,they may be called respectively aﬃne,Cartan-Weyl,Palatini and Møller.In order to recover the right mass for the Schwarzschild solution8Møller did actually rescale arbitrarily the potential derived canonically from the Hilbert action and multiplied by a factor of two[24]the honest one.A linear combination of the energy-momentum tensor and its associated su-perpotential involving an inﬁnitesimal gauge parameter gives3the ordinary Noether current for the one dimensional subgroup along a given gauge di-rection.Ignoring extra terms due to higher derivatives it would have the formJξ=ξρJρ+dξρ∧Uρ(5) This current has its own superpotential Uξ:=ξρUρ.In the Palatini case the latter becomes the Komar superpotential[25]after suitable modiﬁcation by the frame change,it has the property to be a tensor.The Palatini super-potential diﬀers from the aﬃne superpotential by a contribution induced by the choice of coordinate frame:one must compensate the change of cordi-nates by a local linear transformation and this mixes the energy-momentum tensor and the gl(D)current.Finally as explained in the previous section the antisymmetry in the two indicesµandνof Uµνρis spoiled by the presence of higher derivatives present in the second order formalism.The reconcil-iation ofﬁrst and second order formalisms requires also some mixing with another symmetry in the case of the orthonormal frame choice,that is in the Cartan-Weyl formalism:one can check that a compensating Lorentz transformation allows us to relate the superpotentials of the two.The whole picture can be studied for the three theories above Hilbert’s scalar action as was just presented or for the corresponding theories above the Einstein metric action which is noncovariant but has onlyﬁrst order derivatives of the metric.The Einstein action diﬀers from Hilbert’s by a surface term and leads to the Einstein energy-momentum complex some-times called pseudotensor.It was a big surprise when Freud[26]discovered the relevant local superpotential,its origin was clariﬁed by Bergmann but it could have been conjectured by Noether and Hilbert!Surface terms have been considered also in the Hamiltonian formalism[10]and for the path integral quantization they are reviewed in[28].We identify on the Einstein side both the Freud superpotential and the Sparling one in section5.1.In the rest of section5we consider the issue of boundary conditions and surface terms building on the previous examples.Let us recall that the gaugeﬁeld part of the superpotential and hence the corresponding part of the current do vanish either when there is no propagating gaugeﬁeld but only a compensator or when there is a global (bulk)spacelike Killing vector or its analog in a general gauge theory.The9asymptotic symmetry of the set of allowed conﬁgurations (and solutions),at the “end”of spacetime where one does the experiment,is needed to deﬁne global charges because we need distinguished subgroups at inﬁnity.It turns out that the contribution to these charges from the gauge ﬁelds vanishes asymptotically despite their inﬁnite range in the case of spatial Killing vectors deﬁned also in the bulk,at least near inﬁnity.We list along the way some projects for part II.2The general formalism and ﬁrst examples 2.1The general formalismA local action that depends for simplicity on the ﬁelds and their ﬁrst deriva-tives S = M L (ϕ,∂ϕ)may be invariant under a continuous (Lie)transfor-mation.In this case one has:δS =0⇔δL =∂µS µ=∂L∂∂µϕδ∂µϕ(6)Using the fact that ∂µand δcommute,we obtain∂µS µ=[∂L∂∂µϕ]δϕ+∂µ[∂L∂∂µϕδϕ(8)∂µJ µ≈0(9)where ≈means on shell.Note that S µis not uniquely deﬁned without more choices.The classical theorem expresses the conservation of this current as a con-sequence of the Euler-Lagrange ﬁeld equations.In diﬀerential form notations J µhas a Hodge-dual (D −1)form noted by J (where D is the spacetime dimension).J is a local function of the ﬁelds but we can only deduce from its closedness (dJ ≈0)that it is exact (J ≈dU )if spacetime is contractible and for a given solution of the equations of motion,in particular the (D −2)10form U is not guaranteed to be“local”,i.e.,can be written locally in terms of the fundamentalﬁelds of the theory.The total charge Q= V D−1J is conserved given suﬃcient decay at spatial inﬁnity,more covariantly ∂V D J ≈0(V D−1is a space like hypersurface).The addition of a topological term to the Noether current is allowed if its topological charge vanishes so that the Noether charge is unaﬀected.Gauge theory:the cascade equations∆A(ϕ)(13)∂∂µϕUµνA:=ΣµνA−∂L∂µξA(x)[JµA+∂νUνµA≈0](16)ξA(x)[∂µJµA≈0](17) As usual,(...)means symmetrization of the indices and[...]antisym-metrization.Note that theﬁrst equation is an identity whereas the other two are just on-shell equations(as emphasized by the≈symbol).The main result of this computation is that the so-called Noether current JµA is locally exact modulo the equations of motion when the symmetry is local.The corresponding superpotential UνµA has to be antisymmetric and can be computed directly from the Lagrangian of the theory by the use of equation14.This antisymmetric property is particular to the case with at mostﬁrst order derivatives of theﬁelds.The Noether identitiesδϕ ξA∆A+∂νξA∆νA =∂µ ξA JµA+∂νξA UµνA (18) whereδL∆A=∂µJµA (19)δϕ∂µξA δLIf we now replace JµA as given by equation20into19and make use of the antisymmetry of UµνA we easily obtain the Noether identities:δL∆µA(ϕ) (22)δϕNote that the Noether identities do not depend on any surface term because all of them are hidden in JµA.This old result will be used in section 4for a better understanding of the aﬃne gauge theory and its reduction to Einstein theory.Remember that we have just treated the simple case where the decom-positions10and11go only up toﬁrst derivative inξA(x).In a more general case,the above conclusions have to be modiﬁed as we will see in section3 in the speciﬁc example of General Relativity.The abelian cascade:the Uµνξsuperpotential1Actually the subgroup is not really abelian in the case of diﬀeomorphisms but there all changes of coordinates are linearly related or unidimensional.13The conserved charges:This will be the right choice to deﬁne a conserved charge in General Relativity and Yang-Mills theories.Let B1∞and B2∞the bound-aries ofΣ∞at time t1and t2respectively(these are actually D-2dimensional closed manifolds).If we again assume that theﬂux of J vanishes onΣ∞then Stokes law applied to equation24will imply that B1∞U= B2∞U. Then the conserved charge may be deﬁned as the integral of U on the inﬁnite spatial boundary of a time-ﬁxed hypersurface:Q= B∞U(25) This deﬁnition is completely independent of the fact that there exist or not an interior black hole horizon or singularity inside space time.The key point is that this construction never leaves the asymptotic region and is both robust and physical as that is precisely where charges are measured. As we will discuss in more detail in the next examples(Yang-Mills and Grav-itation),there exist relations between the boundary conditions we have to impose on ourﬁelds to deﬁne the variational principle,the form of asymp-totic Killing vectors and the associated gauge invariant conserved charges. In some very special cases(for instance in presence of a global spatial Killing vector),we should be able to use other timelike hypersurface thanΣ∞,for),leading to the notion of quasi-local instance atﬁnite distance(sayΣr14charges.We postpone this discussion to section5.2for the gravitational case.-OnΣ1:Let us start with the usual Yang-Mills Lagrangian,eventually coupled to a matter term:1L Y M=−This Lagrangian is invariant under the local gauge transformation:δξA Aµ=∂µξA+f A BC A BµξC=DµξAδξϕ=ξA R Aϕ⇒δξL Y M=0Of course f A BC are the structure constants and R A the speciﬁc inﬁnites-imal generators in the representation R of the group.We can now use this symmetry in equation8and then rewrite it as in equation12.The useful quantities13and14can now be computed for the Yang-Mills Lagrangian27:JµA=−f C BA A BνFµνC+∂L mat∂∂µϕR AϕOur purpose is to study the fate of conserved quantities in the presence of local invariance so it is important to recognize these equations as conser-vation equations of the type J≈dU.Here we have a superpotential which is not anymore a gauge scalar.In fact the integral at spatial inﬁnity of FµνA does not make any sense as a conserved quantity because it is not gauge invariant.The good gauge independent superpotentials are thus parameter dependent ones and can be obtained by the abelian cascade method.The result is obvious:Uµνξ=FµνAξAJµξ≈∂νUµνξIf we recall the discussion of conserved charges of the previous subsection, we obtain that the gauge invariant conserved charge is(equation25):Q(ξA(x))= B∞Uξ(28) where as usual Uξis the D−2form associated to the Hodge dual of Uµνξ.The point is now that in order to obtain physical charges we have to specify and select whatξA(x)can be.This is treated in the following.The Yang-Mills ChargesFirst we would like to recall an important point which has to be taken into account in a variational principle.A variational principle is deﬁned only when boundary conditions are speciﬁed.In addition,if a boundary condition is chosen,we cannot add anymore an arbitrary total derivative to the Lagrangian because in general when the fundamentalﬁelds of the theory do not vanish on the boundary(say at inﬁnity)the variational principle(i.e.δS=0⇒Equations of motion)will not be satisﬁed.For example,the variational principle for the Yang-Mills Lagrangian27 implies that∂M∂L∂∂µϕδϕ(29) has to vanish for an arbitrary variation.We do not want to analyse here the behaviour of the solutions of this equation say at spatial inﬁnity in terms of power series in1In that case JµξK =0everywhere and so the corresponding charge Q(ξK)=B UµνξK can be computed on any(D-2)dimensional surface outside matter sources.We want to insist here on the following points-The case of Dirichlet conditions30and31is just the simplest solution for the vanishing of equation29.The general solution to this condition has to be treated in the asymptotic regime with the appropriate decrease.-Physical conditions will specify the boundary condition(as in the case of free orﬁxed-ends strings)which will not onlyﬁx part of the surface term of the Lagrangian but also give some conditions on the asymptotically allowed gauge parameters.-Boundary conditions should be gauge invariant.In the case of General Relativity this is made possible by introducing a reference space at inﬁnity (where it is needed).Some well known examples are:-The Maxwell case with matterﬁelds which vanish at inﬁnity.In that case the asymptotic Killing equation just becomes lim r→∞∂µξ=0.The subalgebra which will give a non vanishingﬁnite charge will be I R.Thus the number of charges is just1(the dimension of I R,which is also the number of independent Casimir operators of the subgroup).In addition,ξ=C t is a global Killing parameter and so we recover the well known result that the electric charge can be computed on any closed surface which surrounds the charged matter distribution.-The SU(2)Yang-Mills-Higgs system where a particular solution to the asymptotic Killing equations is justξA=ΦA0(the direction of the Higgs ﬁeld at inﬁnity),see for instance[7].2.3The p-form theoryWe can consider the abelian p-form Lagrangian given essentially byL=G∧∗G(32) Where G=dB,B being the p-form abelian gaugeﬁeld(see[6]).The local gauge invariance is justδξB=dξ,whereξis an arbitrary (p-1)-form gauge parameter.We will not repeat all the computations but just give theﬁnal result which is that the parameter dependent conserved charge is given by:18Q (ξ)=B ∞ξ∧∗G Ifwe again impose Dirichlet type boundary conditions,the analogue of 30and 31for ξis thus:lim r →∞dξ=0It is obvious from the deﬁnition of Q (ξ)and the equations of motion of B (d ∗G ≈0)that when ξ=dβ,the charge will vanish on shell (re-member that B ∞is already a boundary hence is closed and that partial integration can be done without any boundary term).Thus in the case of the p-form,the subgroup which could potentially give some non trivial con-served charge is just the set of (p-1)-forms which are closed but not exact or in other words the (p −1)th De Rham cohomology group H p −1of B ∞.The number of conserved charges will then be given by the (p −1)th -Betti number b p −1(B ∞)=dim H p −1(B ∞) .For example for a spacetime with 2inﬁnite boundaries components (wormhole)we recover 2ordinary charges.The reader interested only in ﬂuid dynamics can now skip to section 6.3The classical case of General Relativity 3.1Second order form of gravitation:the cascade Equationsfor diﬀeomorphismsLet L (g,∂g,∂2g )=1−gR be the scalar Hilbert Lagrangian density of our theory.It is equal to the so-called Einstein Lagrangian up to the surface term that eliminates second derivatives of the metric,see section 5.A variation of L is given byδL =∂L∂∂µg δ∂µg +∂L δg =∂L ∂∂µg +∂ν∂µ∂L ∂∂µg δg −∂ν ∂L ∂∂µ∂νg∂νδg ≈0(34)19。

小学上册第十四次英语第四单元测验卷(有答案)

小学上册英语第四单元测验卷(有答案)英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1.The _____ (金鲨) glides through the ocean with grace. 金鲨优雅地穿梭于海洋中。

2.My dog has a fluffy _______ (毛发).3.What do plants need to grow?A. LightB. WaterC. SoilD. All of the above答案:D4. A ______ is a natural barrier formed by the landscape.5.What do we call a place where you can buy books?A. LibraryB. BookstoreC. MarketD. School答案:B6.The wind is _______ (howling) outside.7.Which gas do we breathe out?A. OxygenB. NitrogenC. Carbon DioxideD. Helium答案:C8. A __________ is a common example of a base.9.The invention of ________ has reshaped modern communication.10.Which word means "happy"?A. SadB. JoyfulC. AngryD. Tired答案:B11.What is the capital city of the Bahamas?A. NassauB. FreeportC. Marsh HarbourD. George Town答案:A12.What do we call the first meal of the day?A. LunchB. DinnerC. BreakfastD. Snack13.What do you call a person who studies geology?A. GeologistB. Earth scientistC. MinerD. Excavator答案:A14.What do we call the time when the sun rises?A. MorningB. NoonC. EveningD. Midnight15.What color is a stop sign?A. YellowB. GreenC. RedD. Blue答案:C16.The element with atomic number is __________.17.How do you say "dog" in Italian?A. CaneB. ChienC. PerroD. Hund18.My mom loves __________ (组织活动).19.We have a ________ (class) project to do.20.What do you call a person who studies plants?A. ZoologistB. BotanistC. GeologistD. Chemist答案:B21.What do we call a person who writes books?A. AuthorB. JournalistC. PoetD. Editor答案:A22.During winter, it sometimes ________ (下雪). I like to make ________ (雪人) with my friends.23.What do we use to write?A. SpoonB. PenC. ForkD. Plate答案:B24.What do we call a place where you can see many trees?A. ParkB. ForestC. GardenD. Orchard25.An ion is an atom with a _____ charge.26.My mom is a __________ (社工).27.The _______ (Wright brothers) are credited with inventing the first airplane.28.I enjoy making ________ (手工艺品) for my family.29.The process of ______ occurs when materials are transported by water.30.What do we call the process of water turning into vapor?A. MeltingB. EvaporationC. FreezingD. Condensation答案:B31. A ladybug has a red ______ (外表) with black spots.32.Chemical bonds hold ________ together in a compound.33.The ocean is _______ (非常清澈)。

tpo32三篇托福阅读TOEFL原文译文题目答案译文背景知识

tpo32三篇托福阅读TOEFL原文译文题目答案译文背景知识阅读-1 (2)原文 (2)译文 (5)题目 (7)答案 (16)背景知识 (16)阅读-2 (25)原文 (25)译文 (28)题目 (31)答案 (40)背景知识 (41)阅读-3 (49)原文 (49)译文 (53)题目 (55)答案 (63)背景知识 (64)阅读-1原文Plant Colonization①Colonization is one way in which plants can change the ecology of a site.Colonization is a process with two components:invasion and survival.The rate at which a site is colonized by plants depends on both the rate at which individual organisms(seeds,spores,immature or mature individuals)arrive at the site and their success at becoming established and surviving.Success in colonization depends to a great extent on there being a site available for colonization–a safe site where disturbance by fire or by cutting down of trees has either removed competing species or reduced levels of competition and other negative interactions to a level at which the invading species can become established.For a given rate of invasion,colonization of a moist,fertile site is likely to be much more rapid than that of a dry, infertile site because of poor survival on the latter.A fertile,plowed field is rapidly invaded by a large variety of weeds,whereas a neighboring construction site from which the soil has been compacted or removed to expose a coarse,infertile parent material may remain virtually free of vegetation for many months or even years despite receiving the same input of seeds as the plowed field.②Both the rate of invasion and the rate of extinction vary greatly among different plant species.Pioneer species-those that occur only in the earliest stages of colonization-tend to have high rates of invasion because they produce very large numbers of reproductive propagules(seeds,spores,and so on)and because they have an efficient means of dispersal(normally,wind).③If colonizers produce short-lived reproductive propagules,they must produce very large numbers unless they have an efficient means of dispersal to suitable new habitats.Many plants depend on wind for dispersal and produce abundant quantities of small,relatively short-lived seeds to compensate for the fact that wind is not always a reliable means If reaching the appropriate type of habitat.Alternative strategies have evolved in some plants,such as those that produce fewer but larger seeds that are dispersed to suitable sites by birds or small mammals or those that produce long-lived seeds.Many forest plants seem to exhibit the latter adaptation,and viable seeds of pioneer species can be found in large numbers on some forest floors. For example,as many as1,125viable seeds per square meter were found in a100-year-old Douglas fir/western hemlock forest in coastal British Columbia.Nearly all the seeds that had germinated from this seed bank were from pioneer species.The rapid colonization of such sites after disturbance is undoubtedly in part a reflection of the largeseed band on the forest floor.④An adaptation that is well developed in colonizing species is a high degree of variation in germination(the beginning of a seed’s growth). Seeds of a given species exhibit a wide range of germination dates, increasing the probability that at least some of the seeds will germinate during a period of favorable environmental conditions.This is particularly important for species that colonize an environment where there is no existing vegetation to ameliorate climatic extremes and in which there may be great climatic diversity.⑤Species succession in plant communities,i.e.,the temporal sequence of appearance and disappearance of species is dependent on events occurring at different stages in the life history of a species. Variation in rates of invasion and growth plays an important role in determining patterns of succession,especially secondary succession. The species that are first to colonize a site are those that produce abundant seed that is distributed successfully to new sites.Such species generally grow rapidly and quickly dominate new sites, excluding other species with lower invasion and growth rates.The first community that occupies a disturbed area therefore may be composed of specie with the highest rate of invasion,whereas the community of the subsequent stage may consist of plants with similar survival ratesbut lower invasion rates.译文植物定居①定居是植物改变一个地点生态环境的一种方式。

2021-2022学年高中英语(人教版必修2)期末质量检测(二) Word版含答案

期末质量检测（二）(时间：120分钟满分：150分)第Ⅰ卷第一部分：听力(共两节，满分30分)第一节(共5小题；每小题1.5分，满分7.5分)听下面5段对话。

每段对话后有一个小题，从题中所给的A、B、C三个选项中选出最佳选项，并标在试卷的相应位置。

听完每段对话后，你都有10秒钟的时间来回答有关小题和阅读下一小题。

每段对话仅读一遍。

(Text 1)M: Jane, could you help me send this email?W: Sorry, Peter. My computer broke down this morning. Why not ask Judy for help?(Text 2)M: Why are you wearing a sweater? It's sunny.W: Although it's not rainy or cloudy like yesterday, I feel very cold since I've got a cold.(Text 3)M: Are you interested in buying that book on ships?W: No, I've just been looking at the pictures. This is the one I'm actually interested in —it's on modern aircrafts. I think you might be interested in this book on future cars.(Text 4)M: Which bed would you like? How about the one with the bedside table and the one with the headboard? They're both very comfortable.W: I'd like the one with the bedside light.(Text 5)M: How on earth did that happen? Did you hit a car or something?W: No. I was driving alone when a stone suddenly hit the windscreen.1．Who will probably send this email?A．Jane. B．Peter. C．Judy.2．What is the weather like today?A．Sunny. B．Rainy. C．Cloudy.3．What is the woman really interested in?A．Ships.B．Modern aircrafts.C．Future cars.4．Which bed does the woman like?A．The bed with the bedside table.B．The bed with the bedside light.C．The bed with the headboard.5．What happened to the man?A．He was hit by another car.B．His windscreen was hit by a stone.C．He broke someone else's windscreen.其次节(共15小题；每小题1.5分，满分22.5分)听下面5段对话或独白。

高中英语真题：2017高考英语阅读理解一轮(九月)精编(五)

2017高考英语阅读理解一轮（九月）精编（五）阅读理解。

阅读下列短文，从每题所给的四个选项(A、B、C 和D)中，选出最佳选项。

A lot of us lose life's tough battles by starting a frontal attack--when a touch of humor might well enable us to win. Consider the case of a young friend of mine, who was on his trapped w ay to work shortly after receiving an ultimatum(最后通牒) about the job. Although there was a good reason for Sam' s being late--serious illness at home--he decided that this by-now-familiar excuse wouldn't work any longer. His supervisor was probably already pacing up and down preparing a dismissal s peech.Yes, the boss was, Sam entered the office at 9:35. The place was as quiet as a locker room; everyone was hard at work. S am's supervisor came up to him. Suddenly, Sam forced a grin and stretched out his hand. "How do you do!" he said. "I'm S am Maynard. I'm applying for a job, which, I understand, beca me available just 35 minutes ago. Does the early bird get the worm?"The room exploded in laughter, except that the supervisor ha d to clamp off a smile and walked back to his office. Sam May nard had saved his job--with the only tool that could win, a laugh.Humor is a most effective, yet frequently neglected, means of handling the difficult situations in our lives. It can be used for patching up differences, apologizing, saying "no", criticizing, g etting the other fellow to do what you want without his losing f ace. For some jobs, it's the only tool that can succeed. It is a way to discuss subjects so sensitive that serious dialog may s tart a quarrel. For example, many believe that comedians on t elevision are doing more today for racial and religious toleran ce than people in any other forum.1. Why was Sam late for his job?A. Because he was seriously ill at home.B. Because he received an ultimatum.C. Because he was busy applying for a new job.D. Because he was caught in a traffic jam.2. What is the main idea of this passage?A. Sam Maynard saved his job with humor.B. Humor is important in our lives.C. Early bird can get the worm.D. Humor can solve racial discriminations.3. The phrase "clamp off" in Paragraph 3 means ________.A. try to hold backB. pretend to setC. send offD. give out4. Which of the following statements can we infer from the pa ssage?A. Many lose life's battles for they are always late.B. Sam was supposed to come to his office at 8:30.C. It wasn't the first time that Sam came late for his work.D. Humor is the most effective way of solving problems.1、答案解析：答案为D。

AdS_3, Black Holes and Higher Derivative Corrections

a r X i v :0705.0735v 2 [h e p -t h ] 19 M a y 2007arXiv:0705.0735AdS 3,Black Holes and Higher Derivative Corrections Justin R.David,Bindusar Sahoo and Ashoke Sen Harish-Chandra Research Institute Chhatnag Road,Jhusi,Allahabad 211019,INDIA E-mail:justin,bindusar,sen@mri.ernet.in Abstract Using AdS/CFT correspondence and the Euclidean action formalism for black hole entropy Kraus and Larsen have argued that the entropy of a BTZ black hole in three dimensional supergravity with (0,4)supersymmetry does not receive any correction from higher derivative terms in the action.We argue that as a consequence of AdS/CFT correspondence the action of a three dimensional supergravity with (0,4)supersymmetry cannot receive any higher derivative correction except for those which can be removed byﬁeld redeﬁnition.The non-renormalization of the entropy then follows as a consequence of this and the invariance of Wald’s formula under a ﬁeld redeﬁnition.1BTZ solution describes a black hole in three dimensional theory of gravity with nega-tive cosmological constant[1]and often appears as a factor in the near horizon geometry of higher dimensional black holes in string theory[2].Furthermore the entropy of a BTZ black hole has a remarkable similarity to the Cardy formula for the degeneracy of states in the two dimensional conformalﬁeld theory[3].For these reasons computation of the entropy of BTZ black holes has been an important problem,both in three dimensional theories of gravity and also in string theory.Initial studies involved computing Bekenstein-Hawking formula for BTZ black hole entropy in two derivative theories of ter this was generalized to higher derivative theories of gravity[4,5,6,7,8,9,10],where the lagrangian density contains arbitrary powers of Riemann tensor and its covariant derivatives as well as gravitational Chern-Simons terms[11],both in the Euclidean action formalism[12]and in Wald’s formalism[13,14,15,16].While the above mentioned formalism tells us how to calculate the entropy of a BTZ black hole for a given action with arbitrary higher derivative terms,it does not tell us what these higher derivative terms are.It was however argued by Kraus and Larsen[5,6]using AdS/CFT correspondence that if the three dimensional theory under consideration has at least(0,4)supersymmetry then the entropy of a BTZ black hole of given mass and angular momentum is determined completely in terms of the coeﬃcients of the gravitational and gauge Chern-Simons terms in the action and hence does not receive any higher derivative corrections.This result is somewhat surprising from the point of view of the bulk theory, since for a given three dimensional theory of gravity the entropy does have non-trivial dependence on all the higher derivative terms.Thus one could wonder how the dependence of the entropy on these higher derivative terms disappears by imposing the requirement of(0,4)supersymmetry.In this note we shall propose a simple explanation for this fact:(0,4)supersymmetry prevents the addition of any higher derivative terms in the supergravity action(except those which can be removed byﬁeld redeﬁnition and hence give an equivalent theory).Our argument is based on the following observation.In AdS/CFT correspondence the bound-ary operators dual to theﬁelds in the supergravity multiplet are just the superconformal currents associated with the(0,4)supersymmetry algebra.The correlation functions of these operators in the boundary theory are determined completely in terms of the central charges c L,c R of the left-moving Virasoro algebra and the right-moving super-Virasoro algebra.Of these c R is related to the central charge k R of the right-moving SU(2)currents2which form the R-symmetry currents of the super-Virasoro algebra and hence to the coef-ﬁcient of the Chern-Simons term of the associated SU(2)gaugeﬁelds in the bulk theory. On the other hand c L−c R is determined in terms of the coeﬃcient of the gravitational Chern-Simons term in the bulk theory.Thus the knowledge of the gauge and gravitational Chern-Simons terms in the bulk theory determines all the correlation functions of(0,4) superconformal currents in the boundary theory.Since by AdS/CFT correspondence[17] these correlation functions in the boundary theory determine completely the boundary S-matrix of the supergravityﬁelds[18,19],we conclude that the coeﬃcients of the gauge and gravitational Chern-Simons terms in the bulk theory determine completely the boundary S-matrix elements in this theory.Now the boundary S-matrix elements are the only perturbative observables of the bulk theory.Thus we expect that two diﬀerent theories with the same boundary S-matrix must be equivalent.(We shall elaborate on this later.)Combining this with the observation made in the last paragraph we see that two diﬀerent gravity theories,both with(0,4)supersymmetry and the same coeﬃcients of the gauge and gravitational Chern-Simons terms,must be equivalent.Put another way,once we have constructed a classical supergravity theory with(0,4)supersymmetry and given coeﬃcients of the Chern-Simons terms,there cannot be any higher derivative corrections to the action involvingﬁelds in the gravity supermultiplet except for those which can be removed byﬁeld redeﬁnition.1 The non-renormalization of the entropy of the BTZ black hole then follows trivially from this fact.The complete theory in the bulk of course can have other matter multiplets whose action will receive higher derivative corrections.However since restriction to the ﬁelds in the gravity supermultiplet provides a consistent truncation of the theory,2and since the BTZ black hole is embedded in this subsector,its entropy will not be aﬀected by these additional higher derivative terms.Our arguments will imply in particular that theﬁve dimensional supergravity action constructed in[21],after dimensional reduction on a sphere,must be equivalent to the three dimensional supergravity action given in eq.(2)below with the precise relationship between the various coeﬃcients as given in eq.(4).This in turn would explain why the analysis of the black hole entropy given in[22,23](see also[24])agrees with the expected result.We should caution the reader however that theﬁeld redeﬁnition needed to arrive at the action given in(2)may not be invertible on allﬁeld conﬁgurations.For example if we take a Chern-Simons action and add to it the usual kinetic term for a gaugeﬁeld then the kinetic term can be removed formally by aﬁeld redeﬁnition.However the theory with the kinetic term has an extra pole in the gaugeﬁeld propagator corresponding to a massive photon which is absent in the pure Chern-Simons theory.This happens because theﬁeld redeﬁnition that takes us from the theory with the gauge kinetic term to pure Chern-Simons theory is not invertible on the plane wave solution describing the propagating massive photon.This however does not aﬀect our argument as long as theﬁeld redeﬁnition is invertible on slowly varyingﬁeld conﬁguration around the AdS3background.In this context we note that suchﬁeld redeﬁnitions are carried out routinely in string theory,e.g. in converting a term in the gravitational action quadratic in the Riemann tensor to the Gauss-Bonnet combination.The former theory typically has extra poles in the graviton propagator which are absent in the latter theory.For completeness we shall now describe this unique(0,4)supergravity action and compute the entropy of a BTZ black hole from this action.The action was constructed in [25,26]by regarding the supergravity as a gauge theory based on SU(1,1)×SU(1,1|2) algebra.3IfΓL andΓR denote the(super-)connections in the SU(1,1)and SU(1,1|2) algebras respectively,then the action is taken to be a Chern-Simons action[30]of the form:S=−a L d3x T r(ΓL∧dΓL+2ΓR∧ΓR∧ΓR ,(1)3where a L and a R are constants.Note that the usual metric degrees of freedom are encodedin the connectionsΓL andΓR[31].Thus there is no obvious way to add SU(1,1)×SU(1,1|2)invariant higher derivative terms in the action involving theﬁeld strengths associated with the connectionsΓL andΓR.From this viewpoint also it is natural that the supergravity action does not receive any higher derivative corrections.The bosonicﬁelds of this theory include the metric G MN and an SU(2)gaugeﬁeld A M(0≤M≤2),represented as a2×2anti-hermitian matrix valued vectorﬁeld.After expressing the action in the component notation and eliminating auxiliaryﬁelds using their equations of motion we arrive at the actionS= d3x √4πǫMNP T r A M∂N A P+2m =12(a L−a R),(3) k R=4πa R=4π 12 ΓR MS∂N ΓS P R+1−det G L(3)0+KΩ3( Γ) ,(6)where L(3)0denotes an arbitrary scalar constructed out of the metric,the Riemann tensor and covariant derivatives of the Riemann tensor.A general BTZ black hole in the three dimensional theory is described by the metric:G MN dx M dx N=−(ρ2−ρ2+)(ρ2−ρ2−)(ρ2−ρ2+)(ρ2−ρ2−)dρ2+ρ2dy−ρ+ρ−where l,ρ+andρ−are parameters labelling the solution.Of these the parametersρ±can be removed locally by a coordinate transformation,so that any scalar combination of the Riemann tensor and metric computed for this metric is a function of the parameter l only.We deﬁneh(l)=L(3)0,(8) evaluated in the background(7),andg(l)=πl3c L q L c R q R2(M−J),q R=1πg(l0).(14)3.The parametersρ±are related to M and J via the relationsM±J=2π(C∓K)4l3(−6l−2+2m2),l0=1m(16)andc L=24π 1m−K =24πa R,(17) where in(17)we have used(3).Using(4)we getc R=6k R,c L=48πK+6k R.(18) Eqs.(11),(12),(18)give the desired expression for the entropy of a BTZ black hole in terms of the coeﬃcients of the gauge and gravitational Chern-Simons terms.By our previous argument addition of higher derivative terms do not change this result as long as they respect(0,4)supersymmetry.Since the crux of our argument has been the relationship between non-renormalization of the boundary S-matrix and the non-renormalization of the classical action,we shall now elaborate on this by examining how this works for the gauge sector of the theory.In this case the Chern-Simons theory has equation of motion F MN=0whereF MN≡∂[M A N]+[A M,A N](19) is the gaugeﬁeld strength.Any additional gauge invariant term in the action will involve the gaugeﬁeld strength and hence will vanish when F MN=0.A standard argument then shows that such terms can be removed from the action using aﬁeld redeﬁnition.We shall now see how the vanishing of the additional terms in the action for F=0 is related to the non-renormalization of the boundary S-matrix.For this weﬁrst review the computation of the boundary S-matrix from pure Chern-Simons theory.We begin by writing the Euclidean AdS3metric in the Poincare patchl2ds2=The gauge ﬁeld action in the Euclidean space takes the form 5S gauge =−i k R3A M A N A P +k RδA (0)a 1z ( z 1)···δA (0)a n z ( z n )e −I [A (0)z ]A (0)z ( z )=0,(24)where ¯J a ( z )are the SU (2)currents of the CFT at the boundary and the A a M are deﬁned throughA M =12(d ΦΦ−Φd Φ)+ (26)whereΦ( z ,x 0)= d 2wK ( z ,x 0; w )B (0)z ( w),(27)K ( z ,x 0; w )=1(x 0)2+|z −w |2 ,(28)and B(0)z is chosen such that(26)satisﬁes the boundary condition(23).Eq.(28)giveslimx0→0∂z K( z,x0; w)=δ(2)( z− w),(29)lim x0→0∂¯z K( z,x0; w)=−1(¯z−¯w)2.(30)Using eqs.(26)-(29)weﬁnd that A z(x0=0, z)is equal to B(0)z( z)toﬁrst order in an expansion in a power series in B(0)z.Thus to this order(23)is satisﬁed for B(0)z=A(0)z. The higher order contributions to B(0)z can be obtained by iteratively solving eq.(23)with the ansatz for A M given in(26)-(28).The result isB(0)z ( z)=A(0)z( z)+1(¯z−¯w) A(0)z( w)A(0)z( z)−A(0)z( z)A(0)z( w) + (31)where···denote higher order terms.We can now substitute the solution given in(26)-(31) into(22)to evaluate the on-shell action I[A(0)z].Evaluation of the boundary contribution is straightforward.In evaluating the contribution from the Chern-Simons term weﬁrst use the equation of motion to express it asik R3ǫMNP∂t T r U−1t∂M U t U−1t∂N U t U−1t∂P U t =ǫMNP∂M U−1t∂t U t U−1t∂N U t U−1t∂P U t (33) and that U−1t∂t U t=Φ,we can express(32)as a pure boundary term−k R4π2 d2zd2w(¯z−¯w)−2A(0)a z( z)A(0)a z( w)−k R2δa1a2(¯z1−¯z2)−2,(36)9¯J a1(¯z1)¯J a2(¯z2)¯J a3(¯z3) =−ik R[5]P.Kraus and rsen,JHEP0509,034(2005)[arXiv:hep-th/0506176].[6]P.Kraus and rsen,JHEP0601,022(2006)[arXiv:hep-th/0508218].[7]S.N.Solodukhin,Phys.Rev.D74,024015(2006)[arXiv:hep-th/0509148].[8]B.Sahoo and A.Sen,JHEP0607,008(2006)[arXiv:hep-th/0601228].[9]P.Kraus,arXiv:hep-th/0609074.[10]Y.Tachikawa,Class.Quant.Grav.24,737(2007)[arXiv:hep-th/0611141].[11]S.Deser,R.Jackiw and S.Templeton,Annals Phys.140,372(1982)[Erratum-ibid.185,406.1988APNYA,281,409(1988APNYA,281,409-449.2000)];Phys.Rev.Lett.48,975(1982).[12]J.M.Maldacena and A.Strominger,JHEP9812,005(1998)[arXiv:hep-th/9804085].[13]R.M.Wald,Phys.Rev.D48,3427(1993)[arXiv:gr-qc/9307038].[14]T.Jacobson,G.Kang and R.C.Myers,Phys.Rev.D49,6587(1994)[arXiv:gr-qc/9312023].[15]V.Iyer and R.M.Wald,Phys.Rev.D50,846(1994)[arXiv:gr-qc/9403028].[16]T.Jacobson,G.Kang and R.C.Myers,arXiv:gr-qc/9502009.[17]J.M.Maldacena,Adv.Theor.Math.Phys.2,231(1998)[Int.J.Theor.Phys.38,1113(1999)][arXiv:hep-th/9711200].[18]S.S.Gubser,I.R.Klebanov and A.M.Polyakov,Phys.Lett.B428,105(1998)[arXiv:hep-th/9802109].[19]E.Witten,Adv.Theor.Math.Phys.2,253(1998)[arXiv:hep-th/9802150].[20]A.Dabholkar, A.Sen and S.P.Trivedi,JHEP0701,096(2007)[arXiv:hep-th/0611143].[21]K.Hanaki,K.Ohashi and Y.Tachikawa,Prog.Theor.Phys.117,533(2007)[arXiv:hep-th/0611329].11[22]A.Castro,J.L.Davis,P.Kraus and rsen,arXiv:hep-th/0702072.[23]A.Castro,J.L.Davis,P.Kraus and rsen,arXiv:hep-th/0703087.[24]M.Alishahiha,arXiv:hep-th/0703099.[25]M.Nishimura and Y.Tanii,Int.J.Mod.Phys.A14,3731(1999)[arXiv:hep-th/9904010].[26]J.R.David,Mod.Phys.Lett.A14,1143(1999)[arXiv:hep-th/9904068].[27]A.Achucarro and P.K.Townsend,Phys.Lett.B180,89(1986).[28]A.Achucarro and P.K.Townsend,Phys.Lett.B229,383(1989).[29]A.Giacomini,R.Troncoso and S.Willison,arXiv:hep-th/0610077.[30]E.Witten,Commun.Math.Phys.121,351(1989).[31]E.Witten,Nucl.Phys.B311,46(1988).[32]P.Kraus and rsen,JHEP0701(2007)002[arXiv:hep-th/0607138].[33]S.Elitzur,G.W.Moore,A.Schwimmer and N.Seiberg,Nucl.Phys.B326,108(1989).12。

A_Conservative_Revolutionary_(I)_一个保守的革命者(上)

A Conservative RevolutionaryI am delighted to have this opportunityto sing the praises of my old friendand colleague Frank Yang. The title ofmy talk is “A Conservative Revolution-ary”. The meaning of the title will become clear at the end of the talk.2 One of my favorite books is Frank’s“Selected papers 1945–1980 with com-mentary”, published in 1983 to celebratehis sixtieth birthday. This is an anthologyof Frank’s writings, with a commentarywritten by him to explain the circumstances in which they were written.There was room in the book for only onethird of his writings. He chose which pa-pers to include, and his choices give a fartruer picture of his mind and characterpoint in my career. I learned more from Fermi in twenty minutes than I learned from Oppenheimer in twenty years. In 1952 I thought I had a good theory of strong interactions. I had organized an army of Cornell students and post-docs to do calculations of meson-proton scat-tering with the new theory. Our calcula-tions agreed pretty well with the cross-sections that Fermi was then measuring with the Chicago cyclotron. So I proud-ly traveled from Ithaca to Chicago to show him our results. Fermi was polite and friendly but was not impressed. He said, “There are two ways to do calcula-tions. The first way, which I prefer, is to have a clear physical picture. The second way is to have a rigorous mathematical formalism. You have neither”. That was the end of the conversation and of our theory. It turned out later that our theory could not have been right because it took no account of vector interactions. Fermi saw intuitively that it had to be wrong. In twenty minutes, his common sense saved us from several years of。

Hierarchy of massive gauge fields

Aµ → ΩAµΩ+ + ∂µΩΩ+
(1)
where Ω(x) is an element of the gauge group, for example Ω(x) = exp{iT aϕa(x)}, and T a are the generators of the corresponding algebra. The group coordinates ϕa
Dcµν (k)
=
gµν
− k2
kµkν m−2 − m2
(3)
1
Using the gauge invariance of the action one can impose another gauge condition,
for example the Lorentz condition ∂µAµ = 0. It improves the ultraviolet asymptotics
However, contrary to the model discussed above, in the Lorentz gauge the propagators of all ﬁelds decrease at inﬁnity as k−2, and the interaction includes only the four point vertex without derivatives and the three point vertex with one derivative. Such a model is renormalizable [6], and due to gauge invariance the renormalized scattering matrix for a proper choice of counterterms coincides with the scattering matrix in the unitary gauge [7], [8], [9], [10].

Gauge Symmetries And Noether Currents In Optimal Control

Applied Mathematics E-Notes,3(2003),49-57c ISSN1607-2510 Available free at mirror sites of .tw/∼amen/Gauge Symmetries And Noether Currents In OptimalControl∗Delﬁm F.M.Torres†Received7June2002AbstractWe extend the second Noether theorem to optimal control problems which are invariant under symmetries depending upon k arbitrary functions of the indepen-dent variable and their derivatives up to some order m.As far as we considera semi-invariance notion,and the transformation group may also depend on thecontrol variables,the result is new even in the classical context of the calculus ofvariations.1IntroductionThe study of invariant variational problemsMinimize J[x(·)]=]b a L(t,x(t),˙x(t))dtin the calculus of variations was initiated in the early part of the XX century by Emmy Noether who,inﬂuenced by the works of Klein and Lie on the transformation properties of diﬀerential equations under continuous groups of transformations(see e.g. [2,Ch.2]),published in her seminal paper[13,14]of1918two fundamental theorems, now classical results and known as the(ﬁrst)Noether theorem and the second Noether theorem,showing that invariance with respect to a group of transformations of the variables t and x implies the existence of certain conserved quantities.These results, also known as Noether’s symmetry theorems,have profound implications in all physical theories,explaining the correspondence between symmetries of the systems(between the group of transformations acting on the independent and dependent variables of the system)and the existence of conservation laws.This remarkable interaction between the concept of invariance in the calculus of variations and the existence ofﬁrst integrals (Noether currents)was clearly recognized by Hilbert[6](cf.[12]).Theﬁrst Noether theorem establishes the existence ofρﬁrst integrals of the Euler-Lagrange diﬀerential equations when the Lagrangian L is invariant under a group of transformations containingρparameters.This means that the invariance hypothesis ∗Mathematics Subject Classiﬁcations:49K15,49S05.†Department of Mathematics,University of Aveiro,3810-193Aveiro,Portugal4950Gauge Symmetries and Noether Currents leads to quantities which are constant along the Euler-Lagrange extremals.Extensionsfor the Pontryagin extremals of optimal control problems are available in[19,21,20].The second Noether theorem establishes the existence of k(m+1)ﬁrst integrals when the Lagrangian is invariant under an inﬁnite continuous group of transformations which,rather than dependence on parameters,as in theﬁrst theorem,depend upon k arbitrary functions and their derivatives up to order m.This second theorem is not as well known as theﬁrst.It has,however,some rather interesting implications.If for example one considers the functional of the basic problem of the calculus of variations in the autonomous case,J[x(·)]=]b a L(x(t),˙x(t))dt,(1)the classical Weierstrass necessary optimality condition can easily be deduced from the fact that the integral(1)is invariant under transformations of the form T=t+p(t), X=x(t),for an arbitrary function p(·)(see[11,p.161]).The second Noether the-orem is related to:(i)parameter invariant variational problems,i.e.,problems of the calculus of variations,as in the homogeneous-parametric form,which are invariant un-der arbitrary transformations of the independent variable t(see[1,p.266],[11,Ch. 8],[3,p.179]);(ii)the singular Lagrangians and the constraints in the Hamiltonian formalism,a framework studied by Dirac-Bergmann(see[4,5]);(iii)the physics of gauge theories,such as the gauge transformations of electrodynamics,electromagnetic ﬁeld,hydromechanics,and relativity(see[3,pp.186—189],[11,p.160],[10],[17]).For example,if the Lagrangian L represents a charged particle interacting with a electro-magneticﬁeld,oneﬁnds that it is invariant under the combined action of the so called gauge transformation of theﬁrst kind on the charged particleﬁeld,and a gauge trans-formation of the second kind on the electromagneticﬁeld.As a result of this invariance it follows,from second Noether’s theorem,the very important conservation of charge. The invariance under gauge transformations is a basic requirement in Yang-Millsﬁeld theory,an important subject,with many questions for mathematical understanding (cf.[7]).To our knowledge,no second Noether type theorem is available for the optimal control setting.One such generalization is our concern here.Instead of using the orig-inal argument[13,14]of Emmy Noether,which is fairly complicated and depends on some deep and conceptually diﬃcult results in the calculus of variations,our approach follows,mutatis mutandis,the paper[19],where theﬁrst Noether theorem is derived almost eﬀortlessly by means of elementary techniques,with a simple and direct ap-proach,and it is motivated by the novelties introduced by the author in[21].Even in the classical context(cf.e.g.[10])and in the simplest possible situation,for the basic problem of the calculus of variations,our result is new since we consider symmetries of the system which alter the cost functional up to an exact diﬀerential;we introduce a semi-invariant notion with some weightsλ0,...,λm(possible diﬀerent from zero);and our transformation group may depend also on˙x(the control).Our result hold both in the normal and abnormal cases.D.F.M.Torres51 2The Optimal Control ProblemWe consider the optimal control problem in Lagrange form on the compact interval [a,b]:Minimize J[x(·),u(·)]=]b a L(t,x(t),u(t))dtover all admissible pairs(x(·),u(·))1(x(·),u(·))∈W n1,1([a,b];R n)×L r∞([a,b];Ω⊆R r),satisfying the control equation˙x(t)=ϕ(t,x(t),u(t)) a.e.t∈[a,b].The functions L:R×R n×R r→R andϕ:R×R n×R r→R n are assumed to be C1with respect to all variables and the setΩof admissible values of the control parameters is an arbitrary open set of R r.Associated to the optimal control problem there is the Pontryagin Hamiltonian H:[a,b]×R n×Ω×R×(R n)T→R which is deﬁned asH(t,x,u,ψ0,ψ)=ψ0L(t,x,u)+ψ·ϕ(t,x,u).(2) A quadruple(x(·),u(·),ψ0,ψ(·)),with admissible(x(·),u(·)),ψ0∈R−0,andψ(·)∈W1,1([a,b];R n)(ψ(t)is a covector1×n),is called a Pontryagin extremal if the following two conditions are satisﬁed for almost all t∈[a,b]:The Adjoint System˙ψ(t)=−∂H(t,x(t),u(t),ψ,ψ(t));(3) The Maximality ConditionH(t,x(t),u(t),ψ0,ψ(t))=maxu∈ΩH(t,x(t),u,ψ0,ψ(t)).(4) The Pontryagin extremal is called normal ifψ0=0and abnormal otherwise.The celebrated Pontryagin Maximum Principle asserts that if(x(·),u(·))is a minimizer of the problem,then there exists a nonzero pair(ψ0,ψ(·))such that(x(·),u(·),ψ0,ψ(·)) is a Pontryagin extremal.Furthermore,the Pontryagin Hamiltonian along the extremal is an absolutely continuous function of t,t→H(t,x(t),u(t),ψ0,ψ(t))∈W1,1([a,b];R),and satisﬁes the equalitydH(t,x(t),u(t),ψ0,ψ(t))=∂H(t,x(t),u(t),ψ0,ψ(t)),(5)for almost all t∈[a,b],where on the left-hand side we have the total derivative with respect to t and on the right-hand side the partial derivative of the Pontryagin Hamil-tonian with respect to t(cf.[16].See[18]for some generalizations of this fact).1The notation W1,1is used for the class of absolutely continuous functions,while L∞represents the class of measurable and essentially bounded functions.52Gauge Symmetries and Noether Currents 3Main ResultTo formulate a second Noether theorem in the optimal control setting,ﬁrst we need to have appropriate notions of invariance and Noether current .We propose the following ones.DEFINITION 1.A function C (t,x,u,ψ0,ψ)which is constant along every Pon-tryagin extremal (x (·),u (·),ψ0,ψ(·))of the problem,C (t,x (t ),u (t ),ψ0,ψ(t ))=k ,t ∈[a,b ],(6)for some constant k ,will be called a Noether current .The equation (6)is the conser-vation law corresponding to the Noether current C .DEFINITION 2.Let C m p :[a,b ]→R k be an arbitrary function of the indepen-dent ing the notationα(t ).= t,x (t ),u (t ),p (t ),˙p (t ),...,p (m )(t ) ,we say that the optimal control problem is semi-invariant if there exists a C 1trans-formation groupg :[a,b ]×R n ×Ω×R k ∗(m +1)→R ×R n ×R r ,g (α(t ))=(T (α(t )),X (α(t )),U (α(t ))),(7)which for p (t )=˙p (t )=···=p (m )(t )=0corresponds to the identity transformation,g (t,x,u,0,0,...,0)=(t,x,u )for all (t,x,u )∈[a,b ]×R n ×Ω,satisfying the equationsL (g (α(t )))d dt T (α(t ))= λ0·p (t )+λ1·˙p (t )+···+λm ·p (m )(t ) d L (t,x (t ),u (t ))+L (t,x (t ),u (t ))+d dtF (α(t ))(8)d dt X (α(t ))=ϕ(g (α(t )))d dt T (α(t )),(9)for some function F of class C 1and for some λ0,...,λm ∈R k .In this case the group of transformations g will be called a gauge symmetry of the optimal control problem.REMARK 1.We use the term “gauge symmetry”to emphasize the fact that the group of transformations g depend on arbitrary functions.The terminology takes origin from gauge invariance in electromagnetic theory and in Yang-Mills theories,but it refers here to a wider class of symmetries.REMARK 2.The identity transformation is a gauge symmetry for any given opti-mal control problem.THEOREM 1(second Noether theorem for Optimal Control).If the optimal control problem is semi-invariant under a gauge symmetry (7),then there exist k (m +1)D.F.M.Torres53Noether currents of the formψ0#∂F (α(t ))∂p j 0+λi j L (t,x (t ),u (t ))$+ψ(t )·∂X (α(t ))∂p j 0−H (t,x (t ),u (t ),ψ0,ψ(t ))∂T (α(t ))∂p j 0(i =0,...,m ,j =1,...,k ),where H is the corresponding Pontryagin Hamiltonian (2).REMARK 3.We are using the standard convention that p (0)(t )=p (t ),and the following notation for the evaluation of a term:(∗)|0.=(∗)|p (t )=˙p (t )=···=p (m )(t )=0.REMARK 4.For the basic problem of the calculus of variations,i.e.,when ϕ=u ,Theorem 1coincides with the classical formulation of the second Noether theorem if one puts λi =0,i =0,...,m ,and F ≡0in the De ﬁnition 2,and the transformation group g is not allowed to depend on the derivatives of the state variables (on the control variables).In §4we provide an example of the calculus of variations for which our result is applicable while previous results are not.PROOF.Let i ∈{0,...,m },j ∈{1,...,k },and (x (·),u (·),ψ0,ψ(·))be an arbitrary Pontryagin extremal of the optimal control problem.Since it is assumed that to the values p (t )=˙p (t )=···=p (m )(t )=0it corresponds the identity gauge transformation,di ﬀerentiating (8)and (9)with respect to p (i )j and then setting p (t )=˙p (t )=···=p (m )(t )=0one gets:λi j d L +d ∂F (α(t ))∂p j 0=∂L ∂T (α(t ))∂p j 0+∂L ·∂X (α(t ))∂p j 0+∂L ∂u ·∂U (α(t ))∂p (i )j 0+L d dt ∂T (α(t ))∂p (i )j 0,(10)d ∂X (α(t ))∂p j 0=∂ϕ∂T (α(t ))∂p j 0+∂ϕ·∂X (α(t ))∂p j 0+∂ϕ·∂U (α(t ))∂p j 0+ϕd ∂T (α(t ))∂p j 0,(11)with L and ϕ,and its partial derivatives,evaluated at (t,x (t ),u (t )).Multiplying (10)by ψ0and (11)by ψ(t ),we can write:ψ0#∂L ∂T (α(t ))∂p j 0+∂L ·∂X (α(t ))∂p j 0+∂L ·∂U (α(t ))∂p j 054Gauge Symmetries and Noether Currents+L d dt ∂T (α(t ))∂p (i )j 0−d dt ∂F (α(t ))∂p (i )j 0−λi j d dt L $+ψ(t )·#∂ϕ∂t ∂T (α(t ))∂p (i )j 0+∂ϕ∂x ·∂X (α(t ))∂p (i )j 0+∂ϕ∂u ·∂U (α(t ))∂p (i )j 0+ϕd dt ∂T (α(t ))∂p (i )j 0−d dt ∂X (α(t ))∂p (i )j 0$=0.(12)According to the maximality condition (4),the functionψ0L (t,x (t ),U (α(t )))+ψ(t )·ϕ(t,x (t ),U (α(t )))attains an extremum for p (t )=˙p (t )=···=p (m )(t )=0.Therefore ψ0∂L ∂u ·∂U (α(t ))∂p (i )j 0+ψ(t )·∂ϕ∂u ·∂U (α(t ))∂p (i )j 0=0and (12)simpli ﬁes to ψ0#∂L ∂t ∂T (α(t ))∂p j 0+∂L ∂x ·∂X (α(t ))∂p j 0+L d dt ∂T (α(t ))∂p j 0−d dt ∂F (α(t ))∂p j 0−λi j d dt L $+ψ(t )·#∂ϕ∂t ∂T (α(t ))∂p j 0+∂ϕ∂x ·∂X (α(t ))∂p j 0+ϕd dt ∂T (α(t ))∂p j 0−d dt ∂X (α(t ))∂p j 0$=0.Using the adjoint system (3)and the property (5),one easily concludes that the above equality is equivalent to d #ψ0∂F (α(t ))∂p j 0+ψ0λi j L +ψ(t )·∂X (α(t ))∂p j 0−H ∂T (α(t ))∂p j 0$=0.4ExampleConsider the following simple time-optimal problem with n =r =1and Ω=(−1,1).Given two points αand βin the state space R ,we are to choose an admissible pair (x (·),u (·)),solution of the control equation˙x (t )=u (t ),D.F.M.Torres55 and satisfying the boundary conditions x(0)=α,x(T)=β,in such a way that the time of transfer fromαtoβis minimal:T→min.In this case the Lagrangian is given by L≡1whileϕ=u.It is easy to conclude that the problem is invariant under the gauge symmetryg(t,x(t),u(t),p(t),˙p(t),¨p(t))= p(t)+t,(˙p(t)+1)2x(t),2¨p(t)x(t)+(˙p(t)+1)u(t) ,i.e.,underT=p(t)+t,X=(˙p(t)+1)2x(t),U=2¨p(t)x(t)+(˙p(t)+1)u(t),where p(·)is an arbitrary function of class C2([0,T];R).For that we choose F=p(t),λ0=λ1=λ2=0,and conditions(8)and(9)follows:L(T,X,U)dT=d(p(t)+t)=dF+L(t,x(t),u(t)),ϕ(T,X,U)ddtT=[2¨p(t)x(t)+(˙p(t)+1)u(t)](˙p(t)+1) =ddt k(˙p(t)+1)2x(t)l=d dt X.From Theorem1the two non-trivial Noether currentsψ0−H,(13)2ψ(t)x(t),(14) are obtained.As far asψ0is a constant,the Noether current(13)is just saying thatthe corresponding Hamiltonian H is constant along the Pontryagin extremals of the problem.This is indeed the case,since the problem under consideration is autonomous (cf.equality(5)).The Noether current(14)can be understood having in mind themaximality condition(4)(∂H∂u=0⇔ψ(t)=0).5Concluding RemarksIn this paper we provide an extension of the second Noether’s theorem to the optimal control framework.The result seems to be new even for the problems of the calculus of variations.Theorem1admits several extensions.It was derived,as in the original work by Noether[13,14],for state variables in an n-dimensional Euclidean space.It can be formulated,however,in contexts where the geometry is not Euclidean(these extensions can be found,in the classical context,e.g.in[8,9,15]).It admits also a generaliza-tion for optimal control problems which are invariant in a mixed sense,i.e,which are56Gauge Symmetries and Noether Currents invariant under a group of transformations depending uponρparameters and upon k arbitrary functions and their derivatives up to some given order.Other possibility is to obtain a more general version of the second Noether theorem for optimal control prob-lems which does not admit exact symmetries.For example,under an invariance notion up toﬁrst-order terms in the functions p(·)and its derivatives(cf.the quasi-invariance notion introduced by the author in[20]for theﬁrst Noether theorem).These and other questions,such as the generalization of theﬁrst and second Noether type theorems to constrained optimal control problems,are under study and will be addressed elsewhere.Acknowledgment.This research was supported in part by the Optimization and Control Theory Group of the R&D Unit Mathematics and Applications,and the pro-gram PRODEP III5.3/C/200.009/2000.Partially presented at the5th Portuguese Conference on Automatic Control(Controlo2002),Aveiro,September5—7,2002. References[1]R.Courant and D.Hilbert,Methods of Mathematical Physics.Vol.I.IntersciencePublishers,Inc.,New York,N.Y.,1953.[2]A.Dur´a n,Symmetries of diﬀerential equations and numerical applications,Uni-versidade de Coimbra,Departamento de Matem´a tica,Coimbra,1999.[3]I.M.Gelfand and S.V.Fomin,Calculus of variations.Dover Publications,Mine-ola,NY,2000.[4]X.Gr`a cia and J.M.Pons,Canonical Noether symmetries and commutativityproperties for gauge systems.J.Math.Phys.,41(11)(2000),7333—7351.[5]M.Guerra,Solu¸c˜o es generalizadas para problemas L-Q singulares.PhD thesis,Departamento de Matem´a tica,Universidade de Aveiro,2001.[6]D.Hilbert,Grundlagen der physik,Math.Ann.,92(1924),258—289.[7]A.Jaﬀe and E.Witten,Quantum Yang-Mills theory,Problem Description of theYang-Mills Existence and Mass Gap Millennium Prize Problem,The Clay Math-ematics Institute of Cambridge,Massachusetts(CMI).[8]J.Komorowski,A modern version of the E.Noether’s theorems in the calculus ofvariations,I.Studia Math.,29(1968),261—273.[9]J.Komorowski,A modern version of the E.Noether’s theorems in the calculus ofvariations II,Studia Math.,32(1969),181—190.[10]J.D.Logan,On variational problems which admit an inﬁnite continuous group,Yokohama Math.J.,22(1974),31—42.[11]J.D.Logan,Invariant Variational Principles,Academic Press,New York,1977.[12]D.Lovelock and H.Rund,Tensors,Diﬀerential Forms,and Variational Principles,Wiley-Interscience,New York,1975.D.F.M.Torres57[13]E.Noether,Invariante variationsprobleme,G¨o tt.Nachr.,pages235—257,1918.[14]E.Noether.Invariant variation problems,Transport Theory Statist.Phys.,1(3)(1971),186—207.English translation of the original paper[13].[15]B.F.Plybon,New approach to the Noether theorems.J.Mathematical Phys.,12(1971),57—60.[16]L.S.Pontryagin,V.G.Boltyanskii,R.V.Gamkrelidze and E.F.Mishchenko,TheMathematical Theory of Optimal Processes,Interscience Publishers,John Wiley &Sons,Inc.New York-London,1962.[17]M.A.Tavel,Milestones in mathematical physics:Noether’s theorem,TransportTheory Statist.Phys.,1(3)(1971),183—185.[18]D.F.M.Torres,A remarkable property of the dynamic optimization extremals,Investiga¸c˜a o Operacional,22(2)(2002),253—263.[19]D.F.M.Torres,Conservation laws in optimal control.In Dynamics,Bifurcationsand Control,volume273of Lecture Notes in Control and Information Sciences, pages287—296.Springer-Verlag,Berlin,Heidelberg,2002.[20]D.F.M.Torres,Conserved quantities along the Pontryagin extremals of quasi-invariant optimal control problems,Proc.10th Mediterranean Conference on Con-trol and Automation—MED2002(invited paper),10pp.(electronic),Lisbon, Portugal,July9-12,2002.[21]D.F.M.Torres,On the Noether theorem for optimal control.European Journalof Control,8(1)(2002),56—63.。

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arXiv:0802.3702v1 [hep-ph] 26 Feb 2008
Introduction.— There is by now an established history of extra-dimensional model building, in which fourdimensional (4D) eﬀective theories are embedded into higher-dimensional theories. Depending on the goals of the model, ﬁelds are assumed to propagate either through the higher-dimensional bulk, or else along lowerdimensional branes. The interest in such models is motivated by phenomenologically desirable properties of higher-dimensional theories that are absent in 4D theories, as in 5D models of electroweak symmetry breaking that address the hierarchy problem of the standard model [1, 2]. In extra-dimensional models, the lighest Kaluza-Klein (KK) modes give rise to an eﬀective 4D theory, while the masses of the excited KK modes determine the scale above which the extra dimensions become manifest. Below the KK scale, a 5D theory with gauge group G typically describes a 4D theory with gauge group G or smaller. However, it is possible that the ﬁrst few massive modes are much lighter than the rest, so that the eﬀective 4D description is that of a larger, product gauge group that is spontaneously broken to a subgroup of G. Alternatively, the higher-dimensional gauge invariance may lead to additional global symmetries in the 4D theory. For example, a 5D SU(2) gauge invariance may contain both the weak SU(2) gauge group and an SU(2) custodial symmetry that protects the ρ parameter. We describe a framework for realizing scenarios of this type, with enhanced gauge or global symmetries in the low-energy effective theory. The basic mechanism that leads to a hierarchy of KK scales is the existence of multiple ultraviolet (UV) regions in the spacetime geometry. Models with multiple UV regions and ultralight modes have been studied in various contexts in the past [3, 4]. In the case of holographic QCD models, the induced geometry on the D8-branes in the D4-D8 system [5] has two UV regions, as does the deconstructed holographic QCD model of Son and Stephanov [6]. Holographic technicolor models based on the D4-D8 system have been constructed [7, 8], and are the most direct analogy to the models we are considering. A toy example.— Consider a 5D SU(2) gauge theory in the background of two slices of 4D anti-de Sitter space
Extra Gauge Invariance from an Extra Dimension
Christopher D. Carone, Joshua Erlich, and Marc Sher
Particle Theory Group, Department of Physics, College of William and Mary, Williamsburg, VA 23187-8795 (Dated: February 2008) We describe higher-dimensional theories whose low-energy 4D descriptions contain larger gauge or global symmetry groups. As an example, we construct a Higgsless SU(2)×U(1)B−L model of electroweak symmetry breaking. The 5D SU(2) gauge invariance contains both the weak SU(2) gauge group and a custodial symmetry that protects the rho parameter. Fermions obtain isospinviolating masses while maintaining universal gauge couplings among all three generations. As a further example, we construct a model of chiral color based on a single SU(3).
ν a ∂z w(z )∂z Aa µ (x, z ) = w(z ) ∂ν ∂ Aµ (x, z ).
(4)
With a common abuse of notation, the mode solutions are of the form
a a Aa µ (x, z ) = Aµ (x) ψ (z ) ,
2
SU(2)
(z L ,L τ) τ U(1)B−L
பைடு நூலகம்
z=0
z
(z R,0)
τ , Bµ (x, τ ) ≈ Bµ (x). The boundary conditions on the SU(2) gauge bosons Aa µ (x, z ) and Bµ (x, τ ) are determined by beginning with Neumann boundary conditions FA zµ = FB τ µ = 0 and considering the decoupling limit of a Higgs doublet with U(1) charge 1/2, localized at the D4A -D4B intersection (τ = 0, z = zR ). (The SU(2) gen1 ab erators are normalized such that Tr T a T b = 2 δ .) The large Higgs vacuum expectation value (vev) enforces g5 A3 µ (x, z = zR ) − g5 Bµ (x, τ = 0) = 0 ,2 A1 µ (x, z = zR ) = 0 . (9)
joined together in the infrared (IR). The metric is ds2 = 1 ηµν dxµ dxν − dz 2 , (1 − |z |)2 (1)
where µ, ν run over 0, . . . , 3 and −1 + ǫL < z < 1 − ǫR , in units of the IR length scale. The geometry has two ultraviolet (UV) regions, the neighborhoods of zL = −1+ ǫL and zR = 1 − ǫR . In the following we will deﬁne the conformal factor as w(z ) ≡ (1 − |z |)−1 . (2)
a d4 x dz w(z ) FMN F a MN ,
(3)
where a is the gauge index; the Lorentz indices M, N run over 0,1,2,3,z and their contractions are with the ﬂat 5D metric. Boundary conditions are chosen to eliminate extra massless scalars (often referred to as the A5 modes). We will generally choose the gauge Aa 5 = 0 [10]. The equations of motion for the SU(2) gauge ﬁelds Aa µ (x, z ) are,