Gauge and Gravitational Anomalies and Hawking Radiation of Rotating BTZ Black Holes

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.。

a rXiv:h ep-ph/2180v116Fe b2UCRHEP–T270A short course in effective Lagrangians.∗Jos´e Wudka †Physics Department,UC Riverside Riverside CA 92521-0413,USA Abstract These lectures provide an introduction to effective theories concentrating on the basic ideas and providing some simple applications I.INTRODUCTION.When studying a physical system it is often the case that there is not enough information to provide a fundamental description of some of its properties.In such cases one must parameterize the corresponding effects by introducing new interactions with coefficients to be determined phenomenologically.Experimental limits or measurement of these parameters then (hopefully)provides the information needed to provide a more satisfactory description.A standard procedure for doing this is to first determine the dynamical degrees of freedom involved and the symmetries obeyed,and then construct the most general Lagrangian,the effective Lagrangian for these degrees of freedom which respects the required symmetries.The method is straightforward,quite general and,most importantly,it works!In following this approach one must be wary of several facts.Fist it is clear that the relevant degrees of freedom can change with scale(e.g.mesons are a good description of low-energy QCD,but at higher energies one should use quarks and gluons);in addition,physics at different scales may respect different symmetries(e.g.mass conservation is violated at sufficiently high energies).It follows that the effective Lagrangian formalism is in general applicable only for a limited range of scales.It is often the case(but no always!)that there is a scaleΛso that the results obtained using an effective Lagrangian are invalid for energies aboveΛ.The formalism has two potentially serious drawbacks.First,effective Lagrangian has an infinite number of terms suggesting a lack of predictability.Second,even though the model has an UV cutoffΛand will not suffer from actual divergences,simple calculations show that is is a possible for this type of theories to generating radiative corrections that grow withΛ,becoming increasingly important for higher and higher order graphs.Either of these problems can render this approach useless.It is also necessary verify that the model is unitary.I will discuss below how these problems are solved,an provide several applications of the formalism.The aim is to give aflair of the versatility of the approach,not to provide an exhaustive review of all known applications.II.F AMILIAR EXAMPLESA.Euler-Heisenberg effective LagrangianThis Lagrangian summarizes QED at low energies(below the electron mass)[1].At these energies only photons appear in real processes and the effective Lagrangian will be then constructed using the photonfield Aµ,and will satisfy a U(1)gauge and Lorenz invariances. Thus it can be constructed in terms of thefield strength Fµνor the loop variables A(Γ)=2FIG.1.Graph generating the leading terms in the Euler-Heisenberg effective Lagrangian ΓA·dx.The latter are non-local,so that a local description would involve only F,namely1 L eff=L eff(F)=aF2+bF4+c(F˜F)2+dF2(F˜F) (1)One can arbitrarily normalize thefields and so choose a=−1/4.The constants b,c and d have units of mass−2.Note that the term∝d violates CP.Though we know QED respects C and P,it is possible for other interactions to violate these symmetries,there is nothing in the discussion above that disallows such terms and,in fact,weak effects will generate them.For this system we are in a privileged position for we know the underlying physics,and so we can calculate b,c,d,....The leading effects come form QED which yields b,c∼1/(4πm e)2at 1loop[1].The parameters b and c summarize all the leading virtual electron effects.(see Fig.1).Forgetting about this underlying structure we could have simply defined a scale M and taken b,c∼1/M2(so that M=4πm e),and while this is perfectly viable,M is not relevant phenomenologically speaking as it does not corresponds of a physical scale.In order to extract information about the physics underlying the effective Lagrangian from a measurement of b and c we must be able to at least estimate the relation between these constants and the underlying scales.In addition we also know that d∼ξ/(4πv)with v∼246GeV andξis a very small constant proportional to the Jarlskog determinant[2].The effective Lagrangian can holdterms with radically different scales and limits on some constants cannot,in general,translate to others.In this case the terms are characterized by different CP transformation properties, and it is often the case that such global symmetries are useful in differentiating terms in the effective Lagrangian.The point being that a term violating a given global symmetry at scaleΛwill generate all terms in the effective Lagrangian with the same symmetry properties through radiative corrections.The caveat in the argument being that the underlying theory might have some additional symmetries not apparent at low energies which might further segregate interactions and so provide different scales for operators with the same properties under all low energy symmetries.When calculating with the effective Lagrangian the effects produced by the new terms proportional to b,c are suppressed by a factor∼(E/4πm e)2,where E is the typical energy on the process and E≪m e.Thus the effects of these terms are tiny,yet they are noticeable because they generate a new effect:γ−γscattering.B.(Standard)SuperconductivityThis is a brief summary of the very nice treatment provided by Polchinski[3].The system under consideration has the electronfieldψas its only dynamical variable(the phonons are assumed to have been integrated out,generating a series of electron self-interactions),it respects U(1)electromagnetic gauge invariance,as well as Galilean invariance and Fermion number conservation.Assuming a local description,thefirst few terms in the effective Lagrangian expansion are(neglecting those containing photons for simplicity)L eff= kψ∗k[i∂t−e k+µ]ψk+ ψ∗kψlψqψ∗pδ(k−l−q+p)V klq+ (2)In this equation the relation e k=µdetermines the Fermi surface,while V∼(electron-photon coupling)2on the Fermi Surface(FS)if e k=µ,if p is near the FS one can write p=k+ℓˆn(with e k=µ).Scaling towards the FS impliesℓ→sℓwith s→0.Then assumingψ→s dψthe quadratic terms in the action will be scale invariant provided d=−1/2.The quartic terms in the action then scales as s and becomes negligible near the FS except when the pairing condition q+l=0is obeyed.In this case the quartic term scales as s0and cannot be ignored.In fact this term determines the most interesting behavior of the system at low temperatures(see[3]for full details).C.Electroweak interactionsAgain I will follow the general recipe.I will concentrate only on the(low energy)inter-actions involving leptonfields,which are then the degrees of freedom.Since I assume the energy to be well below the Fermi scale,the only relevant symmetries are U(1)gauge and Lorenz invariances.In addition there is the question whether the heavy physics will respect the discrete symmetries C,P or CP;using perfect hindsight I will retain terms that violate these symmetriesAssuming a local description I have[1]L eff= ¯ψi(i D−m i)ψi+ f ijkl ¯ψiΓaψj ¯ψkΓaψl + (3)where the ellipsis indicate terms containing operators of higher dimension,or those involving the electromagneticfield.The matricesΓare to be chosen among the16independent basis Γa={1,γµ,σµν,γµγ5,γ5}The coefficients for thefirst two terms are befixed by normalization requirements.While a SM calculation gives f∼g2/m2W=1/v2(v≃246GeV)and is generated by tree-level√graphs(see Fig.II C)because of this the scale1/be observed(or bounded)despite the E≪v condition because they generate new effects: C and P(and some of them chirality)violation.FIG.2.Standard model processes generating four fermion interactions at low energies(e.g.. Bhaba scattering)D.Strong interactions at low energiesIn this case we are interested in the description of the interactions among the lightest hadrons,the meson multiplet.The most convenient parameterization of these degrees of freedom is in terms of a unitaryfield[9]U such that U=exp(λaπa/F)whereπa denote the eight mesonfields,λa the Gell-Mann matrices and F is a constant(related to the pion decay constant).The symmetries obeyed by the system are chiral SU(3)L×SU(3)R,Lorenz invariance,C and P.With these constraints the effective Lagrangian takes the formL eff=a tr∂U†.∂U+ b tr∂µU†∂νU∂µU†∂νU+... + (4)I can set a∼F2by properly normalizing thefields.In this case the leading term in the effective Lagrangian will determine all(leading)low-energy pion interactions in terms of the single constant F.The effects form the higher-order terms have been measured and the data requires b∼1/(4π)2.This result is also predicted by the consistency of this approach which requires that radiative corrections to a,b,etc.should be at most of the same size as their tree-level values.6III.BASIC IDEAS ON THE APPLICABILITY OF THE FORMALISMBeing a model with intrinsic an cutoffthere are no actual ultraviolet divergences in most effective Lagrangian computations.Still there are interesting renormalizability issues that arise when doing effective Lagrangian loop computations.Imagine doing a loop calculation including some vertices terms of(mass)dimension higher than the dimension of space-time.These must have coefficients with dimensions of mass to some negative power.The loop integrations will produce in general terms growing withΛthe UV cutoffwhich are polynomials in the external momenta2and will preserve the symmetries of the model[4].Hence these terms which may grow withΛcorrespond to vertices appearing in the most general effective Lagrangian and can be absorbed in a renormalization of the corresponding coefficients.They have no observable effects(though they can be used in naturality arguments[5].Effective theories will also be unitary provided one stays within the limits of their appli-cability.Should one exceed them new channels will open(corresponding to the production of the heavy excitations)and unitarity violating effects will occur.This is not produced by real unitarity violating interactions,but due to our using the model beyond its range of applicability(e.g.it the typical energy of the process under consideration reaches of exceeds Λ).One can,of course,extend the model,but this necessarily introduces ad-hoc elements and will dilute the generality gained using effective theories.For example consider W W Z interactions with an effective Lagrangian of the formL eff=λ(p,k)Wµν(k)Wνρ(p)Zρµ(−p−k)+···;(5) (where Vαβ=∂αVβ−∂βVα)One can then chooseλto insure unitarity is preserved(at least in some processes),for example[6]λ0λ(p,k)=Another common situation where effective Lagrangians appear occurs when some heavy excitations are integrated out.This can be illustrated by the following toy model3S= d n x ¯ψ(i∂−m)ψ+12Λ2φ21+fφ¯ψψ (9) whereφis heavy.A simple calculation givesS eff= d n x ¯ψ(i∂−m)ψ+1+Λ2¯ψψ (10) andL eff=¯ψ(i∂−m)ψ+f2Λ2 n¯ψψ(11) Note that terms with large number of derivatives will be suppressed by a large power of the small factor(E/Λ),if we are interested in energies E∼Λthe whole infinite set of vertices must be included in order to reproduce theφpole.A.How to parameterize ignoranceIf one knows the theory we can,in principle,calculate L eff(or do a full calculation).Yet there are many cases where the underlying theory is not known.In these cases an effective theory if obtained by writing all possible interactions among the light excitations.The model then has an infinite number of terms each with an unknown parameter,and these constants then parameterize all possible underlying theories.The terms which dominate are those usually called renormalizable(or,equivalently,marginal or relevant).The other terms are called non-renormalizable,or irrelevant,since their effects become smaller as the energy decreasesThis recipe for writing effective theories must be supplemented with some symmetry re-strictions.The most important being that the all the terms in the effective Lagrangian mustrespect the local gauge invariance of the low-energy physics(more technically,the one re-spected by the renormalizable terms in the effective action)[7].The reason is that the presence of a gauge variant term will generate all gauge variant interactions thorough renor-malization group evolution.a.Gauge invariantizing Using a simple argument it is possible to turn any theory into a gauge theory[8]and so it appears that the requirement of gauge invariance is empty.That this is not the case is explained here.Ifirst describe the trick which grafts gauge invariance onto a theory and then discuss the implications.Consider an arbitrary theory with matterfields(spin0and1/2)and vectorfields V nµ, n=1,...N.Then•Choose a(gauge)group G with N generators{T n}.Define a covariant derivative Dµ=∂µ+V nµT n and assume that the V nµare gaugefields.•Invent a unitaryfield U transforming according to the fundamental representation ofG and construct the gauge invariant compositefieldsV nµ=−tr T n U†DµU(12) Taking tr T n T m=−δnm,it is easy to see that in the unitary gauge U=1,V nµ=V nµ.Thus if simply replace V→V in the original theory we get a gauge theory.Does this mean that gauge invariance irrelevant since it can be added at will?In my opinion this is not the case.In the above process all matterfields are assumed gauge singlets(none are minimally coupled to the gaugefields).In the case of the standard model,for example,the universal coupling of fermions to the gauge bosons would be accidental in this approach.In order to recover the full predictive power commonly associated with gauge theories,the matterfields must transform non-trivially under G which can be done only if there are strong correlations among some of the couplings.It is not trivial to say that the standard model group is10SU(3)×SU(2)×U(1)with left-handed quarks transforming as(3,2,1/6),left-handed leptons as(1,2,−1/2),etc.,as opposed to a U(1)12with all fermions transforming as singlets[10].B.How to estimate ignoranceA problem which I have not addressed so far is the fact that effective theories have an infinite number of coefficients,with the(possible)problem or requiring an infinite number of data points in order to make any predictions.On the other hand,for example,if this is the case why is it that the Fermi theory of the weak interactions is so successful?The answer to this question lies in the fact that not all coefficients are created equal,there is a hierarchy[9,10].As a result,given any desired level of accuracy,only afinite number of terms need to be included.Moreover,even though the effective Lagrangian coefficients cannot be calculated without knowing the underlying theory,they can still be bounded using but a minimal set of assumptions about the heavy interactions.It is then also possible to estimate the errors in neglecting all but thefinite number of terms used.As an example consider the standard model at low energies and calculate two processes: Bhaba cross section and the anomalous magnetic moment of the electron.For Bhaba scat-tering there is a contribution due the Z-boson exchange(see Fig.II C)e+e−→Z→e+e−generates O=14In addition the coefficient is suppressed by a factor of m e since it violates chirality.11eeeFIG.3.Weak contributions to the electron anomalous magnetic moment The point of this exercise is to illustrate the fact that,for weakly coupled theories,loop-generated operators have smaller coefficients than operators generated at tree level.Leading effects are produced by operators which are generated at tree level.C.Coefficient estimatesIn this section I will provide arguments which can be used to estimate(or,at least bound) the coefficients in the effective Lagrangian.These are order of magnitude calculations and might be offby a factor of a few;it is worth noting that no single calculation has provided a significant deviation from these results.The estimate calculations should be done separately for weakly and strongly interacting theories.I will characterize thefirst as those where radiative corrections are smaller than the tree-level contributions.Strongly interacting theories will have radiative corrections of the same size at any order51.Weakly interacting theoriesIn this case leading terms in the effective Lagrangian are those which can be generated at tree level by the heavy physics.Thus the dominating effects are produced by operators which have the lowest dimension(leading to the smallest suppression from inverse powers ofΛ)and which are tree-level generated(TLG)operators can be determined[11].When the heavy physics is described by a gauge theory it is possible to obtained all TLG operators[11].The corresponding vertices fall into3categories,symbolically •vertices with4fermions.•vertices with2fermions and k bosons;k=2,3•vertices with n bosons;n=4,6.A particular theory may not generate one or more of these vertices,the only claim is that there is a gauge theory which does.In the case of the standard model with lepton number conservation the leading operators have dimension6[12,11].Subleading operators are either dimension8and their contribu-tions are suppressed by an additional factor(E/Λ)2in processes with typical energy E. Other subleading contributions are suppressed by a loop factor∼1/(4π)2.Note that it is possible to have situations where the only two effects are produced by either dimension8 TLG operators or loop generated dimension6operators.In this case the former dominates only whenΛ>4πE.a.Triple gauge bosons The terms in the electroweak effective Lagrangian which describe the interaction of the W and Z bosons generated by some heavy physics underlying the standard model has received considerable attention recently[13].In terms of the SU(2)and U(1)gaugefields W and B and the scalar doubletφthese interactions areL eff=1information about the heavy physics.2.Strongly interacting theoriesI will imagine a theory containing scalars and fermions which interact strongly.Gauge couplings are assumed to be small and will be ignored.This calculation is useful for low energy chiral theories but not for low energy QCD[14,15,9].A generic effective operator in this type of theories takes the formO abc∼λΛ4 φΛψ3/2 b ∂(4π)2/3Λ,Λφ=116π2(17)In terms of U∼exp(φ/Λφ),the operators take the formO abc=1For the case whereφrepresents the interpolatingfield for the lightest mesons PCAC impliesΛφ=fπ[14,9].Thenψ4∝116π2ψ2∂2U2∝1Λ2 ¯ψγµψ ¯ψγµψ + (20)whereψdenotes the electronfield.The calculation is illustrated in Fig.IV D where the loops involving the4-fermion oper-ator are cut-offat a scaleΛ.The SM and new physics(NP)contributions are,symbolically,_4πg ()2_1v 2_1v2_2f Λ_4π()2Λ_2f Λ_4π()2Λ_2f Λ_4π()2Λ_2f Λ_2f Λ_2fΛ+++. . . FIG.4.Radiative corrections to Bhaba scattering in the presence of a 4-fermion interactionSM:116π2+··· NP:f16π2+ (21)Note that this consistent behavior (that the new physics effects disappear as Λ→∞)results form having the physical scale of new physics Λin the coefficient of the operator.Had we used f ′/v 2instead of f/Λ2the new physics effects would appear to be enormous,and growing with each new loop.It is not that the use of f ′/v 2is wrong,it is only that it is misleading to believe f ′can be of order one;it must be suppressed by the small factor (v/λ)ing these results we see that this reaction is sensitive to Λprovidedf (v/Λ)2>sensitivity.If the sensitivity is,say 1%this corresponds to Λ/√V.APPLICATIONS TO ELECTROWEAK PHYSICSWith the above results one can determine,for any given process,the leading contributions (as parameterized by the various effective operator coefficients).Using then the coefficient estimates one can provide the expected magnitude of the new physics effects with onlyΛas an unknown parameter,and so estimate the sensitivity to the scale of new physics.It is important to note that this is sometimes a rather involved calculation as all con-tributing operators must be included.For example,in order to determine the heavy physics effects on the oblique parameters one must calculate not only these affecting the vector bo-son polarization tensors,but also this which modify the Fermi constant,thefine structure constant,etc.as these quantities are used when extracting S,T and U from the data[18].A.Effective lagrangianIn the following I will assume that the underlying physics is weakly coupled and derive the leadingoperators that can be expected form the existence of heavy excitations at scale Λ.The complete list of dimension6operators was cataloged a long time ago for the case where the low energy spectrum includes a single scalar doublet[12]7.It is then straightfor-ward to determine the subset of operators which can be TLG,they are[11]•Fermions: ¯ψiΓaψj ¯ψkΓaψl•Scalars:|φ|6,(∂|φ|2)2•Scalars and fermions:|φ|2×Yukawa term•Scalars and vectors:|φ|2|Dφ|2,|φ†Dφ|2•Fermions,scalars and vectors: φ†T n Dµφ ¯ψi T nγµψjwhere T denotes a group generator andΓa product of a group generator and a gamma matrix.Observables affected by the operators in this list provide the highest sensitivity to new physics effects provided that the standard model effects are themselves small(or that the experimental sensitivity is large enough to observe small deviations).I will illustrate this with two(incomplete)examplesB.b-parityThis is a proposed method for probing newflavor physics[19].Its virtue lies in the fact that it is very simple and sensitive(though it does not provide the highest sensitivity for all observables).The basic idea is based on the observation that the standard model acquires an additional global U(1)b symmetry in the limit V ub=V cb=V td=V ts=0(given the experimental values0.002<|V ub|<0.005,0.036<|V cb|<0.046,0.004<|V td|<0.014, 0.034<|V ts|<0.046,deviations form exact U(1)b invariance will be small).Then for any standard model interaction a reaction to the typen i b−jet+X→n f b−jet+Y(22) will obey(−1)n i=(−1)n f(23) to very high accuracy.The number(−1)#of b jets defines the b-parity of a state(it being understood that the top quarks have decayed).The standard model is then b-parity even,and the idea is to consider a lepton collider8 and simply count the number of b jets in thefinal state;new physics effects will show up as events with odd number of b jets.The standard model produces no measurable irreducible background,yet there are significant reducible backgrounds which reduced the sensitivity toΛ.To estimate these effects I define•ǫb=b−jet tagging efficiency•t c=c−jet mis tagging efficiency(probability of mistaking a c−jet jet for a b−jet •t j=light-jet mis tagging efficiency(probability of mistaking a light-jet for a b−jet so that the measured cross section with k-b-jets is¯σk= u+v+w=k n u ǫu b(1−ǫb)n−u m v t v c(1−t c)m−v ℓw t w j(1−tj)ℓ−w σnmℓ(24) whereσnmℓdenotes the cross section for thefinal state with n b-jets,m c-jets,andℓlight jets.Note that n u ǫu b(1−ǫb)n−u is the probability of tagging u and missing n−u b-jets out of the n available.As an example considerL eff=L sm+f ijLimits from e+e−→t¯c+¯t c+b¯s+¯bs→1b−jet+Xsǫb=50%ǫb=70%2.5fb−1 1.5TeV500GeV 5.0TeV 5.5TeV200fb−110.0TeVThese results are promising yet they will be degraded in a realistic calculation.First one must include the effects of having t c,j=0.In addition there are complications in using inclusive reactions such as e+e−→b+X since the contributions form events with large number of jets can be very hard to evaluate(aside from the calculational difficulties there19are additional complications when defining what a jet is).A more realistic approach is to restrict the calculation to a sample with afixed number of jets(2and4are the simplest) and determine the sensitivity toΛfor various choices ofǫb and t j using this population only.C.CP violationJust as for b-parity the CP violating effects are small within the standard model and so precise measurements of CP violating observable might be very sensitive to new physics effects.In order to study CP violations it is useful tofirst define what the CP transformation is. In order to do this in general denote the Cartan group generators by H i and the root gener-ators by Eα,then it is possible tofind a basis where all the group generators are real and, in addition,the H i are diagonal[20].Define then CP transformation by Transformationsψ→Cψ∗(fermions)φ→φ∗(scalars)A(i)µ→−A(i)µ,(i:Cartan generator)A(α)µ→−A(−α)µ,(α:root)it is easy to see that thefield strengths and currents transform as Aµ,while Dφ→(Dφ)∗.It then follows that in this basis the whole gauge sector of any gauge theory is CP conserving; CP violation can arise only in the scalar potential and fermion-scalar interactions using this basis.In order to apply this to electroweak physics I will need the list of TLG operators of dimension6which violate CP,they are given by9¯ℓe ¯dq −h.c.(¯q u)ε(¯q d)−h.c. ¯qλA u ε ¯qλA d −h.c.¯ℓe ε(¯q u)−h.c. ¯ℓu ε(¯q e)−h.c.|φ|2 ¯ℓeφ−h.c.|φ|2 ¯q u˜φ−h.c. |φ|2(¯q dφ−h.c.)|φ|2∂µ ¯ℓγµℓ|φ|2∂µ(¯eγµe)|φ|2∂µ(¯qγµq)|φ|2∂µ(¯uγµu)|φ|2∂µ ¯dγµdO1= φ†τIφ D IJµ ¯ℓγµτJℓO2= φ†τIφ D IJµ ¯qγµτJ qO3= φ†εDµφ (¯uγµd)−h.cAll operators except O1,2,3violate chirality and their coefficients are strongly bounded by their contributions to the strong CP parameterθ;in addition some chialiry violating operators contribute to meson decays(which again provide strong bounds for fermions in thefirst generation)and,finally,in natural theories some contribute radiatively to fermion masses and will be then suppressed by the smaller of the corresponding Yukawa couplings. For these reasons I will not consider them further.Moreover,since I will be interested in limits that can be obtained using current data,I will ignore operators whose only observable effects involve Higgs particles.With these restrictions only O1,2,3remain;their terms not involving scalars areO1→−igv22 ¯νL W+e L−h.c.O2→−igv22 ¯u L W+d L−h.c.O3→−igv28 ¯u R W+d R−h.c.The contributions from O1,2can be absorbed in a renormalization of standard model coef-ficients whence only O3produces observable effects,corresponding to a right-handed quark current.Existing data(fromτdecays and m W measurements)impliesΛ∼>500GeV One can also determine the type of new interactions which might be probed using these operators[11].The heavy physics which can generate O3at tree level is described in Fig.10. If the underlying theory is natural we conclude that there will be no super-renormalizableSR coupling(unnatural)FIG.5.Heavy violating operators.Wavy lines denote vectors,solid lines fermions,and dashed ones scalars.Heavy lines denote heavy excitations.couplings;in this case O3will be generated by heavy fermion exchanges only10 Notefinally that these arguments are only valid for weakly coupled heavy physics.For strongly coupled theories other CP violating operators can be important,e.g.f10It is true that vertices involving light fermions,light scalars and heavy fermions produce mixings between the light and heavy scales,but this occurs at the one loop level.In contrast cubic terms of orderΛin the scalar potential would shift v at tree level.。

a r X i v :0807.0331v 1 [h e p -p h ] 2 J u l 2008Gravitational Contributions to the Running of Gauge CouplingsYong Tang and Yue-Liang WuKavli Institute for Theoretical Physics China,Institute of Theoretical PhysicsChinese Academy of Science,Beijing 100190,ChinaGravitational contributions to the running of gauge couplings are calculated by using different regularization schemes.As the βfunction concerns counter-terms of dimension four,only quadratical divergencies from the gravitational contributions need to be investigated.A consistent result is obtained by a regularization scheme which can appropriately treat the quadratical divergencies and preserve non-abelian gauge symmetry.The harmonic gauge condition for gravity is used in both diagrammatical and background field calculations,the resulting gravitational corrections to the βfunction are found to be nonzero and regularization scheme independent.PACS numbers:11.10.Hi,04.60.–mIntroduction :Enclosing general relativity into the framework of quantum field theory is one of the most interesting and frustrating questions.Since its coupling constant κis of negative mass dimension,general rel-ativity has isolated itself from the renormalizable the-ories.Renormalization is tightly connected with sym-metry and divergency of loop diagrams.It was until the invention of dimensional regularization[1]that diver-gencies of gravitation were not investigated systemati-cally.Although,pure gravity at one-loop is free of phys-ically relevant divergencies [2],if higher-loop diagrams are considered,higher dimensional counter-terms are needed to add to the original lagrangian.With the hope that quantum gravity,coupled with other fields,might show some miraculous cancelations,Einstein-Scalar sys-tem was investigated by using dimensional regulariza-tion which makes all divergencies reduced to logarith-mic ones,and found to be non-renormalizable[3].Later,Einstein-Maxwell,Einstein-Dirac,and Einstein-Yang-Mills systems were studied and shown to be also non-renormalizable[4].In spite of this theoretical inconsis-tence of general relativity and quantum field theory,we cannot deny that gravity has effect on ordinary fields and it contributes to the corrections of physical results since all matter has gravitational interaction.Treating general relativity as an effective field theory provides a practical way to look into quantum gravity’s effects[5].Recently,Robinson and Wilczek [6]calculated gravi-tational corrections on gauge theory and arrived at an interesting observation that at or beyond Planck scale,all gauge theories,abelian or non-abelian,are asymptot-ically free due to gravitational corrections to the running of gauge couplings.The result obtained in ref.[6]has attracted one’s attention.After reconsidering gravita-tional effects on gauge theories,different conclusions were reached by several groups.The result in[6]was derived in the framework of background field method,as such a method has off-shell and gauge-dependent problem,it was shown in [7]that the result obtained in [6]is gauge dependent,and the gravitational correction to βfunc-tion at one-loop order is absent in the harmonic gauge or general ξing Vilkovisky-DeWitt method,it was found in [8]that the gravitational corrections to the βfunction also vanishes in dimensional te on,a diagrammatical calculation for two and three point functions was performed in [9]to obtain the βfunction by using cut-offand dimensional regulariza-tion schemes,as a consequence,the same conclusion was yielded that quadratical divergencies are absent.In all these calculations,the crucial question concerns how to appropriately treat quadratical divergencies.If gravitational corrections have no quadratical divergen-cies,there would be no change to the βfunction of gauge couplings.However,we know that in dimensional reg-ularization all divergencies are reduced to logarithmic ones,and in cut-offregularization the gauge invariance cannot be maintained.To arrive at a consistent con-clusion whether gravitational corrections affect the run-ning of gauge couplings,one should make a careful cal-culation by using a symmetry-preserving and quadratic divergence-keeping regularization scheme.For this pur-pose,we shall apply in this paper the Loop Regulariza-tion (LR)method[10]to evaluate the gravitational con-tributions to the running of gauge couplings.This is because the LR method has been shown to preserve both the non-abelian gauge symmetry and the divergent be-havior of original integrals,it has be applied to consis-tently obtain all one-loop renormalization constants of non-abelian gauge theory and QCD βfunction[11],and to derive the dynamically generated spontaneous chiral symmetry breaking in chiral effective field theory[12],as well as to clarify the ambiguities of quantum chiral anomalous[13]and the topological Lorentz/CPT violat-ing Chern-Simons term[14].In the following,after briefly introducing the Loop Regularization (LR)method proposed in [10],and pre-senting the general formalism needed for considering gravitational effects,we first make a careful check to re-cover the results obtained in [7,8,9]by using dimen-sional and cut-offregularization schemes.Then by using the LR method with both the diagrammatical and back-ground field method calculations,we present new results.Finally,we arrive at a regularization scheme independent conclusion for gravitational contributions.2Loop Regularization :all loop calculations of Feynman diagrams can be reduced,after Feynman parametrization and momentum translation,into some simple scalar and tensor type loop integrals.For diver-gent integrals,a regularization is needed to make them physically meaningful.There are many kinds of regular-ization schemes in literature.For the reason mentioned above,we shall adopt the Loop Regularization method.In the LR method,the crucial concept is the introduc-tionofthe irreducible loop integrals(ILIs)[10]which are defined,for example,at one-loop level asI −2α(M 2)=d 4k1(k 2−M 2)2+α(1)and higher rank of tensor integrals,with α=−1,0,1,....Here I 2and I 0denote the quadratic and logarithmic divergent ILIs respectively.In general,the divergent integrals are meaningless.To see that,let us exam-ine tensor and scale type quadratical divergent ILIs I 2µν(0)and I 2(0),one can always write down,from the Lorentz decomposition,the general relation that I 2µν(0)=ag µνI 2(0)with a to be determined appropri-ately.When naively multiplying g µνon both sides of the relation,one yields a =1/4,which is actually no longer valid due to divergent integrals.To demonstrate that,considering the zero component I 200of tensor ILIs I 2µν,and performing an integration over the momentum k 0for both I 200and I 2,it is then not difficult to found after comparing both sides of the relation that a =1/2which should hold as the integrations over k 0for I 200and I 2are convergent.To check its consistence,considering the vacuum polarization of QED.In terms of the ILIs,the vacuum polarization is given byΠµν=−4e 2dx 2I 2µν(m )−I 2(m )g µν+2x (1−x )(p 2g µν−p µp ν)I 0(m )(2)which shows that only quadratical divergencies violategauge invariance.If an explicit regularization has a prop-erty that the regularized quadratical divergencies satisfy the consistency condition[10]I R 2µν=12g µνI R2,I 0µν=116π2M 2c −µ2[ln M 2c M 2c)] I R=i µ2−γw +y 0(µ2−g14gµαgνβF a µνF aαβ(5)where R is Ricci scalar and F aµνis the Yang-Mills fields strength F µν=∇µA ν−∇νA µ−ig [A µ,A ν].It is hard to quantize this lagrangian because of gravity-part’s non-linearity and minus-dimension coupling constant κ=√−g =√2κh −12h 2)...](8)In fact,the above expansion is an infinite series and the truncation is up to the question considered.These infi-nite series partly indicate that gravity is not renormal-izable.From this perspective,it is more accurate to say that the system is treated as an effective field theory.Af-ter assembling the same order terms in κ,we could get the lagrangian for usual quantization.Diagrammatical Calculation :Let us set ¯g µν=ηµν,where ηµνis the Minkowski metric.h µνis interpreted as graviton field,fluctuating in flat space-time.The la-grangian can be arranged to different orders of h µνor κ.The free part of gravitation is of order unit and gives the graviton propagator[2]P µνρσG (k )=i2∂µh νν=0.Forsimplicity,in the following,the metric g µνis understood as ηµν.The interactions of gauge field and gravity field(a)(b)(c)(d)(e)FIG.1:Feynman diagrams that gravitation contributes atone loop order to gauge two and three point functions.are determined by expanding the second termof the la-grangian (5).And various vertex could be derived.De-tails of these Feynman rules are referred to [9].Using the traditional Feynman diagram calculations,we can com-pute the βfunction by evaluating two and three point functions of gauge fields.These green functions are gen-eral divergent,so counter-terms are needed to cancel these divergencies.The relevant counter-terms to the βfunction areT µν=iδab Q µνδ2,T µνρ=gf abc V µνρqkp δ1Q µν≡q µq ν−q 2g µν(10)V µνρqkp ≡g νρ(q −k )µ+g ρµ(k −p )ν+g µν(p −q )ρThe βfunction is defined as β(g )=g µ∂2δ2−δ1).In gauge theories without gravity,the counter-terms are logarithmical divergent as the quadratical divergencies cancel each other due to the gauge symmetry.However,if gravitational corrections are taken into account,diver-gent behavior becomes different.On dimensional ground,it is known that quadratical divergencies can appear and will contribute to the counter-terms defined above,so that they will also lead to the corrections to the βfunc-tion.Although gravitational corrections contain logarith-mical divergencies,these divergencies are multiplied by high order momentum.So logarithmical divergencies will lead us to introduce counter-terms of six dimension,likeT r [(D µF νρ)2],T r [(D µF µν)2]and T r [F νµF ρνF µρ].These terms do not affect βfunction of gauge coupling constant.So,in later calculations,we will omit the logarithmic di-vergencies and only pay our attention to the quadratical divergencies.For convenience,the gravitational contri-butions are labeled with a superscript κ.∆βκ=g µ∂2δκ2−δκ1)(11)It can be shown that at one-loop level,gravity will contribute to two and three point functions from five di-agrams (see Fig.1).The first two diagrams are corre-sponding to two point function,and the other three dia-grams are for three point function.One can easily show that Fig.1(c)has no quadratical divergencies and only logarithmic divergences are left.For Fig 1(d),as one of the end of graviton propagator could be attached to any gauge external leg,there are two similar contributions which can be obtained with a cycle for the momenta and their corresponding Lorentz index.In harmonic gauge,the two point functions from Fig.1(a)and Fig.1(b)are found in terms of ILIs to beT (a )µν=2κ2dx Q µνI 2+q 2(3x 2−x )I 0+q µq ρI νρ2+q νq ρI µρ2−g µνq ρq σI ρσ2−q 2I µν2(M 2q )T (b )µν=−3κ2Q µνI 2(0)(12)and the three point functions from Fig.1(d)and Fig.1(e)are found,when keeping only the quadratically divergent terms,to beT (d )µνρ=igκ2−V µνρqkpI 2(0)+dx g µνq σI ρσ2−g νρq σI µσ2+q ρI µν2−q µI νρ2 (M 2q )+g νρk σI µσ2−g ρµk σI νσ2+k µI νρ2−k νI ρµ2 (M 2k )+g ρµp σI νσ2−g µνp σI ρσ2+p νI ρµ2−p ρI µν2 (M 2p )T (e )µνρ=3igκ2V µνρqkp I 2(0)(13)with M 2q =x (x −1)q 2.Contraction is performed by using FeynCalc package[15].Now we shall apply the different regularization schemes to the divergent ILIs.In cut-offregularization,when keeping only quadratical divergent terms,one hasI Rµν2=116π2g µνΛ2(14)The resulting two and three point functions areT (a +b )µνcutoff ≡T (a )µνcutoff +T (b )µνcutoff ≈2Q µνκ2dx 116π2Λ2+ i 2i 2g µνI R 2,the two and three point functions arefound to be T (a +b )µνDR≈4κ2Q µνdxI R2(M 2q ),(17)T (d +e )µνρDR =2igκ2dx (g µνq ρ−q µg νρ)I R2(M 2q )(18)+(g νρk µ−k νg ρµ)I R 2(M 2k )+(g ρµp ν−p ρg µν)I R 2(M 2p )where the regularized quadratic divergence in dimen-sional regularization behaves as the logarithmic oneI R2(M2q)|DR=−−iε−γE+1+O(ε)](19) So far,we have shown that in both cut-offregulariza-tion and dimensional regularization,there are no gravita-tional corrections to the gaugeβfunction,which agrees with the ones yielded in[7,8,9].We now make a calculation by using loop regulariza-tion.With the consistency condition I R2µν=12I R2(0)+2I R2(M2q)+q2(3x2−x)I R0(M2q)(20)T(d+e)µνρLR=2igκ2 dx 116π2 M2c−µ2s[ln M2c16π2 M2c−µ2s[ln M2c16π2 ln M2cg2F aµνA aν=0,we then obtain the sameβfunction correction as eq.(24)which has the same sign as the one in[6].Conclusions:we have investigated the gravitational contributions to the running of gauge couplings by adopt-ing different regularization schemes.From the above ex-plicit calculations,it is not difficult to see that the dif-ferent conclusions resulted from different regularization schemes mainly arise from the treatment for the quadrat-ical divergent integrals.In fact,as long as taking the consistency condition for the regularized quadratic di-vergent ILIs that I R2µν=1。

a r X i v :h e p -l a t /0010003v 1 3 O c t 20001Theories with global gauge anomalies on the latticeP.Mitra a∗aSaha Institute of Nuclear Physics,Block AF,Bidhannagar,Calcutta 700064,INDIAA global anomaly in a chiral gauge theory manifests itself in different ways in the continuum and on the lattice.In the continuum case,functional integration of the fermion determinant over the whole space of gauge fields yields zero.In the case of the lattice,it is not even possible to define a fermion measure over the whole space of gauge configurations.However,this is not necessary,and as in the continuum,a reduced functional integral is sufficient for the existence of the theory.Presented at Lattice 2000,Bangalorehep-lat/00100031.IntroductionAnomalies are of two different types.Local or divergence anomalies have been known since 1969[1]:classically conserved symmetry currents cease to be conserved after quantization if there are anomalies of this kind.For example,the the-oryL =¯ψ(i∂/−eA /)ψ−116π2F µνF µν=0.(2)If an anomalous current is associated with a gauged symmetry ,it leads to an apparent problem in quantization because the equations of motion of the gauge fields require the current to be con-served.A treatment of such a theory just like a usual gauge theory shows an inconsistency.This problem can be sorted out by paying proper at-tention to phase space constraints,as suggested by [2].The anomaly itself can be made to vanish in a sense by going to the constrained subman-ifold of classical phase space.However,theories with anomalous gauge currents are to be distin-guished from theories with nonanomalous gauge currents.If a theory is nonanomalous ,it pos-sesses gauge freedom,and is describable in any22.Functional integral in the continuum The full partition function of the gauge theory with fermions may be written asZ= D AZ[A],(3) Z[A]≡e−S eff= D¯ψDψe−S(ψ,¯ψ,A)(4)An anomaly-free theory has Z[A]gauge invariant. If there is a gauge anomaly,Z[A]varies under gauge transformations of A:Z[A g]=e iα(A,g−1)Z[A],(5) whereαmay be regarded as an integral represen-tation of the anomaly.It obeys some consistency conditions(mod2π):α(A,g−12g−11)=α(A,g−11)+α(A g1,g−12)α(A,g−1)=−α(A g,g).(6) The case becomes one of a global anomaly ifαis independent of A,and vanishes for g connected to the identity but not for some g which cannot be continuously connected to the identity.A-independence implies an abelian representation satisfyingα(g2g1)=α(g1)+α(g2).(7) In the SU(2)case,the two components of the gauge group manifest themselves in two possible values of the phase:e iα=±1.In an anomaly-free theory,the partition func-tion factorizes into the volume of the gauge group and the gauge-fixed partition function:Z= D AZ[A]= D AZ[A] D gδ(f(A g))∆f(A)= D g D AZ[A g−1]δ(f(A))∆f(A) = D g D AZ[A]δ(f(A))∆f(A)=( D g)Z f(8)This is the standard Faddeev-Popov argument. Here,δ(f)represents a gauge-fixing operation and∆f is the corresponding Faddeev-Popov de-terminant.This decoupling of gauge degrees of freedom does not occur if a local anomaly is present.For a global anomaly however,the partition function factorizes again:Z= D ge−iα(g) D AZ[A]δ(f(A))∆f(A)(9)As the phase factors form a representation of the gauge group,D ge−iα(g)= D(gh)e−iα(gh)=e−iα(h) D ge−iα(g)(10)where h stands for afixed gauge transformation. If h is not connected to the identity,e−iα(h)=1, and consequently D ge−iα(g)=0,which in turn means that Z=0.Does this mean that the the-ory cannot be defined?Let us look at expectation values of gauge invariant operators.D AZ[A]OD ge−iα(g) D AZ[A]δ(f(A))∆f(A)(11) The expression on the right is of the form0D AZ[A]δ(f(A))∆f(A).(12)The right hand side is precisely what one gets in the canonical approach to quantization where gauge degrees of freedom are removed byfixing the gauge at the classical level and only physi-cal degrees of freedom enter the functional inte-gral.The Faddeev-Popov determinant arises in the canonical approach as the determinant of the matrix of Poisson brackets of what may be called the”second class constraints”,i.e.,the Gauss law3operator and the gaugefixing condition f,which is of course introduced by hand and not really a constraint of the theory.There are both ordinary fields and conjugate momenta,but the latter are easily integrated over.The point is that the full functional integral is not needed in the canonical approach and there is no harm if it vanishes!A trace is left behind by the global anomaly. One may imagine a classification of the gauge-fixing functions f where f,f′are said to belong to the same class if there exists a gauge transfor-mation connected to the identity to go from a con-figuration with f=0to one with f′=0.Then Z f=Z f′.More generally,when such a transfor-mation is not connected to the identity,Z f=e−iα(g0)Z f′,(13) where g0is determined by f,f′.These factors e−iα(g0)occurring in partition functions cancel out in expectation values of gauge invariant oper-ators,so that Green functions of gauge invariant operators are fully gauge independent[4]. There is an assumption in all this:that there is a possibility offixing the gauge.A general the-orem[6]asserts that gauges cannot befixed in a smooth way.For the construction of functional integrals,however,it is sufficient to have piece-wise smooth gauges.It should also be remem-bered that these questions arise even for theories without disconnected gauge groups and are not specific to the context of global anomalies.ttice formulationOn going to the lattice,one starts to use group-valued variables associated with links instead of A defined at points of the continuum.The topol-ogy also changes:the gauge group becomes con-nected on the lattice:it becomes possible to go to any gauge transformation from the identity in a continuous manner.Thus there are no large gauge transformations any more.Does it mean that there is no global anomaly on the lattice? The issue is complicated because chiral symme-try is not straightforward here.Chiral symmetry on the lattice has begun to make more sense in the last few years thanks to the Ginsparg-Wilson relation imposed on D,the euclidean lattice Dirac operator:γ5D+Dγ5=aDγ5D,(14) where a is the lattice spacing.An analogue ofγ5 appears from the above relation:γ5D=−DΓ5,Γ5≡γ5(1−aD).(15) It satisfies(Γ5)2=1,(Γ5)†=Γ5,(16) and can be used to define left-handed projection: P−ψ≡12[1+γ5]=¯ψ.(17) In this way of defining chiral projections,P−,but not P+,depends on the gaugefield configuration. Nontriviality of chirality on the lattice stems from this P−.A fermion measure is defined by specifying a basis of lattice Diracfields v j(x)satisfyingP−v j=v j,(v j,v k)=δjk.(18) One has to integrate over Grassmann-valued ex-pansion coefficients inψ(x)= j a j v j(x).(19)Expansion coefficients also come from the expan-sion of¯ψin terms of¯v j satisfying¯v j P+=¯v j, but these are as usual,i.e.,do not involve gauge fields.Questions of locality and integrability arise be-cause of the gaugefield dependence in P−.Ab-sence of a local anomaly appears to be sufficient to ensure locality[7].Global anomalies are man-ifested as a lack of integrability.Consider,following[5],a closed path in the SU(2)gauge configuration space,with the param-eter t running from0to1.Definef(t)=det[1−P++P+D(t)Q t D(0)†],(20) with D(t)the Dirac operator corresponding to gaugefields at parameter value t,and Q t the uni-tary transport operator for P−(t)defined by∂t Q t=[∂t P−(t),P−(t)]Q t,Q0=1.(21)4Then f(t)is real,positive and satisfiesf(1)=T f(0).(22)HereT=det[1−P−(0)+P−(0)Q1]=±1(23)depending on the topology of the considered path in the gauge configuration space.f changes sign an even or odd number of times along path de-pending on T and while det D(t)is related to f2, det D(t)det D(0)†=f2(t),(24)the chiral fermion determinant det Dχ(t)behaves like f:det Dχ(t)det Dχ(0)†=f(t)W(t)−1, (Dχ)ij≡a4 x¯v i(x)Dv j(x).(25) Here W(t)is a phase factor arising from the gauge field dependence of v j.It is a lattice artifact and may be taken to reduce to unity near the contin-uum limit.Then det Dχchanges sign,i.e.,fails to return to its starting value after transportation along a closed path if the path hasT=−1.(26)Such paths have been shown to exist in the SU(2) theory.A part of such a path lies along a gauge orbit,and a part is non-gauge.2Thus det Dχis multivalued,implying that the fermion measure is not well defined,and hence the functional integral does not make sense.This is roughly similar to the continuum.The Dirac operator is gauge-invariant and its determinant and f can change only on non-gauge portions of the closed path.So the problem of sign change of f occurs once again in non-gauge paths connect-ing gauge-related configurations.However,in the continuum,the sign change occurs between con-figurations which can be connected only by a non-gauge path.On the lattice,the sign change oc-curs when configurations are connected by a non-gauge path,though a connection is also possible。

电影黑洞的英语作文In the realm of science fiction, black holes have long captured the imagination of filmmakers and audiences alike. These cosmic phenomena, characterized by their immense gravitational pull and the mystery surrounding what lies within, have been the subject of numerous films. Theportrayal of black holes in cinema often serves as a metaphor for the unknown, the infinite, and the unexplored.One of the earliest and most iconic representations of a black hole can be found in the 1979 film "The Black Hole." This Disney production was a pioneer in its use of special effects to depict the gravitational anomalies and the eerie silence that surrounds these cosmic bodies. The film's black hole was portrayed as a gateway to another dimension, sparking the curiosity of viewers about the possibilitiesthat lie beyond our known universe.In more recent years, Christopher Nolan's "Interstellar" (2014) took a more scientifically accurate approach to black holes. The film featured a black hole named Gargantua, which was designed in consultation with physicist Kip Thorne. The depiction of Gargantua showcased the effects of gravitational lensing and the distortion of time, providing viewers with a more grounded and scientifically plausible understanding of black holes.The narrative of "Interstellar" revolves around a group ofexplorers who venture through a wormhole near Saturn insearch of a new home for humanity. The film's exploration of love, time, and the human spirit against the backdrop of a black hole adds a layer of emotional depth to the scientific concepts presented.Another notable film that explores the concept of black holes is "Event Horizon" (1997). This science fiction horror film tells the story of a spaceship that disappears into a black hole and returns with a malevolent presence on board. While less scientifically accurate, "Event Horizon" uses the black hole as a plot device to delve into themes of fear, isolation, and the unknown.The portrayal of black holes in cinema has evolved over time, reflecting both the advancements in our understanding ofthese celestial bodies and the creative imaginations of filmmakers. Whether used as a metaphor for the unknown or asa backdrop for exploring the human condition, black holes continue to captivate audiences and inspire filmmakers topush the boundaries of storytelling.。

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

英语作文星际穿越电影Title: Exploring the Phenomenon of Interstellar Travel through Film。

Interstellar travel has long been a subject of fascination, sparking imagination and inspiring scientific inquiry. In the realm of cinema, this fascination has been vividly portrayed through various films, with one notable example being Christopher Nolan's "Interstellar". This movie delves deep into the possibilities and challenges of interstellar travel, weaving together elements of science fiction and emotional drama to create a thought-provoking narrative.Firstly, "Interstellar" captivates audiences with its portrayal of the vastness and beauty of space. Through stunning visual effects and cinematography, viewers are transported to distant galaxies and awe-inspiring celestial landscapes. This portrayal not only serves to entertain but also sparks contemplation about the sheer scale of theuniverse and the potential for exploration beyond our own solar system.Moreover, the film explores the scientific concepts underlying interstellar travel, albeit with a degree of artistic license. Concepts such as wormholes, time dilation, and gravitational anomalies are depicted in a visually compelling manner, prompting viewers to ponder thefeasibility of such phenomena and their implications for space exploration. While some aspects may stretchscientific credibility, the film nonetheless stimulates curiosity and encourages audiences to delve deeper into the mysteries of the cosmos.One of the most intriguing aspects of "Interstellar" is its exploration of the human experience in the face of extreme adversity. The characters grapple with profound questions of love, sacrifice, and the survival of the human species. Their emotional journey adds depth to the film, inviting viewers to reflect on the resilience of the human spirit and the bonds that connect us across space and time.Furthermore, "Interstellar" raises thought-provoking questions about the future of humanity and our place in the universe. As we confront environmental challenges and contemplate the possibility of life beyond Earth, the film serves as a poignant reminder of the importance of exploration and discovery. It challenges us to consider the legacy we leave for future generations and the role of science and technology in shaping our destiny.In conclusion, "Interstellar" offers a compelling exploration of the phenomenon of interstellar travel through the lens of cinema. Its breathtaking visuals, scientific concepts, and emotional depth combine to create a memorable viewing experience that sparks curiosity and contemplation. While it may not provide definitive answers, the film encourages audiences to ponder the possibilities of the cosmos and our place within it. As we continue to push the boundaries of scientific knowledge, films like "Interstellar" serve as both entertainment and inspiration, reminding us of the boundless potential of human exploration.。