An Infinite Dimensional Symmetry Algebra in String Theory

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

我奇怪的想法英文作文The Curious Mind: A Journey Through Unusual Thoughts.In the vast expanse of the universe, our minds are tiny islands floating on a sea of infinity. They are the repositories of our thoughts, dreams, and imaginations, and sometimes, they are the birthplaces of strange and unusual ideas. These ideas, often labeled as "weird" or "strange" by society, are actually the most fascinating aspects of our existence. They push the boundaries of our understanding, challenge our perceptions, and force us to question the world we know.For me, one such strange idea has always fascinated me: the concept of parallel universes. The idea that there could be an infinite number of worlds, each with its own laws of physics, history, and culture, is mind-boggling. What if, somewhere out there, there is a universe where gravity works in reverse, or where the sun shines at night? Or perhaps a universe where history unfolded differently,and the outcomes of major events were entirely different?The concept of parallel universes is not just a figment of my imagination; it has been explored by physicists, philosophers, and writers alike. The idea gained popularity in the 20th century with the development of quantum physics, which suggested that the universe might be made up of multiple realities that coexist simultaneously. This theory, known as the Many-Worlds Interpretation, proposed thatevery possible outcome of a quantum event occurs in a separate universe.While the scientific community is still debating the validity of this theory, it has sparked a wave ofcreativity among writers and artists. It has given us a platform to explore the limitless possibilities ofexistence and to imagine worlds that are entirely different from our own. Novels, movies, and TV shows have beeninspired by the concept of parallel universes, allowing usto escape the confines of our reality and immerse ourselves in exciting new worlds.Another strange idea that intrigues me is the concept of time travel. The idea that we could travel through time, visit the past or future, has fascinated humans for centuries. From the time-traveling heroes of sciencefiction novels to the philosophical debates about the nature of time, this concept has always captivated our imaginations.The possibility of time travel raises a number of fascinating questions. Could we change the course ofhistory by interfering with past events? Would we even be able to recognize the future if we saw it? And what wouldit mean to travel through time and find ourselves in a world that is entirely different from the one we left?These are questions that science has yet to answer, but they are questions that continue to inspire us to push the boundaries of our understanding. The concept of time travel may never become a reality, but it remains a powerful tool for exploring our understanding of the universe and our place within it.In conclusion, strange ideas are not just figments of our imaginations; they are windows to a world beyond our comprehension. They challenge our perceptions, push the boundaries of our understanding, and inspire us to question everything we know. Whether it's the concept of parallel universes or the possibility of time travel, these ideas force us to reevaluate our understanding of the world and our place within it. As we continue to explore the vast expanse of the universe and the infinite possibilities of our minds, these strange ideas will continue to guide us on our journey through existence.。

无穷维hamilton系统的kam定理无穷维Hamilton系统的KAM定理是一个重要的数学定理，在确定性系统研究中有着非常重要的地位。

它给出了无穷维相空间中保持固定动能和固定作用量的区域的存在性，并且表明存在一个稳定奇点附近的不变曲面，上面的动力学行为可以用有限维动力学来描述。

该定理的证明是非常复杂和技术性的，需要利用许多高级数学工具和技巧。

下面将一步一步地介绍无穷维Hamilton系统的KAM 定理。

第一步是介绍Hamilton系统的基本概念和背景知识。

Hamilton系统是由Hamilton函数和Hamilton方程组组成的动力学系统。

在无穷维情况下，Hamilton函数和Hamilton方程组也需要一些特定的结构和条件。

其中一个重要的条件是哈密顿函数的二阶导数在某个拓扑拓扑空间中是有界的。

第二步是介绍KAM定理的主要内容和结果。

KAM定理的一个关键结论是存在一组正则变换，将原系统变换到一个新的坐标系上，使得变换后的Hamilton函数保持固定的动能和固定的作用量。

同时，KAM定理还表明，在动能和作用量给定的情况下，原系统的相空间中存在着固定动能和固定作用量的区域，使得该区域上的动力学行为可以用有限维动力学来描述。

第三步是介绍KAM定理的证明思路和主要技术。

KAM定理的证明主要依赖于对无穷维流形的几何性质和分析性质的研究。

其中一个重要的工具是Kolmogorov-Arnold-Moser(KAM)定理和其相应的延拓。

这些定理描述了一类特殊的可积系统的长期行为和近似行为，并提供了证明KAM定理的基本框架。

此外，还需要使用多项式展开和Baire范数等技巧，来处理无穷维情况下的一些技术性问题。

第四步是介绍KAM定理的具体证明过程。

由于篇幅的限制，无法详细介绍整个证明过程，但可以大致说明一下证明的主要思路。

首先，通过构造一系列连续且可微分的正则变换，将原系统变换到一个新的坐标系上。

然后，通过一系列逼近过程，逐步将动力学图像的细节压缩到越来越小的区域中。

斯蒂芬波尔茨曼定律数学表达式哎呀，小伙伴们，今天咱们来唠唠斯蒂芬波尔茨曼定律呀。

这个定律在物理学里可是相当重要的呢。

它主要是关于热辐射的一个定律哦。

它的数学表达式呢，就像是一把神奇的钥匙，能打开很多关于热辐射相关问题的大门。

它的表达式啊，和黑体辐射有着密切的关系。

黑体呢，就是那种能吸收所有照射到它上面的辐射而无反射的理想物体。

这个定律告诉我们，黑体的辐射力是和它的绝对温度的四次方成正比的。

具体的表达式就是E = σT⁴。

这里面的E就是黑体的辐射力，单位是瓦特每平方米（W/m²），T就是黑体的绝对温度，单位是开尔文（K），而σ呢，就是斯蒂芬 - 波尔茨曼常数啦，这个常数的值大概是5.67×10⁻⁸ W/(m²·K⁴)。

这个定律在很多实际的场景里都有用到呢。

比如说在研究恒星的辐射能量的时候，恒星就可以近似看成是一个黑体，通过这个定律就能大概算出它的辐射能量。

还有在工业上的高温炉，也可以用这个定律来研究它的热辐射情况。

而且这个定律在大气科学里也有它的身影。

大气里的很多热量交换过程，热辐射是很重要的一部分，这个时候斯蒂芬波尔茨曼定律就能帮助我们更好地理解热量是怎么在大气里传递的。

再比如说在建筑的热学设计方面，如果我们要考虑一个建筑表面的热量吸收和散发，这个定律也能给我们提供很好的理论基础呢。

我们可以根据这个定律来选择合适的建筑材料，让建筑在夏天不会吸收太多热量变得很热，在冬天又能很好地保持室内的热量。

在航天领域也离不开它哦。

卫星在太空中会受到太阳的热辐射，通过这个定律就可以计算出卫星表面受到的辐射能量，从而设计合适的散热或者隔热措施，确保卫星的正常运行。

所以说呀，斯蒂芬波尔茨曼定律虽然看起来就是一个简单的数学表达式，但它的作用可真的是超级大呢。

它就像一个小齿轮，在很多大的科学和工程的机器里，都起着不可或缺的作用。

《非线性发展方程的无穷维Hamilton方法》篇一一、引言非线性发展方程是数学物理领域中一类重要的研究对象，它们广泛存在于各种自然现象和社会现象中。

无穷维Hamilton方法是解决这类问题的一种有效方法，它通过引入无穷维的相空间和辛结构，将非线性发展方程转化为Hamilton系统，从而便于分析和求解。

本文将介绍非线性发展方程的无穷维Hamilton方法的基本思想、基本步骤和应用。

二、无穷维Hamilton方法的基本思想无穷维Hamilton方法是一种将非线性发展方程转化为Hamilton系统的数学方法。

其基本思想是将非线性发展方程的解空间视为一个无穷维的相空间，通过引入无穷维的辛结构，将原方程转化为一个Hamilton系统。

这个Hamilton系统具有明确的辛结构，使得我们可以利用辛几何的理论和方法来分析和求解原方程。

三、无穷维Hamilton方法的基本步骤1. 定义无穷维相空间和辛结构。

根据非线性发展方程的特点，选择合适的相空间和辛结构，使得原方程可以转化为一个Hamilton系统。

2. 建立Hamilton算子。

根据相空间和辛结构，建立Hamilton 算子，它是相空间中向量场的Frobenius-Poisson括号算子。

3. 求解Hamilton系统的解。

利用辛几何的理论和方法，求解Hamilton系统的解，从而得到原非线性发展方程的解。

四、无穷维Hamilton方法的应用无穷维Hamilton方法广泛应用于各种非线性发展方程的求解和分析中。

例如，在流体力学、量子力学、光学等领域中，许多重要的物理现象都可以用非线性发展方程来描述。

通过应用无穷维Hamilton方法，可以有效地分析和求解这些非线性发展方程，从而更好地理解这些物理现象的本质和规律。

五、实例分析以KdV（Korteweg-de Vries）方程为例，介绍无穷维Hamilton方法的具体应用。

KdV方程是一种重要的非线性发展方程，广泛应用于水波、等离子体等物理现象的研究中。

(0,2) 插值||(0,2) interpolation0#||zero-sharp; 读作零井或零开。

0+||zero-dagger; 读作零正。

1-因子||1-factor3-流形||3-manifold; 又称“三维流形”。

AIC准则||AIC criterion, Akaike information criterionAp 权||Ap-weightA稳定性||A-stability, absolute stabilityA最优设计||A-optimal designBCH 码||BCH code, Bose-Chaudhuri-Hocquenghem codeBIC准则||BIC criterion, Bayesian modification of the AICBMOA函数||analytic function of bounded mean oscillation; 全称“有界平均振动解析函数”。

BMO鞅||BMO martingaleBSD猜想||Birch and Swinnerton-Dyer conjecture; 全称“伯奇与斯温纳顿－戴尔猜想”。

B样条||B-splineC*代数||C*-algebra; 读作“C星代数”。

C0 类函数||function of class C0; 又称“连续函数类”。

CA T准则||CAT criterion, criterion for autoregressiveCM域||CM fieldCN 群||CN-groupCW 复形的同调||homology of CW complexCW复形||CW complexCW复形的同伦群||homotopy group of CW complexesCW剖分||CW decompositionCn 类函数||function of class Cn; 又称“n次连续可微函数类”。

Cp统计量||Cp-statisticC。

数学上的高阶无穷小英文Mathematics of Higher-Order InfinitesimalsThe realm of mathematics is a vast and intricate landscape, where the exploration of concepts often leads to unexpected and profound discoveries. One such intriguing area within the mathematical universe is the study of higher-order infinitesimals. These infinitesimal quantities, which are smaller than any finite number yet larger than zero, have captivated the minds of mathematicians and philosophers alike, unveiling a deeper understanding of the nature of infinity and the fundamental principles that govern the universe.At the heart of this inquiry lies the notion of differentiation, a foundational concept in calculus that allows us to analyze the rate of change of a function at a specific point. Traditionally, the study of differentiation has been limited to first-order infinitesimals, where the derivative represents the slope of the tangent line to a curve at a given point. However, as mathematicians delved deeper into this field, they recognized the potential for exploring higher-order derivatives and the corresponding infinitesimal quantities.The concept of higher-order infinitesimals emerged from the work ofrenowned mathematicians such as Gottfried Wilhelm Leibniz, Isaac Newton, and Augustin-Louis Cauchy, among others. These pioneers of calculus recognized that the study of higher-order derivatives could unlock a deeper understanding of the behavior of functions and the underlying mathematical structures.One of the key insights that drove the exploration of higher-order infinitesimals was the realization that the traditional approach to differentiation, which relied on the concept of limits, could be expanded to encompass more complex scenarios. By considering the behavior of functions as they approached a specific point, mathematicians were able to define higher-order derivatives, each representing a distinct rate of change.The second-order derivative, for instance, describes the rate of change of the rate of change, providing valuable information about the curvature of a function. Similarly, third-order derivatives shed light on the rate of change of the curvature, and so on, leading to a hierarchy of higher-order infinitesimals.The significance of higher-order infinitesimals extends far beyond the realm of pure mathematics. In the fields of physics and engineering, the study of these infinitesimal quantities has proven invaluable in modeling complex systems and predicting their behavior. For example, in the analysis of mechanical vibrations,higher-order derivatives are used to capture the nuances of oscillatory motion, enabling more accurate predictions and the design of more efficient systems.Furthermore, the exploration of higher-order infinitesimals has opened up new avenues of research in areas such as optimization theory, where the study of higher-order derivatives can lead to more efficient algorithms for solving complex optimization problems. In the realm of quantum mechanics, the application of higher-order infinitesimals has contributed to a deeper understanding of the behavior of subatomic particles and the fundamental forces that govern the universe.Despite the profound implications of higher-order infinitesimals, their study is not without its challenges. The conceptual and technical complexities involved in working with these infinitesimal quantities have often posed significant barriers to their widespread adoption and understanding. Mathematicians and scientists have had to grapple with issues such as the convergence of infinite series, the properties of higher-order derivatives, and the interpretation of these quantities in the context of various mathematical and physical theories.Nevertheless, the pursuit of knowledge and the desire to unravel the mysteries of the universe have driven mathematicians and scientiststo overcome these challenges, continuously expanding the boundaries of our understanding. The study of higher-order infinitesimals has become a testament to the power of human ingenuity and the relentless pursuit of deeper insights into the fundamental nature of our world.As we delve deeper into the realm of higher-order infinitesimals, we are confronted with the realization that the universe is not merely a collection of discrete entities, but rather a tapestry of interconnected processes and relationships. The study of these infinitesimal quantities has the potential to reveal the underlying patterns and structures that govern the behavior of complex systems, from the subatomic to the cosmic scale.In conclusion, the mathematics of higher-order infinitesimals represents a captivating and ever-evolving field of study, one that continues to challenge and inspire mathematicians, scientists, and thinkers alike. By exploring these infinitesimal quantities, we gain a deeper understanding of the fundamental principles that govern the universe, opening up new avenues for discovery and the advancement of human knowledge. As we continue to push the boundaries of our understanding, the study of higher-order infinitesimals promises to uncover even more profound insights, shaping the future of our scientific and technological landscape.。

a r X i v :m a t h -p h /0411040v 1 10 N o v 2004Abstract .—Building upon Dyson’s fundamental 1962article known in random-matrix theory as the threefold way ,we classify disordered fermion systems with quadratic Hamil-tonians by their unitary and antiunitary symmetries.Important physical examples are af-forded by noninteracting quasiparticles in disordered metals and superconductors,and by relativistic fermions in random gauge field backgrounds.The primary data of the classification are a Nambu space of fermionic field opera-tors which carry a representation of some symmetry group.Our approach is to eliminate all of the unitary symmetries from the picture by transferring to an irreducible block of equivariant homomorphisms.After reduction,the block data specifying a linear space of symmetry-compatible Hamiltonians consist of a basic vector space V ,a space of en-domorphisms in End (V ⊕V ∗),a bilinear form on V ⊕V ∗which is either symmetric or alternating,and one or two antiunitary symmetries that may mix V with V ∗.Every such set of block data is shown to determine an irreducible classical compact symmetric space.Conversely,every irreducible classical compact symmetric space occurs in this way.This proves the correspondence between symmetry classes and symmetric spaces con-jectured some time ago.Keywords :disordered electron systems,random Dirac fermions,quantum chaos;repre-sentation theory,symmetric spaces1.IntroductionIn a famous and influential paper published in 1962(“The threefold way:algebraic structure of symmetry groups and ensembles in quantum mechanics”[D ]),Freeman J.Dyson classified matrix ensembles by a scheme that became fundamental to several areas of theoretical physics,including the statistical theory of complex many-body systems,mesoscopic physics,disordered electron systems,and the area of quantum chaos.Being set in the context of standard quantum mechanics,Dyson’s classification asserted that “the most general matrix ensemble,defined with a symmetry group that may be completely arbitrary,reduces to a direct product of independent irreducible2P.HEINZNER,A.HUCKLEBERRY&M.R.ZIRNBAUERensembles each of which belongs to one of three known types.”These three ensembles, or rather their underlying matrix spaces,are nowadays known as the Wigner-Dyson symmetry classes of orthogonal,unitary,and symplectic symmetry.Over the last ten years,various matrix spaces beyond Dyson’s threefold way have come to the fore in random-matrix physics and mathematics.On the physics side,such spaces arise in problems of disordered or chaotic fermions;among these are the Eu-clidean Dirac operator in a stochastic gaugefield background[V2],and quasiparticle excitations in disordered superconductors or metals in proximity to a superconductor [A2].In the mathematical research area of number theory,the study of statistical cor-relations in the values of the Riemann zeta function,and more generally of families of L-functions,has prompted some of the same extensions[K].A brief account of why new structures emerge on the physics side is as follows. When Diracfirst wrote down his famous equation in1928,he assumed that he was writing an equation for the wavefunction of the ter,because of the insta-bility caused by negative-energy solutions,the Dirac equation was reinterpreted(via second quantization)as an equation for the fermionicfield operators of a quantum field theory.A similar change of viewpoint is carried out in reverse in the Hartree-Fock-Bogoliubov mean-field description of quasiparticle excitations in superconduc-tors.There,one starts from the equations of motion for linear superpositions of the electron creation and annihilation operators,and reinterprets them as a unitary quan-tum dynamics for what might be called the quasiparticle‘wavefunction’.In both cases–the Dirac equation and the quasiparticle dynamics of a superconduc-tor–there enters a structure not present in the standard quantum mechanics underlying Dyson’s classification:the fermionicfield operators are subject to a set of conditions known as the canonical anticommutation relations,and these are preserved by the quantum dynamics.Therefore,whenever second quantization is undone(assuming it can be undone)to return fromfield operators to wavefunctions,the wavefunction dy-namics is required to preserve some extra structure.This puts a linear constraint on the allowed Hamiltonians.A good viewpoint to adopt is to attribute the extra invariant structure to the Hilbert space,thereby turning it into a Nambu space.It was conjectured some time ago[A2]that extending Dyson’s classification to the Nambu space setting,the relevant objects one is led to consider are large families of symmetric spaces of compact type.Past understanding of the systematic nature of the extended classification scheme relied on the mapping of disordered fermion problems tofield theories with supersymmetric target spaces[Z]in combination with renormalization group ideas and the classification theory of Lie superalgebras.An extensive review of the mathematics and physics of symmetric spaces,covering the wide range from the basic definitions to various random-matrix applications,has recently been given in[C].That work,however,offers no answers to the question as to why symmetric spaces are relevant for symmetry classification,and under what assumptions the classification by symmetric spaces is complete.SYMMETRY CLASSES OF DISORDERED FERMIONS3 In the present paper,we get to the bottom of the subject and,using a minimal set of tools from linear algebra,give a rigorous answer to the classification problem for disor-dered fermions.The rest of this introduction gives an overview over the mathematical model to be studied and a statement of our main result.We begin with afinite-or infinite-dimensional Hilbert space V carrying a unitary representation of some compact Lie group G0–this is the group of unitary symmetries of the disordered fermion system.We emphasize that G0need not be connected;in fact,it might be just afinite group.Let W=V⊕V∗,called the Nambu space of fermionicfield operators,be equipped with the induced G0-representation.This means that V is equipped with the given representation,and g(f):=f◦g−1for f∈V∗,g∈G0.Let C:W→W be the C-antilinear involution determined by the Hermitian scalar product , V on V.In physics this operator is called particle-hole conjugation.Another canonical structure on W is the symmetric complex bilinear form b:W×W→C defined byb(v1+f1,v2+f2):=f1(v2)+f2(v1).It encodes the canonical anticommutation relations for fermions,and is related to the unitary structure , of W by b(w1,w2)= Cw1,w2 for all w1,w2∈W.It is assumed that G0is contained in a group G–the total symmetry group of the fermion system–which is acting on W by transformations that are either unitary or antiunitary.An element g∈G either stabilizes V or exchanges V and V∗.In the latter case we say that g∈G mixes,and in the former case we say that it is nonmixing. The group G is generated by G0and distinguished elements g T which act as anti-unitary operators T:W→W.These are referred to as distinguished‘time-reversal’symmetries,or T-symmetries for short.The squares of the g T lie in the center of the abstract group G;we therefore require that the antiunitary operators T representing them satisfy T2=±Id.The subgroup G0is defined as the set of all elements of G which are represented as unitary,nonmixing operators on W.If T and T1are distinguished time-reversal operators,then P:=T T1is a unitary sym-metry.P may be mixing or nonmixing.In the latter case,P is in G0.Therefore,modulo G0,there exist at most two different T-symmetries.If there are exactly two such sym-metries,we adopt the convention that T is mixing and T1is nonmixing.Furthermore, it is assumed that T and T1either commute or anticommute,i.e.,T1T=±T T1.As explained throughout this article,all of these situations are well motivated by physical considerations and examples.We note that time-reversal symmetry(and all other T-symmetries)of the disordered fermion system may also be broken;in this case T and T1are eliminated from the mathematical model and G0=G.Given W and the representation of G on it,the object of interest is the real vector space H of C-linear operators in End(W)that preserve the canonical structures b and , of W and commute with the G-action.Physically speaking,H is the space of ‘good’Hamiltonians:thefield operator dynamics generated by H∈H preserves both4P.HEINZNER,A.HUCKLEBERRY &M.R.ZIRNBAUERthe canonical anticommutation relations and the probability in Nambu space,and is compatible with the prescribed symmetry group G .When unitary symmetries are present,the space H decomposes by blocks associ-ated with isomorphism classes of G 0-subrepresentations occurring in W .To formal-ize this,recall that two unitary representations ρi :G 0→U (V i ),i =1,2,are equiv-alent if and only if there exists a unitary C -linear isomorphism ϕ:V 1→V 2so that ρ2(g )(ϕ(v ))=ϕ(ρ1(g )(v ))for all v ∈V 1and for all g ∈G 0.Let ˆG 0denote the space of equivalence classes of irreducible unitary representations of G 0.An element λ∈ˆG 0is called an isomorphism class for short.By standard facts (recall that every represen-tation of a compact group is completely reducible)the unitary G 0-representation on V decomposes as an orthogonal sum over isomorphism classes:V =⊕λV λ.The subspaces V λare called the G 0-isotypic components of V .Some of them may be zero.(Some of the isomorphism classes of G 0may just not be realized in V .)For simplicity suppose now that there is only one distinguished time-reversal sym-metry T ,and for any fixed λ∈ˆG0with V λ=0,consider the vector space T (V λ).If T is nonmixing,i.e.,T :V →V ,then T (V λ)⊂V must coincide with the isotypic component for the same or some other isomorphism class.(Since conjugation by g T is an automorphism of G 0,the decomposition into G 0-isotypic components is preserved by T .)If T is mixing,i.e.,T :V →V ∗,then T (V λ)=V ∗λ′,still with some λ′∈ˆG 0.Now define the block B λto be the smallest G -invariant space containing V λ⊕V ∗λ.Note that if we are in the situation of nonmixing and T (V λ)=V λ,thenB λ= V λ⊕T (V λ) ⊕ V λ⊕T (V λ) ∗.On the other hand,if we are in the situation of mixing and T (V λ)=V ∗λ,thenB λ= V λ⊕T (V ∗λ) ⊕ V ∗λ⊕T (V λ) .The block B λis halved if T (V λ)=V λresp.T (V λ)=V ∗λ.Note that if there are two distinguished T -symmetries,the above discussion is only slightly more complicated.In any case we now have the basic G -invariant blocks B λ.Because different blocks are built from representations of different isomorphism classes,the good Hamiltonians do not mix blocks.Thus every H ∈H is a direct sum over blocks,and the structure analysis of H can be carried out for each block B λseparately.If V λis infinite-dimensional,then to have good mathematical control we truncate to a finite-dimensional space V λ⊂V λand form the associated block B λ⊂W .The truncation is done in such a way that B λis a G -representation space and is Nambu.The goal now is to compute the space of Hermitian operators on B λwhich commute with the G -action and respect the canonical symmetric C -bilinear form b induced from that on V ⊕V ∗;such a space of operators realizes what is called a symmetry class .SYMMETRY CLASSES OF DISORDERED FERMIONS5 For this,certain spaces of G0-equivariant homomorphisms play an essential role, i.e.,linear maps S:V1→V2between G0-representation spaces which satisfyρ2(g)◦S=S◦ρ1(g)for all g∈G0,whereρi:G0→U(V i),i=1,2,are the respective representations.If it is clear which representations are at hand,we often simply write g◦S=S◦g or S=gSg−1.Thus we regard the space Hom G0(V1,V2)of equivariant homomorphisms as thespace of G0-fixed vectors in the space Hom(V1,V2)of all linear maps.If V1=V2=V,then these spaces are denoted by End G0(V)and End(V)respectively.Roughly speaking,there are two steps for computing the relevant spaces of Hermi-tian operators.First,the block Bλis replaced by an analogous block Hλof G0-equi-variant homomorphisms from afixed representation space Rλof isomorphism classλand/or its dual R∗λto Bλ.The space Hλcarries a canonical form(called either s or a)which is induced from b.As the notation indicates,although the original bilinear form on Bλis symmetric,this induced form is either symmetric or alternating.Change of parity occurs in the most interesting case when there is a G0-equivariantisomorphismψ:Rλ→R∗λ.In that case there exists a bilinear form Fψ:Rλ×Rλ→Cdefined by Fψ(r,t)=ψ(r)(t),which is either symmetric or alternating.In a certain sense the form b is a product of Fψand a canonical form on Hλ.Thus,if Fψis alternating,then the canonical form on Hλmust also be alternating.After transferring to the space Hλ,in addition to the canonical bilinear form s or a we have a unitary structure and conjugation by one or two distinguished time-reversal symmetries.Such a symmetry T may be mixing or not,and both T2=Id and T2=−Id are possible.The second main step of our work is to understand these various cases, each of which is directly related to a classical symmetric space of compact type.Such are given by a classical Lie algebra g which is either su n,usp2n,or so n(R).In the notation of symmetric spaces we have the following situation.Let g be the Lie algebra of antihermitian endomorphisms of Hλwhich are isometries(in the sense of Lie algebra elements)of the induced complex bilinear form b=s or b=a.This is of compact type,because it is the intersection of the Lie algebra of the unitary group of Hλand the complex Lie algebra of the group of isometries of b.Conjugation by the antiunitary mapping T defines an involutionθ:g→g.The good Hamiltonians(restricted to the reduced block Hλ)are the Hermitian oper-ators h∈i g such that at the level of group action the one-parameter groups e−i th satisfy T e−i th=e+i th T,i.e.,i h∈g must anticommute with T.Equivalently,if g=k⊕p is the decomposition of g intoθ-eigenspaces,the space of operators which is to be computed is the(−1)-eigenspace p.The space of good Hamiltonians restricted to Hλthen is i p. Since the appropriate action of the Lie group K(with Lie algebra k)on this space is just conjugation,one identifies i p with the tangent space g/k of an associated symmetric space G/K of compact type.It should be underlined that there is more than one symmetric space associated to a Cartan decomposition g=k⊕p.We are most interested in the one consisting of the6P.HEINZNER,A.HUCKLEBERRY&M.R.ZIRNBAUERphysical time-evolution operators e−i th;if G(not to be confused with the symmetry group G)is the semisimple and simply connected Lie group with Lie algebra g,this is given as the image of the compact symmetric space G/K under the Cartan embedding into G defined by g K→gθ(g)−1,whereθ:G→G is the induced group involution. The following mathematical result is a conseqence of the detailed classification work in Sects.3,4and5.Theorem1.1.—The symmetric spaces which occur under these assumptions are ir-reducible classical symmetric spaces g/k of compact type.Conversely,every irre-ducible classical symmetric space of compact type occurs in this way.We emphasize that here the notion symmetric space is appliedflexibly in the sense that depending on the circumstances it could mean either the infinitesimal model g/k or the Cartan-embedded compact symmetric space G/K.Theorem1.1settles the question of symmetry classes in disordered fermion systems; in fact every physics example is handled by one of the situations above.The paper is organized as follows.In Sect.2,starting from physical considerations we motivate and develop the model that serves as the basis for subsequent mathemati-cal work.Sect.3proves a number of results which are used to eliminate the group of unitary symmetries G0.The main work of classification is given in Sect.4and Sect.5. In Sect.4we handle the case where at most one distinguished time-reversal operator is present,and in Sect.5the case where there are two.There are numerous situations that must be considered,and in each case we precisely describe the symmetric space which occurs.Various examples taken from the physics literature are listed in Sect.6, illustrating the general classification theory.2.Disordered fermions with symmetries‘Fermions’is the physics name for the elementary particles which all matter is made of.The goal of the present article is to establish a symmetry classification of Hamilto-nians which are quadratic in the fermion creation and annihilation operators.To mo-tivate this restriction,note that any Hamiltonian for fermions at the fundamental level is of Dirac type;thus it is always quadratic in the fermion operators,albeit with time-dependent coefficients that are themselves operators.At the nonrelativistic or effective level,quadratic Hamiltonians arise in the Hartree-Fock mean-field approximation for metals and the Hartree-Fock-Bogoliubov approximation for superconductors.By the Landau-Fermi liquid principle,such mean-field or noninteracting Hamiltonians give an adequate description of physical reality at very low temperatures.In the present section,starting from a physical framework,we develop the appropri-ate model that will serve as the basis for the mathematical work done later on.Please be advised that disorder,though advertised in the title of the section and in the title of paper,will play no explicit role here.Nevertheless,disorder(and/or chaos)are theSYMMETRY CLASSES OF DISORDERED FERMIONS7 indispensable agents that must be present in order to remove specific and nongeneric features from the physical system and make a classification by basic symmetries mean-ingful.In other words,what we carry out in this paper is thefirst step of a two-step program.Thisfirst step is to identify in the total space of Hamiltonians some lin-ear subspaces that are relevant(in Dyson’s sense)from a symmetry perspective.The second step is to put probability measures on these spaces and work out the disor-der averages and statistical correlation functions of interest.It is this latter step that ultimately justifies thefirst one and thus determines the name of the game.2.1.The Nambu space model for fermions.—The starting point for our considera-tions is the formalism of second quantization.Its relevant aspects will now be reviewed so as to introduce the key physical notions as well as the proper mathematical language. Let i=1,2,bel an orthonormal set of quantum states for a single fermion. Second quantizing the many-fermion system means to associate with each i a pair of operators c†i and c i,which are called fermion creation and annihilation operators, respectively,and are related to each other by an operation of Hermitian conjugation †:c i→c†i.They are subject to the canonical anticommutation relationsc†i c†j+c†j c†i=0,c i c j+c j c i=0,c†i c j+c j c†i=δi j,(2.1) for all i,j.They act in a Fock space,i.e.,in a vector space with a distinguished vec-tor,called the‘vacuum’,which is annihilated by all of the operators c i(i=1,2,...). Applying n creation operators to the vacuum one gets a state vector for n fermions.A field operatorψis a linear combination of creation and annihilation operators,ψ=∑i v i c†i+f i c i ,with complex coefficients v i and f i.To put this in mathematical terms,let V be the complex Hilbert space of single-fermion states.(We do not worry here about complications due to the dimension of V being infiter rigorous work will be carried out in thefinite-dimensional setting.) Fock space then is the exterior algebra∧V=C⊕V⊕∧2V⊕...,with the vacuum being the one-dimensional subspace of constants.Creating a single fermion amounts to exterior multiplication by a vector v∈V and is denoted byε(v):∧n V→∧n+1V.To annihilate a fermion,one contracts with an element f of the dual space V∗.In other words,one applies the antiderivationι(f):∧n V→∧n−1V given byι(f)·1=0,ι(f)v=f(v),ι(f)(v1∧v2)=f(v1)v2−f(v2)v1,etc.In that mathematical framework the canonical anticommutation relations readε(v)ε(˜v)+ε(˜v)ε(v)=0,ι(f)ι(˜f)+ι(˜f)ι(f)=0,(2.2)ι(f)ε(v)+ε(v)ι(f)=f(v).8P.HEINZNER,A.HUCKLEBERRY&M.R.ZIRNBAUERThey can be viewed as the defining relations of an associative algebra,the so-called Clifford algebra C(W),which is generated by the vector space W:=V⊕V∗over C. This vector space W is sometimes referred to as Nambu space in physics.Since we only consider Hamiltonians that are quadratic in the creation and annihi-lation operators,we will be able to reduce the second-quantized formulation on∧V to standard single-particle quantum mechanics,albeit on the Nambu space W carrying some extra structure.Note that W is isomorphic to the space offield operatorsψ.On W=V⊕V∗there exists a canonical symmetric C-bilinear form b defined by b(v+f,˜v+˜f)=f(˜v)+˜f(v)=∑i(f i˜v i+˜f i v i).The significance of this bilinear form in the present context lies in the fact that it en-codes on W the canonical anticommutation relations(2.1),or(2.2).Indeed,we can view afield operatorψ=∑i(v i c†i+f i c i)either as a vectorψ=v+f∈V⊕V∗,or equivalently as a degree-one operatorψ=ε(v)+ι(f)in the Clifford algebra acting on∧V.Adopting the operator perspective,we get from(2.2)thatψ˜ψ+˜ψψ=f(˜v)+˜f(v)=∑i f i˜v i+˜f i v i .Switching to the vector perspective we have the same answer from b(ψ,˜ψ).Thusψ˜ψ+˜ψψ=b(ψ,˜ψ).Definition2.1.—In the Nambu space model for fermions one identifies the space offield operatorsψwith the complex vector space W=V⊕V∗equipped with its canonical unitary structure , and canonical symmetric complex bilinear form b. Remark.—Having already expounded the physical origin of the symmetric bilinear form b,let us now specify the canonical unitary structure of W.The complex vector space V,being isomorphic to the Hilbert space of single-particle states,comes with a Hermitian scalar product(or unitary structure) , V.Given , V define a C-antilinear bijection C:V→V∗byCv= v,· V,and extend this to an antilinear transformation C:W→W by the requirement C2=Id. Thus C|V∗=(C|V)−1.The operator C is called particle-hole conjugation in physics. Using C,transfer the unitary structure from V to V∗in the natural way:f,˜f V∗:=SYMMETRY CLASSES OF DISORDERED FERMIONS9Proof .—Given an orthonormal basis c †1,c 1,c †2,c 2,...this is immediate from C ∑i (v i c †i +f i c i )=∑i (¯v i c i +¯f i c †i )and the expressions for , and b in components.Returning to the physics way of telling the story,consider the most general Hamilto-nian H which is quadratic in the single-fermion creation and annihilation operators.Assuming H to be Hermitian,using the canonical anticommutation relations (2.1),and omitting an additive constant (which is of no consequence in physics)this has the form H =12∑i jB i j c †i c †j +¯B i j c j c i ,where A i j =¯Aji (from H =H †)and B i j =−B ji (from c i c j =−c j c i ).The Hamiltonians H act on the field operators ψby the commutator,ψ→[H ,ψ]≡H ψ−ψH ,and the time evolution is determined by the Heisenberg equation of motion,−i ¯h d ψ¯h A B −¯B−¯A.To recast all this in concise terms,we need some further mathematical background.Notwithstanding the fact that in practice we always consider the Fock space represen-tation C (W )→End (∧V )by w =v +f →ε(v )+ι(f ),it should be stated that the primary (or universal)definition of the Clifford algebra C (W )is as the associative algebra generated by W ⊕C with relationsw 1w 2+w 2w 1=b (w 1,w 2)×Id(w 1,w 2∈W ).(2.3)The Clifford algebra is graded byC (W )=C 0(W )⊕C 1(W )⊕C 2(W )⊕...,where C 0(W )≡C ,C 1(W )∼=W ,and C n (W )for n ≥2is the linear space of skew-symmetrized degree-n monomials in the elements of W .In particular,C 2(W )is the linear space of skew-symmetric quadratic monomials w 1w 2−w 2w 1(w 1,w 2∈W ).From the Clifford algebra perspective,a quadratic Hamiltonian H is viewed as an operator in the degree-two component C 2(W ).Let us therefore gather some standard facts about C 2(W ).First among these is that C 2(W )is a complex Lie algebra with10P.HEINZNER,A.HUCKLEBERRY&M.R.ZIRNBAUERthe commutator playing the role of the Lie bracket(an exposition of this fact for the case of a Clifford algebra over R is found in[B3];the complex case is no different). Second,in addition to acting on itself by the commutator,the Lie algebra C2(W) acts(still by the commutator)on all of the components C k(W)of degree k≥1of the Clifford algebra C(W).In particular,C2(W)acts on C1(W).Third,C2(W)turns out to be canonically isomorphic to the complex orthogonal Lie algebra so(W,b)which is associated with the vector space W=V⊕V∗and its canonical symmetric complex bilinear form b;this Lie algebra so(W,b)is defined to be the subspace of elements E∈End(W)satisfying the conditionb(Ew1,w2)+b(w1,Ew2)=0(for all w1,w2∈W).The canonical isomorphism C2(W)→so(W,b)is given by the commutator action of C2(W)on C1(W)∼=W,i.e.,by sending a∈C2(W)to[a,·]=E∈End(W);the latter indeed lies in so(W,b)as follows from the expression for b(Ew1,w2)+b(w1,Ew2) given by the canonical anticommutation relations(2.3),from the Jacobi identity [a,w1]w2+w2[a,w1]+w1[a,w2]+[a,w2]w1=[a,w1w2+w2w1],and from the fact that w1w2+w2w1lies in the center of the Clifford algebra.To describe so(W,b)explicitly,decompose the endomorphisms E∈End(V⊕V∗) into blocks asE= A B C D ,where A∈End(V),B∈Hom(V∗,V),C∈Hom(V,V∗)and D∈End(V∗).Let the adjoint(or transpose)of A∈End(V)be denoted by A t∈End(V∗),and call an element C in Hom(V,V∗)skew if C t=−C,i.e.,if(C v1)(v2)=−(C v2)(v1).Proposition2.3.—An endomorphism E= A B C D ∈End(V⊕V∗)lies in the com-plex orthogonal Lie algebra so(V⊕V∗,b)if and only if B,C are skew and D=−A t. Proof.—Considerfirst the case B=C=0,and let D=−A t.Thenb E(v+f),˜v+˜f =b(A v−A t f,˜v+˜f)=˜f(A v)−A t f(˜v)=A t˜f(v)−f(A˜v)=−b(v+f,A˜v−A t˜f)=−b v+f,E(˜v+˜f) .Using B t=−B and C t=−C,a similar calculation for the case A=0givesb E(v+f),˜v+˜f =b(B f+C v,˜v+˜f)=C v(˜v)+˜f(B f)=−f(B˜f)−C˜v(v)=−b(v+f,B˜f+C˜v)=−b v+f,E(˜v+˜f) . Since these two cases complement each other,we see that the stated conditions on E∈End(W)are sufficient in order for E to be in so(W,b).The calculation can equally well be read backwards;thus the conditions are both sufficient and necessary.SYMMETRY CLASSES OF DISORDERED FERMIONS11 Let us now make the connection to physics,where C(W)is represented on Fock space and the elements v+f=w∈W becomefield operatorsψ=ε(v)+ι(f).Fixing orthonormal bases c†1,c†2,...of V and c1,c2,...of V∗as before,we assign matrices with matrix elements A i j,B i j,C i j to the linear operators A,B,C.A straightforward computation using the canonical anticommutation relations then yields: Proposition2.4.—The inverse of the Lie algebra automorphism C2(W)→so(W,b) is the C-linear mapping given byA B C−A t →12∑i j(B i j c†i c†j+C i j c i c j).Now recall that C2(W)acts on the degree-one component C1(W)by the commuta-tor.By the isomorphisms C2(W)∼=so(W,b)and C1(W)∼=W,this action coincides with the fundamental representation of so(W,b)on its defining vector space W.In other words,taking the commutator of the Hamiltonian H∈C2(W)with afield op-eratorψ∈C1(W)yields the same answer as viewing H as an element of so(W,b), then applying H= A B C−A t to the vectorψ=v+f∈W byH·(v+f)=(A v+B f)+(C v−A t f),andfinally reinterpreting the result as afield operator in C1(W).The closure relation[C2(W),C1(W)]⊂C1(W)and the isomorphism C1(W)∼= W make it possible to reduce the dynamics offield operators to a dynamics on the Nambu space W.After reduction,as we have seen,the generators X∈End(V⊕V∗) of time evolutions of the physical system are of the special formiX=。