On covariant Poisson brackets in classical field theory

Transcription

1 On covarant Posson brackets n classcal feld theory Mchael Forger and Máro O. Salles Ctaton: Journal of Mathematcal Physcs 56, ); do: / Vew onlne: Vew Table of Contents: Publshed by the AIP Publshng Artcles you may be nterested n Classcal gauge theory n Rem J. Math. Phys. 52, ); / Hamltonan multvector felds and Posson forms n multsymplectc feld theory J. Math. Phys. 46, ); / The Günther s formalsm n classcal feld theory: momentum map and reducton J. Math. Phys. 45, ); / Lagrangan Hamltonan unfed formalsm for feld theory J. Math. Phys. 45, ); / k-cosymplectc manfolds and Lagrangan feld theores J. Math. Phys. 42, ); /

2 JOURNAL OF MATHEMATICAL PHYSICS 56, ) On covarant Posson brackets n classcal feld theory Mchael Forger 1 and Máro O. Salles 1,2 1 Insttuto de Matemátca e Estatístca, Unversdade de São Paulo, Caxa Postal 66281, BR São Paulo, SP, Brazl 2 Centro de Cêncas Exatas e da Terra, Unversdade Federal do Ro Grande do Norte, Campus Unverstáro Lagoa Nova, BR Natal, RN, Brazl Receved 9 Aprl 2015; accepted 8 September 2015; publshed onlne 21 October 2015) How to gve a natural geometrc defnton of a covarant Posson bracket n classcal feld theory has for a long tme been an open problem as testfed by the extensve lterature on multsymplectc Posson brackets, together wth the fact that all these proposals suffer from serous defects. On the other hand, the functonal approach does provde a good canddate whch has come to be known as the Peerls De Wtt bracket and whose constructon n a geometrcal settng s now well understood. Here, we show how the basc multsymplectc Posson bracket already proposed n the 1970s can be derved from the Peerls De Wtt bracket, appled to a specal class of functonals. Ths relaton allows to trace back most f not all) of the problems encountered n the past to ambgutes the relaton between dfferental forms on multphase space and the functonals they defne s not one-to-one) and also to the fact that ths class of functonals does not form a Posson subalgebra. C 2015 AIP Publshng LLC. [ I. INTRODUCTION The quest for a fully covarant hamltonan formulaton of classcal feld theory has a long hstory whch can be traced back to the work of Carathéodory, 3 De Donder, 7 and Weyl 34 on the calculus of varatons. From a modern pont of vew, one of the man motvatons s the ssue of quantzaton whch, n tradtonal versons lke canoncal quantzaton as well as more recent ones such as deformaton quantzaton, starts by brngng the classcal theory nto hamltonan form. In the context of mechancs, where one s dealng wth systems wth a fnte number of degrees of freedom, ths has led mathematcans to develop entre new areas of dfferental geometry, namely, symplectc geometry and then Posson geometry, whereas physcsts have been motvated to embark on a more profound analyss of basc physcal concepts such as those of states and observables. In the context of relatvstc) feld theory, however, ths s not suffcent snce, besdes facng the formdable mathematcal problem of handlng systems wth an nfnte number of degrees of freedom, we have to cope wth new physcal prncples, most notably those of covarance and of localty. The prncple of covarance states that meanngful laws of physcs do not depend on the choce of local) coordnates n space-tme employed n ther formulaton: extendng the axom of Lorentz nvarance n specal relatvty, t s one of the cornerstones of general relatvty and underles the modern geometrcal approach to feld theory as a whole. Equally mportant s the prncple of localty, statng that events ncludng measurements of physcal quanttes) localzed n regons of space-tme that are spacelke separated cannot exert any nfluence on each other. Clearly, a mathematcally and physcally correct hamltonan formalsm for classcal feld theory should respect these prncples: t should be manfestly covarant and local, as s the modern algebrac approach to quantum feld theory; see, e.g., Ref. 2. As an example of a method that does not meet these requrements, we may quote the standard hamltonan formulaton of classcal feld theory, based on a functonal formalsm n terms of Cauchy data: there, the mere necessty of choosng some Cauchy surface spols covarance from the very begnnng! To avod that, a dfferent approach s needed /2015/5610)/102901/26/$ , AIP Publshng LLC

3 M. Forger and M. O. Salles J. Math. Phys. 56, ) Over the last few decades, attempts to construct such a dfferent approach have produced a varety of proposals that, roughly speakng, can be assembled nto two groups. One of these extends the geometrcal tools that were so successful n mechancs to the stuaton encountered n feld theory by treatng spatal dervatves of felds on the same footng as tme dervatves: n the context of a frst order formalsm, as n mechancs, ths requres assocatng to each feld component, say ϕ, not just one canoncally conjugate momentum π = L/ ϕ but rather n canoncally conjugate momenta π µ = L/ µ ϕ, where n s the dmenson of space-tme. In mechancs, tme s the only ndependent varable, so n = 1.) Identfyng the approprate geometrcal context has led to the ntroducton of new geometrcal enttes now commonly referred to as multsymplectc and/or polysymplectc structures, and although ther correct mathematcal defnton has only recently been completely elucdated, 11 the entre crcle of deas surroundng them s already reasonably well establshed, formng a new area of dfferental geometry; see Refs. 4, 16 21, and for early references. A dfferent lne of thought s centered around the concept of covarant phase space, 5,6,35 defned as the space of solutons of the equatons of moton: usng ths space to substtute the correspondng space of Cauchy data elmnates the need to refer to a specfc choce of Cauchy surface and has the addtonal beneft of provdng an embeddng nto the larger space of all feld confguratons, allowng us to classfy statements as vald off shell.e., on the entre space of feld confguratons) or on shell.e., only on the subspace of solutons of the equatons of moton). Each of the two methods, the multsymplectc formalsm as well as the covarant functonal formalsm, has ts own merts and ts own drawbacks, and experence has shown that best results are obtaned by approprately combnng them. As an example to demonstrate how useful the combnaton of these two approaches can become, we shall n the present paper dscuss the problem of gvng an approprate defnton of the Posson bracket, or better, the covarant Posson bracket. From the pont of vew of quantzaton, ths s a queston of fundamental mportance, gven the fact that the Posson bracket s expected to be the classcal lmt of the commutator n quantum feld theory. Moreover, quantum feld theory provdes compellng motvaton for dscussng ths lmt n a covarant settng, takng nto account that the non-covarant) equal-tme Posson brackets of the standard hamltonan formulaton of classcal feld theory would correspond, n the sense of a classcal lmt, to the non-covarant) equal-tme commutators of quantum feld theory, whch are known not to exst n nteractng quantum feld theores, due to Haag s theorem. Unfortunately, n the context of the multsymplectc formalsm, the status of covarant Posson brackets s hghly unsatsfactory. Ths may come as a bt of a surprse, gven the beautful and conceptually smple stuaton prevalng n mechancs, where the exstence of a Posson bracket on the algebra C P) of smooth functons on a manfold P s equvalent to the statement that P s a Posson manfold and, as such, qualfes as a canddate for the phase space of a classcal hamltonan system: for any such system, the algebra of observables s just the Posson algebra C P) tself or, possbly, an approprate subalgebra thereof, and the space of pure states s just the Posson manfold P tself. In partcular, ths s true n the specal case when P s a symplectc manfold, wth symplectc form ω, say, and where the Posson bracket of two functons f,g C P) s the functon { f,g} C P) defned by { f,g} = Xg X f ω = ωx f, X g ), 1) where X f XP) denotes the hamltonan vector feld assocated wth f C P), unquely determned by the formula X f ω = df. 2) Ths stuaton changes consderably, and for the worse, when we pass to the multsymplectc settng, where ω s no longer a 2-form but rather an n + 1)-form and the hamltonan vector feld X f s no longer assocated wth a functon f but rather wth an n 1)-form f, n beng the dmenson of space-tme. It can then be shown that Equaton 2) mposes restrctons not only on the type of vector feld that s allowed on ts lhs but also on the type of dfferental form that s allowed on ts rhs. Indeed, the valdty of an equaton of the form X ω = df mples that the vector feld X must be

4 M. Forger and M. O. Salles J. Math. Phys. 56, ) locally hamltonan,.e., we have L X ω = 0, but t also mples that the form f must be hamltonan, whch by defnton means that ts exteror dervatve d f must vansh on all multvectors of degree n whose contracton wth the n + 1)-form ω s zero, and ths s a non-trval condton as soon as n > 1. It s trval for n = 1 snce ω s assumed to be non-degenerate.) Thus t s only on the space Ω n 1 H P) of hamltonan n 1)-forms that Equaton 1) provdes a reasonable canddate for a Posson bracket. And even wth ths restrcton, we get a Le algebra Ω n 1 H P) whch, n contrast to the stuaton n mechancs, has a huge center, contanng the entre space Z n 1 P) of closed n 1)-forms on P, snce the lnear map from Ω n 1 H P) to X LHP) that takes f to X f s far from beng one-to-one: ts kernel s precsely Z n 1 P). Stll, the argument suggests that the transton from mechancs to feld theory should somehow nvolve a replacement of functons by dfferental forms of degree n 1 whch s not completely unreasonable when we consder the fact that, n feld theory, conservaton laws are formulated n terms of conserved currents, whch are closed n 1)-forms. Unfortunately, ths replacement leads to a whole bunch of serous problems, some of whch seem nsurmountable. Frst and foremost, there s no reasonable canddate for an assocatve product on the space Ω n 1 H P) whch would provde even a startng pont for defnng a Posson algebra. Second, as has been observed repeatedly n the lterature, 16,22 27 the condton of beng a locally hamltonan vector feld or a hamltonan n 1)-form forces these objects to depend at most lnearly on the multmomentum varables, and moreover we can easly thnk of observables that are assocated to forms of other degree such as a scalar feld, correspondng to a 0-form, or the electromagnetc feld strength tensor, correspondng to a 2-form): ths by tself provdes enough evdence to conclude that hamltonan n 1)-forms consttute an extremely restrcted class of observables and that settng up an adequate framework for general observables wll requre gong beyond ths doman. And fnally, as has already also been noted long ago, 16,17,20,27 29 the multsymplectc Posson bracket defned by Equaton 1) fals to satsfy the Jacob dentty. In the case of an exact multsymplectc form.e., when ω = dθ), ths last problem can be cured by modfyng the defnng equaton 1) through the addton of an exact hence closed) term, as follows 12 : { f,g} = Xg X f ω + d Xg f X f g Xg X f θ ). 3) However, ths does not settle any of the other two ssues, namely, the lack of an assocatve product to construct a Posson algebra; the restrcton to hamltonan forms and forms of degree n 1, whch leads to unreasonable constrants on the observables that are allowed, excludng some that appear naturally n physcsts calculatons. It should be mentoned here that these are long-standng problems: they have been recognzed snce the early stages of development of the subject see, e.g., Refs. 16 and 27 and also Refs. 25 and 26) but have so far remaned unsolved. A smple dea n ths drecton that has already been exploted s based on the observaton that dfferental forms do admt a natural assocatve product, namely, the wedge product, so one may ask what happens f, n the above constructon, vector felds are replaced by multvector felds and n 1)-forms by forms of arbtrary degree. As t turns out, ths leads to a modfed super-posson bracket, defned by a formula analogous to Equaton 3). 13,14 But t does not help to overcome ether of the aforementoned other two ssues. On the other hand, n the context of the covarant functonal formalsm, there s an obvous assocatve and commutatve product, namely, just the pontwse product of functonals, and apart from that, there also exsts a natural and completely general defnton of a covarant Posson bracket such that, when both are taken together, all the propertes requred of a Posson algebra are satsfed: ths bracket s known as the Peerls De Wtt bracket. 8 10,15,32 Thus the queston arses as to what mght be the relaton, f any, between the covarant functonal Posson bracket, or Peerls De Wtt bracket, and the varous canddates for multsymplectc Posson brackets that have been dscussed n the lterature, among them the ones wrtten down n Equatons 1) and 3). That s the queston we shall address n ths paper.

5 M. Forger and M. O. Salles J. Math. Phys. 56, ) In the remander of ths ntroducton, we want to brefly sketch the answer proposed here: detals wll be flled n later on. Startng out from the paradgm that, mathematcally, classcal felds are to be descrbed by sectons of fber bundles, suppose we are gven a fber bundle P over a base manfold M, where M represents space-tme, wth projecton ρ : P M, and suppose that the classcal felds appearng n the feld theoretcal model under study are sectons φ : M P of P.e., maps φ from M to P satsfyng ρ φ = d M ), subject to approprate regularty condtons: for the sake of defnteness, we shall assume here that all manfolds and bundles are regular n the sense of beng smooth, whle the regularty of sectons may vary between smooth C ) and dstrbutonal C ), but whenever t s left unspecfed, we wll tactly be assumng that we are dealng wth smooth sectons. To fx termnology, we defne, for any secton f of any vector bundle V over P, ts base support or space-tme support, denoted here by supp f, to be the closure of the set of ponts n M such that the restrcton of f to the correspondng fbers of P does not vansh dentcally,.e., supp f = x M f Px 0. 4) Usng the abbrevaton supp for the base support rather than the ordnary support, whch would be a subset of P, consttutes a certan abuse of language, but wll do no harm snce the ordnary support wll play no role n ths paper.) Now suppose that f s a dfferental form on P of degree p, say, and that s a closed p-dmensonal submanfold of M, possbly wth boundary, subject to the restrcton that and supp f should have compact ntersecton, so as to guarantee that the followng ntegral s well defned, provdng a functonal F, f on the space of sectons of P, F, f [φ] = φ f, 5) where φ f s of course the pull-back of f to M va φ. Note that the aforementoned restrcton s automatcally satsfed f s compact and also f f has compact base support; moreover, gven an arbtrary dfferental form f on P, we can always construct one wth compact base support by multplyng wth a cutoff functon,.e., the pull-back to P of a functon of compact support on M.) Regardng boundary condtons, we shall usually requre that f has a boundary, t should not ntersect the base support of f, supp f =. 6) Ths smple constructon provdes an especally nterestng class of functonals for varous reasons, the most mportant of them beng the fact that they are local, snce they are smple ntegrals, over regons or more general submanfolds of space-tme, of local denstes such as, e.g., polynomals of the basc felds and ther dervatves, up to a certan order. More specfcally, dervatves up to order r, say, of felds that are sectons of some fber bundle E over M are ncorporated by takng P to be the rth order jet bundle J r E of E.) Ths s an ntutve noton of localty for functonals of classcal felds, but as has been shown recently, t can also be formulated n mathematcally rgorous terms. 1 Ether way, t s clear that the product of two local functonals of the form 5) s no longer a local functonal of the same form: rather, we get a blocal functonal assocated wth a submanfold of M M and a dfferental form on P P. Therefore, a mathematcally nterestng object to study would be the algebra of multlocal functonals whch s generated by the local ones, much n the same way as, on an ordnary vector space, the algebra of polynomals s generated by the monomals. But the pont of man nterest for our work appears when we assume P to be a multsymplectc fber bundle 11 and M to be a Lorentz manfold, usually satsfyng some addtonal hypotheses regardng ts causal structure: more specfcally, we shall assume M to be globally hyperbolc snce ths s the property that allows us to speak of Cauchy surfaces. In fact, as s now well known, M wll n ths case admt a folaton by Cauchy surfaces defned as the level sets of some smooth tme functon. However, t s often convenent not to fx any metrc on M a pror snce, n the context of general relatvty, the space-tme metrc tself s a dynamcal entty and not a fxed background feld. Wthn ths context, and for the case of a regular frst-order hamltonan system where felds are sectons of a gven confguraton bundle E over M and the dynamcs s obtaned from a regular

6 M. Forger and M. O. Salles J. Math. Phys. 56, ) frst-order lagrangan va Legendre transform, t has been shown n Ref. 15 that, usng ths extra structure, one can defne the Peerls De Wtt bracket as a functonal Posson bracket on covarant phase space. Here, we want to show how, n the same context, multsymplectc Posson brackets between forms, such as n Equatons 1) and 3), can be derved from the Peerls De Wtt bracket between the correspondng functonals. For the sake of smplcty, ths wll be done for the case of n 1)-forms, but we expect smlar arguments to work n any degree. Concretely, we shall prove that gven a fxed hypersurface n M typcally, a Cauchy surface) and two hamltonan n 1)-forms f and g, we have F, f, F,g = F,{ f,g }, 7) where the bracket on the lhs s the Peerls De Wtt bracket of functonals and the bracket { f,g} that appears on the rhs s a multsymplectc pseudo-bracket or multsymplectc bracket gven by a formula analogous to Equaton 1) or to Equaton 3). A more detaled explanaton of ths result wll be deferred to the man body of the paper manly because of several techncal ssues that appear when tryng to formulate t wth the requred amount of mathematcal rgor. In fact, the constructon turns out to nvolve both types of multphase space that appear n feld theory and that we refer to as ordnary multphase space and extended multphase space, respectvely: they dffer n that the latter s a one-dmensonal extenson of the former, obtaned by ncludng an addtonal scalar energy type varable. Geometrcally, extended multphase space s an affne lne bundle over ordnary multphase space, and the hamltonan H of any theory wth ths type of feld content s a secton of ths affne lne bundle. Moreover, each of these two multphase spaces comes equpped wth a multsymplectc structure whch s exact.e., the multsymplectc form s, up to a sgn ntroduced merely for convenence, the exteror dervatve of a multcanoncal form), naturally defned as follows. Frst, one constructs the multsymplectc form ω and the multcanoncal form θ on the extended multphase space by means of a procedure that can be thought of as a generalzaton of the constructon of the symplectc structure on the cotangent bundle of an arbtrary manfold. Then, the correspondng forms on the ordnary multphase space are obtaned from the prevous ones by pull-back va the hamltonan H: therefore, they wll n what follows be denoted by and by θ H to ndcate ther dependence on the choce of hamltonan. We can express ths by sayng that the multsymplectc structure on extended multphase space s knematcal, whereas that on ordnary multphase space s dynamcal. Correspondngly, we shall often refer to the brackets on extended multphase space, defned by Equaton 1), together wth Equaton 2), or by Equaton 3), as knematcal multsymplectc Posson) brackets, and to the brackets on ordnary multphase space, defned by the analogous equatons together wth { f,g} = Xg X f = X f, X g ), 8) X f = df 9) or { f,g} = Xg X f + d Xg f X f g Xg X f θ H ) 10) as dynamcal multsymplectc Posson) brackets. In both cases, the brackets defned by the smpler formulas 1) and 8) are really only pseudo-brackets because they fal to satsfy the Jacob dentty, and the correcton terms that appear n Equatons 3) and 10) are ntroduced to cure ths defect. As we shall see, what appears on the rhs of Equaton 7) above s the dynamcal bracket on ordnary multphase space and not the knematcal bracket on extended multphase space n accordance wth the fact that the Peerls De Wtt bracket tself s dynamcal. We conclude ths ntroducton wth a few comments about the organzaton of the paper. In Secton II, we set up the geometrc context for the functonal calculus n classcal feld theory, ntroduce the class of local functonals to be nvestgated and gve an explct formula for ther frst functonal dervatve. In Secton III, we present a few elementary concepts from multsymplectc geometry, whch s the adequate mathematcal background for the covarant hamltonan formulaton of classcal feld theory. In Secton IV, we combne the two prevous sectons to formulate, n

7 M. Forger and M. O. Salles J. Math. Phys. 56, ) ths context, the varatonal prncple that provdes the dynamcs and derve not only the equatons of moton De Donder Weyl equatons) but also ther lnearzaton around a gven soluton lnearzed De Donder Weyl equatons), wth emphass on a correct treatment of boundary condtons. Although much of ths materal can be found n the lterature, t s ncluded here to set the stage, explan mportant parts of the necessary background and fx the notaton, so as to make the paper reasonably self-contaned. In Secton V, we present the classfcaton of locally hamltonan and exact hamltonan vector felds on both types of multphase space; t s stated n Theorem 1 for extended multphase space and n Theorem 2 for ordnary multphase space. We would lke to pont out that to our knowledge, Theorem 2 s new and as explaned above, t s ths verson that covers the case of nterest here), whereas Theorem 1 has been known for some tme; n fact, an even more general classfcaton, namely, of locally or exact) hamltonan multvector felds rather than just vector felds on extended multphase space has been establshed n Ref. 14. Stll, we found t worthwhle to nclude both cases here, manly to facltate comparson between the two stuatons and to llustrate to what extent the statements are parallel and where devatons occur. In Secton VI, we complete our dscusson of the background by brefly revewng the central theorems of Ref. 15, and then pass on to present the man result of ths paper, specfyng a precse formulaton of the connecton between multsymplectc Posson brackets and the functonal Posson bracket of Peerls and De Wtt. Fnally, Secton VII provdes further dscusson of ths result, ts mplcatons and perspectves for future nvestgatons. The paper presents a substantally revsed and expanded verson of the man results contaned n the PhD thess of the second author, 33 whch was elaborated under the supervson of the frst author. II. GEOMETRIC SETUP FOR THE FUNCTIONAL CALCULUS We begn by collectng some concepts and notatons that we use throughout the artcle. As already mentoned n the ntroducton, classcal felds are sectons of fber bundles over space-tme, so our startng pont wll be to fx a fber bundle P over the space-tme manfold M not necessarly endowed wth a fxed metrc, as mentoned before), wth projecton ρ : P M. The space of feld confguratons C s then the space of smooth) sectons of P, or an approprate subspace thereof, C Γ P), 11) whose elements wll, typcally, be denoted by φ. Formally, we can vew ths space as a manfold whch, at each pont φ, has a tangent space T φ C and, smlarly, a cotangent space Tφ C. Explctly, denotng by V φ the vertcal bundle of P, pulled back to M va φ, V φ = φ VerP), 12) and by Vφ ts twsted dual, defned by takng the tensor product of ts ordnary dual wth the lne bundle of volume forms over the base space, V φ = V φ n T M, 13) we have that, accordng to the prncples of the varatonal calculus, T φ C s the space of smooth sectons of V φ, or an approprate subspace thereof, T φ C Γ V φ ), 14) whose elements wll, typcally, be denoted by δφ and called varatons of φ, whereas Tφ C s the space of dstrbutonal sectons of Vφ, or an approprate subspace thereof, TφC Γ Vφ ). 15) The reader wll note that n Equatons 11), 14), and 15), we have requred only ncluson, rather than equalty. One reason s that the system may be subject to constrants on the felds whch cannot be reduced to the smple statement that they should take values n some approprate subbundle of the orgnal bundle ths case could be handled by smply changng the choce of the bundle P).

8 M. Forger and M. O. Salles J. Math. Phys. 56, ) But even for unconstraned systems, whch are the only ones that we shall be dealng wth n ths paper, there s another reason, namely, that we have not yet specfed the support propertes to be employed. One obvous possblty s to set T φ C = Γc V φ ), TφC = Γ Vφ ), 16) whch amounts to allowng only varatons wth compact support. At the other extreme, we may set T φ C = Γ V φ ), TφC = Γc Vφ ). 17) And fnally, there s a thrd opton, specfcally adapted to the stuaton where the base space s a globally hyperbolc lorentzan manfold and adopted n Ref. 15, whch s to take T φ C = ΓscV φ ), TφC = Γtc Vφ ), 18) where the symbols sc and tc ndcate that the sectons are requred to have spatally compact support and temporally compact support, respectvely. These optons correspond to dfferent choces for the support propertes of the functonals that wll be allowed. Generally speakng, gven a functonal F on C, we defne ts base support or space-tme support, denoted here smply by supp F, as follows 1 : x supp F There exsts an open neghborhood U x of x n M such that for any two feld confguratons φ 1, φ 2 C satsfyng φ 1 = φ 2 on M \ U x, F[φ 1 ] = F[φ 2 ]. Ths defnton mples that supp F s a closed subset of M snce ts complement s open: t s the largest open subset of M such that, ntutvely speakng, F s nsenstve to varatons of ts argument localzed wthn that open subset. It also mples that the functonal dervatve of F f t exsts) satsfes 19) F [φ] δφ = 0 f supp F supp δφ =. 20) For later use, we note that the functonal dervatve wll often be expressed n terms of a formal) varatonal dervatve, F [φ] δφ = d n x δf δφ [φ]x) δφ x). 21) M Typcally, as always n dstrbuton theory, the functonal dervatve wll be well defned on varatons δφ such that supp F supp δφ s compact. Thus f no restrctons on the space-tme support of F are mposed, we must adopt the choce made n Equaton 16). At the other extreme, f the space-tme support of F s supposed to be compact, we may adopt the choce made n Equaton 17). And fnally, the choce made n Equaton 18) s the adequate one for dealng wth functonals that have temporally compact support,.e., space-tme support contaned n the nverse mage of a bounded nterval n R under some global tme functon: the typcal example s that of a local functonal of the form gven by Equaton 5) when s some Cauchy surface. More generally, note that for local functonals of the form gven by Equaton 5), we have supp F, f = supp f. 22) However, t should not be left unnotced that the equalty n Equatons 16)-18) and, possbly, n Equaton 22), can only be guaranteed to hold for non-degenerate systems, snce n the case of degenerate systems, there wll be addtonal constrants mplyng that we must return to the opton of replacng equaltes by nclusons, as before. In what follows, we shall make extensve use of the fact that varatons of sectons can always be wrtten as compostons wth projectable vector felds, or even wth vertcal vector felds, on the total space P. To explan ths, recall that a vector feld X on the total space of a fber bundle s called projectable f the tangent map to the bundle projecton takes the values of X at any two ponts n the same fber to the same tangent vector at the correspondng base pont,.e., T p1 ρ Xp 1 ) = T p2 ρ Xp 2 ) for p 1, p 2 P such that ρp 1 ) = ρp 2 ). 23)

9 M. Forger and M. O. Salles J. Math. Phys. 56, ) Ths s equvalent to requrng that there exsts a vector feld X M on the base space whch s ρ -related to X, X M m) = T p ρ Xp) for p P such that ρp) = m. 24) In partcular, X s called vertcal f X M vanshes. Now note that gven any projectable vector feld X on P, we obtan a functonal vector feld X on C whose value at each pont φ C s the functonal tangent vector X[φ] T φ C, denoted n what follows by δ X φ, defned as or more explctly, δ X φ = Xφ) Tφ X M ), 25) δ X φm) = Xφm)) T m φ X M m)) for m M. 26) Conversely, t can be shown that every functonal tangent vector can be obtaned n ths way from a vertcal vector feld X on P,.e., gven a secton δφ of φ VerP), there exsts a vertcal vector feld X on P representng t n the sense that δφ s equal to δ X φ. To do so, we can apply the mplct functon theorem to construct, for any pont m n M, a system of local coordnates x µ, y α ) for P around φm) n whch ρ corresponds to the projecton onto the frst factor, x µ, y α ) x µ, and φ corresponds to the embeddng n the frst factor, x µ x µ,0). Moreover, n these coordnates, δφ s gven by functons δφ α x µ ), whereas X s gven by functons X α x µ, y β ), so we may smply defne an extenson of the former to the latter by requrng the X α to be ndependent of the y β, settng X α x µ, y β ) = δφ α x µ ) n a neghborhood of the orgn n y-space and then usng a smooth cutoff functon n y-space.) Of course, the reader may wonder why, n ths context, we bother to allow for projectable vector felds rather than just vertcal ones. The pont s that although vertcal vector felds are entrely suffcent to represent varatons of sectons, we shall often encounter the converse stuaton n whch we consder varatons of sectons nduced by vector felds whch are not vertcal but only projectable, such as the hamltonan vector felds appearng n Equatons 1)-3) and 8)-10). Regardng notaton, we shall often thnk of a projectable vector feld as a par X = X P, X M ) consstng of a vector feld X P on the total space P and a vector feld X M on the base space M, related to one another by the bundle projecton: then Equatons 25) and 26) should be wrtten as and δ X φ = X P φ) Tφ X M ), 27) δ X φm) = X P φm)) T m φ X M m)) for m M, 28) respectvely. The same argument as n the prevous paragraph can then be used to prove the followng. Lemma 1. Let φ be a secton of a fber bundle P over a base manfold M. Gven any vector feld X M on M, there exsts a projectable vector feld X φ P on P whch s φ-related to X M,.e., satsfes X φ P φ) = TφX M), and then we have φ φ X α) = XM φ α), for any dfferental form α on P. P As an example of how useful the representaton of varatons of sectons of a fber bundle by composton wth vertcal vector felds or even projectable vector felds can be, we present explct formulas for the frst and second functonal dervatves of a local functonal of the type consdered above begnnng wth a more detaled defnton of ths class of functonals. Defnton 1. Gven a fber bundle P over an n-dmensonal base manfold M, let be a p-dmensonal submanfold of M, possbly wth boundary, and f be a p-form on the total space P such that the ntersecton of wth the base support of f s compact. The local functonal assocated to and f s the functonal F, f : C R on the space C Γ P) of feld confguratons defned by F, f [φ] = φ f for φ C. 29) These functonals are dfferentable, and ther dervatve s gven by a completely explct formula.

10 M. Forger and M. O. Salles J. Math. Phys. 56, ) Proposton 1. Gven a fber bundle P over an n-dmensonal base manfold M, let be a p-dmensonal submanfold of M, possbly wth boundary, and f be a p-form on the total space P such that the ntersecton of wth the base support of f s compact. Then the local functonal F, f assocated to and f s dfferentable, and representng varatons of sectons of P n the form δ X φ where X = X P, X M ) s a projectable vector feld, ts functonal dervatve s gven by the formula F, f [φ] δ X φ = φ L XP f L XM φ f ) for φ C, δ X φ T φ C, 30) where L Z denotes the Le dervatve along the vector feld Z. Remark 1. Under the boundary condton that the ntersecton of wth the base support of f s empty, Equaton 30) can be rewrtten as follows: F, f [φ] δ X φ = φ XP df XM φ df ) for φ C, δ X φ T φ C. 31) The same equaton holds when ths boundary condton s replaced by the requrement that δ X φ should vansh on. Proof. Recall frst that for any functonal F on C, ts functonal dervatve at φ C along δφ T φ C s defned by F [φ] δφ = d dλ F[φ λ ] λ=0, where the φ λ C consttute a smooth one-parameter famly of sectons of P such that φ = φ λλ=0, δφ = d dλ φ λ Fxng φ and δφ and choosng a projectable vector feld X = X P, X M ) that represents δφ as δ X φ, accordng to Equaton 25), consder ts flow, whch s a local) one-parameter group of local) automorphsms Φ λ = Φ P, λ,φ M, λ ) such that X P = d dλ Φ P, λ and X λ=0 M = d dλ Φ M, λ. λ=0 Ths allows us to take the one-parameter famly of sectons φ λ of P to be gven by the oneparameter group of automorphsms Φ λ, accordng to φ λ = Φ P, λ φ Φ 1 M, λ. Now we are ready to calculate F, f [φ] δ X φ = d ) φ λ dλ f λ=0 d = dλ φ λ f λ=0 = d = φ Φ P, λ dλ f ) λ=0 = φ L XP f L XM φ f ). λ=0. d ΦP, dλ λ φ Φ 1 M, λ f λ=0 + d dλ Φ 1 M, λ φ f ) ) λ=0 To derve Equaton 31) from Equaton 30), t suffces to apply standard formulas such as L Z = d Z + Z d and the fact that d commutes wth pull-backs, together wth Stokes theorem, and use the boundary condton 6) to kll the resultng two ntegrals over. The same argument works when δ X φ s supposed to vansh on, snce we may then arrange X M to vansh on and X P to vansh on P.

11 M. Forger and M. O. Salles J. Math. Phys. 56, ) III. MULTIPHASE SPACES AND MULTISYMPLECTIC STRUCTURE As has already been stated before, the bundle P appearng n Secs. I and II, representng the multphase space of the system under consderaton, wll be requred to carry addtonal structure, namely, a multsymplectc form. There has been much debate and even some confuson n the lterature on what should be the rght defnton of the concept of a multsymplectc structure, but all proposals made so far can be subsumed under the followng. Defnton 2. A multsymplectc form on a manfold P s a dfferental form ω on P of degree n + 1, say, whch s closed, dω = 0 32) and satsfes certan addtonal algebrac constrants among them that of beng non-degenerate, n the sense that for any vector feld X on P, we have X ω = 0 = X = 0. 33) Of course, ths defnton s somewhat vague snce t leaves open what other algebrac constrants should be mposed besdes non-degeneracy. One rather natural crteron s that they should be suffcent to guarantee the valdty of a Darboux type theorem. Clearly, when n = 1, the above defnton reduces to that of a symplectc form, and no addtonal constrants are needed. But when n > 1, whch s the case pertanng to feld theory rather than to mechancs, ths s no longer so. An mportant aspect here s that P s not smply a manfold but rather the total space of a fber bundle over the space-tme manfold M, whch s supposed to be n-dmensonal, so one restrcton s that the degree of the form ω s lnked to the space-tme dmenson. Another restrcton s that ω should be n 1)-horzontal, whch means that ts contracton wth three vertcal vector felds vanshes, X Y Z ω = 0 for X,Y, Z vertcal. 34) And fnally, there s a restrcton that, roughly speakng, guarantees exstence of a suffcently hgh-dmensonal lagrangan subbundle of the tangent bundle of P, but snce we shall not need t here, we omt the detals: they can be found n Ref. 11. The man advantage of the defnton of a multsymplectc form as gven above s that we can proceed to dscuss a number of concepts whch do not depend on the precse nature of the addtonal algebrac constrants to be mposed. For example, a vector feld X on P s sad to be locally hamltonan f L X ω = 0, 35) whch accordng to Cartan s formula L X = d X + X d to be used wthout further menton throughout ths paper) s equvalent to the condton that X ω s closed, and s sad to be globally hamltonan, or smply hamltonan, f X ω s exact, that s, f there exsts an n 1)-form f on P such that X ω = df. 36) Recprocally, an n 1)-form f on P s sad to be hamltonan f there exsts a vector feld X on P such that Equaton 36) holds: ths condton s trvally satsfed when n = 1 but not when n > 1. Note that due to non-degeneracy of ω, X s unquely determned by f and wll therefore often be denoted by X f, whereas f s determned by X only up to addton of a closed form: despte ths partal) ambguty, we shall say that X s assocated wth f and f s assocated wth X. In the specal case when ω s exact,.e., we have ω = dθ, 37) where θ s an approprate n-form on P called the multcanoncal form, a vector feld X on P s sad to be exact hamltonan f L X θ = 0. 38) In ths case, of course, the assocated hamltonan form can be smply chosen to be X θ, snce d X θ = L X θ X dθ = X ω. In partcular, ths happens when P s the total space of a vector bundle

12 M. Forger and M. O. Salles J. Math. Phys. 56, ) over some base space E, say, whch n turn wll be the total space of a fber bundle over the space-tme manfold M), provded that ω s homogeneous of degree one wth respect to the correspondng Euler vector feld or scalng vector feld,.e., snce we may then defne θ by L ω = ω, 39) θ = ω. 40) Moreover, we can then employ to decompose vector felds and dfferental forms on P accordng to ther scalng degree, and as we shall see below, ths turns out to be extremely useful for the classfcaton of hamltonan vector felds whether locally or globally or exact) and of hamltonan forms. The standard example of ths knd of structure s provded by any frst order) hamltonan system obtaned from a frst order) lagrangan system va a non-degenerate covarant Legendre transformaton. In ths approach, one starts out from another fber bundle over M, denoted here by E and called the confguraton bundle: the relaton between the two bundles E and P s then establshed by takng approprate duals of frst order jet bundles. Namely, consder the frst order jet bundle of E, denoted smply by JE, whch s both a fber bundle over M wth respect to the source projecton) and an affne bundle over E wth respect to the target projecton), together wth ts dfference vector bundle, called the lnearzed frst order jet bundle of E and denoted here by JE, whch s both a fber bundle over M wth respect to the source projecton) and a vector bundle over E wth respect to the target projecton), and ntroduce the correspondng duals: the affne dual J E of JE and the usual lnear dual J E of JE. In what follows, we shall refer to the latter as the ordnary multphase space and to the former as the extended multphase space of the theory. As t turns out and has been emphaszed snce the begnnng of the modern phase of the development of the subject n the early 1990s see, e.g., Ref. 4), both of these play an mportant role snce not only are both of them fber bundles over M and vector bundles over E, but J E s also an affne lne bundle over J E, and the dynamcs of the theory s gven by the choce of a hamltonan, whch s a secton H : J E J E of ths affne lne bundle. Moreover, and ths s of central mportance, both of these multphase spaces carry a multsymplectc structure. Namely, J E comes wth a naturally defned multsymplectc form of degree n + 1, denoted here by ω, whch up to sgn) s the exteror dervatve of an equally naturally defned multcanoncal form of degree n, denoted here by θ,.e., ω = dθ, and f we choose a hamltonan H : J E J E, we can pull them back to obtan a correspondng multcanoncal form and a correspondng multsymplectc form θ H = H θ 41) = H ω 42) on J E, so agan, = dθ H. The man dfference between the extended and the ordnary multphase space s that θ and ω are unversal and knematcal, whereas θ H and are dynamcal. In terms of local Darboux coordnates x µ,q, p µ ) for J E and x µ,q, p µ, p) for J E, nduced by the choce of local coordnates x µ for M, q for the typcal fber Q of E and a local trvalzaton of E over M, we have and so that wrtng H = H d n x, and θ = p µ dq d n x µ + p d n x, 43) ω = dq dp µ d n x µ dp d n x, 44) θ H = p µ dq d n x µ H d n x, 45) = dq dp µ d n x µ + dh d n x, 46)

13 M. Forger and M. O. Salles J. Math. Phys. 56, ) or more explctly, = dq dp µ d n x µ + H q dq d n x + H dp µ d n x, 47) where d n x µ s the local) n 1)-form obtaned by contractng the local) volume form d n x wth the local vector feld µ / x µ ; for more detals, ncludng a global defnton of θ that does not depend on any of these choces, we refer to Refs. 4, 15, and 19. It may be worthwhle to note that whereas the form ω s always non-degenerate, the form s degenerate n mechancs,.e., for n = 1, but s non-degenerate n feld theory,.e., for n > 1.) IV. VARIATIONAL PRINCIPLE AND EQUATIONS OF MOTION The fundamental lnk that merges the functonal and multsymplectc formalsms dscussed n Secs. II and III nto one common pcture becomes apparent when the constructon of functonals of felds from forms on multphase space outlned n the ntroducton s appled to the multcanoncal n-form θ H on ordnary) multphase space: ths provdes the acton functonal S of the theory, defnng the varatonal prncple whose statonary ponts are the solutons of the equatons of moton. Indeed, the acton functonal s really an entre famly of functonals S K on the space C of feld confguratons φ see Equaton 11)), gven by S K [φ] = φ θ H, 48) where K runs through the compact submanfolds of M whch are the closure of ther nteror n M and have smooth boundary K. Wthn ths setup, a secton φ n C s sad to be a statonary pont of the acton f, for any compact submanfold K of M whch s the closure of ts nteror n M and has smooth boundary K, φ becomes a crtcal pont of the functonal S K restrcted to the formal) submanfold = φ C φ K = φ K 49) C K,φ of C, or equvalently, f the functonal dervatve S K[φ] of S K at φ vanshes on the subspace K T φ C K,φ = δφ T φ C δφ = 0 on K 50) of T φ C. As s well known, ths s the case f and only f φ satsfes the correspondng equatons of moton, whch n the present case are the De Donder Weyl equatons; see, e.g., Refs. 15, 18, and 19. Globally, these can be cast n the form or even φ X ) = 0 for any vertcal vector feld X on P, φ XP = 0 for any projectable vector feld X = X P, X M ), 51) 52) whereas, when wrtten n terms of local Darboux coordnates x µ,q, p µ ) as before, they read µ ϕ = H ϕ, π), µ π µ = H ϕ, π), 53) q where P = J E, φ = ϕ, π) and H = H d n x. Smlarly, gven such a soluton φ, we shall say that a secton δφ n T φ C s an nfntesmal statonary pont of the acton f t s formally tangent to the formal) submanfold of solutons, or equvalently, f t satsfes the correspondng lnearzed equatons of moton, whch n the present case are the lnearzed De Donder Weyl equatons. Globally, representng δφ n the form δ X φ where X = X P, X M ) s a projectable vector feld, these can be cast

14 M. Forger and M. O. Salles J. Math. Phys. 56, ) n the form or even φ Y L XP ) = 0 for any vertcal vector feld Y on P, φ YP L XP ) = 0 for any projectable vector feld Y = Y P,Y M ), 54) 55) whereas, when wrtten n terms of local Darboux coordnates x µ,q, p µ ) as before, they read µ δϕ 2 H = + q j p µ ϕ, π) δϕ j 2 H + p ν ϕ, π) δπ ν j pµ j, µ δπ µ = 2 H q j q ϕ, π) δϕj 2 H p ν ϕ, π) j q δπν j, 56) where P = J E, φ = ϕ, π), δφ = δϕ, δπ), and H = H d n x. Proof. For the frst part concernng the dervaton of the full equatons of moton from the varatonal prncple), we begn by specalzng Equaton 30) to vertcal vector felds X on P.e., settng X M = 0 and replacng X P by X), wth f = θ H, and usng standard facts such as the formula L Z = d Z + Z d or that d commutes wth Le dervatves and pull-backs, together wth Stokes theorem, to obtan that, for any vertcal vector feld X on P, S K [φ] δ X φ = φ X + φ X θ H. K Obvously, condton 51) mples that ths expresson wll be equal to zero for all vertcal vector felds X on P whch vansh on P K. Conversely, t follows from Lemma 2 below that f ths s the case, then condton 51) holds. Moreover, t s easly seen that ths condton s really equvalent to the condton K φ XP XM φ = 0 for any projectable vector feld X = X P, X M ), but t so happens that the form φ s dentcally zero, for dmensonal reasons. Fnally, a smple calculaton shows that Equaton 51), when wrtten out explctly n local Darboux coordnates, can be reduced to the system 53). For the second part concernng the lnearzaton of the full equatons of moton around a gven soluton φ), we proceed as n the proof of Proposton 1: fxng φ and δφ, suppose we are gven a smooth one-parameter famly of sectons φ λ C of P such that φ = φ λλ=0, δφ = d dλ φ λ as well as a projectable vector feld X = X P, X M ) that represents δφ as δ X φ, accordng to Equaton 25), together wth ts flow, whch s a local) one-parameter group of local) automorphsms Φ λ = Φ P, λ,φ M, λ ) such that λ=0, X P = d dλ Φ P, λ and X λ=0 M = d dλ Φ M, λ allowng us to take the one-parameter famly of sectons φ λ of P to be gven by the one-parameter group of automorphsms Φ λ, accordng to λ=0, φ λ = Φ P, λ φ Φ 1 M, λ.

15 M. Forger and M. O. Salles J. Math. Phys. 56, ) Then for any vertcal vector feld Y on P, we have d dλ φ λ Y ω λ=0 H = d ΦP, dλ λ φ Φ 1 M, λ) Y ω λ=0 H = d dλ φ Φ P, λ Y ) ) λ=0 + d dλ Φ 1 M, λ ) φ Y ) ) λ=0 = φ L XP Y L XM φ Y ) = φ Y L XP + φ [XP,Y ] L XM φ Y ), where the last two terms vansh accordng to Equaton 51), and the same argument holds wth Y replaced by Y P where Y = Y P,Y M ) s any projectable vector feld. Fnally, an elementary but somewhat lengthy calculaton shows that Equaton 54), when wrtten out explctly n local Darboux coordnates, can be reduced to the system 56), provded φ satsfes the system 53). The lemma we have used n the course of the argument s the followng. Lemma 2. Gven a fber bundle P over an n-dmensonal base manfold M, let φ be a secton of P and α be an n + 1)-form on P such that, for any compact submanfold K of M whch s the closure of ts nteror n M and has smooth boundary K, and for any vertcal vector feld X on P that vanshes on P K together wth all ts dervatves, the ntegral φ X α) K vanshes. Then the form φ X α) vanshes dentcally, for any vertcal vector feld X on P not subject to any boundary condtons). Proof. Suppose X s any vertcal vector feld on P and m s a pont n M where φ X α) does not vansh. Choosng an approprately orented system of local coordnates x µ n M around m, we may wrte φ X α) = a d n x where a s a functon that s strctly postve at the coordnate orgn correspondng to the pont m) and hence, for an approprate choce of suffcently small postve numbers δ and ϵ, wll be ϵ on B δ here we denote by B r and by B r the open and closed ball of radus r around the coordnate orgn, respectvely). Choosng a test functon χ on M such that 0 χ 1, supp χ B δ and χ = 1 on B δ/2, lfted to P by pull-back and multpled by X to gve a new vertcal vector feld χx on P that vanshes on P B δ together wth all ts dervatves, we get φ χ X α) = d n x χx)ax) d n x ax) ϵ vol B δ/2 ) > 0, B δ B δ B δ/2 whch s a contradcton. To summarze, the space of statonary ponts of the acton, denoted here by S and consdered as a formal) submanfold of the space of feld confguratons C, can be descrbed n several equvalent ways: we have or or or S = φ C φ s statonary pont of the acton, 57) S = φ C φ satsfes the equatons of moton 53), 58) S = φ C φ XP ) = 0 for any projectable vector feld X = X P, X M ), 59) S = φ C φ X ) = 0 for any vertcal vector feld X on P. 60) Ths space S plays a central role n the functonal approach: t s wdely known under the name of covarant phase space. Moreover, gven φ n S and allowng X = X P, X M ) to run through the projectable vector felds, the formal) tangent space T φ S to S at φ s T φ S = δφ T φ C δφ s nfntesmal statonary pont of the acton, 61)

16 M. Forger and M. O. Salles J. Math. Phys. 56, ) or or T φ S = δφ T φ C δφ satsfes the lnearzed equatons of moton 56), 62) or T φ S = φ δ X φ T φ C YP L XP ) = 0, 63) for any projectable vector feld Y = Y P,Y M ) T φ S = φ δ X φ T φ C Y L XP ) = 0. 64) for any vertcal vector feld Y on P V. LOCALLY HAMILTONIAN VECTOR FIELDS The results of Sec. IV, n partcular, Equatons 63) and 64), provde strong motvaton for studyng projectable vector felds X = X P, X M ) on ordnary) multphase space whch are locally hamltonan that s, such that X P s locally hamltonan), snce they mply that each of these provdes a functonal vector feld X on covarant phase space defned by a smple algebrac composton rule: X[φ] = δ X φ for φ S. 65) As we shall see n Sec. VI, ths functonal vector feld s also hamltonan wth respect to the natural symplectc form on covarant phase space to be presented there. But before dong so, we want to address the problem of classfyng the locally hamltonan vector felds and, along the way, also the globally hamltonan and exact hamltonan vector felds on multphase space. For the case of extended multphase space, usng the forms ω and θ, ths classfcaton has been avalable n the lterature for some tme, 13,14 even for the general case of multvector felds. But what s relevant here s the correspondng result for the case of ordnary multphase space, usng the forms and θ H, for a gven hamltonan H. As we shall see, there s one basc phenomenon common to both cases, namely, that when n > 1, the condton of a vector feld to be locally hamltonan mposes severe restrctons on the momentum dependence of ts components, forcng them to be at most affne lnear + constant): ths appears to be a characterstc feature dstngushng mechancs n = 1) from feld theory n > 1) and has been notced early by varous authors see, e.g., Refs. 16 and 27) and repeatedly redscovered later on see, e.g., Refs , 25, and 26). However, despte all smlartes, some of the detals depend on whch of the two types of multphase space we are workng wth, so t seems worthwhle to gve a full statement for both cases, for the sake of comparson. We begn wth an explct computaton n local Darboux coordnates x µ,q, p µ ) for J E and x µ,q, p µ, p) for J E nduced by the choce of local coordnates x µ for M, q for the typcal fber Q of E and a local trvalzaton of E over M, supposng that we are gven a hamltonan H : J E J E whch we represent n the form H = H d n x, as usual. Startng out from Equatons 43)-47), we dstngush two cases. A. Extended multphase space Usng Equatons 43) and 44) and wrtng an arbtrary vector feld X on J E as X = X µ x µ + X q + X µ + X 0 p 66) we frst note that X wll be projectable to E f and only f the coeffcents X µ and X do not depend on the energy varable p nor on the multmomentum varables p κ k and wll be projectable to M f and only f the coeffcents X µ depend nether on the energy varable p nor on the multmomentum varables p κ k nor on the poston varables qk. Next, we compute