ACCADEMIA NAZIONALE DEI LINCEI ANNO CCCLXXIII - 1976 CONTRIBUTI DEL CENTRO LINCEO INTERDISCIPLINARE DI SCIENZE MATEMATICHE E LORO APPLICAZIONI N.22 i\NI)REW I-L.A.NSO:0J - ]'ULLIO REGGI~ -- CLi\UDIO TEI]'ELBOIlVI CONSTR.Lt\INED HAMILTONIAN SYSTEMS CICLO DI LEZIONI TENUTE DAL 29 APRILE AL 7 1vIAGGIO 1974 ROMA ACCADEMIA NAZIONALE DEI LINCEI 1976 --------.---------~---_._---,._- (, S (~ s- Dott. G. Hardi, Tipografo dell' Accademia :\azionale dei Lincei }{O:\IA, 19i(1 TABI~E OF CONTENrrS FORE\VORD Page 5 CHAPTER 1. 1) IRAC'S GE~ERAL ~IETHOD FOR CO~STRAI~EDHAl\lILTONIA~ SYSTEMS 7 ..c-\.. Forlnal Introduction . 8 B. !-lanzilton Variational Princzple 'zvitll Constraints 13 C. Extension to Infinite Degrees of j;reedo)Jz 16 D. Other Poisson Bracket Surfaces. ..... 18 E. IJynalnics on Curved Surfaces. ...... 20 F. Quantlun 7lzeory and Canonical l//ariables 24 CHAPTER 2. RELATIVISTIC POIXT PARTICLE. 27 ~r\. .J.Vo Gauge Constraint. 2i B. Gauge Constraint .. 29 C. Quantuln J11"ecllanics 31 CHAPTER 3. RELATIVISTIC SPI~NING PARTICLE 33 A. Revie'zv of Lagrangian ~4jJjJroacll to Top. 33 B. Constraints on Top Lagrangian 38 C. Dirac TreatlJzent of Top COllstraints 4° I). Quantum A1echanics. 47 CHAPTER 4. STRI~G MODEL. 51 A. System 'Zvithout Gauge Constraints 53 B. Ortllonorlnal Gauge Constraints 57 C. Dirac Brackets ........ 59 I). Fourier COlnponents of T/'ariables 68 E. Quantlan .J.~fecllanics 71 CHAPTER 5. :YIAXWELL ELECTRO:\L\GXETIC FIELD 73 A... /?lectnnn'lgnttic [{anziltonian 7oitlzout (7auge Constraillts . 74 B. Radiation Gauge Constraints 78 C. ~4 xial Gauge .... 80 D. iVull-Plane Brackets. 82 E. iVull-Plane Radiation Gauge ....... 86 CHAPTER 6. YANG-MILLS GAUGE FIELD 89 A.. Lie Groups ........ 89 B. Systenz lvithout Gauge Constraints 9° C. Radiation Gauge Constraints ... 94 I). ..4lternate Radiation (;auge Techniques. 97 E. ..4xial Gauge ............ 98 CHAPTER 7. EIKSTEI:r\'S THEORY OF GRAVITATION. lor A. General Form of tile Hanziltonian 102 B. The Lagrangian . l°S C. The Halniltonian ........ 1°7 D. ~4.~ymjJtotical(v Flat Space, .C;ur.face Integrals, Inzproved Hanliltoniall. Poincare Illvariance at .'::Jpacelike Infillity .. 1°9 -4- . Page 115 E. F'ixation of tile Sjacetinlc Coordinates (C'auge) I. ()pen Spaces: 118 (a) i\.Dl\I's "1'----T'" Gauge . 121 (h) Dirac's ":\laxilnal Slicing" (~augc o­ 1 -) 1. Closed Spaces: York's Gauge 128 ~Ietric . ..:'\PPE\"DIX .:\. Conventions. A.PPE\"\)IX B. 12 1. Extrinsic Cur\"aturc (llHl the Etnhedcling Equations of (~a uss and Codazzi 9 2. Prouf of Eq. (i. IS )................. 13° 3. :VIOll1entull1 and A.ngular :YIolllcntuln of t he l~rasitational Field. 13° 2 4. Relation of Eq. (-1-.16) to Eq. (1.83) .............. 13 13·+ I<.. EFERE :\CES .............................. F()R l~ \V()RIJ ]'his \v'ork is an outgro\vth of a serIes of lectures gi ven by one of us Cf. 1\..) under the auspices of the i\.ccaden1ia Nazionale dei Lincei in 1\.0111e in the spring of 1974. It is intended to help fill the need for a unified treat111cnt of Dirac's approach to the canonical Han1iltonian forn1ldation of singular I..a­ grangian systen1s. \\7c have attelnpted as far as possible to refer to the original literature on the subjcct, but there have undoubtedly been S0111C inad\'crtant 0111issions, for \vhich \ve apologize. \\ie \vish especially to thank Peter Goddard and Giorgio l)onzano for their essential participation in the for111ulation of the " string lTIodcl " gi \"en here, and Karel Kuchar for per111itting us to use parts of his unpublished lecture notes at Princeton in Chapter 7. C. T'. is grateful to J....~. \\lhceler for much encouragcn1ent, and to the National Science foundation for support under grant GP 30799X to Prin­ ceton lJni vcrsity \vhile i\.. ]. 1-1. thanks the Institute for i~d \'anced Study, the )Jational Science ~---oundation, and the U. S. i\t0111ic Energy COn1111ission for their support of various phases of this project. \\7e are indebted to .i\caclen1ic Press, Inc, the publishers of ...~nnals of Physics C:\". \T.), for perlnission to usc various sections of I-1anson and l\.eggc (1974) and Regge and Teitelboin1 (1974) in this ,,"ork. T\\'o of us (...-\. J. H. and 1'. R.) arc grateful to the ...~ccaden1ia X azionalc dei L.incci for the congenial hospitality enjoyed \vhile this \,'ork ,,'as being prepared. ..~. J. I-I A:\SO:\ ]'. REGCE C. ]'EITELBOI~1 I. I)IR~-\C'S GENI~RAr--J ~/II~TI-IOl) FOR (~O~SI"I{~-\I:\El) Hi\MILI"ONI~~X S\TSTElVIS Constrained canonical systen1s occur \vith ren1arkahlc frequency in phy­ sics. :\Iax\vell's theory of electrol11agnctisl11, I~instein's theory of gTa vitation, and nun1crous Inanifestly Lorentz in\·ariant 111echanical S)1'stCIT1S possess constraints \vhich invalidate the strictly canonical classical systen1s. It is clear that a correct Ha111iltonian for111ulation of a constrained classical systcln is interesting in its o-\\'n right, as \\'ell as being quite useful in de\·cloping a valid canonical quantization procedure for the syste111. Our purpose here is to introduce the reader to the systclnatic trcatll1cnt of constrained I-Ia1l1iltollian systelns devel0ped i11 itially by I)irac (I 9 50 ) and to ind icatcits rcIati() n to quantum 111echanics \vhere kno\vn. T'his chapter \vill deal \vith the for111al aspects of constrained 1-1 all1ilto­ nian systelns. rrhc ren1aining chapters are devoted to specific applications of the 11lethods. In particular, \VC \vill exan1ine relativistic spinless particles, relativistic spinning particles, the relati\,istic string, vector fields \\,ith ~\hclian and non-...\helian gauge groups, and gravitation. rrhe reader \vho v\·ishes to acquire a gcncr?l feeling for the applic~lti()n of these 11l,ethods \vithout getting bogged do\\'n in details is adyiscd to skinl Chapter I, and then carefully study a falTIiliar systenl. (e.g. ("hapter 2, 4 or 5), referring back to Chapter I \vhen necessary. \\Te define a singular Lagrangian L (ql 'Yl") as one for \vhich thc \'cloci­ tcs (jz" cannot be expressed uniquely in tern1S of the canonical 11101nenta pI" = I.:L/2Y 1o due to the existence of constraints a1110ng the canonical coordinates and 1110n1enta fol1o\ving fro1l1 the for111 of the Lagrangian alone. rrhe pl-O hlerll of developing a consistent classical H a111iltonian d ynalnics corresponding to a singular Lagrangian systen1 \,'as apparently attacked first by Dirac (1950). Subsequently Dirac (195 I), ...r\.nderson and Bergn1ann (1951), Bergll1ann and Goldberg (1954), Dirac (1958a) and })e\'/itt (1959) refined Dirac's original n1ethods. ...r\.n expanded treatlnent of the general constrained Hamiltonian systelTI appears in Dirac's lectures on quantun1 mechanics (1964); sec also Dirac (1969). IZundt (1966) and Shann1ugadhasan (1963, 1973) rcvic\v some fine points. Finally, \ve ll1ention several other approaches to the quantization of sin­ gular Lagrangian systelTIs. Schvvinger (1951 (l, I95 1 b, 1953) and l)eierls (195 2 ) utilize variational techniques; SY1l1anzik (1971) gives an extended treatlnent (see also the Appendix of rron1boulis (1973)). DeWitt (1967 b), l,"addce\' and Popov (1967 a, 1967 b) and Faddeev (1969) ll1ake use of Feynlnan path integrals to undcrstand singular systen1s. No attempt vvill be 111ade to treat these methods here. -8- '1'he forn1ulations of both classical and quantuln Inechanics have under­ gone profound changes in recent tilDes through the usc of modern mathernatical language and advanced techniques of functional analysis. \\"'"e think that ultin1atcly these concepts should be introduced into our treatlTIent to gi ve a less heuristic vic\v of the subject than \,'"C have succeeded in developing so far. V\TC have not pursued this lTIattcr in vic\\'" of the practical character of the present notes. Their usefulness, \,'"e feel, is in providing a set of direct guidelines to setting up a consistent canonical forlnalis111 for an an1azing variety of physically significant systelTIS \vhile a voiding lTIany COn11l10n pitfalls. ...\. f~OR:\IAL I XTRODLCTIO\, ()ur forn1aI discussion of constrained systcnls hegins \\'jth the consider­ ation of an action functional ..b (1.1) c= J ciT L(qi' q,) \vhere qi('r) is a canonical coordinate and qi = dq//d-: is a crlnonical \-clocity. We confine ourselyes to I-Jagrangians ",'"ithollt explicit 7-dependcncc. I)cfining the canonical n10n1enta as dL pi -- (1.2) dlo:. , Jl \ve find the l~ulcr equations ctpi 8L 0 (1·3) c1--r Jqi hy reqUIrIng the variation of the action S to be stationary. If \,'"e choose for our l)oisson brackets the con\'"entl011 ~~~\ 813 2:\. 2B ciqi 3pi cqi \ve have __ 61~ (1. 5) .I \\' here 0;° is the Kronecker delta. Hereafter, repeated indices \vill be sun1111ed over unless other\yise stated. The canonical I-IalTIiltonian (1.6) then forn1ally generates the I-Iamilton equations of motion • -- f !-]"} - -~!!~ q£ - tq£, ~c - djJ1: .' {' H} dH c pt == pt , c == - ~i~- · 9 N O\V \ve suppose that I ... (qi , qi) is singular, so that there is no unique solution qz" (q , p) expressing the velocities in terms of the canonical coordinates and 1110111enta. ..-\ necessarv and sufficient condition that I---t be singular is 'l'his is a sIgn that there exist certain prz'llzalJ! constraints (1.8) 11l ==-= I".., 1\1 follo\:ving fron1 the forn1 of the I ... agrangian alone. "fhe synlhol « ~' 0 » is read d \veakly zero" and 111eans that 9m 111ay have non\'anishing canonical Poisson brackets (1.4) \vith S0111C canonical variahles. l'he canonical Han1iltonian (1.6) is no\v not unique. \\7C 111ay in fact replace it by the effecti ve I-I anliltonian -~ -L 1! (r'j f lJ),~ (1.9) H -- I-I c im I JJl \ q 'L lIe.. II generates lle\\' equations of lTIotion replacing (1.7), L~I-I ( "'I l l',?m ql' {q/ , I-I j ~ 'dpz" 11 m lY;i (1.