ACCADEMIA NAZIONALE DEI LINCEIANNO 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\INEDHAMILTONIAN SYSTEMS
CICLO DI LEZIONI TENUTE DAL 29 APRILE AL 7 1vIAGGIO 1974
ROMA
ACCADEMIA NAZIONALE DEI LINCEI
1976
--------.---------~---_._---,._-
(, S(~ s-
}{O:\IA, 19i(1Dott. G. Hardi, Tipografo dell' Accademia :\azionale dei Lincei
TABI~E OF CONTENrrS
CHAPTER 2. RELATIVISTIC POIXT PARTICLE.
~r\. .J.Vo Gauge Constraint. .B. Gauge Constraint . .C. Quantuln J11"ecllanics
CHAPTER 3. RELATIVISTIC SPI~NING PARTICLE
A. Revie'zv of Lagrangian ~4jJjJroacll to Top.B. Constraints on Top LagrangianC. Dirac TreatlJzent of Top COllstraintsI). Quantum A1echanics.
..c-\.. Forlnal Introduction .B. !-lanzilton Variational Princzple 'zvitll ConstraintsC. Extension to Infinite Degrees of j;reedo)JzD. Other Poisson Bracket Surfaces. . . . . .E. IJynalnics on Curved Surfaces. . . . . . .F. Quantlun 7lzeory and Canonical l//ariables
CHAPTER 4. STRI~G MODEL. .
A. System 'Zvithout Gauge ConstraintsB. Ortllonorlnal Gauge ConstraintsC. Dirac Brackets . . . . . . . .I). Fourier COlnponents of T/'ariablesE. Quantlan .J.~fecllanics
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51
53575968
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7478808286
89
89
9°949798
lor
102
l°S1°7
1°9
Inzproved Hanliltoniall.
1) IRAC'S GE~ERAL ~IETHOD FOR CO~STRAI~EDHAl\lILTONIA~SYSTEMS
FORE\VORD
CHAPTER 7. EIKSTEI:r\'S THEORY OF GRAVITATION.
A. General Form of tile HanziltonianB. The Lagrangian .C. The Halniltonian . . . . . . . .D. ~4.~ymjJtotical(v Flat Space, .C;ur.face Integrals,
Poincare Illvariance at .'::Jpacelike Infillity . .
CHAPTER 5. :YIAXWELL ELECTRO:\L\GXETIC FIELD
A... /?lectnnn'lgnttic [{anziltonian 7oitlzout (7auge Constraillts .B. Radiation Gauge ConstraintsC. ~4 xial Gauge . . . .D. iVull-Plane Brackets.E. iVull-Plane Radiation Gauge . . . . . . .
CHAPTER 6. YANG-MILLS GAUGE FIELD
A.. Lie Groups . . . . . . . .B. Systenz lvithout Gauge ConstraintsC. Radiation Gauge Constraints . . .I). ..4lternate Radiation (;auge Techniques.E. ..4xial Gauge . . . . . . . . . . . .
CHAPTER 1.
E.
-4-
F'ixation of tile S jacetinlc Coordinates (C'auge)
I. ()pen Spaces:
(a) i\.Dl\I's "1'----T'" Gauge .
(h) Dirac's ":\laxilnal Slicing" (~augc
1. Closed Spaces: York's Gauge
. . Page 115
118
121
1o-)
..:'\PPE\"DIX .:\. ~Ietric Conventions. . . 128
A.PPE\"\)IX B.1. Extrinsic Cur\"aturc (llHl the Etnhedcling Equations of (~auss and Codazzi 12
9
2. Prouf of Eq. (i. IS ) . . . . . . . . . . . . . . . . . 13°
3. :VIOll1entull1 and A.ngular :YIolllcntuln of t he l~rasitational Field. 13°
4. Relation of Eq. (-1-.16) to Eq. (1.83) . . . . . . . . . . . . . . 132
I<..EFE RE :\CES.............................. 13·+
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 thespring of 1974. It is intended to help fill the need for a unified treat111cnt ofDirac's approach to the canonical Han1iltonian forn1ldation of singular I..agrangian systen1s. \\7c have attelnpted as far as possible to refer to the originalliterature on the su b jcct, but there ha ve undou btedly been S0111C inad \'crtant0111issions, for \vhich \ve apologize.
\\ie \vish especially to thank Peter Goddard and Giorgio l)onzano for theiressential participation in the for111ulation of the " string lTIodcl " gi \"en here,and Karel Kuchar for per111itting us to use parts of his unpublished lecturenotes 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 Princeton lJni vcrsity \vhile i\.. ]. 1-1. thanks the Institute for i~d \'anced Study,the )J ational Science ~---oundation, and the U. S. i\t0111ic Energy COn1111issionfor their support of various phases of this project.
\\7e are indebted to .i\caclen1ic Press, Inc, the publishers of ...~nnals ofPhysics 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 congenialhospitality 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 physics. :\Iax\vell's theory of electrol11agnctisl11, I~instein's theory of gTa vitation,and nun1crous Inanifestl y Lorentz in \·ariant 111echanical S)1'stCIT1S possess
constraints \vhich in validate the strictly canonical classical systen1s. It isclear that a correct Ha111iltonian for111ulation of a constrained classical systclnis 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 isto introduce the reader to the systclnatic trcatll1cnt of constrained I-Ia1l1iltollian
systelns de vel0ped i11it ially by I) irac (I 9 50 ) and to ind icatcits rcIat i() 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,relati vistic 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 ad yiscd 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 \'clocitcs (jz" cannot be expressed uniquely in tern1S of the canonical 11101nentapI" = I.:L/2Y1o due to the existence of constraints a1110ng the canonical coordinatesand 1110n1enta fol1o\ving fro1l1 the for111 of the Lagrangian alone. rrhe pl-O hlerll
of developing a consistent classical H a111iltonian d ynalnics corresponding toa 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 originaln1ethods. ...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\vsome 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) gi ves 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 pathintegrals to undcrstand singular systen1s. No attempt vvill be 111ade to treatthese methods here.
-8-
'1'he forn1ulations of both classical and quantuln Inechanics ha ve under
gone profound changes in recent tilDes through the usc of modern mathernatical
language and ad vanced 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
(1.1)
..b
c= JciT L(qi' q,)
\vhere qi('r) is a canonical coordinate and qi = dq//d-: is a crlnonical \-clocity.
We confine oursel yes to I-Jagrangians ",'"ithollt explicit 7-dependcncc. I)cfining
the canonical n10n1enta as
(1.2) pi dL--
dlo:. ,Jl
\ve find the l~ulcr equations
(1·3)ctpi 8L
c1--r Jqi0
hy reqUIrIng the variation of the action S to be stationary.If \,'"e choose for our l)oisson brackets the con\'"entl011
\ve ha ve
~~~\ 813ciqi 3pi
2:\. 2B
cqi
(1. 5)__ 61~
.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 cpt == pt , c == - ~i~- ·
9
NO\V \ve suppose that I ... (qi , qi) is singular, so that there is no uniquesolution 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
(1.9) H -~ I-I -L 1! (r'j f q lJ),~ lIe.-- c im I JJl \ 'L .
II generates lle\\' equations of lTIotion replacing (1.7),
(1. 10)
L~I-I ( "'I
ql' { q/ I-I l l',?m, j ~ 'dpz" 11 m lY;i
rrhcse are the 1110st general equations of n10tion consistent \vith variations
'dqi , 'Dpz' \\·hich preser\·e the constraints 9m ~ o.In order to ha \·e a consistent system, \\"e require the '7 deri \·ati ves of the
constraints (1.8) to be zero, or to be linear cOITIhinations of the constraints sothey are \veakly zero:
Yu :--== { 9n , H}~ {91/ , 1-Ic }
I f I~q. (1.1 1) is not alread y true as a consequence of the original prin1aryconstraints (1.8), t\\TO possibilities occur. First, \\·e n1ay find that Eq. (1. J J
gives no ne\\T inforn1ation, but sin1ply ilTIpOSeS conditions on the forn1 of 11m ,
Second, Eq. (1.1 1) Inay in1ply a ne\v relation an10ng the p's and q's, independent
of 11m , These are seco/lc!ary constraints and n1ust be adjoined to the originalconstraints (1.8). N O\V \ve repeat the process, requiring the '7 deri \Tati ves of
the secondary constraints to vanish, and so on, until all independent constraintsand conditions on U m have been found. If K additional constraints result,
\ve add them to the IVI prilnary constraints and sUInmarize the complete set as
(1.12) CPa(q ,p) ~ 0 , a =::: 1 " . " K + M == T.
Finally, consistency of all the constraints \vith the equations of n10tion requiresthat there exist a solution for U m as a function of q and p
-10-
so that H itself is expressible in tern1S of q and p;
H == H(q ,p) .
f~ollo\ving I)irac, \VC novv define a function l~ (q , p) as a first (lass quan
tity if
(1. I 3) { l~ , gJa } ~ 0 , tl==I T',
1{ (q , P) IS defined as secon(! class if
{ R , gJa 1~ 0
for at least one d. (Sccond class quantities are obviously anlbiguous up tolinear conl,hinations of first class quantities. :\ote that a second class constraint
squared is first class). ...~ll of our constraints I. 12) can no\v be di\'ided into
t\VO sets, one consisting of all the linearly independent ,/irst class constrain!.",
(1.15 a) Yi (q , p) .~ 0 , 1== 1"",1.
and the ocher of the rcnlaining N" == 'r --- I seeoJhi (lass (:o/lstrllints
I. 1 5 b) ~.x (q , p) ~ 0 ,
Xotc that hoth '~i and epx 111ay include secondary constraints as \yell as
pri1nary constraints.Dirac (1964) has proven that the second class constraints \\'ill gi\'c 14 1SC
to a nonsingular :\ X X lnatrix of Poisson brackets \\'hieh \\'C \\'rite
(I. 16)
(\\?hcn COlllpllting (~xr~ , it is clear that onc 11111st not usc the constraint cquatiollSuntil after calculating the l)oisson bracket.) Sincc the dctcrnlinant of an
antisynlnlctric nlatrix vanishes if the dinlension is odd, \\'C conclude that thenunlbcr N of second class constraints ll1USt be e'veu. Since CY.r~ is nonsing'ular,
its in \'erse (~~~l exists and satisfies
\Ve novv proceed to construct fron1 any dynatnical variable ...-\. a lle\V variable
...-\.' \vhich has vanishing brackets 'lvitlz all secoJt(l class constraints. \\Te define
(I. 18)
and observe indeed that
A, -;\ { A· 1 C-- 1- ...J.. - , ~a f a~ q;",
(I. 19)
-{' m l--- f ' mt.-o- ..I.J.., rY J l ...tl.. , rY J - .
Note that {A', ~i} IS not necessarily \veakly zero.
- 11-
No\v \VC sin1ply postulate that the Poisson bracket of t\VO quantities i\ and13 111USt be replaced by the Poisson bracket of their priJnc(1 variables,
1.20) {A, B}~{-L;\" B'}.
Xote that although -L~' ~ .i\ , B' ~ B, the Poisson bracket {A', 13'} IS not\veakly equal to {A, B}. If \ve define the Dirac bracket as
(r.2r) { A B }* - {,\ B} --- r " I. C· -1 f r B· 1, - 1""1., l .Il.. , CPa J a~ l q;~, J ,
then \ve easily see that
(1.22) { '\ 13 (* (.\' B' I :--..-' {" \' 1·3 I ;::"'..: r '\ 1) 'l-Ll.., . f ~ 'l ..i.1.. , f'~ ...l.., f '-=- l"'l.., ) f·
If all l)oisson brackets are no\v replaced hy IJirac brackets, Eq. (1.22)tells us that \ve ha \"e e[fecti \'ely chosen to deal only \\Tith first class constraints.\Ve can set all second class constraints strollg(v to s('ro beca.use the l)irac 1)rackctof anything \'lith a second class constraint \"anishcs:
F'r0111 Eq. (1.22) and the definition (r. 18) of the prin1ed variahle, \\"C inllllcdiately sec that {..i.~, { B , C }* }* ~ {i\', { B', C' }}, so the Jacobi identity
(1.23) {..i.~ , {B , C }* }* { B , {C , -L~ }* }* ~r- {C , { ...-\ ,13 }* }* ~. 0
is satisfied \\Tcakly by the I)irac bracket. ~'1oreo\"er, using the definition (1.21)one can sho\v directly that (r.23) is actually a strong equation.
\Ve note here also the iterative property of the Dirac bracket. I f thenUlnhcr of constraints is large, \\"e 1l1ay a \"oid invcrting large 111atriccs by
taking a sn1aller second class subset of the constraints and C0111puting theprclin1inary bracket (1.21). 1'h('n S01l1e second class su hset of the rcnlainingconstraints is used in l~q. (1.21) \\·ith all brackets on the right hand side replaced by the prelin1inary bracket. Repeating this procedure until all constraints are strongly valid gi ves the same results as C0111puting the final IJirac1)rackct in a single step.
N O\V \\Te are equipped to understand n10rc clearly the nature of the effccti veHan1iltonian A in Eg. (r .9). If \ve set
(1. 24) H == fI' == He - {He, cPr.(} c;rl qJ(),
then
u~ (q , p) == - { He , cp~ } c;rl
and H(q ,P) is the physical first class replacc111ent for I-Il·, \vhich coulcl have
1)cen second class.
\Vith the choice I.2-l-) for A, it is clear that the Hatniltonian is still notcOlnpletely determlned: the equations of motion of the constraints are unaltered if \ve add to H an y linear con1binatiop of the I first class constraints ~t'
12 -
IS Eq. ~ I . J 5 a). \Vc therefore take as our total !-fani iltonian
(1. 2 5) 'l}' dJ, (q p)l Tl \' ,
\\·here Ii has vanishing hrackcts\\yith all constraints c\yen though it contains
I arbitrary functions '(I", Since the '~i do not necessarily havc vanishing brackets
\\yith canonical variahles \vhich are not constraints, \ve ha vc nc\y cquations
of 111otion
1.26)Ii i === {q£ , H } ..~ { qi , I-I' } + 'lj' { q£ , Yj }pi == {pi, 1-1 } ~ {pi, H' } -1- 'lj' { pi , Yj }
\vhich explicitly involve the 'l}," S0111e restrictions on the functional forIn of
the 'l',' and their tin1e derivatives in tcr111S of the rj,. and Pi 111ay fol!o\y fron1
1.26) (see, for cxanlp1e, I~q. (5.22)).
l'hero1e 0 f the Yi in Eq. I . 2 5) is to gcnerate infi nites in1 a I contact t ran
sfornlations of the p's and q's that do not affect thc physical state of the SYStC1l1
',l)irac, 1964; 13erg111ann and Goldberg, 1935). Hereafter \\ye \\'ill refer to
such transfornlations as gauge transforn1ations.
]'he arhitrary functions 'l'i in H occur hecause the original L,agrangian
possessed I /{tllf<g-l' (Iegrccs oj.free{lo711 associated \vith the first class con·:;traints
'~t" \Ve 111ay fix the \'alues of the 'l'i C:) by choosing explicit fOr111S for each
gauge
~~1.27) Ii (q , P , 7) ~ 0 , I == J " • "
and inlposing thC111 as constraints llot follo\ving fr0111 the Lagrangian.
The choice of gauges (1. 2 7) should bc~ 111ade in such a \va y that the con
straints f~,. \vill cease to be first class: the Inatrix { '~i , Ii} should be \vell-defined
and nonsingular. Then \vhen \ve replace all brackets in the theory by those
consistent \vith y" === 0 , Ij === 0, the arbitrariness due to the 'l'i in Eq. (r .26)\vill disappear. \,rc note that S0111e traditional gauge conditions in \'01 \'e velo
cities \vhich apparently cannot be recxpl-essed in tcrll1S of canonical coorcli
dinates and InOll1cnta; in such cases, \ve do not kno\\' ho\\r to conlputc the
n1atrix {~~i , 'Yi} and it seems that the L)irac 111ethod cannot be used.
In the end, \\'e obvious] y \vant to express the systeln in terll1S of the truly
independent canonical variables alone. It Inay happen that the obvious gauge
choices do not cOlnpletely reduce the phase space a vailable for particle Illa
tion dO\\'n to the size implied by the Euler equations. .~\dditional constraints
necessary to define cOlupletely the physical system ll1ay occur disguised in
the form of invariant relations. \\7e define the~i (q, p) to be invariant relations if
(r .28 a)
and
at 'T = 0
(r .28 b)
- 13-
(In Eq. (1.28 h), thc~ sign 111eans that all other constraints besides the ~i ha\~e
been set to zero). \\'e thus conclude fro111 Eq. (1.28 b) that ~l' vvill re111ain ~ 0
for all ~ if (1.28 a) holds.In variant relations differ from constants of the motion in an essential
\\'ay. Constants of the motion are generally used to specify particularsolutions of the equations of n10tion, \vhile in variant relations are insteadconditions \vhich lllust be satisfied in order for a solution to be considereda physical one.
In \Tariant relations can be \Tie\\~cd as ordinary secondary constraints
by using a Lagrange n1ultiplier )'i corresponding to each invariant relation~i ~ O. \\Tc sin1ply express the 1110nlenta in tern1S of the velocities, so that
;/ (q ,P) can he \vrittcn as a nc\v function;i (q ,~), and take
as our nc\v Lagrangian. The Euler equations arc
d ClL 8L+ ~ [\ ~i (ti 8~i ?~£) -f--~lT l'5q} 8~J
-8q)
l
d),z" y-c,i""' . dl:· d~,__ ]+ ~.~l + )'i
~t
- 0cit "-:z "". dt :l'all} ell]j
d 8L 2Lp~-
I[~i r~cit 8),i dAi
- ~ 02
These gi ve back the original equations of n10tion, except that no\v the constraints;i ~ 0 occur as secondary constraints follo\ving from the primary constraintsP;. =~ 2L/2~i ~ o. l'he net effect is to make the ~i into ordinary constraints\\'hile the )'i disappear froIn the dynamics.
I-tet us no\v suppose we wish to in1pose all a vailable gauges and invariantrelations, elinlinate all arbitrary functions froln the equations of motion anddescribe the systeln only in terrns of the truly independent phase space variables. If \ve have properly chosen the gauge constraints and invariant relations,all constraints vvill now be second class.
Redefining epC( to include all of the constraints, gauges, and invariantrelations, vve should find that the matrix C-x(3 == { epa , ep(3} is nonsingular. ThenEq. (1.2 I) gi ves the form of the Dirac bracket consistent vvith setting all constraints strongly zero and using only the proper variables to describe the Han1iltonian dynaInics of the systen1. For sufficiently simple systems, this finalDirac bracket provides the starting point for canonical quantization of thesyStCll1. This point is discussed further in Section I. F.
B. I-IAMILTON VARIATIONAL PRINCIPLE \VITI--I CO~STRAINTS
\Ve no\v develop the ideas of the previous section frorn a slig-htly differentvie\vpoint.
- 14-
We begin by considering the Hamilton variational principle In phasespace, \vith the pfsand qfs considered as independent variables
o = ~S = ~f (~ 1" dq, - H d-r) =
=~f(~ ~ CPidq,-qidP')-Hd-r).
\Ve then find I-Ianlilton's equations,
8Hdqi
No\:v consider the entire set of constraints, gauges and invariant relations\vhich restrict the phase space a vailable for the particle 111otion. rrhe 2 n
canonical variables pi , q/ can be reexpressed in ter111S of the 2,12 independent
variables
Z," , z" == I ,. . ., 2 'IJl
and the 21Z-27Jl contraints
i == 27/'l 1 , ... , 2/l .
l'hus \ve n1a v express pl" and qi as functions of the z's and of 7.,
(1.3 1)
Note that while pi and qi are by definition not explicit functions of 7, the 2 2"
111a y be expIicitIy 7 - dependent ; the expIicit ,:'sin I~q. (1 ..3 I) are necessar y tocompensate for any 7-dependence of the Zi.
)Jo\:v we consider the z/sand '7 = Zo as a set of independent variables,neglecting for the 1110nlent the vanishing of the constraints. rrhen \ye find
(1.3 2 )
where
(1·33)
n 211
~ 2: (pi dqi - qi dpi) == 2: Ca dzcxi=l a=O
\Ve see ilnmediatcly that
(1·34)
is just the Lagrange bracket of the ne\v set of variahles Za, including constraints,\vith respect to the old sct of canonical variables qi and pi. If \ve no\v define
-15-
the Poisson bracket as
(1.35)
we find the follo\ving properties:
2n
2: {Zk , Zi } (Zk , Zj) == ~ijk=l
(1·37)
Equation (1.37) fo11o\vs fron1 the fact that
(1.3 8)
The equations of nlotion for Zi are no\v
(1·39)
\vhere (qi , pi ,'1") are treated as the independent variables \vhen conlputingthe right-hand side of the equation.
l'he action principle now can be written
N C\V, ho\\~ever, let us require that the constraints tJi ~ 0 hold throughout thevariation, so
~CPk-2m == OZl. == 0 ,
'The restricted action principle is thus
k == 2 JJZ - t- 1 ,. . " 2 II .
o = as = 0 j·(~ Ci dZi + (Co - If) Ch) .
The variables Zi, i == 1 , ... , 27Jt are independent variables \vhose Lagrangebrackets are given by Eq. (1.34).
Next \ve sho\v that the Poisson brackets of the independent Z/ are justthe Dirac brackets. If \ve define the Inatrix
a , b == 27Jl 1 , ... , 2lZ ,
\ve find that the Dirac brackets are
(I .43)21Z
{t;:I Z t * - {z ry t -- ~ { t;:I t;:I t C-b
I{ <'":' t;:I t"'"' i, j J - i , 4Jj J ~ .v i , "-' tl j '(. "-' b , "'"'j j •
a,b=:!.nz-j- 1
-16-
No\v \\"C ll1ultiply by the L.agrange bracket, so
~m ~n
~ (Zi, Zk) {Zi , Zj }* == ~ (z£ , Zk) {Zi , Zj}* ==i=1 i=1
2n
ajk - 2: akaC~l{Zb,Zj}==a,b=~m+l
== ;)jl.' ,
\vhcre k, j == 1 " .. , 2,ll and the sun1 can be extended fron1 2JIl to 2n h~cause,
hy :Eq. (1.43), {Zi' £j}* ~= 0 \vhen i == 2JIl I, .. " 2n. I'hus the I)iracbrackets are the in \Terse 0 f the restricted Lagrange brackets follo\ving fronl I~q.
(r .42), \vith only 2 III variables; by definition, the Dirac brackets lTIUst therefore be the Poisson brackets of the restricted systen1. In other \vords, a simplerestriction in the nun1bcr of variables appearing in the Lagrange bracketscauses drastic chang-es in the inverse of the Lagrange bracket nlatrix; thecanonical I)oisson 1)rackets are changed to Dirac brackets, \vhich can beexpressed in terms of the original canonical Poisson brackets only by using
Eq. (1.43).It is no\v tri vial to pro ve the iterative propert y of the Dirac brackets
mentioned earlier. Indeed, successi ve restrictions on the range of variablesof the Lagrange brackets gi ve the san1e final restricted Lagrange brackets,and hence the san1e inverse.
C. EXTENSION TO INFINITE DEGREES OF I~REEDOlVI
\Ve no\v establish our conventions for dealing \vith classical field theories
(see, for exalnple, Goldstein, 1950; Kundt, 1966). The discrete label i on qi (t)no\v becomes a continulun label x plus additional discrete labels i\, so qi
can be replaced by the field cDA (t , x),
(I.4S) L (qi , qi) --+ 2" ( <I>A(t , x) ;
T'he Lorentz-invariant action functional is then the integral over theL.agrangian density:
S [<I>A(x)] = rd4x2 (<I>A (x) , 21-' <I>A(x» .
.i\t this point, \\"C lnust decide on a n1etric convention for treating co variant and contravariant vectors in four-dilnensional space-tin1e. \\7.e choose
(T .46)
-1 0 0 0
o 100
o 0 I 0
000 I
x[J. === (t , x) X!J.. == g[J.'J XV == (- t , x)
(r .48)
- 17-
i\l1 of our conventions are listed in i\ppendix i\., \vhere thev are corn pared
to other con ventions in comn10n usage.
I'he canonical I110n1enta arc no\v
and the Euler equations are \vritten
"'\ 32~' !~~(8~--<i5A)
If the Lagrangian is translation-invariant, \ve find that the canonical
energy-lTIOmentU111 tensor
n~lV _V c --
is conserved,
(r .50)
If \ve take the canonical Poisson brackets to be given symbolically by
(1.5 1)
\vhere 0 (x - )1) is the Dirac delta function, then the 11: amiltonian IS
(r .52) H - /.. d3 eGO - (. d3 (A ( ) d<f)A _ (fJ ~\c - X c - X 1t t ,x 3t e>o-.. ) •
U sing the Poisson brackets (1.5 I) ,\ve see that He generates the titHe c\"olutinn
of the canonical variahIes through functional deri vati \"('s
dnA (t ,x) _ { A H") _ . BH/--d-I---- - it , c J - - ~cDA (t ,~
N O\V suppose the systen1 has constraints
a = I,"', I',
following fro111 the fOr111 of the Lagrangian and fron1 the equations of motion
of the primary constraints. I'hen the arguments given earlier lead us to con
clude that the energy-1110tTIenturn tensor is an1biguous. Defining O;w as thefirst class energy-1110mentum tensor, \ve replace the canonical tensor by the
total energy-1110mentum tensor
(r .55)
-
2
- 18-
\vhere the Yi arc the I first-class constraints. ...;.\s before, agreenlent \vith theequations of motion nta y rcstric1- the v~~'J (x) some\vhat. F'ixing the gaugedegrees of freedom \vill fix the (l)~~') and elinlinate the arbitrariness due tothe l/~·l') fronl the equations of motion
No\v \ve \vrite the ren1aining N == T' --- I second class constraints as
rt. == I,···, N
"[hen \ve can define consistent Dirac brackets by conlputing
(I .36) C:l f3 (.'i- ,y) == { q:>~ (t , x) , q:>~ (t , y)}
and its InyerSC, \vhich obeys
(I .57) rd3 zc~tex , z) cy~ (z ,y) = Jd3 zCx"y (x , z) c:;l (z , y) === ~xi3 ~3 (x -- y).~J
Equation (1.22) is then replaced by
(I .58) {A (t , x) , B (t , y) }* == {A (t , x) , B (t , y) } -
rd3 zd3 w{A(t, x) ,rp,,(t, z)} C;~\z ,w) {rpf;(t, w), B (t, y)} .• J
(1.60)
I'he brackets for the systenl \vith all gauges and invariant relations imposedcan of course be conlputcd in the Stune ll1anner.
I t should be noted that the canonical generators of other synl111etry transfornlations of the Lagrangian may also require the addition of multiples ofthe first class constraints if gauge conditions are not ilTIposed.
F'inally, \ve observe that one Inust generally supplement the definitionsof the continuous constraints (1.54) viith appropriate boundary conditions,lest Cx ;3 (x ,_v) become singular. Our treatment of such nlatters here \vill bepurely praglnatic,\vith no attenlpt to rigorously define the nature of the functional spaces invol 'led.
D. OTHER POISSON BRACKET SURFACES
Instead of con1puting the Poisson brackets (1.51) at equal times, \ve mayHI fact choose a variety of surfaces (Dirac, 1949 a; I(ogut and Soper, 1970;F'ubini, I-Ianson and ]acki\v, 1973). Let
specify· a suitable surface. Then the Lagrangian should Le re\vritten in terlnsof d).\ === 2<PA (X)/27 and the action functional should be expressed as
S = ( d" d3 cr!£' (cD,\ , <DA , 2<DA /2o) = rch L'__' oJ
-19-
\vhere the (i'S are a suitable set of three variables \vhich span four-space \vhencon1bined \vith 7. rrhc canonical mon1enta are
(I .6 I)
l'he Poisson brackets thus take the forIn
(1.62)
In order to understand the nature of the Hamiltonian for a system usinga general Poisson-bracket surface l~ (x) =-= '"'=', we must generalizc the treatn1entof thc generators of spacetilne transformations. Let f;~ (x) generate a spacetin1ctransformation labeled by the index a,
(1.63)
For the present discussion, let us a void the complications of "t'-dependentHamiltonians by assulning that f;~ (x) is a sYlnmetry transformation of theLagrangian, so that \ve ha vc a canonical ~oether current e~ (x) \vhich is conserved, 2!J. O;~ (x) =-= o. 1'hen the canonical generator of the transforn1ationcan be shov~'n to l)e
where
(1.65)dQad=r==o.
\Ve remark as usual that before we impose gauge constraints, e~ is ambiguousand can be replaced by
(1.66)
when generating the equations of motion.The Hamiltonians or (iynanzical generators of our system consist of those
space-time syn1metry generators \vhich change the Poisson bracket surfaceF (x). The other, kine1natical generators are those which leave the surfaceF (x) unaltered. For exalTIple, in the conventional equal-time forn1ulationwe have
surface: F (x) =-= .-:r0
dynamical generators: po , 1\1°£ (boosts)
kinelnatical generators: pi, lVI(j (rotations).
The Hamiltonian is takcn to be H === po because po generates the transforrna
tion from one surface x O == "r to another.
- 20-
In the null-plane forn1ulation of the dynan1ics (Kogut and Soper, 1970; seealso ...'-\ppenclix A), \VC have
surface: F (x) == :r - ==2
cl ynan1ical generators: p- ==
kinen1atical generators: P+-
3 0 -Z" I 3· o·2 (1' - l' ) ,M = yz (M' I - M ')
2
M+ i = /2 (M 3i + MOi) , M'j',
\vhere i and j take on only the values 1 or 2. The generator M+- == MO:3
is technically dynamical because it generates a scale change of x"; hovvever,it is son1ctin1es gi ven special treatlnent since it lea yes x === 0 unchanged.
E. DVNAl\fICS ON CURVED SURFACES
Dirac's technique is particularly suited for studying the Hamiltonianforn1 of a theory in \vhich states are defined on a general spacelike surfaceand not just on the special surfaces considered in the previous section (IJirac,1951, 196-.+). l'hc basic idea is to introduce a systenl of curvilinear spacctinlccoordinates uP- :=.: (1£0 , u1 , Zi 2 , u 3) into the theory in such a \vay that the equationU O =:-:: constant defines a generic spacelike surface. One then forn1ulates allthe d ynanlics in ternlS of the up· instead of the original coordinates x p-. Note
that one is also introducing arbitrary coordinates 1£1 , u 2 , u 3 on the surface.
l'his last step becomes C01l1pulsory if one \vorks (as in Chapter 7) in a generalRienlannian nlanifold \\?here no natural choice for the spatial coordinates(such as uf == xz" in flat spacetime) exists.
The procedure (" pararnetrization") used for incorporating arbitraryspacetin1e coordinates into a field theory parallels very closely the forn1tdationof the d ynan1ics of a particle \vith respect to an arbitrary tin1e scale. Considerfor sinlplicity the case of a non-relati vistic free particle for \v hich the action is
(1.67) t I ( dX)2 !'
S [x (t) ] = Jdt 2- m di = JL, dt .
1'0 express the problen1 In an arbitrary tinle scale U, one re,vrites (1.67) Inthe forn1
(I .68) , I'· I ( dx ) 2( cl t ) -1 Jr.S [X (u) t (1{)] == du -- JIl ------- ---. == L" du
\ ," ..J :2 d u du u,
thereby introducing the original tin1e coordinate t as a nc'Zu (lyJlaflzical l'ar/able
on the san1C footing \vith the position x.
----21 -
The action in the fornl (r .68) is no\v invariant under reparanletrizationsu --->- feu). .i\s a consequence, the I-Iamiltonian
(r .69) H _. eltIt - PI du
vanishes identically. (T'hc I110tnenta in (I.69) are defined hy PI == ?L'l)2(dt/du), etc.... ). ()ne also gets fro111 (1.68) the primary first class con
straint
(I .70) K [I , x ,Pxl ~ 0
(r .72)
\vhere In this si111plc case the quantity K is just
I( == + P.;/2 11Z •
rrhe constraint (1.70) 111USt be added---I11ldtiplied by an arbitrary function--
to the (vanishing) canonical Harniltonian (1.09) to get the total I-Ia111iltonian\vhich then reads just
H == N£
and vanishes \veakly due to (1.70). l'hc arbitrary function N describes then
the rate of change of the physical tin1e 1 \vith respect to the arbitrary para
111cteru.1"he steps taken to paran1ctrize a field thcory follcnv the pattern rc\·ic\\Ted
for the particle case. One thus introduces the four l\Iinko\vskian coordinatesy1.L as nC\\T fields on the SaI11e footing \\Tith the original fields <VA of the theory
at hand by re\\Triting the action
S [<I>A (x")] = Jd4 x2>, ($,\ ; 2$J,/2x IJ.J
in the forn1
(1·73)
\vith
S [$A (u") , Xl' (u")] = Icf4 ug;,.J
rrhe action \vrittcn in the forlll (Y. 73) is invariant under reparan1ctrizatiollsz/Y. ~ fX ancl the theory has t;1US Lecon1e " generally co\"ariant ". ...-\.s a
consequence of this in variance the Hanliltonian is
(1. 75)
and \';lJ1ishcs identically just as it did in the particle casco (The momenta
appearing 111 (I, 75) arc defined by Tell. = i) [J J.t;, d 3U 1/ i) (2xu j2uO) , . .. etc.).
- 22-
...t\lso, one no\\7' gets not just one primary constraint as in (r. 70) but fourprilnary constraints per space point vvhich are of the forn1 (4<-)
(r .76) 7tfL(U) + KfL(U) [<PA , 7tA
] ~ 0.
A 11lore convenient, but conlpletel y equi valent, fornl of the constraints (I. 76)is obtained by projecting thetu into one norn1al c01l1ponent
(r .77 a)
(Here nfL is the unit norn1al to the surface, a functional of the x fL) and threetangential ones:
(r .77 b)
r[he ad vantage of the projected version (r .77) of the constraints is twofold.l~irst of all, \ve replace the high] y arhitrary description of the motion in ternlSof the coordinates u'X by a description in tern1S of deforn1ations of the surfaceparallel to itself (governed by "~.) and orthogonal to itself (governed by ~;ttl')
\vhich has an invariant geoluetrical rneaning. Secondly, the change in thefield variables under a displacen1ent of the surface parallel to itself consistsonly of the response of the field to a change of coordinates in the surface andhas no dynamical content, being determined cornpletely" by the transforn1ationcharacter of the field. \Ve separate in this ~Tay the part of the problcrn thatis trivial fronl the truly dynalnical part \vhich is contained in ~YtJ.. l\loreover,\vhcn the constraint YtJ. ~ ° is in1posed as a restriction on the HanliltonJacobi principal functional S in the classical theory or on the state functional,~ in the quantum theory then, thanks to Eq. (r .83 a) belo\v, the constraints.~. ~ ° follo\v as a cOJZsequence of YtJ. ~ 0, as has been sho\\rn hy IVloncriefand Teitelboinl. (r 972). 1"'his situation is to be contrasted \vith the forn1ulationbased on (1.76) in \vhich the t\t\TO aspects of the problein are mixed and onehas to deal \vith four equations of the same degree of complexity.
1'0 obtain the extended Hanliltonian \ve add no\v the constraints (1.77)to the original (zero) Harniltonian (1.75). The Hamiltonian now reads
(I .78) N£ Ye.)1 •
'The arbitrary functions N1(" lapse") and N i
(" shift") describe the \vayin \vhich the initial surface is deforn1ed into another infinitesin1ally close one(fig. I. I). SO, if ~~ is an arbitrary functional of the canonical variables ofthe theory (\t\Thich include the x P-) one has
F[a t] - F[a] == ouo {F, H}.
(*) In the particle case the quantity K appearing in (1.70) is the total energy. For afield, the K p_ appearing in (1.76) are given by appropriate cOlnponcnts of the energy-momentum tensor. We refer the reader to Dirac (I949b) and Kuchar (1974) for an analysis ofthis point.
--.---
23 -
1'he I~oisson bracket in (1.79) is defined as
(1. 80) ,r 3 ( of{F G} == (I u ---------, .J ' o(I)A (u) -t-
of oC; 'F' ("" )-"--,- -- ( <--)]"! •07:"~ (It) \ )
In particular, \vhen applied to the canonical variables the111scl \·cs equation
(1.80) gives
(1.81 a)
(1.81 b)
(1.81 c)
r _A (0 i) B (0 ,'i\} __l I\, U ,u ,TC \1£ ,1t ) - 0 ,
(ih / 0 ,") __ R ( 0 , I '", I ~H ~ ( t' , 'i)l 'vA (J£ ,U ,I" U, It ) J -- 0 A 0 iJ£ , U
-
and sin1ilarlv for the x!.L and itp_. l'he a-function in (1.8 I c) IS dcf1nccl hv
for an arbitrary scalar testing function! and it lllay be considered to transfornl
as a scalar at u and as a density at u f• ( ...\ctually since a has point suppurt
the only' thing that 111attcrs here is that the sun1 of \veights at u and u' he unity
the allocation of ',veights to both points is other\vise arbitrary).
8 u0 Nr (u) ~r (u)
Fig. I. I. IJefonnatll1l1 of a coorclinatized surface. Starting fronl a given surface 0'. of constant uO-tiIne, on \\-hich a coordinate systcnl is defined, one goes to an infinitesill1ally
tlisplacecl surface cr' which corresponds to a snLtlI change ouo by lneans of a clefonll;1tionN (u) = :\' 1 (u) n (u) :-:,r(u) er , \vhere e) is the tangent vector to the r-th coordinate line.
N ate that the defonnation defines cr' not only in the geoJnetrical sense but also sets a coordinate
systell1 on cr' by the prescription of giving the saIne spatial coordinates to the points at thetail and at the tip of the deforn1ation.
The hrackets 1.81) are equal tin1e brackets in the tin1C UO \\'hich 111('ans,
for cxan1ple, that (DA and (DB have \·anishing hrackets for an arhitrary spacclike
separation.
Equation (I.Ro) can also be used in principle to find the brackets of thecanonical variahles for nonspacelike separations. '1'0 do this, one first solvesthe equations of llH)tion and then expresses (D A (u'O) and 7t
B (1£'0) as functionals
of the " initial c()~lditions" (DA Cuo) and itB (uO).
Let us finally n1ention a fundan1cntal property of.Y~ and <.Y~. defined
in (r. 77), nan1ely'" their Poisson hrackets (P.B.). It turns out that \vhereasthe form of the ~.L varies, of course, from theory to theory, there is one important
- 24-
feature COlTInl0n to all such ,Yt'(J.' naITIely the fact that the P. B. of any two ofthcn1 is a linear combination of the Y~1. themsel ves and this linear cOllzbinatio71/s t//t sante for all tlzeories. What \ve arc en1phasizing is not the fact that theP. B. of any t\\'O constraints is a linear conlbination of the constraints (firstclass property)-this merely guarantees the preservation of the constraintsduring the c\'olution of the SystClTI. \\That is remarkable is that the coefficientsin this linear con1bination (" structure constants ") are universal. As a matterof fact one can derive the brackets of the ,~1. fron1 only tvvo assumptions (Teitclhoin1 1973 a, b), nan1ely: (i) The oY(u. are first class (other\vise the theoryis inconsistent to start with) and (ii) IIamilton's equations are integrable,that is the change in the canonical variables during the evolution fron1 a gi veninitial surface to a gi yen final surface is independent of the particular sequenceof intcrtl1ecliate surfaces used in the actual evaluation of this change. (A consistt~ncy' requirelnent tern1cd by ICuchal" "path independence of dynamicalevolution "). The result is (I)irac, 1948, 195 I, 1964; Sch\vinger, r962 b)
(1.83 a)
(r.83 c)
{ ,YC;. (x) , ,~(x')} === ~ (x) 0 ,r (x , x') ,
{oY~(x) ,~(x')} === o~(x') 0 ,s (x ,x') 4£J (x) ~ (,- v')Jl S \ 0 ,r \~\, ,A, •
rrhe quantity .Yt'r in (1. 83 a) is defined by ;Ytf === grs ,Y~ \vhere grs is the metric
of the uO === const. surface. (~ote that grs depends on the canonical variablesX!1. \'ia c£[rs == "'Jet.r3 (2x 'l./2u}') (2x(~ !-;u5
) , ,vhere "lJcx;3 === diag (- r, 1, I, r).Lastly, it should IJC cll1phasized that the abo\'c mentioned \vay of derivingthe hrackets (1.83) sho\vs directly that those equations apply equally \vellfor any generally co\'ariant field theory defined on a Riell1annian spacetin1e.\\le observe that I~qs. (r .83) hold also in theories \vhich are" already parameterized " (i.e. generally covariant to start \vith) such as general relati vity,for \vhich the generators ,Yf'(J. turn out not to be of the general form (r. 77),as \ve shall see in Chapter 7.
I:;. QUANTU~1 rrI-IEOR Y .AXD CANONICAL VARIABLES
()ne of the I11ain 1110ti vations for de veloping thc !-1 an1iltonian d ynan1icsof a constrained classical systenl is the desire to deduce the analogous quantun1t11cchanical systC1l1; there is often a very close connection bet\veen the formof the I)irac brackets and the quantu111-111echanical conl111utators. I-Io\vever,in practice, the usual prescription
I.8-1-)
Inay he plagued hy ordering an1biguities on the right -hand side of the expression for the IJirac hracket. Suppose, as in Section 1.13, \ve let the Zi (r) be the2 III independent variables in terlTIS of \vhich the Dirac brackets are expressed.
- 25-
Then, according to general theorenls on canonical systetns (Jost, 1964; Kunzlc,1969), in the neighborhood of any point l~ in phase space, there exists a localset of 2 1JZ variables
(1. 85) f/(Z,7)
\vhich ohevs canonical brackets
(1 .86) { _. - }* ~ipt, qi == - OJ •
.:\ glohal treatnlcnt of phase space raises nontri vial questions \vhich canbe solved only in the fralne",rork of algebraic topology (Abraham and Marsden,1967). Ho\vcver, here \vere interested only in presenting a heuristic discussionof the forn1al aspects of the theory, and so \vill assun1e for sirnplicity thatpl' and qt' may be treated as global phase space coordinates. (T'he extended)J e\\~ton-\Vigncr coordinates gi ven in Chapter 3 arc an exan1ple of a situationin \vhich the glohal prohlct11S can be handled \vith kno\vn techniques).
In general, the 2 lIZ independent z/ s \\~ill he certain 7-dependent functionsof the canonical coordinates.
In ter111S of pk and qk, the action principle (1.42) can he \vritten
(1.87) o = oS = o. / · (-;- A~ CP d qk ".~ q'" dP) .._- Hd 7 ) •
Repeating the entire argu111cnt of Eqs. (1.32-40), \ve have also
(1.88)
\vhere no\v everything is expressed in terrns of the ne\v \~ariables qk and jk.Thus
c === c === I ~ ( - k drJ~ _ - d;k ).ex C( 2 ~ P Jz qk Jz
k=l ex ex
H === Co -- Co + H ,
where ty, === 0 " . " 2 1JZ and Zo :=: 1'. T'he integrands of }:qs. (1.87) and (1.88)diff~r at 1110St by an exact differential which can be re1110ved by a suitablecanonical transformation on qk and pk.
A specific choice of the variables qk and pk detern1ines Co and hence,fro111 (I. 89), H. T'his choice fixes the explicit 1'-dependence of the Zi appearingin the equation of motion (1.39) \vith (q, P , H) ~ (q , p ,H).
Conversely, \ve Inay always perform a suitable 1'-clependent canonicaltransformation on the variables qk and pk \vhich changes the Hamiltonianinto any desired function. If the Dirac brackets of the z/s do not dependexplicitly on ":", then the best choice for the I-Iamiltonian is clearly the one\vhich assigns no explicit "r-dependence to the z/ s.
- 26
'The I--Ianliltonian J-I generates the equations of motion
-~~- = { qi ' 11 }* = -~11.9°)
Since all time-dependence of the canonical variables IS itnplicit, one Inight
call this a "Heisenberg picture" forln of the equations of 1110tion.
T'hc identification of q£ and p£ is in general necessary to solve the ordering
proble111s inherent in the transition fron1 classical mechanics to quantum 111echa
nics \"ia the correspondence principle. 1'he final goal, then, \vould be to find
a co\yering of pbase space by a set of neighborhoods, each one possessing a
sct of regular coordinates qi' j£ obeying I~I .86), together \vith the canonical
transfor111atiot1s relating the coordinates in the intersection of di fferent neighborhoods .
.:\ Inethod \\"hich s0111etin1Cs \\·orks in practice is to seck a c0111plctc sct
of independent d yna111ical variahles ...-\~ \\"hich have vanishing [rlnonz"cal bra
ckets ,\'ith all first class constraints follo\\"ing fron1 the forIn of the Lagrangian.
L..et us as~;;ume for the sake of the present argurnent that there are no second
class constraints~or, equivalently, that \ve are \vorking \vith a prelilninary
set of hrackets consistent \vith setting all constraints to zero except the first
class constraints and their gauges::' rrhen if \\"C dcnote by '~i the I independent
fi.rst class constraints and by y" the I corresponding gauges, \':c \\·;-111t variables~.-\: such that
,. \<I .9I)
:\O\V let
1.92 ;
?rx he the sct of 2 I ~'s and y's:
/1' "-?rx == ('1" 1 , ... , ''iI' Y1 , ... , YI)
]'hcn since the Yz" are first class, the in \~crsc of C",;3 === { ?'l. , CPi~} can be \\·ritten
in 1;< I blocks of the forIn
-1 [DllCar, ~ 11,
-- 21
I~--l on1 Eqs. (I.9 I) and (r .93), \\"C ilntnediately see that
{ ;\ * .\ * 1* cv { /\ * " * ")...'l..(l , ....'l..b j ~ .Ll..a, ...l..b r
1'"lgar{llcss 0..1 the gauge clzoice. (Xote the contrast \vith the properties of ~-\' clefi.nedlJy ~"I.I8~;'
In practice there may be a particular gauge choice for \vhich there exist
variables ~-\: satisfying
(r .95)
-- 27-
in addition to (1.91). Yet another possibility \vhich son1etinles occurs is that
neither (1.91) nor (1.93) is satisfied, hut C;r31 is arranged so that (1.94) holdsfor a particular choice of independent variables.
We see that the .L~: are a logical set of variables to usc in d~"\fining thequantization procedure. The ordering probleI11s \\Thich occur in I11aking the
transition froll1 classical brackets to con1mutators are likely to be much less
severe for variables \vhich obey (1.94). t T
nfortunately, no general procedurefor finding the appropriate variables is a vailable at this tin1c .
.,.~ different procedure \vhich a \·oids entirely the usc of Dirac bracketsis to consider a I-lilhert space :Yt', not necessarily \vith positive-definite n1etric,of \Jvhich the I-lilbert space ~~of physical states is a subspace. ()n ,Ye' one defines
operators for each unconstrained variable, \vith c0I11mutation relations corre
sponding to the original Poisson brackets. The constraints epee no\v appearas non vanishing operators on .Y~' \vith \?anishing Inatrix elements hct\vcen
any t\\?O physical states \ a , b
< ai W I bIIX o.
'rhe constraint conditions 1l1ay appear also in the stronger forn1s
ep~ a == 0 or < a i qJx == 0 .
I30th Eqs. '~ 1.96) and (1.97) arc referred to as su hsicliarv conditions. Suchconditions are inlposed on states of ,Yt)f in order to select the suhspace :If of
physical states \;\,:ith positi \'e norn1. 1"'he resulting theory has a ver:y synlnlctric
appearance, hut the price paid for simple cOlumutation relations is the introduction of a larger manifold states.
c~s our first application of the Dirac approach to constrained Han1ilto
nian systenls, \ve examine the relati vistic spinless point particle in the manifestly Lorentz-invariant formalism. The action is taken proportional to thepath length,
(2. I)
2
S = - m Jcis == - m jI I
It 1
dx~ dx' )2 ,
\vherc \\'"C recall that our rnetric convention 15 I == -- gOO == gIl == g22 == g33
I'hen \ve choose an arbitrary monotonic paranleter 7 laheling the particle'sposition on its \vorld line and define
(2.2) dxP- (-r)u~==--- .
d't"
- 28-
'rhus the Lagrangian is
and the action
L== 'Ill
IS in \-ariant under reparan1etrizations 7 ~ 7' I t is easy to check that
Det { :2 IJ'u!J. 2u" } = 0, so that L is indeed a singular Lagrangian requiringthe usc of Dirac's methods to define the Hatniltonian dynan1ics.
I'he canonical 1110n1enta are
'[he J~ulcr equations and their solutions are
(2.6 (7)
or eqlli\-alcntl~'
:2.6 b) cl2X~71Z --- ---- == 0
ds2
})p.
IIIs ,
\\-hcre s is the path length defined hy the integral (2. I),rrhe Poisson brackets for variables at equal 7 are defined as.
so that
(2.8 a)
(2.8 b)
{ I)~ (-) 'l~ (.-\} _ ~'L\. ,..1.--'J \ ~,) - --- 0v •
{ P~ ('-'1 p'J (,-\} == 0\W), \~)
Since we know the solutions (2.6) of the equations of motion, we may considerEqs. (2.8 (7, b, c) to give the brackets at ':" == S == 0 and C0I11pute the bracketsfor variables at different points. Only (2.8 c) is changed:
(2.8 d) { Xl). (0\) x'J (s ')} == gP.') (_~__ ) •" /' \, / Jll ,
='Jcnv \ve observe that the canonical Han1iltonian 'Zh7Jlz's,!zes,
(2.9)
1IIIIIIIIi_---------------------- 29-
and so I-I c is identically the prilned Hamiltonian, I-I c == H' == o. 1'he vanishing
of J--I c is attributable to the fact that the Lagrangian is h01110gcncous of dcgreeone in the \"clocities. l'hercforc the m0111entu111 p:t is homogeneous of degree
zero in the velocities and no unique solution u!J. (x , P) exists. 1'his is typicalof a singular Lagrangian; the homogeneity of l)f
t is seen fro111 Eq. (2.5) to lead
dircctly to the first class primary constraint
(2.lO)
'The total Iialniltonian rnay then be taken to be
(2. I I) 21n)~0.
H correctly gcnerates I-Ian1iIton's equations of 1110tion In the arbitrary paratcr 7:
(2.l2 a)
(2.IZb)
Exa111ining Eq. (z. l2 a), \VC sce that v can be expressed in ter111S of the velocitiesuY', so that H finally heco111cs
I-I ==21lZ
There is, ho\vever, still an arbitrary function in the system because the scaleof 7 (and hence the scale of u P-) has not heen fixed relati ve to x!.L.
B. GAUGE CO~STRAI~T
X O\V let us 111ake use of the gauge freedo111 in the action to fix the scaleof 7 and elilninate all arhitrary functions fro111 thc system. \'/e choose
27Jl ~ 0
Note that other choices of the gauge ep2 may be equally suitable (Dirac, 1949).Our old first class constraint p2 1n2~ 0 is now second class. We n1aythus C0111pute the matrix
(2. I 5)
and its In verse,
oCe<[3 == { epa , CPi3 } == 2 pO
-2 pO
o
(2. l6) c-1ai3 == 2 pO
- 30-
N OvV \VC replace the Poisson brackets (2.7) \vith the Dirac hrackets
(2.17 a)
\vhich are consistent \vith setting the constraints (2.14) strongly zero. Inparticular, we have at equal T == xO
(2.17 b){ Plt V}* ~I.V + fLO Pv
" x == -g g pO
{ P!.L PV}* == { 1.L v }* ==, x ,x o.
We can no\v elin1inate the extra variables in the canonical systenl by usingthe constraints (2.14) to set
(2. I 8)
The choice of H == !Jo to replace the original vanishing Hamiltonian is justifiedby the fact that it generates I-Ian1.ilton's equations of Illotion in the nc\\? evolution variable t:
dA.
dt
The velocities are thus gi ven by
JA. -L f.~ H 1*dt i l"" f·
(2.20 a)
\vhile
(2.20 b)
(2.20 c)
cL1;£ == { z" H}* == { . i pO}* == pz"elt x, x, po
__?pt~ == {pt' H J'l.* == 0dt ' ,
T'huspi (t) == I)i (0)
Xi (t) == Xi (0)pi (0) t
(p2(0) + nl2)~ ,
and the unequal tin1e bracket system IS
{ pz" (0) , xi (t)}* == D1j'
(2.21) { pi (0) , pi (t)}* == 0
{xi(o) , x.i(t)}* == t(o~i(p2 1· JJz'!.) _ pz·pi)j(p2 -t-- ?n'!.):)/'2.
-31-
\Vc note for cOlllpleteness that our system is Poincarc-co\yariant, 'fhetranslation generators p,L are constants of the n10tion, as are the con1ponents
11°'· == tpt" - Xi 11 == tpi - Xi (p2 1n2)~-
l\l ij == Xi pi _ xi pi
of the Lorentz transformation generators M!-1'J. (Note that {Mo i, H}* =1== ° but
dlYIOijdt == 0 due to the explicit tilTIe derivative in Eq. (2.19).) Defining the
Lorentz group structure constant
\ve verify the I)oincare group algebra:
{ pP., p'J}* == °{MClr3, p!-L}* == g~La. pr3 _ g!J.r3 pC(
{l\l fL'J, lVIC(r3 }* == C~~C(~ M°'r.
C. QUANTUlYl lVIECITANICS
]'he system of quantul1l,-ITIcchanical operators corresponding to the classical
relati vistic spinless particle can be deduced directly fro111 the Dirac brackets
of the previous Section. \Ve first define the Hilbert space nor111 as
(2.26) (rpJ rp) = Jrp*(x) rp(x) d3 x.
'Then adopting the convention (1.84), vve take Xi and pi to be operators satis
fying the equal tinle C0111111utator
(2.27)
(2.28)t == ,.,yO == parameter == c-number
H == pO == + (p'2 -t- 7JZ2)k == q-number .
J-Ian1ilton's equations hecon1e
(2.29)dAdt
i [1-1 , -L~] ,
so that the Heisenberg-picture equations of lTIotion are
(2.30 )
- 32-
Since the algebra (2.27) n1ay be realized in the Schr6dinger picture as
\ve find the following nonlocal Schrodinger equation:
Hm (t , x) :::= (-_ n 2 2)~ (t ) . 89 (I, x)T V nz c.p ,x == z 81 .
Iteration of Eq. (2.32) gives an equation of the same form as the local KleinGordon equation.
Requiring the I-Jorentz group generators to be hernlitian vvith respectto the norn1 (2.26) gives
\Ve find that the J)oincare algebra holds,
'i [MQ.~, p!J.] == _gP.rx p~ _+_ g!J.f3 pC(
and in addition
Equations (2.35) and (2.36) are taken by definition to lnean that x j transforinsas the space part of a four-vector (Jordan and l\1ukunda, 1963).
rrhe extra ternl in Eq. (2.36) beyond that required by the pure Lorentztransforlnation properties of x!J. occurs also in the classical Dirac bracketsand is interpretable as follo\vs. I--he gauge choice
is not Lorentz invariant, and yet the Dirac bracket procedure forces the constraint equation to be strongly valid in all Lorentz frames. l"'his requiren1entcan be n1ade physically consistent only if a Lorentz boost to a ne\\7" fraIne,
P'!J. ==:: p!J. w p•v Pv
IS accompanied by an infinitesinlal gauge transformation
,- _" ::. (_ ",..) -_ ~ I /-\ ~~ /'" \, ~,.\, - ~TU~
consistent with the equations of n10tion. Classically, the change in x P. n1aybe \vritten
x'p. C':) == x!J. (~)
- 33
Our consistency condition is therefore
and so
Therefore, for a pure boost,
~. . . dxj· '0(2.41) oxJ == x'J - x J == - d~ W Ot Xi + W J Xo
. pi . ( )== + Wo; t -- H WO t x,' t .
Accounting for appropriate orderings of quantun1-mechanical operators,this agrees exactly \'lith the change in .-:t.j generated by the boost operation (2.36).
3. RELATIVISTIC SPINNING PARTICLE
...t\. REVIE\V OF LAGRANGIAN ApPROACH TO Top
We next consider the Lagrangian approach to classical relati vistic spinning particles developed by Hanson and Regge (1974). The treatment gi \'enhere will be slightly n10re general than that in the original paper, \vhich dealtonly with spherical tops. In order to ensure Poincare-inyariance, unphysicaldegrees of freedon1 are introduced into the Lagrangian. \\lhen \ve in1poscconstraints to eliminate the un\'lanted yariahles, \ve must use the Dirac formalism to find a consistent I-Iamiltonian systen1 and a consistent quantumsystem corresponding to the original Lagrangian.
\Ve begin, as \ve did for the spinless particle, by considering a particle\vorld-line with points labeled by an arbitrary n10notonic paran1eter '7. I-Io\veverno\v \ve associate with each value of '7 not only a position x!J. ('7) but also aLorentz 'lnatrix A!J.v ('7) obeying
A I.t AAV _ !J.VA -g
A !J.AAV _ !l.VA-g
AOo > I.
Denoting 't'-derivatives by an overdot, we \'lrite the most general possihleaction as
't':J
S = Jd"t" L(xll-, :til-, All-v, All-v) .
't't
Now \\re argue that invariance of the systen1 under Poincare transformations of the external reference fralne severeI y restricts the form of the Lagran-
3
- 34-
gian. I ...et us \\Trite a generic elen1ent of the l)oincarc group as ~fj == (x!J., AP·v)
and choose the group ITIultiplication conventions
Right transforln: (x', A') == (x , A) . (a , M) == (1\1-1 x -+- a , A1\!I)
Left transform: (x', A') == (a ,1\11) . (1: , A) == (A-1 a + x , lV1A) .
With the convention (3.3), the right index of j\.!J.v refers to the external sjJaceaxes of the system, \vhile the Ifft index refers to the internal body-fixed-axesof the top. [I f the conventions for right and left multi plication in (3.3) \verereversed, the physical lneanings of the indices of j\~LV \\Tould be interchanged].I t is no\v easy to see that
A!J.v
XV } are right-Poincare-invariantA!J.AAVA
(3·5).!J. + AA!J.A- V}% I,V X
are left-Poincare-invariantj\I,P. AAv
Our description of the top \\Till therefore be manifestly in variant \vithrespect to space-axis Poincare transforlnations if we allo\\" the Lagrangianto depend OJ1~V on the ten \'ariahles (3.4):
(3. 6) L (, ~ ,.~ A :\ - L (A p. ..'J "'\- rxy A[3 \\, .1: ,.t, ,~) - \J l. v .x , -'- _ y) .
Since the derivative of (3.1) in1pIies j\.../\T is an antisymmetric 4X4 matrix, onlysix cornponents of j\j\T can be independent. Sin1ilarly, Eq. (3. I) sho\vs thatthe sixteen variables j\r~J are rcally functions of only six independent variables\vhich \ve \\"rite as CPt', i == I , .. " 6. 1'he canonical momenta may then hetaken as
The Euler equations become
(3. 8)
\vhile the canonical Poisson brackets are \\Tritten
We 'no\\! digress briefly to develop an intuiti vel y a ppealing notation forthe angular mOlnenta and their equations of motion. First \ve recall that in
- 35
llonrelati vistic 111echanics, the constant bod y-fixed coordinate x' of a pointin a top IS
\vhere RtJ" is an orthogonal rotation matrix
relating x' to the space-axis coordinate x. x' is taken to be a time-independentconstant, so the tirne derivative of (3.10) gives
(3. 12)
here we define
With CJ:/ == ! 2/ik w jk , (3. I 2) becomes the usual equation v == w Xx.
If we now take the 7-deri vati ve of Eq. (3. I), \\"e discover the generalizedangular velocity
which has SIX independent components and reduces to (Jij == u/i \;vhenj\'P'v is a pure rotation. Let us also define an angular change by
(3. 15) oOp·v == J\./'!J. 01\/ == --- Ol\.J·!J. J\./,v == - ~(f[l·.
\vhere
'I'hen vve can express G[J.v and ~O!J.'J in ter111S of the independent angular variables
~i as follo\vs:!J.V [l.V ( ) •
(J == ai ~ <:Pi,
~n!J.V [.LV ( ) ~au == ai ~ a~i,
or
or
Compatibility \vith Eq. (3.16) den1ands that afv satisfy
darV
-Ji--
(3. 20)
a~'V (~) generally possesses an Inverse function \vith the properties
!.LV bc<[3 pc< vB p.[3 vc<ai i == g g --- g g .
Using Eqs. (3. 18) and (3. 19), one can sho\v that
db~[3 db~Vbl.t') _~_ ~ b!Y.[3 L __ CflV c<[3 bC:"C
J d?j J - en' t
- 36-
\vhcrc C:~~e<~ is the Lorentz group structure constant (2.24). Equation (3. 20)
n1cans that b~·") is a realization of the adjoint representation of the Lorentzgroup L.ie algebra (Racah, 1965).
• l' '1'XO\V the Lagrangian depends on CPi only through i\J\ === j\crJ\. l'hus
if vVC define
,ve 111a y ,vri te
T l dL I d(j!J..V dL-
dePt"--
-8~-1:- 8cr!J..v2I ~lV ( ) 52 at' CP p.')'
1-1"'ro111 I~q. (3. 19), \ve ha ve also
sIt') is thus a con1hination of canonical coordinates and canonical 1110111cntason1c\\"hat sin1ilar to the spinless Lorentz group generator p') _ XV p'l .
l;sing l~qs. (3. 14) to (3.23), ,ve can no,v express the Poisson hrackets;:3.9) in tcnllS of S1(,\) and j\'~). l'he result IS
{~~,B}dB 8A. 8B
8..\
8Se<f1 '
\vhere .\ and S arc to he considered as independent \"ariablcs. \\"e then findthe canonical })oisso!l brackets
(3. 2 5)
{ P''', x') } === _ g~l')
(S'l') 5 IX [3 I. === + C'l'iIXi~ So,,;l "-, J '0"; "-
{51l') I\IX[3 I _ A IX') !l[3 A. a~l v[3, J J - J\. g -- j\. g
{pll P')} == { Jl .•') l == { A' la ;\ v[3 I. == 0 ., x ,~t J j \. ,J \. J
]'hercfore S'W obeys the Lorentz algebra by itself and generates the right(space-axis) Lorentz transformation on j\ IX [3 • Note that
{S'lV, AO:[3 AY[3} == o.
Thus the brackets (3.24) are con1patible \'lith the strong validity of the constraints (3. r) even when all of the s,t') and -L\IX[3 are treated as independent va
riables vvhen taking partial deri\"ativcs. In effect, we ltazJe sirlestepperf OJle stage
oj tlte Dirac prorc{!ure by starting ,vith brackets \vhich are consistent \vithI~q. (3. I).
~cxt, \\Te sho\v ho\v to re\vrite the Euler equations (3. 8) in tcrn1S of S~l')
and r;!J.'). Using Eqs. (3.17) and (3.22), ,\~e find
t
I "'I p.'). ...o == 2- '-'.I ai 9j Sp.,)
I C 'lV Aa [3y---- ;:)'1.') eli - .. r'3 ai2 ;
3L8~\~;- -
_._-
- 37-
N"0 \V I~qs. (3. I 8) and (3. I 9) 1et us rcex prcssthis as
(3. 26)
One can verify explicitly that the choice (3.6) for the forn1 of the L,agrangianin1plies
I I I 8I,___ s't , crf~ -1- crll . s/ ~\ .!t8j\\
'rhus the spin Euler equation can also be \vritten
(3. 28)
In order to forge a connection hetvveen I~qs. (3.26-28) and the fcuniliar equations
for the rotator (Goldstein, 1950).. .
Lspacc === L body OJspacc X Lspacc === (external torque),
\\'C definc the hc)(l y'"-fixed SpIn
1'hcn \vhen \\'c refer S'l') to the space coordinate systcn1, \ve find that
C~on1paring to l~q. (3.26), \ve find that the n10tion of the hod Y'-fixed ~pin illits o\vn fran1c is
/. '?)\3·3~
l~inally, \ve ohserve that if \VC define
cll ')...:-1 ,
thcn the Euler equations (3.8) and (3.28) in1ply
d~1!J.'J------- == °.d"t'
\Ve no\v sho\v that l\1'l') is precisely the conserved generator of right (space
axis) Lorentz transforrnations. lJndcr a right Lorentz transforn1ation, theLagrangian is invariant, so
(3·35)
_ c~_ (P ~ .)t + 'l... i ~ .j- d"t' !J. Ow .1 au) CPu ,
- 38-
"",here \ve used the Euler equations (3.8). Novv the infinitesin1al right Lorentztransformation Ow has the effect
aw x/.L == (Uar3 (xC( g0tJ. - x f3 gatJ.)
aCJ) A~'" == (U\X~ (AP,a g',l';3 _ A!l~ g",a) .
From Eq. (3. I 7), we find
For arbitrary (Da~, Eq. (3. 35) in1pIies Eq. (3. 34) and:\I IW is the conserved Noethercurrent of right Lorentz transforlnations. By explicit con1putation using Eqs.(3.25), we see that MWJ and p~ obey the Poincare algebra (2.25).
B. CONSTRAIXTS ON Top LAGRANGIAN
We now exan1ine the constraints which must appear in the top Lagrangianif it is to describe a reasonable physical systeln. First \ve define the set of tenright Poincare invariants
(Ao .\1 Al .\1 ,\2.',1 A3 .\1az"== "'x, \lX,J.\.',IX' ',IX,
A· 0", Al A·0'1 \2 ;-\ 0'1 A3 A·2'1 A3 A-:-3\1 Al A·Iv ,\ 2 )',I , j \I, .... \. 'I , ',I , 'I , i\. v
conlposcd of the variables (3.4) \vhich appear in the Lagrangian. The deri vati ves of L with respect to these variables will be written
7-reparan1etrization invariance of the action is assured by the requirement thatL be h07nogeneous of degree one in the velocities; L (a) is thus taken to satisfyEuler's differential equation for h01110geneous functions,
10
L(a) == ~ aiLi == aiLl"i=1
Since the ten mOlnenta pl-t, S'W are homogeneous of degree zero in the velocitiesaz", at most nine momenta can be independent; there must exist a trajector.yconstraint relating the momenta,
We will restrict ourselves in \vhat follo\vs to constraints of the form
(3.42 a)
- 39-
\vhere :\12 is not a constant but is a function of the body-fixed angular I110n1Cntum S == AS,A.T satisfying
(3.42 b) l\ '1 2 /-Sij -S()z\ .lV ('-, )-
rrhe moti vation for this choice \vill becnn1e apparent shortly.Next, we n1ust force our top to ha ve only the usual three spin degrees
of freedom in the nonrelativistic limit. In order to eliminate three of the sixcomponents of S!).v, UJe hereafter restrict ourselves to Lagrangians jar 'leJhich therelation
and Eq. (3.42) jollo'lv jront the jornz oj the Lagrangian alone. (Since PtJ.\7·Il ==-= 0,
Eq. (3.43) contains only three independent constraints). By choosing theconstraint (3.43), \ve effectively pick the physical meaning of the coordinate xtJ.of our spinning particle. .l\ccording to I)ryce (1948), Eq. (3.43) identifies xtJ.as a body's rest ~fralne center oj 1/z011Zentul1z, \vhich transforn1s as a four-vector.
I~quation (3.43) tTIust for consistency have a \veakly vanishing derivative:
Con1bining this infortnation \vith the l~uler equations (3.8) and (3.28), \vefind
(3·45)
Putting this back into Eq. (3.28), \ve have
S~LV ~ ° == (external torque),
thus cornpleting the connection bet\vecn Eqs. (3. 29) and (3.3 I).N ow we note the general identity
\vhere
S*fW == ,.~ E!).'JI.a S2 1.0 ,
and EWJi.cr IS the totally antisYlntnetric pseudo-tensor, \vith 2°123 == I.
lVlultiplying by Pi).' \ve find that the constraint (3.43) directly implies
(3·47) Sf-LV s*"- ~v ~ o.
Thus, for exan1plc, fron1 Eqs. (3.45) and (3.26),
------
(3.48)
* Au, 8LSf-Lv A . --) ~ °,
dA ~
- 40-
(Nate that
(-srJ-'J ,,* _ I rJ.WJ.r1 A SYO A " == ArJ.(X (_~_ ,.. sYo) AV (3\"- ) - 2 E:xy (3ij 2 ~:x(3y8
due to .L~p,C( .1\'1(3 A)"Y j\08 E:x(3yo == Det ..A. ErJ.'JI,O == ErJ.\Ji'o).
\\'re will not deal here with the problem of finding Lagrangians satisfyingl~q. (3.43); procedures for doing so are given in f-Ianson and Regge (1974)for the case of the spherical top. .i\.. sin1ple exan1ple of a suitable sphericaltop Lagrangian is
(3.49 a) L = - 2- l {Au2 ~ BO'· 0' [CA u2~ BO'· 0')2_
- 8 B (.L~UO'crlt - 2 B Det O')]~}~
\vhere u 2 == UrJ. u[.L , 0' . 0' == O'!.L'J crwJ , uO'cru == u[.L cr[.L'J crV/, ul. , Det cr == (O'rJ.'J 0':'1)2/ 16,
and A and B are constants. Direct computation confirms that the constraints
(3.49 b)
follo\v from the form of the Lagrangian alone. Lagrangians also exist for \vhichthe mass and spin are separate unrelated constants:
(3· 50)I S S~lV B2 '-WJ ~ •
C. DIRAC TREATMENT OF Top CONSTRA INTS
\\Te no\v consider the dynamics of a relati vistic top whose Lagrangiangi ves rise to the following \veak constraints among the canonical momenta:
(3.5 I)
"fhe canonical Hamiltonian IS
2 -M (s) ~ °
y'J. == S~" Pv ~ 0 .
He == p~l, U!.L T £ • L p!L I I S~'J L~ i -- == UrJ. T "- (jWJ - == °
2
and so may be taken as the initial first class Hamiltonian. Ho\vever, the constraints (3.5 I) are not all first class, since
{p2 M 2, \7"1,} ~ 0
{y"", Vv} ~ S[.LV p2 .
-41-
1"'here are nevertheless t\yO first class combinations of the constraints
(3.51), \vhich \ve take to be
We observed in Eq. (3.47) that 92 ~ 0 fo11o\vs directly frorTI \7'1 .~. o. ()urfirst class I-I an1iltonian n1ay novv be taken to be
(3 r'r')\. .))
Since consistency requIres
V (",) ~'J2 \ ~ T2'
'·il t === {VT[.L !-1 t~1v 'J .-....,:, 0 ,
any 111ultiples of \:/t \vhich 111ig-ht be added to H 111ust ha ve vanishing coefficients and there are no secondary constraints.
Defining the tensor
(3· 56)
\ve compute the velocities to be
(3·57)uy· === {x!.L , H } === 2vI p!.L
[.LV _ \I,[.L {A v H} _ r[.LV + ('J S~*'l'J(J - j ), , - VI 4 V2 •
...\s in the spinless case, \ve n1ay set VI === (- U2)§)(2M), and similarly restrict
the form of V2' Nevertheless, there relTIains an arbitrariness in VI and V2 \\Thichis eliminated only \\Then \ve choose gauge constraints.
Equations (3.5 I) represent four independent constraints. ]'hc t\'/o con1binations (3.54) are first class \vhile the t\VO rernaining independent con1binations, say
(3· 58)
are secoJl{i class:
Clj === {\1£ , \Tj} == p [.L pIt
Strictly speaking, \ve lTIUst replace all Poisson L)rackets by the corresponding Dirac brackets (1.2 I) even in our preliminary first-class Hamiltoniansystem.
- 42-
~O\V let us proceed directly to impose a gauge condition \vhich is effectively the" phase space conjugate" of the constraint V'" ~ 0 on the mon1enta.Physically, \ve \vant A'~ to be a pure rotation in the rest frame of pI", just asS!J.
vbeCOlTIeS a pure three-angular-n10n1cntun1 in the rest fran1e. V\le therefore
conjecture that
(3.60)
IS a consistent constraint choice, \vhere
(3.6 I)
and j\00==1\0
0 == . - )\.00 == I for a pure space rotation. Since
only three of the four constraints (3.60) are independent.
:\'" O\V \VC use Eqs. (3.3 I), (3.46) and (3.37) to sho\v
-S:Oi .~ r (1°/, S-. i _ -SOA 1-. i)"'C. VI ~. I. . ~ A'
h I-U.')\.')''X \')(3 h E ( b h ld d 2 (-S) . d\v ere ' == j ~ 1~t3; . 1"' en if . q. 3.42 ). 0 s an 1\1 IS even un erSOi -~ _ SOi, lOt' (S) n1ust be ocld:
'"'1M 2ell.. _ ~Ioi (-Sij -SOi, _ -loi /-Sij -SO£)---::-- - ~,~) - -.--- \ ,- L •
CiSoi
\\Te conclude that if \ve set
at son1e point 't'o, \VC ha ve also
(3. 66) rO/ == 0 .
l'hen from Eq. (3.63) \ve know that SOi ~ 0 and so SOi vanishes for all T.
Equation (3.65) is therefore an invariant relation provided \VC restrictourselves to trajectories obeying Eq. (3.42).
\Ve no\v define
Since Vi ~ 0 implies
-S1)"~ Pi ~ 0 ,
we must have
- 43-
Fro111 Eq. (3.57), the equation of Inotion for prJ- is
\\Te no\\?" choose the gauge
pO ~ I) == (~ PrL
P1L)~,
so that
orty,. == o.
'Thus fro111 Eq. (3.67), \ve find
'T'he equation of motion of pi IS no\v
Consistency then forces
so that the gauge choice pO ~ P has fixed one of the arbitrary functions Inthe Han1iltonian as expected. Furthermore
It _ A!LV P ~'L P _ ,,0 pp - J 1. v ~ 00 - - g
vlhich is exactly equivalent to Eq. (3.50). We conclude that the constraints
consist of one gauge COJlfiz'tion and t'lRJO invariant relations.
\\le may now compute preliminary Dirac brackets consistent \vith thesix independent constraints
\vhich \ve \vrite as
i === (VI V2 V3 ~1 y2 ~3)ep '" s , C:, , s ,
I'he n1atrices needed to compute the preli111inary brackets are
Z === 1,···,6.
o
- 44-
and its Inverse
o
We Inay put our brackets into a manifestly covariant forn1 by re\vriting(C-1),j' as
(C L T (C- Laf3 rr)'i === £a ) jf3 ,
\vherc
z',j==: I,···,6
a,~==I,···,8
(3· 72 a)
and
(3· 72 b)
rr,{'!l 0rr/ ex ==0 rrk~l
_ pi/pO 0 0
rrk~l == _ p2/pO0 -+- I 0
_ p3/pO 0 0 +1
k== 1,2,3
[1.===0,2,2,3
l'he Inatrix Cexr, and its in verse are sin1ply
cexf3 ===Pg!l\J
/c-1. ex(3\ ~ ) ==:
°
°p
_SWJ
T'kp. is no\v used to can vert Vl~ and ~k into \Jp. and ~!J. hy using the identitiesprJ. V!J. == 0 and p~L~!J. == - p;'t ~!1. as follows:
(3· 75){i\., yk} Tk!J. ~ {A, ykTk!l'} === {A, V rJ.} ,
{A , ;k} T k!J. ~ {A , ~k l'k!J.} ~ {A , ~!l.} .
Defining the eight component object
ex == (y-'-o \Tl \12 V3 ~0;:1 ~2 ;:3)'cp ", ,~,~,~,~ IX==: I,'·', 8,
\ve nla y use Eq. (1.21) to \vrite the prelinlinary Dirac brackets as
(3· 76) { A B ) I t:'..I {;\ B } I {A "' T~L l f ;: B l, f rv ....!J.., - P , v J t ~!1. , J +
+ .._.~- {A , ~!J.} {y!J. , B} + {i\. , ~!J.} S!J.v { ~'J , B} ,
- 45-
\vherc \ve use a prin1c to distinguish the prelirninary brackets fron1 the final
1)rackcts.1~he cxplicit forn11das for the prin1cd brackets of the canonical variables are
( ) I . { Prt, x') ....j' =~= _ g'LV\,3·77 - \. =
2. {x~t ,x'J}' === -S~'J/CPCl. pCl.)
3. {pl-t, p'J }' == 0
4. {P!L, S'JI, }' == 0
5. {x tL , S'JA}' == CS~w pl. _ S(LI, P'))/CPC( pCl.)
6. {StJ.'), SCl.13}' == StLC((g'Jr3 _ p'J p13/p). PA) _ SU13 (g'Jc( __ p'J pcx/PA pi)
+ SCX') (g't13 _ ptL p13/PA I)A) _ S13'J (gILCX _IJ!t pCl./PA
pI,)
7. { j\.11-'), xC( }' == A~l.rj (p'J gC(13 __ pr3 gCl.'J)j(PA
PA)
8. { j\.,t'), p cx }' == 0
9. { j\11. '), j\.C(13 }' == 0
10. { ..;\!l'), SC(13 }' === J\!)''X (g'J13 __ p') I>f3 jP" pA) _ j\.,t13 (g'Jcx __ p'J petIP), pI,)
-i\P'crpG(gC('J p13 _ g13') PC()/(Pj. P").
l~hesc brackets are no\\· con1patiblc \vith setting the constraints V'1. and
~rl identically zero. Xote that the position variables x P' have nOJlzero brackets
(Pryce, 1948). )Jevcrtheless, the Poincare algebra (2.25) is still satisfied in the
prinled bracket systen1 and X IL transforn1s like a four-vector.C' \"",), 1 ylt • 1 I -S·o/ . h I..:JIllCC " au( C;' vanI:) 1 strong y, ~ vanlS es a so:
(3.78) -S·~Ol· === j\.o c lX r3 !\i == _ P sc<13 '\.1' (_ p2 \-J ===~ ex. ,J .. f3 C( .. J 13 ) 0 .
FurtherlTIOre, from l~q. (3.64) \ve find
1-\s a consequence, I cx13 no\v obeys
(3. 80) Iet13P~ == o.
\Ve can forrn a I-Iamiltonian from the remaining first class constraint,
') -0'lVI""' no longer depends on S I because of I1~q. (3.78).
The velocities arc sin1ply
uP. == {X!L, H }' === 2 IV 1)'1
[L'J _ ;\,.,1. { \.'J H l' _ r.' 111.'J() - J . j J, , J ---- LJ •
-46 -
'vVc no\v elitninate all arbitrary functions by selecting a gauge constraintconjugate to the trajectory constraint. IVIaking usc of the iterati ve propertyof the Dirac brackets, we choose the final constraints
and ilnpose then1 on the printed brackets (3· 77).The required rnatrices are
.. [0CtJ == { ((). ([).}' == 0Tt' I} 2P
-1 I [ 0 I]C·· --lJ - 2 po -I 0
Equation (1.2 I) with all Poisson brackets replaced by the primed brackets(3.77) gi ves the fornl of the final Dirac brackets, \vhich \ve no\v denote \vithstars. The results are
(3. 85)
2.
3·
4·
5·
6.
7·
8.
9·
{x'\ x'J}* == S~LV 11\II~ ~ (P'''' SO') _ I)') SO~")/(M2 pO) ==== S~') /1\11 2 _ (P') S~ll' pi _ p~l S'Ji pi)j(:\I2 (poy~)
{P~\ pV}* == 0
{ P~, S'JA}* == 0
{ X''', S'JI'}* == __ .1.. rsf-t'J !J I. _ S,l}. p'J) -t- ~~_ (SOV pI, _ SOl, pv)M2 \ '- POl\1 2 '-
{S~'), SCX r3 }* == S'"CX(g'J~ + pv p~jl\f2) _ S~~(g')CX + pv PCXjM~)
+ Sc('J (g[J.~ -1- p~ P~1M2) __ S~') (g[J.cx -+_ pf-t PCX/M 2)
{ _\'W /\ (X~l.* == _ I ;\~ (PA Ov _ pv OA) AC( f(r3 +j '~J 2 po 1\1 2 .J. 1-, g g y
+ __~__ ACX (Py Or3 _ pr3 Oy) AtJ. II.,)2 po M2 Y \ g g I.
- 47-
The Hamiltonian IS
(3. 86)
\vhilc Han1ilton's equations of Inotion are
4-1- - dt + {A , (p2 + M2(S'J))t}'
Here \ve have used the traditional notation x O == t to en1phasize that xO IS
no\v a parameter and not a canonical variable.]"'he Poincare group generators are no\v \vritten
pO == H
(3. 88)pi == p'"
M Oi == tP'" - ~'tJ H -t- SOi == tp t• - Xi H ~ sii P.ijI-I
1\1 ')" == Xi pi _ xi pi -t- S')" .
]"'he star brackets of these quantities obey the Poincare algebra Eq. (2.23)and the theory is again Poincare-covariant. Applying Eq. (3.87), Vle confirmthat the generators (3.88) are constants of the motion \vith respect to xO == t.
The transfornlation of x P' under the Poincare group is altered in the starhrackets. x O == t is no\v just a parameter, \vhile x transforIns as follo\vs:
{ Po _.i}* - { H i 1,* - _ pi jH,J:, - ,X J -
{ P i .. jt* == _ ~Jj,~t J 0
{ 1\. -1 ij k '\ * ~ i;" j ~ jk l'1 1 ,x)- ==0 X"---o x
rrhe extra tern1 in {lYrOi, x
k }* is just the correction discussed in Chapter 2
\vhich n1aintains the validity of the constraints (3.83) in the ne\v Lorentzfran1e.
D. QUANTU!vl l\TECHANICS
In principle, the system of single-particle positive-energy quantumnlcchanical operators describing the relati vistic top can be deduced directlyfroIn the final Dirac brackets (3.85). In practice, however, this can be quitedifficult unless \VC are able to construct fron1 the original set of variables a reduced set of independent variables (r .85) obeying canonical brackets. F'or thetop, the appropriate variables correspond to the Pryce-Newton-Wigner variables (Pryce, 1935: N c\:vton and ",Tigner, 1949) supplemented by the I~uler
angles. We define
s z' == -A- siik Sjk
- 48-
and take our ne\v variahles to be
qO === ts x P . sOt
q - X -rt- - H (H + lVl) -~. ~-- -I-I"+·KT
J === M S + p (P . S)H H(H + M)
R ij == A ii _ j\iO pilCH + lVI)/j A ik pk pi
=== j\' - H (H + M) .
p===p
pO === H == (P'~ + lY12 (j))~.
2 -i' .Here IVI (S J) can he taken to be a functIon only of
J'" === R"j]i
because
Here we haYC observed that Rij is an orthonormal I11atrix,
R ik R.ik === Rki Rki === ~l".i
and used the identity Ezjk Rai Rbi Rck === subr Det R === S"bc.
l)efining
\ve find by direct C0I11putation fr0111 E:qs. (3.90) and (3.85) the follo\\'ing brackets:
(3·95) I . { pi, r }: = - ai~
{ I-I , qJ} === - 1).1IH
2. { q£, qi }* === 0
3· { p~L, p" }* === 0
4· { p~, Jj } * === 0
5· { qi, ] j }* === 0
6. { ]i, Ji 1.* _ -::-iik ]kJ - .....
7· { R iJ , qA' }* === 0
8. { Rii, I)k }* === 0
{ R ii , H }* === -I R ik sP'/ II
:2 H.
9· { R ij, RIm }* === 0
10.{]i R~j 1.* _ -::-ijl, R fl,
, j -- \000 ~.
-49 -
Equations (3.95) now forn1 the basis for our quantum mechanical system.vVe begin by choosing the Hilbert space norn1
(<I> , <1» = Jd3 x<l>* (x , t) <I> (x , t) .
We then take
t == qO == c-number2 . 2 -
M === - p~l P (1. == IVl (J) == q-nun1ber,
and the Hamiltonian
where the t\velve independent operators are
q£, pi, Ji and three of the R il.
Starting froIn the Dirac brackets (3.95), \VC postulate the comn1utation relations
(3·99) i [P i, qj] == ~ ij
i [pi, Ji] == i [pi, pj] == i [pi, R zO
k ] == 0
i [qi, qi] == i [qi, ]i] == i [qi, Rik ] == 0
i [J i, ] i] == - S iik Jk
i []£, R li ] == _ szjk R1k
i [R zi , RIm] == 0 •
We also observe that
(3. 100) i [J£, Ji] == 0
i [1£, Ji] == + s£ikJki [Ji, R il
] == + Siik Rkl,
(3. 101)
so f COIntTIutes \vith right vectors and transforms left vectors. The Han1iltonian (3.98) has the comtTIutation relations
i [H , q£] = piJH
i [H , pi] == 0
i [H , ]£] = 0
We omit the explicit computation of i[H , RZi] due to nontrivial orderingproblems.
l'he Poincare group algebra is generated by the operators I-I, P, and
lVI'i == qi pi _ qi pt· siik Jk
(3. 102)
- 50-
l'he Han1ilton equations of tTIotion for an operator i\ are
ciA. 8A----- == z [H A] + -- .dt ' 8t
i\n explicit realization of the algebras (3.99) and (3. roo) in terITIS of differential operators is the follo\ving:
})1 == ~_? __i 8ql
rro get explicit expressions for J and J in terI11S of the three independent
Euler angles, \VC first define the 3 X 3 I11atrices
(3· lOS)
realizing the algebra of the Jl ,s, and then write Rtj" as
(3. 106)R(j ( ~ ) - iaLa - if3L 2 - iyL 3a, ,y==e e e
cos (I., -slna 0 cos ~ 0 sin ~ cos y ~sln y 0
SIn 7.. cos 7.. 0 0 0 sIn y cos y 0
0 0 _~ sin ~ 0 cos ~_ 0 0
cos rt.. cos ~ cos y --sin a sin y -cos 7.. cos ~ sin y -sin a cos y cos (/.. sin ~
sin (/.. cos ~ cos y -1-- cos 7.. sin y - sin rt.. cos ~ sin y cos rt.. cos y sin C( sin ~ •
-sin ~ cos y sin ~ sin y cos ~
Then \ve see that J lTIay be \vritten
F = i (- ~~~ ~ :x +- sin y ~ -+ cos y cot ~ :y)'J • ( sin y 8 -L d. d )J'" == Z -.- ---- I cos Y 8S- - sIn y cot ~ ~(
SIll ~ 8rx I I ,
\vhile J IS
-1 . ( 8. 8 cOS'Y. 8)'J == I - cos 7.. cot ~ ~- -SIn rt.. -:y:- + ---:----(.-\ox C!i3 SIn ,j oy
(3. 108)) ( dJ:' == i -- sin (/.. cot ~ ~
-:3 . dJ === ljrx •
-51-
Thus the Schrodinger picture \:va ve equation which IS the counterpart of theI--Ieisenberg picture equation (3.103) is
?] 1(!Ji\ -1- .... (J \) 2 ! == .1. TlV \.) ~ Z 'at '
\vhere j is taken to be the operator (3.1°4) or (3.108). (We make no atten1pthere to resolve ordering problen1s \vhich n1ay arise \:vhen 1\12 (J) is expressedas an operator). I:~quation (3.109) is non10ca1; a local equation, \vhich cculdbe the starting point for a quantu111 field theory of relativistic tops, I11ighttake the form
(3. 110)
We ccnclude \vith the relnark that [)irac's pr·ocedures can also be carriedout for the electrically charged rclati vistic top. I-Io\vever, since this systen1becolnes exceedingly c0I11plicated and is still ilnperfectly understood, \:ve \vil1not atten1pt to treat it here (see Hanson and l~egge, 1974).
4. STRING MODEL
Our examples thus far ha ve dealt on1 y with point-particle mechanics,\,\tThere all canonical variables depend only on a single parameter 7. As ourfirst example of a field theory \vith continuous degrees of freedom, \ve exan1inethe relativistic" string n10del " (Goddard et al., 1973; for a revic\:v, see RelJbi,1974). V\Te take as the action functional the reparametrization-invariantexpression for a surface elnbedded in a ])-din1ensionallV1inko\vski space, analogous to the reparametrization-invariant line elenzent (2.4) chosen as thepoint-particle action. '[he N an1bu action is then
"t'2 G2(T)
S [xP<] == - N _I'~ dTJ" dcr((_~~~?-~~)2 _ (.?-=~_ !~~_) (dXr-t ~v. )\l/:!d"r d(J / d"r v,!, 2cr dcr JJ .
T"l Gler)
N is a normalization \vith dimensions of Planck's constant divided by lengthsquared, and one convcntionally chooses
'[he 1) canonical coordinates ~y!J. (7 , cr) no\v are t\vo-dinlcnsional field densitiesvlith continuous dcgl-ees of frcednI11 labeled by cr. .l\ny transforn1ation of thefornl
.... (-- )(j -r G \" , cr
leaves the integrand of ~4.I) unchanged.
- 52-
~~ rnetric is induced in the (':-, (5) space by its embedding in x~-space.
Ho\vever, \NC can avoid explicit use of this ll1etric by \vorking exclusively\vith '7, (J, and the quantities
(4.4 a)
and
dXP-("t',O')"t~ == v~ === --
d't'
dX~ ("t' , 0')x'!J.=== u~ === --
dO'
(4.4 b) E === v2 , G === u 2 , F===u·v.
rrhen \ve ll1ay \vrite the Lagrangian density of the string as
OJ _ N {( )2 2 2 }1/2~-- u·v -v u
=== - N { F2- EG }1/2.
The canonical 1110mcntun1 &'1 is then \vritten
=== - N2 ~1 Gv!-1 + N:! ~-1 Gu'~.
I t is con venient to define also
N 0\V \ve require the variations of the action functional (4. I) to vanish.For the 1110111ent, \NC continue to allo\v (51 and (52 to depend on T, thus obtaining
-r2 02(-r)
o = (')S = I dTJdIT (')Xf' (- _d~l" - d~r)'t'l 0 1(:)
02(-r)
+JdIT ((')x1" ,91") 1::::
Ol(-r)
't"2
+JLIT {(')Xf' (a2) [ TIl' (IT2) - ~11:2 giL (IT2)] - (')X1" (ITl) [IT" (ITl) - _~1~1- giL (ITl)])
rrhe l~uler equations are therefore
53 -
while the boundary conditions are
~x!J. ("rl , a) === ~x!J. (72 , a) =-= 0
i === I '?, ..., ,
Hereafter, \·\ye \vill treat n1ainly the con ventional case \yith 7-indcpcndentboundaries in the da integral and \vill choose
'[he houndary conditions that \VC need to use ~xplicitly then ])CCOn1C
Il tJ- ("r , a === 0) === 0
Il fl ("r , a === n) === 0 .
i\.. SVSTE!vf 'Vrr:f-lOUT GAUGE CO~STRAI~TS
Froln Eq. (4.6), \ve find the follo\ving t\VO prin1ary constraints,
(4· 13 b)
A d ynalnical relation,
. ~1 === f!lJ. f!lJ
~2 === f!lJ. u
,>N'""u· u ~ 0
;~ o.
follo\vs fro111 Eq. (4.7). Equations (4· 12) and (4.14) in1ply that 1,2 =-= 0 ata === 0 ,n. If \VC define our canonical Poisson brackets to be
{ f!}J!J. ("r , a) , XV ("r , a')} === - g!J.v ~(a - a')
{gJ!J.("r, cr), pjJV(7, cr')} === 0
{ xi! C·r , cr) , x') (r , a')} === 0 ,
then \ve find that '~l and ~2 are fi1~St class:
{ ~l (-, , IT) , ~1 (, , a')} = -+ ~2 (Y2 \' , a) + 'h (, ,a')) -;r; a(a - a ') .~ a
1
(4· I 6) { Y2 (,,: , a) , Y1 (": , a')} === (,~ 1 (,,: , a) Y1 Cr , a')) -~~ ~ (a - (J') ;~ 0
{ ,L (- ~1 ,L (- ~')} - (~'J (- a) + tlJ (- a')) --~--- ~ (~ -- ~') .~ 0't'2 "', \.J) , 't'2 \... ,v - \.. T 2 \. , 'T 2 I., 6() 0 v v ro.,;:, •
(See .l\.ppendix B for a discussion of the relation bet\veen Eqs. (4.16) and Eqs.(1.83)). Hereafter, the explicit dependence of the canonical variables on T
will be dropped if no confusion arises.
- S4-
N O\V \VC cxarnine thc canonical " energy-monlentU111 tensor" in (1", a)space. If we let zit == vt-L == .iP. U)L == 1IY' == ':r'1J. \vith uY· == (utJ. 1/)1) the
1 ' :! " , aI' 2 '
tensor may be taken as
[6 r = }~ u~ - o~ fI!c b dU~
,~!L Vp• ~__ 2
IT'l VtJ.
Formally, the canonical tensor IS conserved,
~ o.
Since [()c](lb ~ ° is obviously first class, ,ve Inay add linear conlbinations ofthe first class constraints (4. I 3) to forn1 the density ,vhich is integrated to gi,'cthe first class generators of gauge-like transforn-lations in 7 and cr. Thus \VC
define the total "energy-Inomentllln tensor" in (1" , cr) space to be
na Uia,f, -L Va IU b == I b 't'l j b !f2 ~ 0 ,
so the generators of ~ and cr transformations arc
1'C
HT = Jda 0\o
1t
H a = ( cia 0\.-Jo
l-L-; and I-I cr ha ve \veakly vanIshIng brackets ,vith one another becausc the Yiarc first class. rrhc brackets \vith J::1J., ho\vever, are nontrivial and \ve I11aV
place restrictions on Uab and '.lab by requiring
"1.'!I:= 'l _ { JL H t - "JUl .. OlJp,~ ---u- X, crj-'" 2;-:;r
Thus the definition (4.6) of g;'" tells us that the choice
VI == I .2
reproduces (4.20).In an ordinary scale-in variant t\vo-dinlcnsional field theory \vith the
canonical tensor [8[]ab
0, there is an infinite class of symrnetry generatorsfound by integrating combinations of [OJah over an arhitrary function. These(nonvanishing) generators produce all of the transforlnations 111apping the t\VO-
- 55-
dinlcnsional space pseudo-confonYlally into itself (see, e.g., r'"'ubini, I-Jansoll and]acki\v, 1973). We 111ay construct a similar class of generators for thc stringusing the zt1eakly zero tcnsor (4. 18).
We begin by defining z£'" (cr) and :?/J~l (a) for - 1t ~ a ~;: 0 l)y
uY (j) === --- U tL ((j)
a) ==
(Rebhi, 1974). rrhen \ve rnay Vv·rite the generator of all T and cr rcparall1ctrizations as the functional
it'
(4. 23 a) L[f]= 4~ (cla{[f(a) ~j(__ a)](Y'2 fN2Z/H-2N[j(a)-j( -cr)jY. u}.1
oTt'
-1:\ I cia j (a) ([J}J NU)2 ,
\vhere
rrhe functionals I~ [fJ forn1 a closed algehra in the Poisson brackets,
{ L [f] , L [g]} == L [fg' - f' g] .
'[he fan1iliar Virasoro functionals are essentially the 11"'ouricr cOlnponcnts of
(4.23) and are defined asit'
L,,== L [ei",,] = -4k (cia eill" (.9' + Nu?-Tt'
Classically, the LJl ohey the algehra
{ Lm , I-'Jl } == -- i (7ll ~ JI) L ,1 -+- m •
\Ve clnphasize that the I-'1l are 'Zueak~v :::,'),"0 hecause they are fOrIllCd froll1 thefirst class pritnary constraints (4. I 3).
\Ve no\v turn to the generators of the Poincare group tranSfC)rnlationsof the fields xP. (7 ,cr). Since the x!l- (7 , a) , l-L == 0 , .. " D - I, are D separatescalar fields in the two-dinlensional (7, a) space, syrnmetries acting onthe index !J.. are IT10re like internal isospin symn1etries than spacetirne S}"111111Ctries of a convcntional field theory.
\Ve con1putc the ~oether currents generating Poincare transfornlationsof the x!-L by cxalnining the corresponding variation of the Lagrangian density,
\vith boundary conditions (4.10):
o == 0P == g;~l. DVp• + Tpl. DUll-
___ 8 / ))j!t ~) 8 (nIt ~ )---- -:;,- (.T oXrl- + .r---- ox".
c~ , c'cr r
56 -
If ox!J. is an infinitesimal translation,
we find the usual Euler equations
dqJfL dill-!'~-8:r~--- + -8~- === 0 .
The --r-independent "charge" generating infinitesimal translations of x p• IS
therefore~2(1') 1t
pl'Cr) = I dcr&J!LCr, cr) = ( dcr&J!'('t', cr).J ~
crIer) 0
\Vc revert for a lTIOment to --r-dependent boundaries for (j and note that boththe Euler equations and the boundary conditions (4.10) enter into the deri\?ation of the T-independence (conservation) of pl-t :
If ox!). is all infinitesimal Lorentz transfoI11ation,
(4.31 a)
we find
(4.31 b) d ( 11, f]lJv V Y'!t) I d ( I" 11v v TPl) _-§r- X -x -r 2cr x -x -0.
'I'his equatioll also follo\vs from Eqs. (4.6-4.7) and the l~uler equations. '1'hcgenerator of infinitesimal Lorentz transformations is thus
it'
M'L>' = {dcr (x" fJJv - XV &J") ..Jo
Using Eq. (4.31 b) and the boundary conditions (4.12), \\?e find that lVI,tV IS
conserved:
cl:Vll-t"·,d-r-- == 0:
The canonical Poisson brackets (4.15) I11ay nc)\v be used to sho\v that thefull l)oincare algebra is satisfied,
{ pC( , pf3 } === 0
{M'tv , pC( } === g~lc( p" _ gV'l. p'"
{MI-tV , lVfxf3 } === c~~ctf3 M°-r
- 57-
\V herc C~~C(r3 is the structure constant (2.24). I t IS also clear the xtJ. trans
forms as
(4·35)
{ pIt, XV (-r , O')} == _ g~L'\'
{j\ 'l!tV C( ("'I"" )} _ !let ." (...,.. ,,_ Vet.!-L ( ".iV ,x w, 0' - g r't ., (J) g x ~,(J),
so that p'l and IVrlv indeed generate the infinitesin1al IJoincarc transfortnationsof X'I (7 , 0'); no integrals of the first class constraints (4. I 3) need to be addedto the canonical expressons for p,t and M!IV.
13. ORTIIONORl\fAL GAUGE CONSTRAINTS
\\Te \\Till no\v choose a convenient set of gauge constraints to fix the scaleof ':" and (J relative to x tJ-. \\rc begin by taking an arbitrary constant vector I,p.
and examining the consequences of the gauge choice
(4.36 a)
(4.36 b)
(pp. )
AtJ. xtJ. (~ , (J) - N':-· ~ 0
N is defined as usual as the normalization appearing in the action (4. I) andp,t is the translation generator (4.29). Note that the conditions (4.36) are
nonlocal in a.If \VC differentiate (4.36 a) \'lith rcspect to ':" and a, \\·e find
1\' u'~ o.
Equation (4.36 b) and the Euler equations in1ply
so the boundary condition (4. I 2) requires
A . IT (~ , (J) ~ 0
for all (J, not just (J == 0 , n.Next \\·c observe that the definitions (4.6) and (4.7) of ,3'" and nIl, together
\\·ith }=q. (4.37), in1ply that
A . {lj> ~ - (A . v) ~2 Gj2
,~ - (/\ . f?/J) XG /fE~l~ 0 .
In addition, Eq. (4.39) requires
A . IT ~ (/\ . v) N2 FIfE ~ 0 .
- 58-
\\TC first conclude that
(4.42 a) 'ZJ' It.~ O.
Secondly, E~q. (4.40) tells us that NGj!f/ ~ -~- I, so \ve discover that
(4.42 b)
By definition, f~qs. (4.42) sho\v that the gaugc choice (4.36) has resulted in anortlzornor7Jzai jJara7J1etrzzatioJl of the surface s\vept out by the string.
I'he value of the Lagrangian can novv be \vritten
(~) NT /F2 EG)1f2 NG + N' 1....",z;===-l'( --- -i I ~--l ~ ..L ,:..
N (1-- C-"~ 2 ~ --T)
so the follo\ving slgn con vention results:
fIlL ~ ~ N U'L == -- Nx',t.
'I'he }~uler equations and boundary conditions novv take the forIn
(4.45 a)
(4.45 b) cr==o,7t,
\vhere \\'e nlust continue to bear in Blind the orthonornlality conditions (4.42).
7he solutions of l~q. (4.45) IZd've 710 plz~vsical JJlea71ing unless they are cOllsistent
l£'itlz tIle constraints.
It is \\'orth\vhile to ren1ark that in1posing l~qs. (4.~r2) alone docs not fixthe gauge completely in thc \va y that (4.36) cIoes. By 111aking a rcparanlctrization
such that
(4.46 a)
'1-' ':J-'ocr C''t''
d(1 J-r
':\'"" 2;;0(1
J"; '\ocr
\vhere
=== 0 ,
- 59-
\\'C 111<1 Y prcscr\'c I~qs. (4.42 ) ,
\vhere J'P' === .y1J. C:- (7 , a) , a (T ,G )). l"hus \ve ll1ay rCll1ain in an orthonortl1al
gauge \\'hile ll1aking a different linear cot11bination of the x P' (-:- , a) proportion
al to -:- in (4.36 a).
C. DIRAC BRACKETS
\\Tc no\\' proceed to develop a set of Dirac hrackets c0111patible v\'ith the
prin1ary constraints (4. I 3), the gaugc choice (4.36), and the boundary conditions
(4. I 2). Only then can \\'C take I~qs. (4.45) to be strongly valid. "le first extend the canonical \'ariables to all \'alues of their argull1ents h~/ defining a
nc\v periodic Poisson bracket
(4·47)
\vhere
and
{ .qJJ~ t (1" , a) , XV (1" , (j')} == _ g' t V ~ (a , a')
~ (a , a') ~~= ~ r (a , cr')
00
~± (0 , a') == 2: [0 (a - a'11=-00
2 U1t) ± 0(a + a' + 27Z1t)]
0/()
da ~± (a, a')f(a) == f(a') , o ~ a's 7t .
\\le \,vill need the follo\ving properties of ~± ,
d~ (0' ,0") _ 2-1_ (cr , cr')---d-O'- - - -~-~d----~--
and2~/2a - ° at a - 0, 1t
2~/2a' - ° at a'== 0, it
L-/- ~/2a 2cr' - ° at (J - ° , it
~2 ~/2(J2=-1= ° at 0,a - 1t
l'hus \ve see that J~q. (4.47) is COll1patible \vith the boundary condition (4.45 b).
- 60-
()ur full set of constraints is no\v
CPl (a) === f!JJ. It ~ 0
cP1 (a) === I. . x ~ 'TA . !)IN1t ~ 0
rrhe Inatrix of Poisson brackets is
(4.5 0 ) Cij (a , a') === { CPi (0") , CPj (a')} ~
o
o
o
o
A·p \.,~o ". tl(cr,cr')
T: C(J
o
A'P dTI: -6cr'..i (cr , a')
') '" 1)... A· A ( ')---.:.....1cr,cr~
oJ (' A ( , I-A~ L1\(J,cr) - .....
I",
o
o
o
o
I-Ierc \Ve ha ve 111ade use of the constraints (4.49) after C0111puting the lH-ackets,as \VC are al ,vays entitled to do.
To comput~ the Dirac brackets, we need the in n~rse matrix Ci/ (v , (;')defined by
.J cIcr" Clk (cr , cr") C,;/ (cr", cr') = 1'1 (v , v') =o
~- (a , a') 0 0 0
0 L1 (a , a') 0 0
0 0 L1 j (a , a') 0
0 0 0 ~ (a , 0")
I'he appearance of ~- in the first diagonal entry of the generalized unit 111atrixI ij (a ,a') and ~ in the ren1aining positions is dictated by the boundaryconditions. If \ve differentiate (4.5 I) \vith respect to a and evaluate the resultat a == 0 or 0' == TC, Eqs. (4.50) and (4.48) imply
d J1
" d "C ( If) c····· 1(" ') J I (') LJa a 14 a, a 41 a ,0' i (j = 0 ,;: === :}cr 11 \ () ,a (J = 0,;: =j 0 .
The corresponding expressions for the other diagonal elements of Iti vanishand they accordingly 111ay be set consistently equal to L1 (a, a').
To determine Ill' \ve first observe that the solution of
Jcicr" C41 (cr , cr') Ci"l (v", cr') = 144 (cr , cr') = ~, (cr , cr')
- 61 -
IS
C- 1 ( ') 1': D ( ') 1': (,~ ( ') cr )/41 a, a === ""), ' P ~J a, a ===), .p ~ a, a 1':.
H ere the generalized step function 0 (() , a') 0 beys
a > a' }, 0 :s;: a :s;: 7t ,
a<a
8·-80- 0 (a , a') === ~ (a , a')
o (a , a') === I
o (a , a') === 0
while the periodic sa\vtooth functions
o :s;: a' < 7t ,
(4· 55)
2:± (a , a') === I", 1: ~ (eiJl(o-O') ± ein(o+o'»)2 Ttl n=fcO n
2: i- (a , a') === 0 (a , a') - :
ha ye the properties
'~( '\ - ,-, (' )~ f- a, a ) - ~ "'-J_ a, a
(4,5 6)2:± (a + 2 7t , a') === 2:± (a , a')
2 2: (a a') === ~ +- (a a') - -~dcr ' " 1':
~ ~
c ~ ( ') _ /\ ( ') _ c ~ (' )-~--- ..;...J_ a, (j - L...l_ a, (j - - '1 ",;;,..,j i_ a, a .ccr ' ocr
But no\v Eq. (4.53) and the requiren1ent that
-1, -1, Tt ( ,,, cr' )C41 (a , a ) == - C14 (a, a) == ·~l)- 2:__ (a , a ) - -i:-
Tell us in1mediately that
III Ccr , crt) = Jdcr" :0' Ll, Ccr , crt) (L:_ Ccr", (J t) -:) = Ll_Ccr , crt).
rrhe other eletuents of the inverse matrix are
Ci21(a , a') == - C~/ (a , a') == 2;:' P ~ (a , a')
(4.5 8)
1t
C-1 ( ') - Tt C-1 (') I Jd "C -1 ( ")13 a, a - 2),' P 14 a, a - -2),~ a 14 \ a , a
o
Tt2
~ ( ')-2-(A-,-P)-2 ",;;,..,j i- a, a
1 1 Tt2C3i (a, a') === - Ci3 (a', a) == .) 2:_ (a , a') .2 (A, Pt
62 ~
The final result for Cij1 is therefore
C -- 1 ( ') 1t'zj' a, a == A' P •
o
ox
_ -~~- 2: (~ a')2A'P _v,
2:;_ (a , a') _ (1'
7t
o
o
~ ~ (a , a')2
o
~ (a, a')2
o
o
~ f (a , a') + :
o
o
o
Equation (1.58) can no\V" he used to con1pute the I)irac brackets compatible\vith the constraints (4.49). 'I'he results are
(4.60 c)
- .)
{x!J. (00) , X" (a')}* = i,~P u!J. (a) ['Lv -t\ g;v (a')] .l":j- (a , a')
+ A~p [AI' - IA~ g;1' (a)] UV(a') ~_ (a , a')
+ NA~ P (AIL f!jJv (a') _ AVf!jJ!L (a)) ,
{ g;" (a) , XV (a')}* = (- g!J.'1 +)~~~ iJjJv (a')) ~ (a , a')
+ ~ [- .1:
2
• &>!t (cr) ~+ (a , a') (--- A2 .qjJ" (cr') I,v ), . P)l~a (A . p)2 it
)"2
1t'2 N 2 u. ')' ~ , 't'N lL ,,]- . u· (cr) u (cr) '"-J_ (a , u ) - ~IJ It' (a) A .(A. p)2 A'
The constraints (4.49) are strongly valid in the Dirac bracket system.We may therefore solve the equations of motion (4.45 a),
(82 32
) u. 82 .:r~t (,,: , a), '~ .) -- ~~ -.,. x· ('r, a) ==-= 4 "( t ) 3() == 0, C"';" O(j~ / 0 "r - (1 't'--- (j
in the orthonornlal systen1 subject to the boundary conditions C4.45 b). rrhcn10st general solution consistent vvith the reflection principle (4.22) takesthe forn1
;tJl (- a) == ~- (Qf l(- -1-- a\) -1- Qll (7 -- u)), \., :2 \ \ ..
- 63-
\vhere \ve 111ay identify
as the coordinate of the end of the string. rrhc momenta arc
\vhere
O 'f-t (Cl) =--: ~q~t (8) _ ",1 /Ji)f-t (8 )""" v-· d8 -N<7 ,0.
(Q"l is often \vritten " })11" in the literature; \ve reserve this syrnbol for the
translation generator (4.29).) By cxanlining
\ve find that the boundary condition
requIres
Q'!l (0 + 27t) -- Q'P (8) == 0 .
Next, \ve notice that the four constraints (4.39) on X!1 and .qlJ~l can be expressed as t\\~O constraints on Qll ,
1(1 == Q,2 (0) == 0
.., P"',') == /... 0 (0) - 0 ,,~,- == ° ./~... -- NT:
~ate also that
... 0' 0\ I, . PA·, (') ==='---;---- •'-' / .i\ it
rro find the brackets of Qll (0) c0111patible with the strong constraints(4.69), \ve insert Eq. (4.62) for xlJ. c-r , (1) into both sides of the Dirac bracket(4.60 a). i\fter identifying ter111S of the satne functional fornl on hoth sidesof the equation, \ve find that
-( QIl (0) , Q" (O')} === - ~ g!J.'1 E (0 - 0')\)
-t- (), . P)-l [8/~'l Q'i{(O') - 0' A" Q'~l (0)]
+~~A:~~~ (AYQ'f' (0) (A · P) Ai' Q'v (0') (), . P) .- A2 Q'i' (0) Q'Y (0')).
- 64-
Here we have defined the functions
p (8) === ~_ ~ I eillO11. n=:pO 7l
(4.72 a) E: (8) === p (8)8
\vith the periodic properties
p (8 + 2 1t) == P (8)
2 U1t)
(4.72 b) E: (8 + 2 1t) == E: (8) + z
~ (8 + Z 1t) == 0 (8) .
One may verify that all of the brackets (4.60) follow from (4.7 1), (4.62)and (4.64). The brackets (4.71) are also con1patible \vith the constraints (4. 69).In fact, it is alTIusing to note that if the " canonical" bracket of Q~L (8) is
taken to be
{ Q~ (8) , QV (at)} == -R- g!J.') E: (8 - Gt) ,
Eq. (4.7 1) is exactly the result one gets by using the Dirac procedure with the
constraints (4.69).Examining Eqs. (4.60) or (4.71), one sees that the Dirac brackets look
like a plausible basis for a quantunl theory provided
).2 == 0
It" === 0 for the D-2 independent degrees
of freedoll1.
\Ve are free to choose the gauge in such a \vay that the theory is as simple as
possible. Therefore \ve take
(4· 75 a) i == I " . " D-z
and define also a c()1nplementary vector I,,: with
A~ === - !"r)---l === - 1IV z(4· 75 b)
so that
•
(4· 75 c) )..... A* == 1 .
- 65-
Hereafter, any vector y~L \\~ill be split into a transverse part Y i, i == I " .. ,
D-2 and
-)(-
(4· 76)
y+ == J,P. y~l == (\7"D- 1 \TO) /V2 == y_
y- == 'A:V!l == (VD-
1- VO)/V2 == y+
In accordance \'lith the null-plane conventions listed in Appendix A.\:Vith the gauge choice (4.75), the nonzero hrackets in Eq. (4.60) becon1c
{ gi (a) , xi (cr')}* == - azj ~ (cr , cr')
{ x+ (cr) , x- (cr')}* == - -J;;
{x- (cr) , x- (o')}* == P+ [u- (0) L:+ (cr , a')IT
+N"~ (,0jJ- (a f) - ,cr (a))]
{ - () £ ( ')}* - --~- ( £ ( , \ ") ( !'X 0, X \ cr - p + \U a) ~_ \ 0 , (j )
( /Jl;- () (}lii ( ')}* _ N 2 rr: i ( ) dL1 ((J , (J')l. ;:r a,;:r a - - P+- It \ a --8-;'-
--.~. (!)Ji (a'))Nrr "
{ eJ- (a) , x- (a f)}* = -;-i-' [,OjJ- (a f) Ll (a , a f)
+}cr (0jJ- (a) L_1 (a , af) -- ~ u- (a))]
( c)/J- () _i ( ')}* _ rr: (}1J i ( ') A ( ')l.J 0 ,x \a - p+J cr Ll o,a
For reference, \ve gi ve also the nonvanishing brackets (4· 7I) \vith the gauge
choice (4· 75):
(4· 78) {Qi (8) , Qj (O')}* == - -~ oij S (0 - 0')
{ Q+ (6) , Q- (6f
)}* = - :-i{Q- (0) ,Q- W)}* =F-i-- [d- W) (p (0 - Of) :) +
+ ci (6) (p (0 - Of) - ~)]
{ Q- (0) , Qi (Of)}* = ;+ Qf i W) (p (0 --- Of) ~}
- 66-
l'he action of the translation group generators (4.29) IS novv
(4· 79)
{ P-, x- (cr)}* == -i~+ ,q;- (a)
{ P- i ( )}* _ rr tJ]J i ( ),X "cr - p+ <:T cr
{ pi , x j (cr)}* == _ ~1"j
\vith all other brackets vanishing. Since
Nrr'T == -p-.+- x+
and p- is the canonical generator of x'+ displacements, \ve may solve the constraint Eq. (4. 13 a) for {!J~ to gi ve the generator of 7-displacenlcnts,
p+ pI-I == -
.:\rr
..
21:\ rdrr (9. & + N 2
U • u) .()
We veri fy that Eq. (4. 80) g i \TCs I-I a tl1 i1ton's equat ions 0 f 111() t i() n In '7 fo r theindependent variables Xl· and f!J/:
(-t. 81 )
\Ve next examine the Lorentz group generators (4.3 2 ). rfhe rotation groupgenerators l\l ij transform Xi as a three-vector.
{ M ij k l.* _ ~ik i _ ~Jk i1 ,Xj-OX oX
'[he boost generator 1V10 i, on the other hand, adds gauge transforn1ations to
the variables so as to prcserve the constraint Eqs. (4.36) in thc nc\v Lorcntzfran1c. We find
{ M-i ~j ( )}* _ ~i.i - ( ) + rrN dxJ (0') ( ,.£ () _'rPi )1 ,.:t cr - -0 x cr -p+~ -~'t cr -1-- Nrr
7t
+ _~.i_~ !f.j (cr) Jf" d 8x i ('"r, x) ~ ( ~P + 3cr lJ. d'"r ~- \ (X , cr) .
o
'rhe first tern1 is expected from Lorentz covariance, while the rernaining terlnsarise froln the fact that a gauge transforlnation (a (7, cr)-reparan1etrization)must be Inade to preserve the gauge in the new Lorentz fran1e. If the infinitcsin1al Lorentz transformation is written'
p'!!' == p,L
- 67-
then the change in x!-L (7 , a) under a Lorentz transformation will invol ve alsoan infinitcsinlal rcparanlctrizatioll 7 , ('j --~ ;; (7 ,a), G(7 , a):
",.rJ. ('-. ~\ " ",.'(1. (.-r- ~\ == ",-!1. (;::. (.- ~) ;:;. (.- ~\. + ,.\!1.') 'V (.- ~),v " .. , V) -r A, \, ~ , u) A I",,, \ .. ,v ,v \," , V) UJ .t"J ", v
(U~J.'J X\J ('7 , a)
l'he value of -17 is fixcd by requiring
-r (P-t I +" I) )(t) v'
Equation (5.85) then immediately gives
Once Ll-:- IS fixed, Lla is con1pletely deten11inecl by the requirenlent (4.46)that the orthonorn1ality of the coordinate systen1, Eqs. (4.42 ), be preserved
by the rcparan1ctrization,
(4. 88),0cr
]'he solution of ]~qs. (4.88) IS
a
N1t(u+V
(' aPv I" , ( ,\)~(J == ----.'--- - da 'l' I '1" a)P+ l\rr 'J\ ,, ~
o
(I
TC
- - _~1tw+v J!" d ',., (.- ') ~ ( ')-- p + . (j (Jv \ ~ ,a ~+ (J ,a .
o
Thus when a I.Jorentz transformation is made, the required change of x tJ• ('1" , a)
\vhich preserves (4.86) and (4.42) is
x'p· ('1" , a) == x P. (-r , a) U.'J ( )(0' Xv '7, (J
N1tw+V J- /til- (-r , a) --~. da' Vv (-r , a') L:+ (a , a') ,
o
Vvhich agrees \vith the result (4.83) given autoJ}latical~y by the Dirac brackets.
- 68-
D. FOURIER COlVIPO~ENTS OF \TARIABLES
\Ve begin the Fourier analysis by observing that the a-a veragcd coordinates and n10Inenta of the string arc
n TI
q" = : JdG Xl' (0 , a) = 217t JdO QI' (0)
o -n
~ ~
pi' = Jda &pI' (0 , a) = N Jda v" (0 , a)
o 0
=== N (Q~ (TC) - Q't (- TC)).
Then "\ve Il1ay \vrite
vvhere one Ina)' deduce fronl (4.68) that I'" is periodic,
Then, [ronl (4.62), the canonical coordinates x P' can he written as
fl.. ft I 't'pf" I fl ft .-X (7, a) == q -;---Ni- + --2 (1 ('"r -t- u) + j (w -- a)).
Next, \ve expand 1 ft in harn10nic oscillators using the DD F FourierC0111pOnents (-1(-) (Del C;iudice, Di \Tccchia and Fubini, 1972 )
(4·95)
TI n
a;:, = ~ I dO Q/f' (fl) exp [im ),. Q (fl) Nn/),· P] ~ ~ JdO Q'" (0) /mO
TC'
I rcia (.qJI' (or , a)2 J
-n
~T It (.- \) im(cr-t--:)l'U \",U) e .
Fron1 (4.92), we see that a~~ Inay be \vritten
7t
!t _ pIt ~ --_. N ~ J' dO'jf l (0') JmOI
am - ( m,O l1Jl..l. 2 {; •
-n
(*) A modified fonn of these oscillators can be used to quantize the stri ng even \vhenthe gauge is time-like (Goddard, I-I a.nson and Ponzano, 197..).).
- 69-
1\1ultiplying by e- im o/nz and sun1ming over 17Z
to find an explicit expression for f!L:0, \ve n1ay ell1ploy I~q. (4.72)
f It (n \ _ i "l I IL - illl 0\J) - -_ .. ~--- ~ ---- am e .
1\~ 1ll-!: 0 17Z
Thus the c0111plete Fourier expansion of x P- is
and &It == N 2xlt /27 , u lt == 2xl"/2cr.
Not all of the Fourier components q''', pIt and a:~l are independent. l~ sing,for exa111ple, the expressions of the Fourier components in tern1S of Qll andthe Dirac brackets (4.71), we find the following nonvanishing Dirac brackets,
{It V}* _ .N / ItV~am ,al1 - - z ~ 7t lJzg 0m,---ll
{p'+-, q-}* == - I
{pi, qi}* == ~_ ()l)"
rp- -}* _ pI.. ,q - p+
( 1>- 1'}* == piI.. ,q p+ '
\vhere It IS the gauge paran1ctcr (4.75) and \\'C bear 111 n1ind that
'rhcse brackets arc c()1npatible \vith the strong constraints (4.49) or (4. 69),\vhich translate into the follo\\Ting constraints on the I~ourier c0111ponents:
qf- == °
T'he expression for a: in terms of the other variables can be deri vcd, forexample, by solving the constraint Eq. (4. I 3 a) for ,q;-,
-70 -
expanding g>'" and u'" in the Fourier cotnponents (4.98), and identifying the
coefficients of cos ncr e- iwr .
Now \ve may take as our independent variables
Z =-== I " . . , }) ----- 2 ,
Z === 1,"',1)--2
1l == -- CX),"', CX)
\vith the nonvanishing brackets
(I)i .i}* _ ~ljl ,q ~- _.. - 0 •
Using (4.101), \ve easily confirn1 that the brackets (4.99) follo\\t~ directly. Note
that
In the literature, one often sees the notation
(4. 106)p+
2! ==--m NT':
\vhere f£,Jl obeys an algebra z'sonzorphic to the \/irasoro algebra (4.26) (Bn)\;ver,
197 2 ):
( UJ (£/ 1* __ . / . , ) (/:Jl'x 1Jl , c-Z ll f - - Z (Jlt - n J.. 1l -1-m .
rrhc in1portant difference, hoY\~c\·cr, is that fE,n is Jlollz1anis!zillg, \vhile the \Tirasoro operators vanish in the Dirac bracket algebra.
We tnay no\v usc the independent variables (4. 103) to \vrite explicitexpressions for the canonical variables and Poincare group generators of thefull y constrained systetTI. The canonical coordinates are
(4. 108)
i ,-, I .----- .t:..J - a- cos ncr e-l11'"~
Nit II FO n Il
i ~ ~ I -il11'£..J ~ -~- am ·an - m COS ncr e2N1tP+ n=l=O allm n
-71-
2: [am·a_- mJJl
I ) -, !: _-- /Il '7+ ...,. /-A (71/ COS l!o e :I. IILU
\:\Tith S0111e labor, one 111ay check that these expressions for .1~ll and «dP~l repro
duce the Dirac bracket algebra (4.60) \'lith the choice (4· 73) for )t.rrhe Fourier decomposition of the Han1iltonian (4.80) is
I-I ==00 (I -., I '));_ ..2: am ·a_- m == -- ~_ p'"
~ J. lJlc.:=-N • I. \
2 ~ amoa '11)lJl=-=l
and continucs to generatc the cquations of 111otioll (4.81). ]'he I--torcntzin \'ariant mass-squared can be expressed In the form
ex:>
1\12== --- 2 p+ P- - p2 == 2 2: am' a_ m •
111=1
rrhe translation generators are P+, 1Ji, and the dependent \'ariable P- gi ven
in (4.101). lTsing the definition (4.32) of the Lorentz transforn1ation generatorl\I'lV together \vith (4. 108), \ve find
1\ -1- If: I pk /... IJj1 i' == q' --- q
Ivrr /..: == _ qk p+
IV1+ == - q- 1)+
lVr- l . =--== q- 1)1.' __ qk p--
\Ill) •
In the classical systen1, the Poincare group algebra (4.34) continues to hold.rrhis question is n10re subtle in the quantuln n1echanical systelTI, \vhich istreated in the next section.
I~. QUA~Tl!l\I IV1ECI--TANICS
]'he quantunl mechanical systen1 corresponding to the classical stringrnodel described in the previous section has been treated extensively in the
literature (Goddard et aI., 1973; I~ebbi, 1974). We vvill give here only abrief summary of the results.
-72 -
The follo\ving variables can be taken as operators \vith con1111utators corre
sponding to their Dirac brackets, using the convention {A, B}* -7 I; [A, BJ:z
[ i k] N ~ik ~am ,a11 === I 7t 1no 0l1,-m
i [P+, q-] == I
i [P i, qi] === () zj'.
i\JI other commutators vanish, with the prOVISO that a£ === pt".The dependent variables are
P - _ - + N1tllO _ I-an ----p+ 2 p+ ( : 1; a m
o a_m :-2N7tiXO)111=-00
a,l. ==-2 p+
00
~ am·an - m111=-00
N1t ifp+ 11' 12 o.
Here the colons : ( ): in P- lnean that the expression is to be norn1al-ordered,\vith the destruction operators a~ \vith 1n > 0 al\vays on the right. Since !Jcontains noncommuting operators, \ve may need to acId for consistency azero-point energy which \ve haye \vritten as N7tcxo/P+. ]'he Lorentz-invariantmass operator is thus
The requirement that ao be expressed in norn1al ordered form causesthe C0111mutator of canonically conjugate pairs [a~, a=m] to pick up aSchyvinger term vvhich \vas not present in the classical theory.
Other COmlTIutators are unaffected:
(4. I 16)
-73 -
'-fhe Ha111iltonian is no\v
p+ p-H == ---- -~---
l\T:
and generates I-Ian1ilton's equations of 1110tion in '7 for the dynatnical varIables. \\Te 0bserve that T is dimensionless and
(4. I 18)3A(--r,cr\ 8A
. [H 1,'] pl/NTZ ,q == li1t'
due to the constraint on x+.
equations of motionWe find that the Fourier con1ponents have
so that the fields x k (T , cr) and 1110111enta g/.: (T , IT) obey
i [H , x k] == g;kIN == 0'1/':
i [H , g;k] = N -::2 xk (-r , cr) = :jk.
l'he I110st unusual feature of the quantized string is of course the factthat the L,orentz group algebra does not close unless certain conditions are111Ct. \,re \\Till not go through the calculation here, but 111crcly state the result.One finds the because of hern1iticity and norn1al ordering requirements, thecomn1utator of the quantu111 operators 1\1/ - and :VI J - takes the for111
i [M1-, M/] = - 2(p+)-~ 1; [172 (1 --- 21 (D -- 2))'111=1 4-
I (1 )] i j j i- ----- (D - 2) - Cl.o (a_ m am - a_ m am) .'J7Z 24
l'his C0111111utator Inust vanish if the Lorentz group algebra is to be satisfied,so \ve conclude that
D == 26
(Xo == 1 .
'-[here are various n1ethods of relaxing these restrictions, but the nor11'1 of somestate al\vays becomes negati ve for D > 26. No clear intuiti ve reason for sucha phenomenon is understood at this time.
5. lVIAXWELL ELEC'-rROlVl.L~GNE1~IC F~IELD
The classic exan1ple of a theory \vith an invariance under an Abeliangauge group is l\1axwell's theory of electrolnagnetism.
-74 -
~~. 1~LECTROl\1AGNETIC I-IAl\1ILTOi\IAN \VITI-lOUT C--;'AUGE CONSTRAINTS
rrhe application of the Dirac nlethod to electronlagnetis111 has heen cxa
lllined by Dirac (1951, 1964). (Sec also Anderson and Bergn1ann, 195 I, andIZundt, 1966). \\Te shall begin by considering a field A!L (t ,x) transforlningas a four-vector on the index [.1 and possessing continuously infinite degreesof frecdonl labeled by the spatial coordinate x, or sonle suitable cOlnbinationof the :tY depending on the type of d ynanlics chosen. The 1110St general fornlfor the action is assulned to be
s = rd 4 x2(A"(x) , 2Al'(X)(2x').
N O\V \\'e \vish to nlake ...-\!L descri be a 111assless \'ector field, \vhich has on 1y
t\VO independent degrees of freedonl, not four. 1'he tinle-tested nlethod fordoing this is to alIo\\'" 2! (up to total eli vergences) to he a function onl y of
1~'IV has the \'irtuc that it IS unchanged by the spacetinle dependent gaugetransfornlation
'rhe freedo111 to choose .:\ (x) Inay be used to clirninate the unphysical C0111ponents of ~-\~t (x). Note that with the convention (5.2), the usual E and Bfields are
13'" ===I '"/; '1 . OJ; 3~\k. slj " .F'.J f< === slj"
:2=== (V X A)1 .
\\1estill ha vc SOI11e latitude a vailablc in choosing the Lagrangian; oneuncon ventional forIn \voldd be the Born- Infeld electrocl ynarnics
(5·5) 2===
\vhich agrees \vith the l\/Iax\vell theory only in the \veak-held lin1it (see Dirac,1960). In \vhat follo\vs, ho\vever, \ve shall be content to in\'estigate the usuall\laxwell Lagrangian,
(5.6)
in Lorentz-Heaviside units.
if === - ~ F~LV F4 (1.')
-75 -
\\Te first nlake the traditional choice of slJecifying the cl ynanlics on con
stant-tiI11c surfaces. l~hen the canonical I110nlenta are
'I'he spatial conlponents of nIL are
(5. 8) :r (x) == -- E (x) == A -t y~r\0 ,
\vhile the tinlC conlponent vanishes \veakly, gi \-ing the prl111ary constraint
(5·9) D( \ evOn X) I"V •
rrhe canonical I)oisson brackets of n'" and l\.v arc given by
(5. 10) { -.T''' ( ) _\ ( )" ~!J. ~3 ( __ 'II. t, x ,- ') \t , Y J - Ov 0 X Y)
and are ob\-iously inconlpatihle v.:ith l~q. (5 ·9)·1~he \-ariation of the fi.elcls ...r\" (x) gives the usual l~~uler equations, \vhich
nlay be only \veakly valid:
(5. 11 ) ') CiS!? == F,t" , == A..",!l __ L\'l," ~ () .L ') 8 (c\) .A.. pJ 'J'" ') ... - ')
The zero C0I11pOnent can be \vritten
(S·12a) .A) ~ o.
LTsing }~q. (5.7), \ve re\~/rite this in the for111
(S·I2b) y.;r --Y·E~o,
\vhich is indeed a \veak equation because it is inconlpatible \vith the l)oisson
brackets (5. 10).
Let us no\v begin to apply I)irac's approach to our syste111. First \ve recallthat the lVlax\vell Lagrangian (5.6) is invariant under the Poincar(~ group ofspacetinle transfornlations [in fact, it is in variant under the full confornlaIgroup of \vhich the Poincare group is a subgroupJ. \i\1e list belo\v the effectof each Poincare group transfornlation on _y'L and the held ~,-\fl (x):
(S·I3 a )
(5.13 b)
where
\IC(~ I f ~ Il (ItB (3 Itx\~\ == _.lorentz trans orm: ox' == (U xi3 \x~ g' 1 -- ,,'tot g' )
~ ;\!L _ [( /X Af3 _ .Jj I_X),"). Afl . i.. !lC( 1\[3 _ !li~:\ lJ.]O~J.. - (.0:x[3 \x g X g '-I. T g... g ~J..
[ lJ. ALL [3 ~ All. ex + ,,----ex[3 I.ll A'V]== (.0.:x[3.X ',-- x ' , k.i),'J g I
- 76 ---
Note that the tensor indices of ...t\'" transforn1 oppositely to those of x'" sinceit is the inverse transforn1ation which relates the old tensor indices to the newcoordinate systen1 .
.i\ canonical conserved N oether current is associated vlith each transforlnation. The canonical currents and the corresponding conserved Poincaregroup generators are given in the equal-time convention by
and
(5.14 b)
t{lf.!lV _ . ~l elf.V _ ~~v O!f.[L __~d£ 2:[1;V\O_.A - X c ~t C ClAj"lf. Ao'" -
== x~t e~v -xv e~!.L l;lf.A ~>:~ ...~o
NO\V \VC take the canonical 1-1 an1iltonian
(5· I 5) LI -- 1)0 - I" i 3 ~ 000 - { i 3 ~ ( !l\r c - - (. x c - (. X n ...lJ..p..
.. '
and add, according to Dirac's prescription, a n1ultiple of the sole prln1aryconstraint nO ~ o. We get
(5. 16) H O = He .1 d3x VI (X) nO (X) =
- rd 3 (I 2 -+ I B2 n i\ 0 I (~'\ 0( ~\ ')- .. X 2:T - 2- - :T' V ..tl.. I VI _1) it :l); .
where \ve have used the identity
\Ve look for secondary constraints by con1puting the bracket of I-Io \vith thepritnary constraint nO ~ 0 and requiring the result to be ~ 0 for consistency:
(5. I 8)
Our t\VO (obviously first class) constraints are thus
t~l == nO ~ 0
~2 == \7 . :T ~ 0 •
-77 -
We no\v add both '~I and ~2 to the Hamiltonian with arbitrary coefficients.Using (5. 17) and doing an integration by parts, \\Te find
~
H = He Jd 3X (VI nO V 2 V . :r) =
= rd 3x (~:r2 +B2
vI nO (V2+ AO) V . :r) -
- J ds· :r AO
•
Soo
I-Iereafter, we \vill neglect surface integrals such as the last term In (5.20).By evaluating the brackets
(5.2 I)
AO == {AD, H} == -VI
Aa £ __ {,,£ H} £ ",0,£ ""1/""1"-== ....""J.., == 1t -.l"""'l. - ~:t1'2 .:x
\ve find that \ve can set VI == - ...~o == AO,D and V2 == 0 to give the final result
(5. 22)
~ote that AO is a basically arbitrary function in the Hatniltonian which iselin1inated fron1 the equations of n1otion only after \ve choose a gauge. 'fhelast t\VO tern1S in (5.22) are in fact the generating functional for infinitesin1algauge transforn1ations with i\ (x) == ...t\0 (x).
l'he functions VI and V2 are not necessarily the san1e for all componentsof the energy-n10Inentun1 tensor. \Ve 111ay in fact write
Arguments parallel to those which led to Eg. (5.22) give
p = fd3x(-:r X B + nO VAo A(V· :r)),
so
We see that Eqs. (5.22) and (5.24) are the appropriate modifications of Eg.(5. 14a) to generate the spacetin1c translations of the theory \vithout gaugeconstraints:
(5. 2 5) {PIt A·V()'l. ~Av/"'\ ., X j ==.: -.~x!l...
The other generators (5. 14) of spacetilne sytnn1etries ll1ay be treated In asimilar fashion.
-78 -
I). RAI)IATION GAUGE C~ONSTRAINT'S
V\re no\v sho\v that the Dirac brackets in the radiation gauge lead directlyto the traditional quantutn-Inechanical structure (e.g. B jorken and l)rell,
1964).Our objective is to use the gauge freedorTI (5.3) in our systenl to fix t\VO
conlponents of .i\f.t so that the first class constraints (5. 19) beCOlTIe second class.Since ito ~ 0, one logical choice is to set ~;.
0~ o. rrhis is accoll1plished hy
the gauge transfornlation
(5. 26)
XO
( dt AU (t , x) ,
o
for \vhich the infinitesinlal generator takes the forIn of the first class, \veaklyvanishing addition to Eq. (5.22):
t 3 • /r i 3 " fl '"'\ 1\L[Al =.1 d' x(A(x)nO(x)-A(x)V· .• (x» =. { lIn ell,! •
No\v the Euler equation (5. I 2) becolnes
(5. 2 7) \7. A' ~. 0,
so that a second tinle-independent gauge transfornlation can be ll1ade tofix A'. ]'he radiation gauge, \7. A" ~ 0, is a con \Tcnicnt choice cOtllpati hIe\vith (5.27), and is achieved by lnaking the gauge transforll1ation
A' -~ A" (x) = A' (x) + Vx ( d 3y 4 IT ; _ y V y ' A' (xo , y)
(5. 28) .1
\ 0' !\ 0" ( ) I d r/3 I n A' (() )11.. == 0 -+ ....l.. X == 0 -1'"'\ {y I IVy' x, y == o.
oxo ~ 4 T: X - Y
Here A' is given by (5.26) and \72 (I/4n I x - y I ) == - 03 (x - Y),\vhile the vanishing of ...;.0" fo11o\vs directly fron1 (5.27). Dropping all pritnes,\ve find the set of constraints
C?l == nO ~ 0
CP2 == V . ~,. ~ 0
C?;{ == AO~ 0
?4 == V·A ~ 0
In the ne\v gauge.
The matrix of Poisson brackets of the constraints (5.29) IS
o
o
--1
o
o
o
o
o V;o 0
o -\7; 0 0
- 79 --
Inlposing the boundary condition that the fields vanish at cx), \ve find thatthe in \-erse of (5.3°) exists and takes the fonll
° 0 __03 (x --- y) 0
0 0 0
(5.3 1) Ci;.1 (x ,_v) == 47": x--y
03 (x - y) 0 0 °0 0 0
47t"
1'hus the Dirac brackets may be conlputed frOITI Eg. (1.58) to be
{rcfl (t , x) , AV(t , y)}* == (- gflV _ g'lO gVo) 03 (x y)
{ ,t .V}* _ {A!l A V1.* _7t,1t - , j-O.
Equations (5.3 2) are COlnpatible \vith setting the constraints (5.29) strongly to zero. 1'herefore, only t\VO of the -,-~i and t\VO of the nz" are independentvariables . .L~lthough \ve n1a y sol ve \7 . A == 0 and \7. ~( == 0 for the rerrlainingt\\TO dependent variables if required, this is unnecessary; in the process ofin verting the nlatrix e ij to fInd the I)irac brackets (5.32), \ve ha ve autolnatically given the dependent \-ariables the SaI11e hrackets they \vould ha \-e\vhen expressed in terI11S of the independent variables.
\'7e now verify Hanlilton's equations of I110tion in the fully constrainedsystem. 1'he Hanliltonian is just
(5 ·33)
1'hus the equations of ITIotion are
(5 ·34){A(x) , H}* == .1 (x) == A
{~l(x) , H}* == - \7 X B== - E.
In deriving Eq. (5.34), \ve have assunled that E vanishes sufficiently rapidlyat CX) to drop a surface integral and ha ve nlade use of Eq. (5. 1 7).
Using Eq. (5.14), \VC find the renlaining Poincare group generators to be
I"
P = .J ti3X(-1/ VA')
(5·35)
- 80-
In the star brackets, pIt and :YIlt'V continue to obey the I)oincare algebra (2.25).Under the action of translations and rotations, the ..:-\i transfornl as
{pi, Aj (x)}* = dj~(X) = 8i Ait
{lVIii, A k (x)}* == (Xi 3i - x.i 2i) A k (x) -t- aik Ai (x) - ajk Ai (x) .
~~s usual, surface integrals have been dropped in Eq. (5.36). Under boosts,however, extra tern1S occur \vhich cannot be neglected. \\7e find
(5·37)
rrhe extra terlTI is required in order for lVI()z" to have vanishing star brackets
\vith V· A. It nlcans that under an infinitesinlal boost paralnetrized by(VOi' ..:-\,t (x) undergoes a gauge transfornlation of order (VOi in order to keep...t\0 == 0 and V· A =--= 0 in the ne\v Lorentz franle:
rro sho\v that A'o == 0, \ve have used the fact that \vith our gauge choice
(5.29), the Euler equations (5. I I) reduce to
and integrated by parts.
c. ...J\XIAL GAUGE
.i\nother an1using gauge \vhich can be treated in the canonical forn1alisn1is the axial gauge (Kumnlcr, 1961; ..:-\rnowitt and Fickler, 1962). We beginby using the gauge invariancc of the theory to set
We see that if \ve choose 'V2 == 0 in I~q. (5.21), then Eq. (5.40 ) \vill hold
for all tin1es anI y if
Equation (5.41) is sinlilar to a secondary contraint follo\ving from thegauge constraint (5.40); it replaces the condition V· A ~ 0 used in the pre-
- 81-
ceding section. Th e full set of constraints is then
(5.42 )CP2 == \7 .:r ~ 0
CP3 == A3~ 0
m 3 AO,3T4 == n - ~ o.
The Poisson bracket matrix is
0 0 0 (8/8x3) 03(x_.y)
0 0 --(8/2x3) 03(x- y) 0
0 -(3/3x3) 03(x-_y) 0 + 03(x-y)
(2/2xa) 03(x-- y) 0 -03(x-- y) 0
We choose boundary conditions so that the inverse of (5.43) is
0 -g (x, y) 0 f(x, y)
eti l (x , y) ==g (x, y) 0 - f(x, y) 0
(5 ·44)0 - f(x, y) 0 0
f (x , y) 0 0 0
\vhere
3g(x ,y) ==f(x, y)dX3
and
(5.46)g (x, y) == ~- 0 (xl_yl) a(x2_y2) \x3 - y 3
1
f (x , )1) == ~- 0 (xl - yl) 0 (x2- y2) E(x3 - y3) .
Here E (x - y) == algebraic sign of (x - y).When we insert Eq. (5.44) into the Dirac bracket formula (r .58), \ve find
(5.47) {n~ (t, x), n V (t, y)}* == 0
(5.48) {i\'l (t , x) , ~~v (t , y)}* == (glLOg3v -t- gVOg31l) f (x , y)
(5.49) {nit (t, x), AV(t, y)}* == (_gllV _gllOg"O) 03 (x- y)
It3 d f ( ~ )+g -3xv
x , Y .
6
- 82-
The nonvanishing brackets are
(5.5 0 ')
{ 7t1
, A1}* == {7t
2, A2
}* == - ~3 (x - y)
{ 7t3 A1}* == -t- _~_\x ,yt, dX1
{ 7t3 A2}* == -+_ 3/(x,zL, d.t-2 ,
so 7t\ 7t2
, A l and ...t\.2 still obey the canonical brackets (5. 10). 1~hese bracketsare 111anifestly con1patible with all the constraints (5.42) and can be used asthe hasis of the axial gauge quantization schen1e.
'rhe Han1iltonian in the axial gauge is
\vherc \ve ha ve set the constraints (5.42) equal to zero. i\.0 is a dependent variable found by sol ving the I~uler equation
, 2L'2 7t
to gi ve
All (x) = - ~ I d~ Ix 3 - ~ I (-\ n 1 (tx1 x 2 ~) + 22 n 2 (tx1 x 2 ~».oJ
U sing just nl, 7t2
, A \ A 2 as the independent variables, \ve easily confirn1all of the brackets (5.5°). ...~ssun1ing appropriate boundary conditions, \VC
find the Han1iltonian equation of n10tion
{7t i, I-I}* == - (V X B)i == - :It i,
{Ai, H}* == .J..~ i, I === I , 2 ,
for the independent canonical variables. ]'he properties of thc rest of ther>oincare group generators are treated as usual.
D. :N"ULL-PLANE BRACKETS
~-\s an illustration of ho\v one can define a Han1iltonian field theory ona surface other than the constant-tin1e surface, \ve exan1ine the null-planetreatIl1cnt of Kogut and Soper (1970). One interesting feature of null-plane
- 83-
dynamics is that the nunlber of independent canonical variables is reducedto half the nunlber prcsent in equal-tinle d ynanlics.
We begin by establishing our notation (see also .l\ppendix ...~). vVedefine
(5.5 1)
') ) _ ")_ _ _I_ (~3 _ ":'0)~o -~. - _ v CJ
Vz
x£ === x£ z == 1 ,2,
so that the nletric IS
-0 0 0
0 0 0
(5.5 2 ) glJ.V ==0 0 0
0 0 0 1_
\vith tL == (+,-, r, 2). Latin indices of vector quantities (e.g. x) \vill takeon only the values i === (r ,2). rrhe variable x+ \vill now be used in place ofx o to define the canonical lTIOlllcnta and the evolution of the systenl. rrheI)oisson brackets and sYlTImetry generators of the theory can then be \vrittenas indicated in Section 1. D.
1'he canonical nlomenta nil- are conjugate to the x t- derivatives of ...J\.(J.'and are therefore defined as
Ltn'
\vhere fi? is still gi ven by (5.6) because the volun1e elenlent is unchanged.1'he canonical Poisson brackets are given on the surface x+ == constant, so
== - o~ 0 (x- -y-) 02 (x - y)
== - o~ 03 ex -y).
Since the nletric is gi ven by (5.52), A-is the canonical conjugate to 7t+, ratherthan A +.
1'he first thing \ve notice is that the definition (5.53) of the canonical1110n1cnta gi yes rise not to one constraint but to three pri17lary constraints
- 84-
\vhich are incon1patible with the canonical brackets (5.54):
(5·55)
(5.5 6)
(5·57)
7t+ ~ 0
1 1"'\ A+ + ""I AlTt - 01 c_ ~ 0
The definition of the fourth momentun1
(5.5 8)
is not a constraint but a dynamical relation between the velocitythe canonical variables Tt- and A-. We observe that
A+ and
so the constraints (5.56) and (5.57) are second class.\Vith null-plane dynamics, the forn1al expression for the canonical energy
n10n1entun1 tensor is
\tvhere the 111etric IS gi ven by (5.52). The canonical conserved translationgenerators are deduced fro111 Section I. D to be
(5.60)../
and formall y generate the transfornlations
(5.6 I)
Since P r == P- generates x+ displacements, the natural object to use as theHan1iltonian is
".
He = - p- = -./ dx- d 2 xot- (x) .
where
Cl + - __ 1 ( -)2 _ 1 F,12 F 12 _ ( -) -L i).\ i\-V c - :2 Tt:2 Tt \.- ITt\. 1) ~'""""l. •
The minus sign in (5.62) is dictated by the convention that the Han1iltonianbe positive definite and the fact that our null plane nletric (5.52) has no n1inussigns as did the ordinary metric (I .46) with gOO == - I.
Now we search for secondary constraints. We forn1 a new Han1iltonianH by adding multiples of the constraints (5.55) - (5.57) to the canonicalHamiltonian,
-- 85 --
Then we require that H ha ve vanishing canonical brackets \vith all of theconstraints (5.55)-(5.57):
(5 .65) { + H.... } - ') - -L ') i ~ 07t , - ,,- 7t i \. z' 7t ~
Equation (5.65) is a genuine secondary constraint, while Eg. (5.66) in1posesconditions on the u/. Of our final set of constraints (5·55)-(5.57) and (5.65),the onI y first class ones are
~1 === n+ ~ 0
Y2 === 3_ 7t- 2£ 7t£ ~ 0 .
1"he relTIaining t\VO,
(5.68)
1.....,1 ;\ +Xl === 7t - ~ ..tl..
2 .....,2." ..'--1.2 == 7t -~. ...'"""1. I
2_.Lt\1~0
2_ A 2 ~ 0
are second class due to F:g. (5.59).NO\V \ve in vert the bracket lllatrix {I./, Xj} gI \Ten In Eq. (5.59) for the
second class constraints ~·ith the result
Ci~.l (x ,y) == [1 0] I S(x- _ y-) 02(x _ y) ,. 014
\vhere s (x- - y-) == sign (x- - y-). Next \ve take H to be the first-classHamiltonian analogous to (1.24),
It is easy to verify that (5.70) contains the explicit solution of Eq. (5.66)for the uJ, as discussed in Section 1 . .L~. 11"'inally, we use Eq. (5.69) to conlputethe preliminary Dirac brackets, which \ve denote with a prime:
(5.71) {AIL (x) , A" (y)}' = (g'"l i v g'"2 g2'') : E(x- - y-) a2(x - y)
{]tf1(x) , ]tv (y)}' = -l"+ gVi V~ : E(X- - y-) a2 (x - y) +I ( ~l+ "z' .....,:r2 g g \.'z'
,,+ III ,.....,.t· ( ~ll Ivg g ~i- g g ~l2 2V) -"X) ~3 ( . \g g di 0 .X - YJ
.l\.ll equations of nlotion are no\\· conlputed using the preliminary brackets(5.7 1) cOlTIpatible with setting the second class constraints (5.68) stronglyto zero.
- 86-
vvhere the ITIultiples of Xi in (5.70) can he dropped if \ve use the prin1edbrackets. Hamilton's equations becon1e
{ A+ H}'==J,A+--:-' 1J, \..-, v_ 2
{A-, H}' == VI
{A i, H}' == Ai 1 ""t."- 2 \,.' 'l'2
{ 7t+, H}' == 2_ 7t- + 2i tt i ~ 0
(5.73 a)
Before choosing any gauge constraints, \ve nlay take our total Hanliltonian to be (5.70) plus arbitrary nlultiples of the first class constraints (5.67);
(5.7 2 )
{ - H}' "" -7t , == ~'+ 7t
{ i H}' J i7t, ==~+7t.
(S· 73 b)
\Ve n1ay therefore choose to express VI and V 2 as
VI == 2 A-
'fhe generating functional of the gauge transfornlation is thus
I~. NULL-PLANE RADIATION G~t\UGE
N O\V let us eliminate the arbitrary functions from our theory by usinggauge invariance to fix the reITIaining t\VO degrees of freedon1 correspondingto the first class constraints (5.67). First it is convenient to recall that astraightforward deri vation of the Euler equations gi ves the following results:
(5·75)
NO\:v let us choose the null-plane analog of the CouloInb gauge by setting tozero the field A+ == A- conjugate to 7t+ ~ o. A gauge transformation \vhichaccomplishes this is
(5.76)
o't·+
\ It.. 1\' II .\11 /) ....,' t. r I A-,/ . --- )...'-1.' _.~ ~'"""\. I == r ' \X _._-~. ( "r \7 ,x ,x
owhere
A'- ~ o.
(5·77)
- 87-
F'r0 n1 I~qs. (5. 75) and (5.58) , \vC fi nd that
2 (2 .i\'+ 2;" .J.~' i)
~ 2: (Tr.- 2i ...A.. /1) ~ 0 .
Exactly as in the equal-tin1e case, \VC n1ay find a gauge transfonnation \\-hich- I ;\ ,,; I '1 . .. '\ ,,- d E ( , rI'hsets TC -:-2;" ...'"""\. tozerO\V11en1alntalnlng~""""\. ~o ueto ~q. 5.77). c
solution is
(5.78)
A' -»- A" = A' (x) - Vx Jd2y III (X4~ y)~ (Vy • A' (y) n- (y))
A '+ i\ "+ _ A'+ ('\ jl'd2 III (x - y)2~,'t. ( -~y\ f; (,.- )\-+.1""1. - X) - Y 47: ~'- TC .:; I1. ~t, Y )
,'" 2
A '--+ :i\"-::..:..= 0-1 d2y In(x-y) ~ (- ':)YA'I'( + .- \)~'"""\. ---- d T- \ n ~iX, X ,y)
, 41t',.;
~ O'
I)ropping the prln1cs, \ve find our final set of constraints
CPl === n+ ~ 0
CP2 == 2_ n- 3; n£ ~ 0
CP3 == ~~+ == ~~- ~ 0
CP4 === n- -t- 2; ...L\.£ ~ 0 .
IT sing the prelin1inary hrackets (5.7 I), \ve con1pute the lnatrix of hrackets
of the constraints (5.79) to he
(5. 80) { m. (-) ~r). ( ,\ t.! -- C'" .J,. !"T I \ ..1 , T J \ J )J -j IJ \ ~1 ,J) -
o 0 --I 0
o 0 OV.~
000
o --V.~· 0 0
y).
rro specify the boundary conditions, \\·c ohseryc that \vhen the (strong) con
straints (5.68) arc con1bined \vith CP4 and inserted into the Euler equation
CP2' Vie fin dn 2 A+ --, (..., .\ 1\V -2~_\L'i~""""\.)~0.
We then choose our boundary conditions (and exhaust all renlalnlng gauge
freedon1) so that the solution of (5.8 I) is
\ ;- ( ) 8 I'· i~ In (z -- x) ~.t"l. \~'t" == 2 ~-= ( Z ----
ClX 41t'.. '
8.\i (x+, x-, Z)
dZ£
1-'hen the inyerse of (5.8) exists and n1ay be written
o 0 03 (x -- y) ()
o
o
o
G (x ,y)
o
o
o
-(; (x ,y)
o
o
- 88-
\vhere
G(x,y)== I ln (x-y)2a(x--y-)4 7t
V; G (X ,y) == a2 (X - y) 0(x- - ~v-) == 63 (x - y) .
.For use in the Dirac brackets we also define
(5.84 b)d 3
~-=- F (x ,y) == a (x - y)CJX
Plugging Eq. (5.83) into Eq. (1.58) with the primed brackets (5.71),yve find the Dirac brackets
(5. 85) {n~l (x) , n V (y)}* === - -~ g~l+ g\!+ V.; F (x ,y) ++ ! [__ (g~ll gV1 + gP2 gV2) g~l+ gVi J~ ++ g~li gV+ 3~] 03 (x __ y)
{A~l(X) , A\' (_v)}* == § (g~ll gV1 + g~l2 gV2) F(x , )') + 2g~l-g"- 2x G (x ,y) +
!--Li v_ x x G ( )+ g g 3i J_ x ,y
1 p+ vi x F ( )- ~-g g 8i X ,y
~Li vj ""x '""Ix G ( \g g di OJ x ,y) -
\vhich are nlanifestly compatible \vith all the constraints (5.68) and (5.79).Using Eq. (5.82) cOlnbined with the constraints (5.68) and (5.79) it is
easy to verify that all of the Dirac brackets (5.85) follow directly from thenonlocal Dirac brackets
(5. 86)
of the two independent canonical variables A l and A 2.
Equation (5.86) serves as the starting point for the quantum theory..l\pplying all of our constraints, \ve find that e+- is simply
( _~ .8-;) G+- - - 1 ((") r\ 2_ ':'1 i\ 1\2 -+- (':'1 A 1 -L ") A2)2)_J - 2 \.- 1 1""1. \., 2 .....~) I \., 1 I\.-2
and the Hal11iltonian is
(5. 88)
The Han1iltonian equations of motion then become
- 89-
Differentiating by x-, \ve find
A i - {":\ At" H}* - _ 1 ":\ ":\ ;\ £- ~--, - 2 ~jVj"''''"i..
so that Eq. (5.89) can be understood very sinlply as the x- integral of theKlein-Gordon equation
(5.90 )~ )~(, i\ i _ ( """\)L" 11.. L ....."'"i.. - 2 0 : L" __ ":\ ~) A.. 1 == 0 .~j ~j .....
We lea ve as an exercise for the reader the development of null-planeelectrodymanics in the gauge A+ ~ 0 chosen by Kogut and Soper (1970).We remark that the choice A + ~ 0 a produces a theory very similar to theequal-tinle electrod ynalTIics in the axial gauge A 3 ~ O.
6. Y.l\NG-l\1ILLS GAUGE F"'IELD
We now turn our attention to theories with a set of vector fields whoseLagrangian is invariant under a non-~-\belian gauge transfortnation, as proposed by )lang and IVlills (1954) [see, e.g., vVeinberg, 1973; i\bers and Lee,1973]. Since the fields in the lllultiplet Inix anlong thelllsel yes under the gaugetransformation, this type of theory is silllilar in SOllle wa ys to the theory ofgravitation gi yen in the next chapter. '1'he yT ang-lVlills fields are in fact selfinteracting fields, again like gravitation. Since vve ha ve until no\v restrictedourselves to free systellls, this Vo/ill be our first exposure to the difficulties ofinteractions.
i\. LIE GROUPS
We begin with a brief digression on Lie groups (Racah, 1965). Consideran n-parameter Lie group f and the associated Lie algebra !F (f). Let '1--a ,
a == I , ... ,'JZ, be a cOlllplete set of linearly-independent elenlents of if (f).The composition rule in f£ (f) for these elements is
(6. I)
The antisymmetry of (6. I) and the ] acobi identity inlply that Cabc satisfies
(6.2)
(6·3)
We can now define quadratic polynomials on the enveloping Lie algebrausing the symmetric second rank "metric tensor"
Det I gab! is non vanishing provided the group is senli-sinlple (has no ...~belianinvariant subgroups). We novv define
~-
L
(6·5)
- 90-
lIsing 1~gs. (6.2) and (6.3) \\TC can sho\v that (~abe is totally antisymrnctricin its indices.
Next, we note that any given Lie group \vill have a specific nunlber of(~asin1ir in variants G,. \vhich are polynomials in the elen1ents I'll and \vhichare In the center of the enveloping algebra:
(6.6) [ T a , G t'] == 0, for all a and all i.
'fhe nl0st fanliliar Casilnir in variant is the quadratic one, 'shieh can bewritten
(6·7)
\vhere gab gbe == 0; and the validity of Eg. (6.6) follows fronl Eq. (6.5).F'inally, \VC note that the JZ X n nlatrices
(6.8)
forn1 the (l{ijoint representation of the Lie algebra. ~Ia satisfles Eq. (6. I)tri\"ially due to the Jacobi identity (6.3).
13. SYSTEM \VITI-IOlJT (~ACGE CO~STRAI~TS
1·"'or any senlisin1ple Lie group J- \vith structure constants cabe
, \ve consideran action functional
,vhere the II \'ector fields ...r\;~' (.1:) , a == I , ... ,ll, are taken to transfornl asfollo\vs under an infinitesin1al gauge transforn1ation:
(6. 10)
~F!J.V == C be /\ F P' vo aa .... b {'.
It is a standard result (Dtiyanla, 1956) that although ~~;~. does not transfornlas the adjoint representation of f on its a-index under the gauge transforInation, the quantity F~v does:
F !J. v - ~p, AV_ ~\' A!.L _ C·~ be ;\ !.L, \'a - v a vag /(l oL'-l.b .L'-l.{O
(6. I I)
For convenience, we define
13 " =--= 1 ~"j;': L,jk. a ~ c,.. r a •
Our objective is no\v to find an action (6.9) \vhich is in\"ariant underthe gauge transformation (6. 10), so that t\VO components of each A;~ are
unphysical, and the A~"s describe massless spin one particles. In principle,
- 91
any Lorentz scalar function of the F'~') yvith the a-indices conlbincd in thefornl of the Casin1ir invariants (6.6) \\Till do. 1'he traditional fornl is
(6. I 3) 2==-
and includes only the terln analogous to the quadratic Casinlir in variant
(6·7)·In the rest of this chapter \VC will restrict ourselves for convenience to
the Lagrangian (6.13) and to the group SU (2). Since SU (2) has only threegenerators and
\vhere c-abc is the total] y antisynlnlctric tensor in three ditnensions, \ve cannovv redefine everything in such a \vay that \ve can forget about raising andlowering group indices. All of \vhat follows can easily be generalized to an yI...iie group by the interested reader.
If \ve no\v specify the d ynanlics on constant-tinle surfaces, the canonical11lonlenta are
(6. I 5)
\Ve sec that
(6. 16) :'T == --- E === --t-- A -,- "AO• (l a I a r Va
l'his is not a constraint equation hut serves to define the d ynanlical properties of .Aa . On the other hand, for [1. == 0, \ve find the expected n prinlaryconstraints:
(6. I 7)
The canonical I)oisson brackets are
(6. I 8) ( p. ( ) A" ( ) t [J.v ~ ~3 ( )l. 1ta t, X, b t, Y J == -- g 0ab 0 X - Y
1'hese are inconlpatible \vith (6. 17).Noyv SInce
8Ie _ I~'!J.\'(;lAa - a
..1-\.l-1.,V
\ve find the Euler equations
(6.20)
1"he zero component can be written
(6.2Ia)
92 -
]~his, as usual, IS also inconlpatible \vith the Poisson brackets. 'rhe spacecomponents of (6.20) are
1
(6.2 I b)
(6.22)
Due to the confornlal space-tinle synlmetry of the systenl, conservedN oether currents exist \vhich generate the full algebra of the confornlal group.I-Iere we shall concern ourselves only with the canonical energy-nl0mentunltensor, \vhich takes the form
8!J.v == _ ~ ..,:\ . v + [J.V se ==c 8Aa a I., g
I" !.J..
1"he canonical Hamiltonian is then
(6.23) I I)- B a • B a - --. ..Ia·:Ta •2 2
We nO\\7 add arbitrary 11lultiples of the prinlary constraints (6. 17) and use(6. 16) to get the preliminary Hanliltonian
...-\s usual, the secondary constraints are found by conlputing
(6.25)
which agrees \vith (6.2 I a). Our 2n first class constraints are now
(6.26)l a 0
tj)1 == :Ta ~ 0
(6.27 a)
\\1e add theln to H! with arbitrary coefficients and integrate by parts to getthe Hamiltonian
H = He Jd3X(V~ ,~~ v~ y~) =
= rd3x (+ :ra • :ra ++B a • B a
~/
+ (v~ A~) (V · .la gO-abe :rb . A e)) -
- ( ds . :ra A~ .~
Soo
- 93-
As before, the surface term will be dropped. I~xamining {i\'\ H}, we findthat the final forn1 of the Halniltonian can be taken to be
(6.27 b) o J 3 ('I IH === P === d x 2:Ta ·:Ta + 2 Ba • Ba -
Parallel argunlents gi ve for the translation generator
(6.28) p= Jd3x(-:TaXBa+7t~VA~+
A a (V . :Ta + gSabc :Tb . AJ) .
Unlike the IVlaxwell field, the present theory has additional Noethercurrents due to the local invariance of the Lagrangian under the non-AbelianLie group. Suppose we Inake a space-tiI11e independent transforI11ation with
n infinitesimal paraI11eters }'£1' so that (6.10) beconles
Then the standard argunlents tell us that the conserved current IS
(6.29)
With constant-tiI11e Poisson brackets, the time-independent generators ofthe synln1etry transforlnations are
(6.30 )
rrhe Qa obey an algebra in the Poisson brackets isonl0rphic to the Lie algebra
(6. I),
(6.3 I)
We see also that the fields .l\~ transfornl under Qa as the adjoint representation of the group:
(6.3 2 )
We lnay easily verify that the infinitesinlal generator of the full gaugetransforlnation (6. 10) is
(6.33 a)
- 94-
and obeys the algebra
I~Io\vevcr, the last t\VO ternl,s of the I-Ian1iltonian (6.27) consist of anothergauge-like tra!lsformation
(6·34) K [Aa ] = g-l Jd3X [7t~ 20 Aa -- (V · :Ta+ gZabc :Tb' A c) Aa ] ~ 0
\vith Aa === - gA~. I-J [./\a] and IZ [Aa] differ by a functional of the forn1
:r r 3 . 0 0Q [Aa ] === L [A(/] - I'.. [Aa ] === g-l I d x (V . (:Ta A a) +g Eabc j\a 7tb A c ) .
,.i
If ~~~ vanishes like r/r at (X) , Q [-- g.(-\~] = 0 and L [-gA?zJ === K [-- g~~~~]~; o.If j\a does not decrease rapidly enough, L [AaJ and Q [l\aJ \vill nol
vanish \veakly. 1~he nonvanishing charge (6.30) generating the symn1etrygroup algebra is jn fact seen to be a particular case of Q [J\al, namely
Q" = Q [Ab = ~ab] = g 1 rd3x (V· .la + gZabc 7t~A~) .
,.1
C. RADIATION GAUGE CONSTRAINTS
The problem of finding a suitable canonical quantization schenle forthe Yang-Mills field in the radiation gauge is \vell-kno\vn to be quite conlplex(Schwinger, 1962a, 1962b; Mohapatra, I97Ia). \Ve now proceed to attackthis problenl by inlposing the gauge constraints in the context of the I)irac111ethod.
\Ve begin by choosing the traditional radiation gauge condition
We will find it useful to define a Green's function Gab (X ,y ,A) solving
\vith the boundary condition that Gab fall like I/r at infinity. While no exactsolution is kno\vn for Gab' it can be conlputed as a po\ver series in g:
G b (x Y A) === Oa~_-:-a , , 4r:lx-y,
I~ 3 I . 8
-g d z ------ E b A~(z).., 41t' x-y ac 4it' z-y!
\Ve now take as our prelinlinary constraints the second class pair
(6.3 8)
(6·39)
- 95-
The Poisson brackets of the constraints (6.38) and (6.39) are
{ cp~ (x) , cpi (y)} === - gEl/be (V·:Tc(X) -i-- gScd~ :Tel(X)' Ae (X)) 03(X - y) ~ 0
{ ?~ (X) , ?~ (y)} == (aab V.: -r- gEl/be A,. (X). Vx) 03 (X - y)
{a b"( 2
CP2 (X) , cpl (Y)J === - (Dab Vx
{ cp~ (X) , cp~ (y)} === 0 .
The inverse of the l1latrix C~}(x ,Y) = {CP~(x), cp5(Y)} can thus be chosento be
[
0[C--1J abC ,
ij X,)/) ==. Gab (.-1: ,)' , .t\)
\vhere C;ab(X ,y ,.L~) is the solution to Eg. (6.36) introduced at the beginningof this section.
Equation (6.4-1) is then used to C0111pute the prelirr1inary Dirac bracketsc0111patible vvith the strong constraints (6.38) and (6.39):
A i ( )] 8G ,lCgEbcd ;/.Y
f\ i ( ) _.I (. )}' _ ~i.i ~ ~3 ( _ )l "'~ll I, X , Il.b ",I ,y - 0 Dab () X Y
f A P'( A V
}'_l llt,X), b(t,y)-O
{ () ( ) v ( )}'7ta I, X ,7tb \1 , Y === 0
{7r;; (I , X) ,A~ (t, y)}' == _gOV Dab 03 (X - y)
{TC~(/, X) ,A~(/, y)}' == _._gOfL Dab D3(x-y)
NO\V \ve l11USt see what becomes of A~ \vhen we in1posc the strong constraints (6.38) and (6.39). l'aking the divergence of Eg. (6.16), \ve find
- 96-
Using Eq. (6.36), \ve solve Eq. (6.43) for A~:
(6-44) A~ (x) R:! - rd3yGab (x ,y , A) gEfJed :Tc (y) · Ad (y)... '
Equation (6.44) is seen to be a 'lveak equation because the right-handside has vanishing brackets (6.42) with 1t~, while A~ does not. It is clear
that our final set of constraints is
(6.45) xi == 1t~ (x) ~ 0
x~ = A~ (x) +g Jd3yGab (x ,y , A) Ebed :Te (y) · Ad (y) R:! 0
The prin1ed brackets (6.42) of these constraints for111 the 111atrix
c~t == { X~ (x) , x~ (y)}' == 0
(6.46) Ci~ == { Xl: (x) , X~ (y)}' == - { X~ (x) , xi (y)}' == + Oab 03(x -- y)
C~~ === { x~ (x) , X~ (y)}' == M ab (;l: ,y)
where M ab (x ,y) can be explicitly COlTIputed as a power series in g if desired.We need not COITIpute M ab (x ,y) here hecause the inverse of C~~.(x ,y) is seen
by inspection to be
We thus find that the final Dirac bracket system is
(6.48) { A~ (t , x) , A~ (t , y)}* == lVlab (x , y)
{A:~ (t , x) , Ai (t , y)}* == 0
{ A~ (t , x) , A'~ (t , y)}* == - gGac (x , y , A) Ecdb A~ (y)
; 3- g.J d zGa, (x , z ,A) Ered A e (z). \7•.
[3r7 ajb + gEjbg A~ (y)] Gdj(z ,y , A)
{ 1t~ (t , x) , 1t~ (t , y)}* == { 7t~ (t , x) , ~>\~. (t , y)}* == 0
{ 1t: (t , x) , 1ti' (t ,y)}* == {7t~ (t , x) , 1tl (t , y)}'
{ 1t~ (t , x) , Al(t , y)}* == { 7t~ (t , x) , Ai' (t ,y)}' .
- 97 ~
We n1ay no\v take the I--Ia111iltonian to be
In the Dirac bracket systerrl (6.48), H indeed generates the correct equationsof 1110tion:
(6.5 0 )
Sch\vinger (I 962b) has argued that the brackets (6.48) produce a Lorentzcovariant theory. We leave this as an exercise.
D. ~L\LTERNATE RADIATION GAljGE TECIINIQUES
~L\ll four sets of constraints (6.39) and (6.45) hold strongly In our Diracbracket systen1 (6.48). Hovve\·er, the field-dependent tern1S on the right
hand sides of the I)irac brackets 111ake it extren1ely difficult to use these bra
ckets as the basis for a canonical quantization schen1e. Several procedures
are a vailable for CirCl1111venting these problenls. ()ne exan1pIe is the direct
path-integral nlethod (f~addeev and Popov, 1967a, 1967b) for developing aI~eynman diagran1 expansion consistent \vith the constraints. The" ghost"particles vvhich occur in this procedure correspond precisely to the integTals
\vhich appear in Gab on the right hand sides of our Dirac brackets.
~>-\nother popular procedure (Sch\vinger, 1962a; IVlohapatra, 197 I a) which\\;·e \vi11 sketch for con1pleteness gives up dealing \vith the strictly canonical
1110n1enta TC;~·. One ignores TC:~ cOlnpletely and transfornls the fields to the
radiation gauge
(6.5 I)
Then :Ta is split into a trans'verse part Pa and a longl·tudi1zal part equal to the
gradient of a scalar field <1>a ,
(6.5 2 ) :Tu == Pa \7<Pa ,
\\·here P" and <Pa ha ve their canonical properties {lefined by
(6·53)
(6·54)
7
-·98 -
l'hus the Euler equation (6.25) and the gauge choice (6.51) imply
(6·55) V·pLJ ~ o.
T'hen one con1putes the Dirac brackets consistent with (6.5 I) and (6.55) finding that Pa and All obey the fan1iliar quantizable l\!laxwell field I)irac brackets(5.32). i\t this point J one C2n use Eqs. (6.16) and (6.2Ia) to find the follo\vingexpressions:
(6·57)
ITsing l~q. (6.36), one then sol ves for <I>a ,
(6.5 R)
and expresses A~ in terms of <I>a ,
o ( 3 2An (x) = . d yG nb (x, y ,A) Vy <l>b(Y)'
Since 7t~ has been ignored, Eqs. (6.58) and (6.59) are taken to define theproperties of <Da and ...-\~~. l'he Han1iltonian can then be written
(6.60)
\,"here the Y<1>u· y<I)a tern1 is con1pared to the instantaneous Coulon1b energy\vhich appears In interacting lVIax\vell electrod ynan1ics in the radiationgauge.
\\Te prefer the techniques of the previous section, \vhere the variablesare all treated in a strictly canonical fashion. l\/loreover, there is a gauge in\vhich the canonical procedure can be used directly to define the quantun1theory: this is the subject of the next section.
E. i\XIAL GAUGE
We no\v explore the properties of the Yang-Mills field in the axial gauge
(6.6 I)
introduced by .l\.rno\vitt and Fickler (1962) (see also IVlohapatra, 1971 b;IZonetschnyand KU111111Cr, 1975). Using the I-Ialniltonian (6.27 b) \ve find that
(6.62)
- 99-
Therefore Eq. (6.6 I) \vill not hold for all tinle unless
\Ve no\v adjoin (6.6 I) and (6.63) to the original constraints (6.26) to forn1the set
'fhe Inatrix of l)oisson brackets is found to be
(6.65) C~~ (x ,y) == { ~:- (x) , q>~ (y)} =
o
o
o
Oab :;~
o
o
o
o
- Oab o
\\Tith appropriate boundary conditions gi ving the explicit solutions for thedependent variables to be \vrittcn later, \\~C find the in verse of (6.65):
(6.66) [C-l]~5 (x ,)1) ==
rlab (x ,--,V) - 0abG (x ,y) H ab (x ,)1) 0ab F (..1: , y)
Oab G (x ,y) 0 - O'-b l~-- (x ,.Y) 0
H ab (x ,y) - 0llb F (x , .1/) 0 0
0ab F (x ,y) 0 0 0
1~he functions F , G , H ab and lub obey
(6.67) F (x ,y) == 2:~ G (x ,y)
2;~ F' (x ,y) == 232~ G (x , y) == 03 (x - y)
- 100-
and are chosen to ha ve the explicit forn1s
(6.68) C; (x ,y) === ~ I x 3- .-v3 I 02 (x - y)
F' (x ,y) === ~ E (.1'3 _y3) ~2 (x - -,v)
lI ab (x ,y) == 2;~' I ilh (x ,y) --l{211bc 7t~ (x) c; (x ,.-V)
\vhere E (x -- )1) is the algebraic sign of (:{~ - y). .Lr\ll ren1aining gauge free
donl is exhausted by the choice of boundary conditions in1plied by (6.68).T'he I )irac brackets corresponding to Eq. (6.66) are
{ .\ tJ. (\ i\ v ()}* ~ ( !to v:~...'l...a X) , ....'-1.b _v === 0ab ,g g
o ( !lO vz'- ·ab g g
ll<) "0) F ( )g,.Jg x,y
II '" '1'0 /\ l' ( )] G ( \g' g ...'-1. 1' X J" X, )')
ItO '1'0 I ( )g' g ab,~1: , )'
(6. 7:J)
(6.7 1)
{ v. () "( )}* (' It z' :~" z' () v.:~ vi i ( )) F ( )7t;z x , 7tb Y === g2ab[ --g' g 7t[ X g' g 7t( _V X,_V
{ 7t~' (x) , ...~~ Cv)}* === (_g,t" _g'lOgVO) Oab 03 (x- y)
gtJ.:~ g"'" Oab 2";-- f~ (x ,y) - g!t1'g"O gSabL' 7t~: (x) G (x ,.-V)
v\le nO\\l take as our four independent \Tariables Ttl: , ...'\(: , i == I ,2, \\Thichha \'C the follo\ving I)irac brackets anl0ng thenlsel \Tes:
(6.7 2 )
{A:~ (x) , }\{ (y)}* === 0
r i () ;\ J ( )}* __ ~ iJ ~ ~3 ( )l 7ta X ,ll"b Y - - ° 0ab 0 X - Y .
]'hese are ranoniral brackets, \vith no fields on the right-hand side. 'l'he
strong constraints 9~ , 9~ and 9~ can no\v be used to show
(6·73)
so that ...f\;; IS a dependent variable:
(6·74)
r..., i ( 0 .1 2 Y)• . ,-' l' 7t1l \. X , .1' , X , C;
rrhen 91 == 0 also itnplies
,... l' (~o 1 .2 .'1;) ;\ i (0 .1 ~2 ';)1/.[c"abL' Ttb \:{, ,x ,x , '-) ...'-1. c X ,:t ,J: ,":' .
(6·75)
1----------------------------------------
-
- 101 -
'fhe Hanliltonian in the axial gauge ITlay be \vrittcn
(6.76)
and \ve Inay conflrnl that it generates the correct equations of J11otion of
the independent variables in the I)irac hrackets:
(6.77)
\Ve \vill forego the usual discussion of I)oincarc covariance, etc.
V\Te obser\~e that the independent canonical variables in the axial gauge
have canonical brackets equal to their I)irac brackets (6.72). l~here are no
fields on the right hand side, and no ghost loops in the 11"eynnlan rules. ()n
the other hand, the canonical variables in the radiation gauge ha\~e fields
on the right hand side of their I)irac brackets and ghost loops appear in the
11"eynnlan rules. I t therefore appears that the canonical quantization proce
dure is best carried out in the axial gauge.
f--inally, \ve renlark that it is also instructive to analyze the \~ang-l\Iills
field in the null-plane fornlalisnl. '1'he gauge choice ~~,; = 0 has properties
sinlilar to the radiation gauge and has gl-:ost loops. 'The gauge "'~a;- = 0, on
the other hand, gi\~es sinlplifications parallel to those of the axial gauge C1'on1
houlis, 1973). In particular, the independent variables have field-independent
I)irac brackets and no ghost loops result. \'le lea ve these analyses as exer
cises for the interested reader .
...~s our last exanlple of the application of I)irac's Inethod to physical
systen1s \vith constraints, \ve treat f:=instcin's theory of gra vitation. ~~n10ng the
references giving elen1ents of the Han1iltonian forlnulation of general relati\'ity
are Pi raniand Schi Id (1 9 50) , Bergn1 ann, Penfi eld, Schi 11er and Z atk is (1 9 50) ,Pirani, Schild and Skinner (1952), Dirac (1958b, 1958c), 13ergmann (1C)62),~\rno\vitt, Deser ~~nd l\lisner (1962), Hojnlan, Kuchar and l"eitelhoinl (1973,
1974), and Regge and Teitelboinl (1974a, b). I~or a n10re con1plete account of
the vast anl0 unt 0 f literatu re, see 1'-.u ch a r (1 974) . \\'e 0 hser ve that in E instei n 's
theory of gra \Titation, it is the expression of the theory in nlanifestly cova
riant forn1 \vhich causes constrair:.ts to exist clIllong the canonical variables:
the gauge-like frceclo111 to 111ake r;eneral coordinate transfornlations can then
be exploited to reduce the nU111hcr of independent degrees of freedon1 on a
spacelike surface to t\VO pairs of canonically conjugate variables per point.
- 102-
Our starting point will be the Hilbert action
(7. I)
for the pure gravitational field, where for convenience the units have beenchosen in such a "vay that G, Newton's gravitational constant, takes on thevalue (I 61t)-1. \Ve deal with a hyperbolic Riemannian spacetime of signature(- , + , + , +). Four-dimensional spacetinle quantities carry an upperleft index (4) whenever it is necessary to distinguish them from the threedimensional quantities to be introduced later. Greek indices run from 0 to3 and Latin indices from I to 3. F our-dinlensional covariant differentiationis denoted by a semicolon, while its counterpart on a three-dimensionalh)"persurface is indicated by a slash. The Rienlann tensor is defined as
vvith
'The Ricci tensor and the scalar curvature are RrJ.'J == R~:t\' and R == R:~ ,respectively. 1'he determinant of the nlctric is denoted by the letter g.
The appearance of the scalar curvature in (7. I) n1akes the actioninvariant under general coordinate transfornlations
which change g!J.') (x) to
I'he theory is thus "already paranletrized " in the sense of Section I. E ..Note that the arbitrary coordinates denoted there by utJ. are called x!J. here
to conform vvith standard practice.I'he action (7. I) describes the pure gravitational field ....-\ddition of n1atter
is straightfovvard provided the matter action density contains no deri vati \~esof g!J.'J, which is actually the case for particles and for the electronlagnetic
field.
A. GENERAL FORM OF THE HAMILTONIAN
Einstein's equations are a second order partial differential systen1 in theten nletric components g!J.'J of a Riemannian spacetime. When looking atthis system from a Hamiltonian point of vic\v, one focuses his attention ona three-dimensional h ypersurface enl bedded in the four-din1ensional space.The state of the system is then gi ven by specifying the value of certain fieldsdefined on the surface; by means of the Hamiltonian, one is able to calculatethe change in the field variables induced by a deformation of the hypersur-
- 103-
face. IfF is an arbitrary functional of state \ve expect its chang{~ under adeforn1ation N!t. of the hypersurface to he of the forn1
(7. 2 )
i O
1-., I d3 "TIL ~ '\f ( \~ === Xl." \x) p. x) ,
according to the general discussion on curved surfaces in Section (I. J~~.). I-JerefrJ. (x) is sonle other functional of the canonical variables, the precise forn1 of\vhich \vill be exanlined in Section 7.C'.
We can no\v use (7.2) to obtain infornlation about the fornl of the I-Ianliltonian by relating the NIL to the ga';3. ]'his is 1110St easily done \\·ith the help
Fig. 7. I. 'The relation bct\veen the :\!J,. and the gG(r3 is obtained by eyaluating the spacetin1Cinterval bet\veen the points ,vith coordinates (xi, t) and (xi -f- dxi , t +- elt) in t,,·o eli fferent
\\'ays and comparing the resulting expressions for arbitrary dx i , clt. This leads to Eqs. (7.5).
of Fig. (7.1) \vhich tells us that the spacetinle distance het\veen the points(Xi ,t) and (Xi dx i , t -t- dt) can be expressed as
ds2 === - (N1)2 dt2 (N i 1g/j (t
On the other hanel, \~TC kno\v
.. idx1
) (N.! dt --+- dx') .
Upon con1parlson \vith }~q. (7.3), Eqo (7.4) sho\vs that
(7· sa)
and
(7.5 b) g NI jJ_ gIi'" - Oi'
Conversely \ve can express K!l in terms of the gWJ as
(7· 6a)
(7.6b)
NJ == gii gOi ,
l\T1 ( 00) - .t1" === -g ~.
- 104-
~ote that g''i in (7.6a) is the inverse of the spatial n1etric g£j (i.e. gk! glnl === O;~l)
and is in general different fron1 (4~zj. In fact the follo\ving useful relations
hold (Arno\vitt, Deser and lVIisner, 1962):
(4) z"./· === ij _ '-T I' N.T jj(NT 1):2'g g ---~... v·
(7. 8)
(7.9 a)
(7.9 b)
(4) [(N1)2 N i NT;.Jgoo == - 1 1-- gl.! .... 1 ,. _
I. 1 1( __ (4)0-)2 - N.T g2\ b _.- ...
l~quations (7.5) tell us that goo and /[Oi nlust enter in the I-Ianliltoniantheory as arbitrary functions. 'I'his is so because the N,t nlust be prescribed
fronl the outside in order to spccifiy the defornlation of the hypersurface;
consequently one cannot expect I-Iall1ilton's equations to restrict then1 at
all. ]'hus \ve learn that the non-tri vial degrees of freedonl are contained in
the gij and their c~)njugates nii.
It \vill save a little \vriting to \vork \vith the N~l defined h)" (7.6) and the
/tl'} as independent variables instead of using the goP. and the gij. '-[his changeof \Tariables is pern1issible because on account of (7.S) and (7.6), one can go
back and forth hct\veen both sets of variables. If \\'"e denote l?y 7t!J. the conjugate to N f
\ then the fact that the Nfl are arbitrary tells us that lt iL ll1uSt enter
the I--Iall1iltonian n1ultiplied by an arbitrary function /,'l which \vill correspond
to the tin1e derivati\Te of :0Jfl• ]'his renlark, tog·ether \vith (7.2), indicates that
\ve should expect the total I-Ianliltonian to be of the forn1
re
I-J - I d3 . ~ ("'\""1 7/£) [ _ij ]L -, ~{ \.1." ./l 1 g ij , I ..
• 1
Since the 7r[L are nlultiplied by arbitrary functions, \ve ll1USt also expectthe (first class) constraints
(7. I I) 7r~)' ~ 0
to hold. Finally, since (7. I I) nlust be preser\Tccl under surface defornlations,
~"e n1ust have {TIfl, I-I} ~ o. Fronl :Eq. (7. IO), \ve find that {n'\ I-I} ~ 0
in1plies the additional constraints
(7. I 2) ·J([L ~ o.
The constraints (7. I 2) rnust be first class in order for the equations of n10tion\vith arbitrary ~~l to he consistent. '-rhus the constraints (7. I I) \vould he
prill1ary constraints \\Thercas (7. I 2) \voldd be secondary constraints.
Note that the abo\'e discussion applies generally for any relativistic
I-fan1iltonian theory in a Rielnannian space. It fo11o\\7s then that \vhen n1atteris present the grayitational and n1atter parts of the H alniltonian should he
separately of the form j N!l.#~. (supplemented of course by any extra first
class constraints characteristic of the ll1atter at hand).
105
\\le sec that yve ha \"C ohtained son1C insight into the structure of the
1-1 an1iltonian on general grounds. T~quation (7.1 I) tells us that \vhen \vorking
fr01l1 the Lagrangian do\vn to the Han1iltonian, \ve should try to add suitable
di vergences to the action density in (7. I) so that the resulting I__ agrangian
\vill contain no ti1l1e derivatives of the N~l (...-\nderson, 1958; De vVitt, 195 8;
I)irac, !958b; ...-\rno\,~itt, Deser and lVIisner, 1962). Furthernlore the Lagrangian should ha \"C a \-ery si1l1ple dependence on the Nfl thenlsel \"es in order
to get a J-Ian1iltonian of the forll1 (7.10). We thus need a \vay to analyze the
spacetiIne curvature \vhich clearly distinguishes the dependence of the cur
vature on the g"j froll1 the dependence on the N P. Such an analysis is best
perforn1ed \,"ith the help of the enlhedding equations of Gauss and Codazzi,,"hich are re\-ie\;vcd in ..:-\ppcndix B.
Follo\\"ing- Kuchal- (1971), \VC no\v dcri\"c the desired Lagrangian hegin
ning \\'ith the I-liJhert action (7. I). l~~quation (}3.5),
relates the extrinsic cur\-aturc to the velocity ga!I,O of the d ynan1ical coordi
nate gab and to the functions N 1 and N" \vhich describe the deforn1ation of
the hypersurface. Note that gab itself also appears in this expression for K a !)
through the Christoffel syn1hol hidden in the covariant deri\"atives Nal h
N b/a • 1'herc are, ho\vc\'er, no tin1e derivatives of N l and N" in (13.5.). rr'hus
if \ve succeed in expressing the action density in ternlS of glib and K ab , \VC \viJl
ha \"C achie\"cd our goal of elin1inating the \'elocities Nl and N/. Let us \vork
in this direction \vith the help of the Gauss-Codazzi equations. In the basis
(n , eJ used in (B. I), \VC have
(7· I 3) (J)R - (oRetr~ - (-t)Ra1 (nRlblb
(l)RlIl __- 'et~~ - ab al ---
== (Ol(abab
- 2 U)Ralal
===
- (-l)R"b __ (I)!yet-- ab 2 \.. letl
lIn l~q. (7. 1 3), \ve ha\"c cxploited the antisyn1n1ctry property R~r~y~ == -R~r)8'(
== R;~('A8'( of the Rien1ann tensor]. ~O\V Vie use (13.12) to transfOrITI (7.13) into
Kr2 __ K I rab -- ') (-ORetab '- ... letl
\"ith K
1'he first three tern1S in (7.14) are alread y of the forn1 sought. \\,.e need
to he concerned on1 y about the renlaining tern1 (-!)R etl~l . Our first step is to
apply to the norn1al nJ. the con11nutation rule for the second covariant deri
vati ves of an arhitrary vector, thus ohtaining
- 106-
~ext note that ll1ultiplication of (7. 15) by n1J. and subsequent contraction Inthe pairs (tL, ~) and ((l, A) can \~erts the right hand side of that equationinto the desired quantity,
(7. 16)
:\0\\' \YC rearrange the left hand side of (7. 16) by 111ca11S of the identities (see
~-\PPC11 cI ix B for a proof)
(7· 1 7 a )
(7· I 7 b)
18 a)
(7.18 b)
r3 y __ (. i~ y \ .. (3 . yn n ;i3y - n n ;~) ;y -- Il : y Il ; r3 ,
r3 y K K abIl :y n ;i3 == - ab ,
r3 KIZ ;~ == - .
I...... Jl· h ( (I))J N~l J I l' hI·(eca Ing t at _. g 2 == 1: g~, \\~e t 1en 0 )taln t e re atlon
(£/ _ ( (-t))J (t)R __ c/},\rrt.j"rt.-z HILBERT - \-- g - eL - ~. ,::t,
\vith
(7. 20)
and
(7.2 I)
I~quation (7.20) sho\vs that by adding to the Hilbert action density theeli \~ergence of (7.2 I), \VC obtain a n10dified action density !f \vhich containsno tinle deri vati ves of N 1 and N i and \vhich contains only first tinlC deri va~
ti \~es of gij' 'rhus the Lagrangian
(7. 22) L c= rd3x !f = ,. d3x N1 i (Kab K ab- K 2 R)
../ ..:
\vill be our starting point for getting the Hanliltonian. Ho\vcver, before doingthat, let us first take a closer look at the di vergence in question. By Ineansof Eqs. (7.5)-(7.9) and (B. 3) one can re\vrite the quantity \;rrt.,c< as (DevVitt,
196 7a)
vet: == 2 (gJ K'! _ [gQ (I{N£ _ g~ gij N1 0)] 'o.,C(. / ,0 ,; ,Z
}~quation (7.23) shows that both types of un\vanted quantities, narnely thefirst tin1e cieri vati yes of N1, N" and the second time deri vati \'es of gij, enter
the Hilbert Lagrangian
L j d3x (_ (4)g)~ (J)RHILBERT == ..
107 -
In the fornl of a total deri vati ve,
and can therefore he elinlinated altogether by omitting (7.25) fronl the
I~agrangian.rIhe fact that one can elinlinate the second tinle cleri vati yes of ~f{/.1 fronl
the action is not surprising and \vas kno\vn long before the I-Ianliltonian fornlulation of general relativity \vas investigated. One can actually do evenbetter and elinlinate all second cieri vati ves (te111poral and spatial) of the ten
,f{!J_'J by adding a suitable di vergence to Y H ILBERT (I~instein's equations arc ofsecond order after alL). ()ne arri yes in this \va y at the so-called Einstein or
" ga111111a-galnnla " Lagrangian (see for exanlple Landau and Lifshitz, 197 I)\vhich \vas used by Dirac (195 8b) as a starting point for finding the gravitational Hanliltonian. What is 11luch 1110re re111arkahle and vital for the I-Ianliltonian fornlalism is the elirnination of the jzrst tinle deri vati yes of N 1 and N i
[recall Section 7.i\J. This possibility \YClS explored hy De\Vitt (1958),...-\nderson (1958), and Dirac (195 8b).
C~. ]'HE I-IAMILTO:\ IAN
\\1e proceed no\v to cIeri ve the exact fornl of the I-I cuniltol1ial1 for the
gra vitational field starting fronl the L,agrangian (7.22). Since no tinlCderivati\~es of the NIt appear \ve inl111ediately get the (first class) prin1aryconstraints
(7. 26) ~o
as \\~C expected. '1'hc 11101nenta 7rJi conjugate to i[ij arc bv definition
" oL'"T'IJ --
i .. - o(~li .
H..ecalling l~q.
(7. 28)
(B.5), \ve find after a sin1ple calculation that
nz'i == _ gQ (K ij _ Kg 1J) ,
sho\\~ing that the conjugate to the first fundanlental fornl gij is closely rela
ted to the second fundan1ental form K ij . Equation (7.28) can be inverted
to express K zj as a function of gij, \vhich anlounts to expressing the velocities(~i.i as functions of the 1110rnenta and the d ynalnical coordinates. ()ne thus
gets fron1 (7.28) that
I T ij _ i (...... 1·j 1..... ij )~ - -g J'.. -:2 /I.g ,
\vhere \ve ha ve defined
(7.3°)i z'j
n == n i == g ij 7t •
There are therefore no prin1ary constraints other than (7.26).
-- 108 -
:\ext, us ing (B. 5), (7.22), (7.26) and (7.29) \ve 0 bta infor the H a111i1to nian
\\'ith
11:1/' 13 ( N·'I., C X 7t!J. 1 ..
and
(7.32 b) --yt/l' =::.: -- 2 (y '/..' rr/<'J . - (2 (y L' . - (Y b' .\ 7tJ,j == ---- 2 7t /, ..<-"> / ,j \ b /,/ . ./ ("-' '<.7,,) 1 !.J
In arn nng at (7·3 I), we ha \T dropped the surface integral 2 fd~ SJ Ttl} ::'\ I
fro111 the right hand side of that equation.
'1'0 obtain the total I-Ian1iltonian, \VC ha\'e to add to (7.31) the first class
prI1l1ary constraints (7.26) 111ldtiplied hy arhitrary functions. \\lc therefore
ha\'c
I-I.,.
I cl0x (~l'Y(l
.J
~/ ,/f;,
In agrcc111cnt \vith the general discussion In Section (7 ..:\).X 0\\' to obtain the lllost general n1otion \\'hich is physically pCrI11issihlc,
\ve ought to add to (7.33) the first class secondary constraints (7.12)
\vith arhitrary coefficients lIY' (x). 1-1o\vc\'er, since the :.J" are arbitrarv to
start \vith, \ve do not get any additional freedo111 by doing this and \ve n1ay
as \ve11 set lIY' == 0 \vithout loss of generality (*). rrhe 1110st general Ha111ilto
nian is then just (7.33).Equation (7.26) tells us that the degrees of frecdo11l descrihed hy the \'aria
hies (7t!.L' N!l) arc not physically i11lportant (7t!L is constrai11ed to he zero and
X~l is arbitrary). vVe can then drop these degrees of freedo111 fro111 the phase
space altogether and treat N I• henceforth as an arbitrary function ef
• (x) \vith
vanishing hrackets. 170r111a11y, such a procedure a1110u11ts to i111posing the
(*) Ceolnetrically speaking \vhat R(v's on here is the follc)\ving: Suppose SOlne spacetirne
coordinate systern .r tJ• is gi\-en. Then the :\!.L ·1'0). part of the H~uniltonian generates the changesin the canonical \'ariables which correspond to passing froln the surface xl) t to the surface
X O t +. dt. :\ O\V the addition of the extra tenn u!.L .1'[!). would tell us that \\'e are not forced
to choose X O = t dt as our next surf-lee, but that \\-c can go frorn .ro t to any inflnitesirnally close surface. .Howc\'er since the spacctinH> coordinate SystClll x:J
• is arbitrary, \ve can
rnake .l-O t dt correspond to an y gi \'en surfa.ce, and conseq uentl y the freedorn inherent
in the u!J· tenn ca.n be considered to be already included in the arbitrariness of :\!)'. A. totally
sinlilar situation occurs \\,ith .:-\0 and the function Vz. in the ::\laxwell held (Section 5..:\). The
analogy is established by associating .\0 \\'ith :\t.L , V2 \vith uP. and by 111aking gauge tran~f()r
rnations (Ai -~ j-\z' .+- Ji .A) take the place of surface defonnations.
- 109-
second class constraint
which n1akes the originally first class equation (7.26) heco111c second class.
rrhc constraints (7.12) rcn1ain first class. ()ne then passes to the I)irac bracket
hy the usual procedure (\vhich in this sin1ple case an10unts to vvorking just
\vith the gz') and It ii ) and takes (7. 26) and (7.34) as strong equations.~~t this stage the Poisson hracket is therefore
\\1hen Eq. (7.35) is applied to the fundan1ental canonical variables then1sel Yes,
\ve find
{ r .. (. ~) k I / ')} _ ~ .. k I ~ ( ,'" ')l, I} J ,It ~X -- a Z} a \~'t ,x ,
as the only non-vanishing Poisson Lracket. l'he syn1bol Oijkl In the right
hand side of (7.36) is a shorthand for
and the Dirac o-function is defined as in (1.82) 'luitflout recourse to the JJl{?fric,
by
for an arbitrary' scalar testing function.
I f l~qs. (7.20) and (7.34) are taken as strong equations, the FI an1iltonian
reads
(7.3 8)
and vanishes vveakly duc to the first class constraints (7.12).The rate of change of an arbitrary functional I~" of gz'j and Te i) IS there
fore given by
so that Eq. (7.2) IS explicitly verified.
D ....~SYMPTOTICALLY FLAT SPACE, SURFACE INTEGRALS, IIVIPROVED
HAlVIILTONIAN, POINCARf~ INVARIANCE .L~T SPACELIKE INFIN1TY
We arrived in the last section at expression (7.38) for the gravitationalI-IaI11iltonian. \\;"'e can no\v check whether I-IaI11ilton's equations correpon
ding to H o (together \vith the constraints (7.12)) reproduce ~~il1stein's equations.
I t turns out that this is indeed the case provided one neglects certain surface
tern1S. For a closed space one is certain that no c0l11plication could possibly
- 110-
arise fron1 neglecting surface integrals, bacause every surface integral vanishes
identically \vhcn the 111anifold has no boundary, So, for a closed space, !-logiven by (7.38) is the correct Ha111iltonian as it stands. Ho\vever for an open,asynlptotica11y flat space certain subtleties arise and one comes to the con
clusion (see below) that in such a case I--I o lnust be supplenlcnted 1)y the addition of certain surface integrals at infinity in order to gi ve the correct equations
of n10tion (Regge and rreitelboinl, 1974a, b). T'he addition of the surface
integrals also pIays a crucial role \vhen one inlposes gauge conditions \vith
the purpose of arri ving at a canonical systenl ha ving just t\VO independentdegrees of freedom per space point.
1~0 see the roll' played by surface integrals, \ve start by observing that
an essential requirclnent which Inust he 111et by an acceptable definition of
the phase space of a d ynanlical systen1 is that all physically reasonable solu
tions of the equations of nl0tion n1ust lie inside the phase space. If this isnot true, the variational problen1
aJ(p,. qi - H) dt = 0 .
has no solutions because the extren1al trajectories are not admitted alTIOng
the original " conlpeting cur\~es " of the variational principle. Once the requircnlent of containing all extrenlal trajectories is lnet, one can enlargethe phase space at \vill but one caJlJlot n1utilate it arbitrarily.
In vaCUUln general relativity, after conditions (7.34) are in1posed, a pointin phase space is represented by t\vcl\'c function variables (gz'j, n k1
). No\\?
any solution of Einstein's equations representing a physically reasonable,asyn1ptotically flat spacetinlc beha vcs at spatial infinity in the Sch\varzschildforn1
(7·39) 1 ') (' 1\1 )' 1')(s:'" "--~~ - 1- " ,.. ct-r -~00 8 rrr (
0 ' . _L 1',I -l·i .:rj )' l~l' d ~j'.I I 8TC' 1'3 C .1 X.
1'he phrase "physically reasonable" here nleans essentially that the totallnass-energy of the systen1 ll1USt be finite. 1'his assulnption \vi11 he satisfied
if the systenl has been radiating (gravitationally or other\vise) during a finitetilne only. I t is quite plausible, ho\vever, that one can adnlit a nlore generalsituation in this context; one could allow a systenl which has been radiating
during an infinite tin1c, hut doing so in such a \vay that the total amount ofradiation renlains finite. 1'he precise forn1 of the line elenlent (7.39) can bealtered by a change of coordinates, but no coordinate systenl exists such that,\vhen 1\1I 0, all conlponents of the Inctric and its first spatial derivati\~es
can be made to decrease at infinity faster than r-1 and r-2 respectively. It
fo11o\\7"s therefore that any definition of phase space lztls to contain 111etric
functions such that
(7.40 a)
and
(7.40 b)
- 111 -
\\1e will need later a nl0re precise definition of phase space, but the essentialpoint is that one cannot do any better than (7.40).
Let us return now to Harnilton's principle, keeping the asynlptotic beha\'ior (7.40) in nlind. When one deals \vith a continuous systenl like the presentone, I-Iall1ilton's equations read
(7.41 a)
ancI
(7·4 Ib) it£j (x) === -- 0 (I-Ian1iltonian)jagij (x) ,
l'he functional cIeri vati ves appearing on the right hand side of (7.41) are,by definition, the coefficients of Ogij and onij in a generic variation of theHan1iltonian, i.e., if
then
(7.43 a)
and
(7.43 b)
~ (I--Iamiltonian)jog,'.i === i\ij
rrhus, in order for I--Ianlilton's equations to be defined at all, the variationof the Hanliltonian nlust he expressible in the form (7.42) for an arbitrarychange in the phase space point (g,") ,nkl ). \V"e shall see now that oH o (\vith110 gi ven by (7.38)) cannot be put in the forn1 (7.42) and that the Hanliltonianhas to be alnendcd by the addition of a surface integral in order for Hanlilton's equations (7.41) to coincide \vith Einstein's equations. Introducingexpressions (7.32) into (7.38) one gets for the change in H o, keeping all terrns,
i d2 (' ijl.: I ('T 1 ~ " T1 ~ )- f SIJ \l~ ogij!l.:-"'~ ,J:ogij -~J
\vhere
(7·45)
1'he coefficients ...r\ii and B ij need not be explicitly \vritten here [they maybe found, for exanlple, fronl the right hand sides of Eqs. (7-3.15 a ,b) of.i-\rnowitt, Deser and Misner, 1962]. We need only to observe that in orderfor I-Ialnilton's equations to reproduce Einstein's equations, one Inust identifyAi} and B ii in Eq. (7.44) with the variational derivatives appearing in (7.43).
- 112-
This tneans that 1/ the slojace integrals /n (7.-+4-) 'Zt'oul{! van/sIt tllen H o wouldthe correct HaJJlilton/an. For a closed space this is indeed the case; hovvever,
for the open, asyn1ptotically flat situation, son1C of the surface tcrn1S in (7.44)do not vanish due to the slo\v asyn1potic decrease characteristic of the gravi
tational field.
1"'0 deal vvith the surface integrals in (7.-+4-) one needs a nlorc con1plete
specification of the asynlptotic behavior than the one given by (7.4-0) .....t\nexhaustive discussion of the boundary conditions in question vvould take
us avvay fron1 our t11ain linc of devT elopn1ent; vve refer the reader to Regge
and 1"'eitelboin1 (197 -+a , b) for a detailed treatnlent of this issue. We indicateexplicitly here only that the lapse and shift functions Nfl are assun1ed to behave
asymptotically as
(7.4-6)
with
It-,~ rx'
r - 70v
~rs == -- ~sr .
[The Greek indices in (f..fl and ~~.l are raised and lovvered vvith the lVIinkowskian
n1ctric 1J~r3 == diag (-- I , I , I , I)].}=.quation (7.4-6) says that \\TC allovv asyn1ptotic spacetinlc translations
(rxfl) , space rotations (~rs) and boosts (~\.) anlong the pern1issi hIe defornla
tions of the hypersurface. \\1hen l~q. (7.46) is inserted into (7.4-4-) and thecorresponding- boundary conditions on l:,j and ~l".i are used, one finds
vvith
1).· .'.I
07:/.1 (x)1
(7.4-8 a)
(7.4-8 b)1'-+00
MIS = -- 2 f d2Sl (x' rels - X S n!T)
1'--;0..00
(7.49 b)
I-Iere v:ve have defined.
(7.50 a)
so that one has
(7· sob)
~lr == --- ~rl
L....- -'--_~__~_~ ~ ~ _
-- 113 -
From (7.47) \ve see that H o has well defined functional derivatives only\vhen the surface integrals (which are in general different fron1 zero) are absentfrom the right side of that equation, i.e., when ~ft == ~ftV == o. I'hus H o is agood Harniltonian only for those defornlations of the hypersurface whichin vol ve neither spacetime translations nor spacetime rotations at infinity.However, Eq. (7.47) itself suggests ho\v to deal with the more general case:one simply passes the surface integrals in (7.47) to the left hand side to obtaina new "inlproved " Hanliltonian
The functional H defined by Eq. (7.5 I) has well defined functional derivativesand generates the correct equations of motion even when asymptotic Poincaretransformations are allo\v"ed arTIong the pern1issible deformations of thehypersurface.
Now H o vanishes weakly on account of (7.12) inlplying that for anysolution of the equations of nl0tion, the nunlcrical value of the H anliltonian
(7.5 I) is
(7.5 2 ) H ~ - rxf-L P!l- + ! ~ftv lVI wJ
This expression IS In general different from zero. When ~(1.'J == rxt" == 0 andrxO == lone has H == PI, which identifies (7.48a) as the energy. Sin1ilarlyproceeding along the familiar lines of N oether's theorem one identifies expression (7.48b) as the linear n10mentunl and (7.49) as the angular momentunl.(See Appendix B.3). I t is important to realize that the identification of P!land M po has been achieved without recourse to any special decomposition ofthe canonical variables or to a specific fixation of the spacetime coordinates,which makes obvious the "gauge independence" of these quantities (compareArnowitt, Deser and Misner, 1962).
Having written expression (7.48) and (7.49) for p ft and lVlpo ' one askshirnself inlnlediately \vhether these quatities have the proper behavior underasyn1ptotic Poincare transforn1ations. The straightforward approach toans\vering this question would be to introduce the lapse and shift functions(7.46) into the equations of nlotion (7.41), but such a procedure would be abit awkward. \Ve prefer to follo\v instead a different route which sheds lighton other aspects of the prohlern. Since the spacetinle is asymptotically Minkowskian, \ve can introduce a system of rectangular spacetime coordinatesat infinity. These coordinates will be denoted by y.\ (A == 0 , 1 , 2 ,3). Thesurface on \vhich the state is defined \vill ha ve, asymptotically, the equationyA == aA b'\ x r, with bAr bAs == Drs (this ensures that the coordinates x r
hecol1le rectangular at infinity). N 0\\'", according to Dirac's procedure for dealingwith curved surfaces described in Section (I. E), one should consider the" surface variables" describing the location of the surface on the same footingas the" truly dynamical" variables then1selves. In the case of the gravitational field, the nonasynlptotic parts of the" surface variables" are alread y
8
-- 114
in1plicitly included in the six canonical pairs (g/j; n ij) , which exceed by fourthe required nunl her for a theory \vith t\VO "truly d ynanlical " degrees of free
doni per point. .L~s a consequence, one has the constraints .ye!J. ~ 0 (Baierlein,
Sharp and \Vheeler, 1962; IZuchar, 1974). Ho\vever, the aSY1TIptotic location of
the surface, governed hy ten independent quantities arnong the sixteen a'\ b"\\ "
(renlelnber b'r b.\s == Ors) [anJlot be {!i?!('rJllin('(! fronl a kncnvledge of glj and n l }.
One nlust therefore introduce (l'\ and b \. together vvith corresponding conju
gate J110nlenta 7:.\ ,itAr as additional canonical variahles on the samc footing
as the g/j ,reIJ . .i\fter this is done, one \vill havc a flanliltonian fornlalisnl \\!hich
is nlanifest1y covariant under Poincar(~ transfornlations at infinity.
\Vhen \ve introduce ten new pairs of canonical variables \ve nlust gain,
una \'oidahly, ten ne\v constraints. J:1=ach of these constraints \vill enter the
f--I~l1l1iltonian \vith an arbitrary Lagrange 111ldtiplier. 1"he Inultipliers in question
\vill descrihe the anlount of hypersurface defornlation at infinity an(l \vill hegi\'cn precisely by the ~~l and ~!l\' appearing in (7.39). 1"he ne\v, extended
Hal11iltonian, 11E , \vill then be of the form
HE = Jd3x N i
' (x) .1t;Jr) r/' (P!L - P!J.) +J ~!l\' (m!,"
== 1--I ~!l PrJ. ~ ~!l\' 'Ill!J.') •
I n addition to the .yt;J. ~ 0 equations, \\"c find the ne\v constraints
and
(7.54 b) 'tIl!J.') l\1tJ. v R::j 0 .
1"he quantities p!J. and 'In!J.') arc constructed froIn the asynlptotic surface variables ([.\ b'\ and their conjugates. 'l'he details of this construction are actually
irrelc\'ant \vhat Inatters are the Poisson brackets that Prj. and 7Jl!J.') satisfy.
()ne nlay in fact shcnv that the integrability conditions (1.83) of a Hanlilto
nian theory on curved surfaces inlply that these quantities satisfy the algehra
of the infinitesinlcd generators of the I\)incare group (I\..cgge and rreite1bDinl.
1974tl ,b). One thus has
(7.55 a)
(7.55 b)
55 [)
\"here lJ!J.') IS the :\1 inkovvski nlctric.
Once \ve kno\v that PP. and JllrJ.v obey the Poincare group brackets, it fol
10\\lS inlnlecliately fronl (7.54) that 1)!J. and 1\11J.') transforrn in the correct \vay
under an asynlptotic Poincare transfornlation; the only \\"ay in \vhich the
constraints C7. 54) could be preserved under surface deforl11ations is for Prj.
- 115-
and P p. (171 tJ. 'J and -- Ivr rJ. 'J) toe \'() Ive in the sanle fa sh i()n . l'hercadcr 111 a y\vonder at this point \\"hy one does not just calculate the I)oisson hrackets
of the p p. and 1V1;:0 atllong then1sel ves instead of \:vorking \vith the extra
variables PIJ. and J71!J.'J" 'fhe ans\yer is that these Poisson brackets do not exist
for the san1e reason that the functional derivatives of H o given by (7.3 8) do
not exist. It is only the sunl (7.5 I) \vhich has \vell defined Poisson brackets (~-)
rrecall (7 ·47)1· Geotlletricall y speaking, the non-existence of functional deri\'ativcs of PrJ. and lVl t: o follo\vs fronl the itl1possibility of continuously defonlling
a surface at infinity \vithout altering its shape else\vherc. (See Section (7 .l~)
belo\v for a discussion related to this issue).
\\7e finish this section \vith the ohser\'ation that thc f-I anliltonian II L
gi ven by (7·53) is to open spaces \vhat H o gi ven by (7.5 I) is to closed spaces.I n fact, both quantities vanish \vcakly and their argunlents contain, in each
case, the full specification of the surface on \vhich the state is defined. C~()rre
spondingly, it is the product of the space of the l:"j, Tel) with the space spanned
hy the aSytllptotic surface variables a'\ b":. and their conjugates \vhich plays
for open three-spaces the role that the (glj, 1t1)) space alone pIa ys for C0t11pactthree-surfaces.
'1'he equatiot1s of tnotion associated \\'ith the I-Iatlliltonian (7.5 I dcscrihc'
the c\¥olution of the SystCt11 under arbitrary defort11ation of the hypersurface
on \vhich the field state is defined. ()ftcn one \\Tants to ans\vcr a 1110rc lit11itcd
question, such as ho\v the Syst(,111 c\'ol\"es along a onc-paralnetC'r fcltllily ofsllrfaces. ]'hcn one cIoes not need the \vhole po\ver of the theor~' hased on
the general f-Iatniltonian containing four arhitrary functions ~1 and :\ /; a
(*) To verify that the constraints (7.5-1-) arc nrst c1~ss, one treats all of then1 at OllCC,
together \\"ith the ,·:10). ~" 0 equation, by \\Titing
IJ E (N)
(The subscript (N) in "Y,Y- and r3 tJ." is intended to retnind us that these quantities are related to
~!J. through the aSYll1ptotic fonnula (7.-1()).) TIle statclnent that all constraints (I.e. ~ytrL .~. 0
and (7·54r arc first class then reaeIs
for any gi\"cn ~ and Ii and sorne;..Actually for the theory to be consi::itcnt with the !\i('lllClll
nian structure of spaectinlc (i.e. for the c\'olution to hc path independcllt in the sense ofSection (I.E.), ; 11111St be gi\'en by (Tcitclboin1. J()73 fl, b)
which is an alternative way of writing Eq (I .83). 1~his is the starting point for deriving (7.55).
116
reduced Hamiltonian \vhich is able to give only information one is askingfor suffices. The reduced I-Iamiltonian theory is obtained by freezing thedegrees of freedom corresponding to an arbitrary deformation of the surface.I t has no constraints and a lesser nun1ber of degrees of freedom than thefull theory. The gauge-like freedom to make arbitrary transformations ofthe spacetime coordinates is thereby destroyed.
It turns out that the procedure for fixing the coordinates is quite different depending on whether the three-space is open or closed; we will therefore treat each case separately.
I. OPEN SPACES
In this case the gauge freedon1 corresponds to the possibility of makingarbitrary deformations of the surface while keeping its asymptotic shapeunchanged.
We then \vant to impose additional constraints so that the previouslyfirst-class equations J/P!J. ~ 0 \vhich constrain the defornlation generators tovanish become second class. After the additional constraints have been imposedthe Hamiltonian will become just a surface integral
(7.56) I === - el't, P!J. + ! ~'t,v M!J.v
according to (7.52). It will be possible to use (7.56) as a meaningful Hamiltonian because, as we shall see belovv, the surface integral (7.56) acquires\ivell defined Poisson brackets (the Dirac brackets) after gauge conditionshave been imposed, even though its functional deri vati yes \vith respect to
gz"j and 1tij do not exist (*).
Instead of immediately evaluating the Dirac brackets for the gaugeconditions of interest from Eq. (r .58), we shall follo\v instead a son1e\vhatindirect procedure. rrhis will enable us to n1ake contact with, and use of,results already available in the literature. The schenle (which includes allcases found so far in practice) runs as follo\vs: One aSSUll1es that the variablesgt'k ,1tik can be separated into two sets, (~'X (x) , 1tcx (x)) and (~A (x) , 1tA(x)),by a bijective, time-independent, functionally differentiable canonical transformation in such a way that
(a) the surface integral (7.56) depends only on the cp'X and the 1tcx
(b) when the 1tc( are prescribed as functions pC( of x in such a way that (**)
(7.57 a) pC( === 0 ,
(*) After coordinate conditions are imposed the deformation of the hypersurface isglobally determined by C/y. and ~!J.v (the freedoln of making arbitrary deformations in the interior has disappe~red). I t then becolnes Ineaningful to ask for the generator of such a deformation. The generator is precisely the surface integral (7.56). This is the geometrical reasonwhy (7.56) acquires \vell-defined Poisson brackets after the gauge has been fixed.
(**) Strictly speaking Eq. (7.S7a) is the \veak equation nC( (x ,t) - jJex (x) ~ o. However in the follo\ving \ve shall not insist on the weak equality sYlnbol unless confusion couldanse.
- 117-
the constraints -nJ. ~ 0, .~. ~ 0 can be sol vcd to express the ([)'X as functionals
(7.57 b)
of the rCll1aining canonical variables. 1~he functional deri vati ves of fX \vith
respect to yA ,itA are assumed to exist.
If the above conditions arc true, then Hanlilton's equations for theI-I at11iltonian
(7.5 8) H red uced [,' A • ] - I [, rj ]y ,itA - g'l' Tt .,~oe = joe
Tt'oe = Poe
together with Eqs. (5.57) are equivalent to F~instein's equations In the parti
cular frame defined by it): == PY. .
Proof. Recalling that Poisson brackets are invariant under canonicaltransfortnations and that the H anliltonian is unchanged if the canonicaltransfornlation IS independent of tit11C, \ve ha ve
(7·59)
()n the other hand
(7.60) 1-1 [c.p'X ; Tt x , ,~A ; itA] . == I [cp'X ; it IX ] = Hreduced [,~A ; itA] •?oe=j): -.?):=joei:oe Poe 'iT:oe = PIX
::\ ext, differentiating (7.60) \vith respect to 'ITA we get
(7.61 )
?('J.=joe-;:oe = Poe
SHredllced
~A(X) •
I-Io\vc\Tcr, by Eq. (7.5ia)
icc< (x) = - a,p:~~y), = 0
:'-?):==-=joeIT:('J. = Poe
whenceSHreduced
STt'A (x)yoe=j('J.Tt'oe = Poe
Equation (7.62) inserted back into (7.59) sho\vs that Hredllced generates
the correct equation of nlotion for yA. In a cot11pletely analogous way one
shows that the correct equation of I110tion is also obtained for itA' The evolution of cp''X as calculated from (7.57b) will agree with the one given hy theunreduced formalism because the constraints are preserved in time.
- 118-
Note that \VC ha ve not allo\\~ed for an explicit dependence of I on ,~A, Tt,\
i.e. a dependence other than the one induced by iJ~ == fX) because such a
1 11 . ] h" I l' . "LI reduced I'" , \epCl1r ence \\'OU C In genera. cause t e varlat10na (er1 vat1 yes or /0',/ ,
;)I-IrcdulcdjoTC:\ to be undefined. 'This fo11o\\'s fronl the fact that 1-I o and I do not
separate] y ha \'C \vcll-··-definecl functional deri vati Yes, only the sun1 flo I
cl()cs. F'or an explicit exanlplc of the issue discussed here see the discussion
of (7, 1°9) belo\\-.
In practice, the new phase space coordinates (ep'Y. ; TCcJ , C'~.\ ; /CA) are not
canonical. Neverthe]ess, one can easily verify along the sanlC lines as theI f I H rcduccd, 'II 1 ' 1 1 ' , 1 l' f ha )()\'C' proo t 1at IS st1 0 )ta1ne( )y InsertIng t 1e so ut10n 0 t c
constraints into I [glj ,reI)] provided that the following restrictions hold:
(h) ]'he "111atrix " {7't x , q/) (x':;} is in vertihlc, I.e.
(7.6.+) I d:3x' { 7't~ (x) , iJij (x')}frj (x') == ° =? frj == ° .
.J
If equations (7,63) and (7.64) hold, then lIrcdllced gi\'en hy (7.3 g) \vill generate
t he correct equations of 1TIotion
i .\ {,,\ Hrcduced'l'y == 'Y , f
• o. -.- {..,.,.. I..J reduced 1re. \ - , .. A' -1 J
pro\'idcd that the Poisson brackets are conlputcd using the general forlnula
C)( ~ l;( ~ 3F )L~Pb 2Qa 2Pb
\vhich holds for a general (not necessarily canonical) set of phase space coordinates (Qil , Pa)'
\\TC no\v proceed to apply the abo\'c procedure to the gauges that ha vc
heen proposed in the literature: the ...-\.rncHvitt, Deser and :VI isner (1962) "'[--'1'
gauge" (\:yhere "'1'-T " stands for " transverse-traceless ") and the so called
" nlaxin1al-slicing " gauge condition proposed by Dirac (I95 8c).
(a) ADlVf's "T-7~" (;auge.
r\rno\vitt, Descr and 1\lisncr 1962) separated the canonical yariahlcs intot\VO sets as follo\vs: (i) (gT; /C" ; 7:'1' ; g;) and (ii) t\\~O (independent) pairs
(g-;/ ; TC1jTT
). 1"'he nc\v variables are defined by applY'ing to both g".i --- 0/.1
anel 7:lj the deconlposition.
(7.6 7)
(7. 68) f/ == . or,) [f,j,/ - I '2 fl.:j'kil] 'v~ 2y'
- 119-
and
(7.7°)
Here 1 !v:!. is the inverse of the flat space Laplacian, \\lith appropriate houndary conditions at infinit:y. r1'he fixation of coordinates is achie\'ed by inlposing
(7.7 1 )
(7.7 2 )
rr;T == 0
T'hus gT and rr;i correspond to the ep'X of Eq. (7.63) and rr; T and g" to the T:x
.
]'he role of (,~A ; rr;:\) is then played by (g-}/' and rr;ijTT.
"[he Poisson brackets of the nc\\! variahles are
(7 73)
(7.75)
{g T ('lA) ...... T I' 'l~')( - 2 ~ (x 'lO')A ,'" \A- J -- () \ ' ,A ,
{ n i (x) ,gk (x')} = I " (aa, - ~I-;; 2i 2,) a(y , x')2 \7- 2 \7-
{g~~r (x) , rr;kITT (x')} == a~'Tkl ~ (x , x') ,
all others heing zero.Th 1 I ~TTl!. /) h' l' a . 1e synl)O 0 lj In (,,7.75 represents t e Intcgro-c 111erentla operator
\vhich projects a synl111etric tensor onto its "1'-'T " part, nanlely,
(7.76) ai/ki~ [( ili! ~- _"~~l) (ajk. - :j\~\) + (il", __ d~~k) (ilj !__ ~~_) ._
_ (akl - d~;!) (ail __ .d~;;)] .
T'he operatot- (7.76) satisfies the foIlc)\ving relations
(7·77)
(7.78)
(7·79)
(7. 80)
(7. 81 )
~Trl! fTT'j' === fTTl!Jl ,
"TT.U TTl!Oii == 2/ ~"m ==-= 0 .
J~quation (7·77) expresses the fact that there are only t\VO independent cano
nical pairs (g~~T; JrkITT) per space point. I~~ronl Eqs. (7.73)--(7.75) one easilychecks that conditions (7.63) and 7.64) are satisfied. ['[he only result thatis not inl111ediately obvious is
:" (a Zk - 2~" 2i 2,.) f' = 0 => f' = 0 ,
'This is easily checked by lllaking appropriate contractions and recalling that,\vith zero boundary conditions at infinity, (I Iv"!.) (0) == 01.
(7. 82)
- 120-
N ext, let us \vrite down more explicitly the reduced Hamiltonian corresponding to this gauge. Once gauge conditions have been imposed, the partof the surface integral (7.56) obtained by setting ct.z" == ~fLr == 0 and ct.1 == I,
namely PI, is what one usually calls "the Hamiltonian". V\Te now writeI .
down P and lea ve PI and 1\;1 p.') to the reader. In the new variables thesurface integral defining pI is
I ., i~ 12 l'P [gij, n l)] :.: J d Sk (g£k,/ - gil',k) == - J d Sk g ,I.:.
Following Arno\vitt, Deser and IVlisner (1962), we assume that, when conditions(7.71) and (7.72) hold, the constraints can be solved to give
(7. 83a) gT == iT [g~t:T , nklTT
] ,
(7· 83b)
Then the red uced generator of pure tilne translations at infinity IS
pI reduced = - ~ d2 Sk gTk
[gTT , n klTT ] .J ' I)
The rate of change of the dynamical variables under a surface deformationwhich IS aSYlnptotically a pure time translation is
.1'1' 1''1' I reduced 1'Tkl 8pI reducedglj == {gtj' ,P } == at)" ~-~~krl~-
(7. 86)8pI reduced
. k/TT == { k/TT pI reduced} == _ aT:rkl _'it n, Z) 8g!.T
I)
with a;;·Tkl and pI reduced given by (7.76) and (7. 84) respectively. The presence
of the" projection operator" a~~:rkl in (7.85) and (786.) ensures that the righthand sides of those equations are transverse and traceless. Equations totallysimilar to (7. 85-86) hold of course for the changes generated by pz" reduced andMreduced..I.. po •
Finally, let us evaluate the Dirac brackets of the original variables gij,n kl in the gauge (7.71 -72). The original definition (1.58) of the Dirac bracketin1plies that, in the phase space coordinates (g1', n T
), (gz" n'), (g~.T, 1tkITT
),
the bracket is obtained sin1ply by dropping the terms involving (gT, 1tT) and
(gi' n') fron1 the original Poisson bracket (7.66). Sin1ultaneously, one shouldconsider the second class constraints (7.7 1), (7.72) and (7.83) as strong equations. This implies that the Dirac bracket between the 1'-1"' variables is givenjust by (7.75). To find the Dirac brackets of the original variables, one thenreturns to the deco111position (7.67) and proceeds as follovv'"s: ...\fter taking(7.7 1), (7.72) and (7.83) to be strong equations one has fron1 (7.67):
'1'1' l'(7. 87) gz"} == gu + it)" + 2 aij ,
(7.88) 7:kl === 1t
kITT -T- fk,l -t-fl,k =~ r:;kITT -+- fUt k)
- 121 -
where /:5 is related to fT by (7.o9). rrhis gives, for exanlplc,
]
(7. 89)
f TT (, ') fTC )1* + {fT (~\ jT (~".l.*-- t g nUl ,X , tj X J' /j \"1), Nm" ~t ) J
where
(7.90 )
and
(7.9 1)
Equations (7.89}--(7.91) express the Dirac hrackets of the g/j \vith the111sel ves in tern1S of the functionals fT defined by (7.83). The renlainingDirac brackets, \vhich \vill be left to the reader, contain also the functional fi'The functionals fT and i/ 1l1ay be obtained in practice to any desiredaccuracy hy the rnethod of successi ve approxi11lations. 1'he 1'-T' quantitiesin the right hand side of (7.89}-(7.91) are to be expressed in ter111S of theg/j , n k1 by 111cans of the projection operator (7.76). The equations ofnl0tion (7.85)-(7.86) can be re\vritten in ternlS of the bracket as
1~=={F,H}*
\vhere H is gi ven by (7.56) and F is an arbitrary functional of the gl"j , n kl.
I t is understood that in evaluating the right hand side of (7.92) one takesthe surface integral appearing in H outside of the bracket and perforn1 s theintegration at the end.
(b) Dirac's" hfaxinzal .Slicing" Gauge
The one paraI1leter fanlily of surfaces on \vhich the evolution IS being observed I1lay be fixed by the " I1laxinlal slicing" condition (*)
7t == nZ: ~ 0 .1-
(*) Equation (7.93) is calleel the" Inaxin1al slicing condition ", because on account of
(7.29) it can be equivalently \vritten asK=o
where K is the trace of the extrinsic curvature tensor. N O\V the change under a deformation
N~t dt of the volu111e ~\T = I d3 .;l~f{1/2 enclosed in a three dill1cnsional region is
. .d (~\T) = d) (gl/2) d3 X = - )1\1 gl/2 Kd3x ,
by (B.S). This shows that if K = 0, the YOIUlnC of the three-surface is unchanged by a surface defonnation. It lnay be shown that for a spacelike hypersurface ernhcddecl on a hyperbolic H.. iclnannian space one is actually dealing \vith a InaxinlUIll.
9
- 122-
C'ondition (7 ·~)3) is in \'ariant under changes of coordinates in the surface and
so docs not restrict the spatial coordinate s ysten1. 'The I-I an1iltonian \vi11
thercfore be only parlially reduced by condition (7.93). It \vi11 still contain
a term / N'If; and will still possess the three constraints .:If', ~ 0 corresponding
to the frecdo111 to 111akc arbitrary tangential deforn1ations in the surface.
'1'0 achieve total reduction one can still in1posc three coordinate conditions,hut \VC shall not \vorry ahout that here (see Dirac, 195 8c).
~-\s })irac (I958c) realized, one 111ay rearrang-e the canonical variables
in the follo\ving \vay: (a) ()nc pair (ep ; rc) and (b) l~ive other pairs (g/i; if/i).'1'he quantities under consideration are defined as fo11o\\'s:
(7 ·9-t)I loc:rgJ h.'
'rhey ha vc the foll()\vi ng l)oisson brackets:
{ ep , it (x')} =--= () ex ,X')
\vith
(7. 1(0)
and
(7· 10r ) { :::"i; (1'\ ::"'J:l (r'\!II.' \_t) ,1\. '\" ) j ,x') .
~~11 other Poisson brackets arc zero.
The quantity l:-/.i appearing in (7.96) and (7.97) is the in\'erse of the confon11al 111etric 1;','.1' i.e. glm kim 6:, and is related to the full 111etric byg,J =:= gl/:~ g,j. 'fhe" conforlnal Kronecker delta" defined hy (7. 100) hasthe follo\ving useful properties:
(7. 102)
(7. 1°3)
(7·roS)
o
123 -
Equation (7.102) says that there arc only fi\'e independent canonical pairs
(1:1.1 ' 7:(i) per space point, In accordance \vith the conditions
(7. 106)
(7. 107)
f1"'rol11 I~qs. (7.~:,R) (7.101) \ve see in1ll1ecliately that the ne\\7 phase space coor
dinates satisfy thc conditions (7.63~;, (7.6-t) of the reduction thcore111 ...:-\ccor
ding to l'~qs. (7.66) and (7.98)-(7.101) the Poisson bracket is COll1puted Inthe n('\\' \'ari a hIes usi ng the equation
(7. 108)
"
• .:3 A [ of oC .~{ F' , G } == d.1 00' oJ:-
6/i -
0(; of'
09 077
oC of'-8~l'; )
I (_ ....-.. 1,/ - 1,/.- ") ... 01.".. O(~.]~lJ gA ..,...1\ (TIJ ..-:;- \. \. . ---- i\. (,:>' - f:::"'z'i o:;:..ld •
,.J ~ Iw • J '"
One can check fronl (7.108; that It and ic ha ve zero Poisson I)rackets \vith
everything and, consequently, they can he set strongly equal to unity and
zero respecti \'ely. 'rhus one can preserve the synl1netry in all indices e\'en
thougb one is dealing \vith n10rc variahlcs than needed [ren1en11>er l<~q s.
(7. I 06) ancl (7. I °7) ] .In order to reduce the H Cll11iltonian, \ve ha \'C to express the surface in
tegral (7.36) as a functional of the the ne\v variables. ~~gain \ve \vill just \vorkout pI and lea \'e pi and ~1(Jp as an exercise for the reader. ()nc gets:
1,"'ron1 (7. 1°9) \ve sec that to ha \Te \vell-defined \'ariational deri \'ati \'CS ofpI reduced \\Tith respect to gl).:, \Ve ha \'C to get rid of thc gik,I' tern1. In other
\\Tords, the reduction of the I-I an1iltonian by 111eans of the nlaxin1al slicing
condition i: == 0 is not possible unless one chooses a n1Dre restricti\Te aSYll1p
totic spatial coordinate condition than in (7.-t0b); \\'C shall require
(7. 110)
rrhe asyn1ptotic forn1 (7.39) IS thus not allo\\Tcd In this context, Inlt the linc
clerncnt
(7.1 I I) (is .---.-----_...._---+ -- (I __;\l __ ') dt~r--;>-CV , KT:r (I ::\1 ) ~ d -1' -1 .J
) o,'J' ~1 (.l',KT:r / -
\vhich is obtained fro111 (7.39) hy a change of coordinates
(7. 112)
satisfies (7. I 10).
(7. I 13)
- 124-
Taking into account (7. 110), the energy heconles
IJl - 2 ~ d2 1/:3- -- J Skg ,k'
()ne ll1a y check that, \vhen inserted into (7. I 13) the n1etric (7. I 1 I) correctl ygi vespl == ~1. ]"'he final step in the reduction procedure is to solve the Han1il
tonian constraint Xl. == 0 and express gl/:3 in (7. I 13) as a functional of the
cR"/j and ir/ j. l'his leads to the follo\ving equation, first exan1ined by Lichne-
ro\vicz (194-+) and recently extensively studied by Choquet-Bruhat (1972,1973) and hy 0' lVlurchadha and ''"ork (197 2 , 1973):
(7. 11 -1-)
\'lith <D =.--.~ ,g-1/12. (Here Tclj = ff£! gil.' ir/k and R, ~ are, respectively, the cur
vature and the Laplacian in the n1etric g£/). 1"'he reduced H anliltonian IS
therefore gi ven hy
(7· I 1 5) 1_Trectu(:Pd == _ ~!l ])"J.l'CclllCC(1 -t- ~_ 9"" "'1r,c<!uced + r 13 ~ "z" ( ) r./£} ( .\1. J. , P H :J. 'J .. (J..i. " "x ./( z" \ A )
. h !)ll'CclllCP(}. 1 / '\ C' '1 . h Id f 1)z" d 1\1 l"'h\Vlt gl ven )y (7. I 13). ,lllnI ar expressIons 0 or' an 1 / ~(J' f'
constraints
(7.T 16)
still hold as first class equations. 'l"'he tangential generator appearing in (7. 1 IS),(7.116) is also understood to be reduced by the condition 1t==0. It n1ay be\vritten in the forn1
(7. 11 7)
\vith the covariant deri vati ve being taken in the nletric (g-z"/.
l~"'inally \ve \vrite the equations of n10tion for il') and fc(j In a n10re
explicit fornl. Fron1 (7.108) these equations are seen to read
(7. 118)reduced
;. _ { "'" HredUCed} _ ~kl oHgl'} - g£j, - a l} oftkr--
• .., ~Hreduced
""' kl == { .., kl Flreduced l == _ ~~~ 0 +1t 1t, J lJ ocr"
0 1)
. h ~k! d H reduced. 1 ( ) d () . 1\Vlt U z"j an gl yen )y 7. 100 an,7. I I 5 respectl ve y.()ne n1ay ohtain the Dirac bracket corresponding to the gauge condition
(7.93) in a \vay totally analogous to the one followed for the rr_T gauge. 1"'hesituati,:)n is sitnpler here because the gauge condition T: == 0 is just an algebraicequation. ]"'he I)irac bracket then turns out to depend on the solution of f~CI'
(7. I 14), \vhich as \ve said above has been exan1ined in detail in the literature.
- 125 -
()nc thus considers (7.93) and (7.114) as strong equations and \\Titcs the l)iracbracket as
(7. I 20 a)
\vhcrc
(7· 120b)
and
(7· 120r)
( l lIz}'
{ (pt
(I)l ( . "\ ") x-, ,.l ) J
-.
J d:1y t~ (y)~(l> I r X')
-~7:01 (~;)
~<1>1 0<1>1 (.1"
~7:{'d
-t- ~-I' l:~ l-ilh(. '\ ::--(r!C)' n \JJ) g3
0<1>4 iX')
oi:(t! l'~, '
\\-here 11~qs. (7.99)~(7.101) and (7.108) have heen used. Cfhe righthand
side of (7. I 20) n1ay he expressed as a functional of the original \-ariahJesit!'j ,r;/i hy n1cans of the equations gij === gl!:~ gz'j and Tel) =-= cg- I !:~ nil). '1'he
other Dirac hrackets arc ohtained in a sin1ilar rnanncr and \\-ill he left to the
reader. rrhc equations of n10tion (7. I 18), (7.119) then reaeI, in tert1ls of theoriginal \-ariahles,
(7. 121 ) nl'j == { TI IJ , H }*
\vith H gi\-cn hy (7. 115)...:\gain in (7. 121) as in (7.92), one deals \\-ith the surface tern1 in H hy taking the surface integral outside of the Poisson hracket.
'1'he constraints (7.116) ren1ain first class in the starred hracket.
2. CLOSED SPACES: York's G'auge.
\\7hen the three-surface on \vhich the state is defined is cOlnpact, all
inf()rlnation ahout the" location" of the surface in the en veJoping spacctilllC
is contained in the CR"ii ,TI(i). rrhere is in this case no aSYl11ptotic region and
no set of aSYl11ptoti~ surface \-ariahlcs tZ"\ b'\; consequently no surfaceintegral ever enters into the H anliltonian, \\-hich reads sin1ply
(7. 122)
~~lso, there arc no problcn1s no\v \vith H o not ha \'ing \vell-defined \'ariatiol1alderi \-ati \·cs.
I t is clear that the procedure used for fixing the gauge in the open case
will not \vork for the C0I11pact case since there are no surface integrals to deal
with. l\Ioreover, a gauge condition such as TI == 0 \vill not be a satisfactory
- 126
condition in this case as it \vould lead to no dynanlics at all (-1<-). To elinlinatc
the frcedolll to lllakc arhitrary defornlations of the surface, one nlust usc
second-class constraints \\'hich arc explicitly tin1e-depenclent. I-Io'Never,
there is no general schenle a \-ailable for deri ving the reduced Han1iltonian
in this case. ()ne kno\vs only that \vhen the gauge condition is of the forn1
qO ~ -: then the red uced I-I anliltonian is -- Po a s111ay be \'erified by ohserving
that the right equations of nlotion are obtained.
F'ortunately, the only gauge condition so far proposed for cOlnpact spaces
is of the sinlple forn1 qO ~ '7 and one can then carryon thc reduction proce
dure \vithout difficulty.
York (1971, 1972) has proposed the condition
(7. I 23)
to fix the spacetinle slicing. ?\ ote that, as happened \\-ith the 7t ~ 0 condition
for open spaces, l~q. (7. I 32) is in \'ariant under changes of coordinates in the
surface and therefore the constraints .j~. ~ 0 \vill still relllain first class. ()nc
nlay get rid of these constraints hy ilnposing three nlore conditions besides
(7. I 23). \Ve \vill not carry out this step hl~rC'.
In order to deal \vith condition (7. I 23) one needs a slightly n10difiec1
versIon of Dirac's \'ariables (7 ·94)--(7 .97); one uses
T===.2
«({-II:! 7t3
(7. 12 5) p === _ gt:!
instead of ? and It gi \-en by (7.94) and (7.93 '1'he rest of the \'ariables renlain
the sanlC'. One then has
(7. 126) { '1' (x) , P (x')} === a(x , x')
instead of (7.98) and again, all other brackets relllain the sanlC. 'rhe })oisson
bracket is then gi \-en hy (7. 108) \\Tith ? replaced by T' and 7t replaced hy P.
'1'0 elinlinatc the extra degrees of freedonl one sol ves the equation .f(l =-~ 0
and expresses P as a functional of the renlaining canonical variables and of
the tin1e x O (the feasibility of this step is the rnain test of \vhether or not (7.123)is a good gauge condition). Before gauge conditions arc inlposed the Poisson
bracket is gi ven by (7. 108) \'lith ? --~ l' and 7t ~ P. \Vhen the gauge condition
C*) The eq uation 'Y~l ~. 0 ,,·ould becoIl1e a second class constraint after the T: ,~ 0
condition is irnposed. :\ o"\v since the gauge condition is tilne independent, the lIalniItonianwould be unchanged; this Ineans that the equation~ of lnotion \\·oldcl be giycn by
II' /' l:~ '\' / 4/ 1*l " (X ~" J(. /j ,
showing that there \\'oldd be no d ynalnics left except for changes of coordinates on the surface.Thus 7': ,~ 0 cannot be usecl to fix the gauge for cOlnpact spaces.
- 127-
(7· I 23) IS enforced, one drops the P and '1' tcrn1S and the hracket 1)('C()l1H'S
\7. 12 7) oC 81" ,8;", cx:::./!)
,.... './ 0 , •
'1'hc reduced I-Ianliltonian then reads
(7.1 2 R) I 1:3 " T t ,u'( X ~ \ ,/{ z' •
l'~q uation (7. 1 28) tells us that thc d ynalnicall y ilnportant part of the II anl il
tonian is the \Tolun1c of the surface ./(;. is connected only \vith changes of
coordinates in the surface). In l·~q. (7.128) gJ is a functi )l~al of ,{://and 7:/:/, and dcpl\nds explicitly on the tin1C .1'0.
In order to express ~f[J as a functional of the relnaining- \"ariahle's, onere\\Trites the ,YfJ. .~ 0 constraint in the fonn of a non-Ii ncar partial di ffcrenti alequation:
\vhich is a generalization of (7. I I-l-), (\·ork, 1972; 0' :\Iurchadha and \·ork197 2 ') .
...-\s \ve cIllphasized at the heginning, the' constraints ./(, ~ 0 still hold
as first class equations and can still he expressed in the fonn 7. 1Ii;. '1'he
l)irac hrackets corresponding to the gauge condition (7. 123) arc the sanle
as those exenlpliflcd hy (7.120) corresponding to the it' ~ 0 condition. '1'he
only change is that no\v the functional <t> appearing in (7.120) sol \·cs (7. I 29)instead of (7. 1 14).
- 128-
i\.PPENDIX i\.
:\lE'I'RIC CO:\\T:EN'fIONS
()ne of the lTIOSt ,"cxing problell1s encountered In \Vrltlng about relati
vistic physics is the choice of a ll1etric convention. \Ve have chosen to usc. 11 ')'):J;3 00 .. .
the ll1etrlc 1 === g === g-~ == g == - g In the 1naln text because the tranSI-
tion between nonrelati vistic and relati vistic canonical 1110111enta seems n10r('
straightfor\vard. This con vention also agrees \,'ith n10st of the current litera
ture on general relativity. On the other hand, it disagrees \vith the latest edition
of Landau and Lifshitz's Classical Tl1eoJ~Y of Fieltls (1971) and \vith Bjorken
and Drell (1964). VV"e therefore hope that the follo\ving table sho\ving the
relation of our con\"entions to those of Landau and Lifshitz and of 13 jorken
and Drell \vill help in S0111e slnall \vay to n1ake this \vork 1110re useful to those
\vho prefer other ll1etric con ventions. Our con ventions for general relati vity
agree with those of lVlisner, '"[horne and \Vheeler (1973) pro\"ided N c\vton'sgravitational constant G is set equal to (16n)-1. r1'he reader 111ay also find in
this reference an exhausti ,"e con1parison \vith other con ventions used in the
literature.
~ === 0,1,2,3
0 0 0
,tV0 -I 0 0
g -0 0 -I 0
0 0 0 -I
13JOI{I{E~ 1\:\1> IJRELL(Li\:\l)j\l' .A~l) LIFSI-IITZ, IF I)IF
FERE:\T)}-IERE
~ == 0, 1 ,2,3
-1 0 0 0
ILV0 0 0
g'0 0 0
0 0 0
p === (_po pI p2 p:3)lJ· \ ",
p == (pO _ pi _ p2 _ pa)p. \' , ,
'\C
ds ===U d ,1')dx' xp.J·- d ( , d II d \ 1/'~
S == \ i- X· - x p)
..:\.",'1 __ ~~'l, \'
I Ezjk Fjk == B i2
~ zliA- Fik = B 1 (+ zli
k Fik = _ B')
- 129-
\Vc hayc also chosen a different null-plane nlctric frol11 that of I(ogut
and Sopcr (1970). ()ur con vention has the ad \'antage that all cOll1ponents
of g!J.\J are positi yc even though \VC arc in a :\Iinko\vski space.
I-IERE
!-L -- I , 2
0 0 0
IlV0 0 0
(g-'0 0 0
0 0 0
pll P!J. == 2 p; p-
IlV
/t
p±
pP p,) .
...-\PPE~DIX B
KOc;CT A:\lJ SOPER
1.1 - I , 2 , -
0 0 0
0 --I 0 0
0 0 0
0 0 0
2
2 pip - -- p~ = ; }l~
Consider a spacelike hypersurfacc clnhcdded in a four-di1l1ellsional
spacetin1e of hyperholic signature. Let the unit norn1al to the surface he n P'.
Denote hy the suffix 1 the projection of a spaceti111e tensor of any rank 011
the norn1al (i.e. "'-\1 == ...A..!.L n!l == _ ...-\1 and sin1ilarl y for higher order tensors)
and hy latin indices the projection onto the surface. rrhe foIlo\\'ing relations
the hold in the basis (n , eJ \vhere e/ denotes the tangent \'cctor to the i-thcoordinate line on the surface:
(B. I a) (!)R Iabet! == \..ubed
(B. I b)
I-i:qu at ions (B. I) are k n 0 \V n as the Gauss-Cod a zzi equat ion s (see, fo rexan1pIcEisenhart, 1926).
l'he syrnnletric space tensor K ab appearing in (B. I) is calle the extrin
sic curvature or Sl!COJll! jUll{!afJZenta! jorJll of the surface and it has, fron1 the
I-Ian1iltonian point of yie\v, the inlportant property of being detern1ined con1
pletcly once the surface is giyen. rThe extrinsic curvature is geon1ctrically
defined by the paraJIcl transport of the norn1a1. n P' along the surface. Since n!J·
has unit length, the difference I)n het\veen n:J. at the point Xl' dx·l ' and the
vector ohtained hy parallel transporting Jl:J. fro111 Xl to Xl d,,1.J : along the
surface lies on the surface. One then defines Kab hy \vriting:
(B.2)
- 130-
'rhe left-h an:l side of (B. 2) n1a y be expressed in terl11S of the lapse and
shift functicH1s as fo11o\vs: First one nDtices froll1 (7.5)-(7.8~) that
ll, =[.I. ~1, 0 ,0 , 0)
~'rhe rninus sign In llo ensures that l/J. points in the direction of increasing Xu),
IT si ng I;~C] s. (7. 5~:i(7.8) \\'(' fou ncl that (B. 3) inl plies
I)' 0 )1-) 0 )1..,0. I} ---).~) . llb - lib llO C.1 --
=--= (2 :\1)--1 [(-- gab,O gOa"S l{Ob.il) _:\c gab." /Slil,b gch,a)J =-=
== (2 :\1)-1 (-- gah,O :\11.b ~b,a -- 2 r~~b NJ =--=
=== - (2 l\d)-l (--- <-f{ab.O -i- N(/,b ~ Ilia) .
(B·5) ,!JIlII. () N 1,b
\\'hich In particular sh()\vs that 1'(//1 IS syn1111ctric.
'rhe follo\ving relations hold In the hasis ,n, ell):
(8.6)
(B.7)
r).(1 I). (J{/ II (J
g === C;l e - It J!,
ll, llt). == -- I). ,
l~quation (13.7) is just (13.2) \\'rittcn in general coordinates.f---roll1 (1).6), \ve get
(13.10)
Since the co\'ariant derivative of (13.8) is gi\'cn by
(B. I I)
1~C]. (B. 10) red uces to
(B. 12)
'rhe last step in (B. 12) follc)\vs fron1 (B.7). 'This pro\'es (7.18a). Equation
(7.18b) is proved In an entirely analogous manner.
3. Jl0Jll.'lltUJJZ auel ~illglt!ar j·l0lJlflllUJlI of the (J~ra(l'itatioJla! F/('!e!.
~~s an illustration, let us sketch ho\v (7.~8b) is iclentificcl as the linear1110111cntuln. rrhc linear 1110lnentU111 is norn1allv defined as the conserved
quantity associated \vith in \'ariance of the action under translations. ~~ tran-
- 131 -
slation is an operation \vhich is defined only for a 11at space it a1110unts
to displacing the fields fron1 the point x k to the point x k 2 7', \vith Sk fixed
(independent of x) and \,~here the J.~,.{' are cartesian coordinates. 'rhe correspon
ding (nlore general) operation for a cur\"cd space is to displace the fields fron1
the point x k to the point x A' ~,.{' (x) in a coordinate-in\"ariant \,"ay. 'l'his is
achieved by substracting fronl a field quantity its L-ic deri \"ati \"C along the
vector field ~A' (x). ~O\V the action
c __ Ii" d \ )1' 13 ~ ( 1)' " T' I ,// '\.-, --- .tiC ..1: 7t g/j ---- ... , ,/7 11-)
•./ ..
is invariant under such a transfornlation c\"en if ~k behaves as ri1 at infinity
(\vhich is the case for an asynlptotic rotation). :\ote that \ve ha ve set ~p === 6iland ~'l\' == 0 in (7.) I) in order to get just a pure I~Ianliltoniangenerating changes
corresponding to a l\Iil1ko\vskian tinle c1isplacclnent at infinity.
Kno\,"ing that the action is in \"ariant, the next step is to follo\v the lines
of X oether's theorct11 and rearrange the ,"ariation of the action in the fornl
~S ~ I 1 cl I 13 ~ ( if a•: C t elt , ( x\7t gil?
II
1f \ve then insert
(13. I 3)
into (B. 13), \ve obtain
tcnl1S vanishing hy the equations of 111otion/ .
'1'he second tern1 in (13.14) vanishes due to constraint -Yf;, ~ o. \'That IS left
n1ay he transfornled into a surface integral so that as reads
(l~. 1 5)
If \ve set ~k -;=~~--? Sk, a constant, \,"C arc dealing \vith an aSYlnptotic
translation. \\7'e can then \vrite
(B.16)
I~ 1 d2 ~. === ~ 2 J Si it .
Fron1 (13.16) \ve identify pl giycn hy (B. 1 r) as the total lincar 1l10nlcntu111
of the systenl. 1t is a constant of the nlotion hecause the action is in variant
under the transforn1ation in consideration, so DS in (B.16) is zero. It is iln
portant to realize that only Cl == ~A' ((X)) provides nontri vial inforn1ation.
- 132-
'1'hc in\'ariancc of the action \vith respect to arbitrary ~k (x) in the interior
is inlrnaterial in this context; ~k enters in the equations al\vays nlultiplied
by ,10> as in (B.14), \vhich again reflects the close relation betv:een the invari
ance of the action under arbitrary changes of the spacetime coordinates and
the occurrence of constraints in the Hanliltonian theory. The fact that only
~,4' (ex» 11latters is the reason that the linear nl0111entU111 in general relati vity
is referred to as the conserved quantity associated \vith /f spatial translations
at infinity". Sinlilarly one speaks of the energy as being associated \vith
tinlC translations at infinity, and one associates the angular 1110111entU111
\vith spatial r()tations at infinity. 'fo find the angular nlonlentuln Lone
just sets~k r ---~r-v ...~ S;"/I oqi xi in the ahove reasoning and replaces (B. 16)
hy ~S == oepi (Li (/2) - L j (11)). 'l'his discussion sho\vs also that the concepts
of energy, nl0111entunl and angular 1110111cntunl have no nleaning for a
closed uni \·erse.
-to Relation (~I /!.q. (4. 16) 10 l~'q.
'r0 reI ate J<~ q s . (4. I 6) t 0 l~~qS . I .8J) 0 n c not iccs fi rs t () fall that the fun c-
tion '~~ is the generator of reparall1ctrizatiollS a -~f (a). 'rhus in the notation
of (I.R 3) \\ye ha "C
(B. I R)
On the other hand the generator of dcfonnatiolls along the nonnal to the string
eli ffers fronl ,~l. hy a factor \\'hich is dctcrlnincd up to a con vcntional sign
frOlll the condition:
for an arhitrary function !\ (a) .1'his gi "es
(B.20)
\vhere :\ IS the nor111alization factor appearing in (4. I).
'l'he functions ,.IIi. and ,Jf'l ha ve the follo\Ning equal-7 hrackets
,(13.2 I) {£J. (a) , ,1{1 (a')} == [(u~)--l (a) ,.leI (a) + (u2)-1 (a') .;~ (a')] a: 0 (a - a') -
-- 2 :\---1 l(u~)--:)/~(0) ,lel (0") '~l (a) -t-
(l~. 22)
(13.23) {dili (a) ,~ (a')} === [-.IIi (a)':)
~~ (a')] ~ 0 (0" - a') .ocr
- 133-
Eqs. (B.22) and (B.23) have the san1e forIn as (r .83 b ,c). l'his should l)c thecase since these relations just characterize .YlJ. and ~y~ as heing respccti\-elyscalar and vector densities in the one-dinlcnsional space of the string. If \ve
notice no\v that the n1etric tensor along the string has just one c0111ponentgll given by
11)-1 2gIl == (g == u ,
\\7e see that (B. 2 I) agrees \\T i th (1.83 a) but does so 0 n~y UJeak~Y . 1'hc discrepancy con1es fron1 the second terITI in the right hand side of (B.2 I):
(B.25) "'\r-1 [( 2 ( ))-:~/'2 ,,/£J ( ) ,,/fl ( )- 2 ..L' \U (J Jl 1 (j~, 1 (J
') , -'~/,) , ,d ,+(u- ((j )) .... e~l ( (j ) ·yt'l((j )] 2cr 0 ((j - (J ) •
Note ho\vever that espression (B.25), being quadratic in the constraints, has\veakly vanishing brackets \vith c\-crything. 'rhus the hracket relations(B.2 I )-(B.23) still ensure that thc d ynanlical evolution of the string is "pathindependent" in the sense of Section (I. ~~). ..:\. sinlilar analysis sho\v5 thatone can replace the factor (u~)-l in (!).20) by - (.1J·~/N:!.)-l (obtained by
solving y~, == 0 for u 2), and get an equally good ./fl. Such a 111odification ineJfl. results in fact in the addition to the right hand side of (B.2 I) of anotherternl quadratic in the constraints.
- 134-
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