Constrained Hamiltonian Systems

135 Pages·1976·6.928 MB·English

by  Andrew Hanson, Tuilio Regge, Claudio Teitelboim

ACCADEMIA NAZIONALE DEI LINCEI
ANNO CCCLXXIII - 1976
CONTRIBUTI DEL
CENTRO LINCEO INTERDISCIPLINARE
DI SCIENZE MATEMATICHE E LORO APPLICAZIONI
N.22
i\NI)REW I-L.A.NSO:0J - ]'ULLIO REGGI~ -- CLi\UDIO TEI]'ELBOIlVI
CONSTR.Lt\INED
HAMILTONIAN SYSTEMS
CICLO DI LEZIONI TENUTE DAL 29 APRILE AL 7 1vIAGGIO 1974
ROMA
ACCADEMIA NAZIONALE DEI LINCEI
1976
----
----.---------~---_._---,._-
(, S
s-
(~
Dott. G. Hardi, Tipografo dell'Accademia :\azionale dei Lincei
}{O:\IA, 19i(1
TABI~E OF CONTENrrS
FORE\VORD Page 5
CHAPTER 1. 1)IRAC'SGE~ERAL~IETHODFORCO~STRAI~EDHAl\lILTONIA~SYSTEMS 7
..c-\.. Forlnal Introduction . 8
B. !-lanzilton Variational Princzple 'zvitll Constraints 13
C. Extension to Infinite Degrees of j;reedo)Jz 16
D. Other Poisson Bracket Surfaces. . . . . . 18
E. IJynalnics on Curved Surfaces. . . . . . . 20
F. Quantlun 7lzeory and Canonical l//ariables 24
CHAPTER 2. RELATIVISTIC POIXT PARTICLE. 27
~r\. .J.Vo Gauge Constraint. . 2i
B. Gauge Constraint . . 29
C. Quantuln J11"ecllanics 31
CHAPTER 3. RELATIVISTIC SPI~NING PARTICLE 33
A. Revie'zv ofLagrangian ~4jJjJroacllto Top. 33
B. Constraints on Top Lagrangian 38
C. Dirac TreatlJzent of Top COllstraints 4°
I). Quantum A1echanics. 47
CHAPTER 4. STRI~G MODEL. . 51
A. System 'Zvithout Gauge Constraints 53
B. Ortllonorlnal Gauge Constraints 57
C. Dirac Brackets . . . . . . . . 59
I). Fourier COlnponents of T/'ariables 68
E. Quantlan .J.~fecllanics 71
CHAPTER 5. :YIAXWELL ELECTRO:\L\GXETIC FIELD 73
A... /?lectnnn'lgnttic [{anziltonian 7oitlzout (7auge Constraillts . 74
B. Radiation Gauge Constraints 78
C. ~4xial Gauge . . . . 80
D. iVull-Plane Brackets. 82
E. iVull-Plane Radiation Gauge . . . . . . . 86
CHAPTER 6. YANG-MILLS GAUGE FIELD 89
A.. Lie Groups . . . . . . . . 89
B. Systenz lvithout Gauge Constraints 9°
C. Radiation Gauge Constraints . . . 94
I). ..4lternate Radiation (;auge Techniques. 97
E. ..4xial Gauge . . . . . . . . . . . . 98
CHAPTER 7. EIKSTEI:r\'S THEORY OF GRAVITATION. lor
A. General Form oftile Hanziltonian 102
B. The Lagrangian . l°S
C. The Halniltonian . . . . . . . . 1°7
D. ~4.~ymjJtotical(v Flat Space, .C;ur.face Integrals, Inzproved Hanliltoniall.
Poincare Illvariance at .'::Jpacelike Infillity . . 1°9
-4-
E. F'ixation oftile Sjacetinlc Coordinates (C'auge) . . Page 115
I. ()pen Spaces:
118
(a) i\.Dl\I's "1'----T'" Gauge .
121
(h) Dirac's ":\laxilnal Slicing" (~augc
1. Closed Spaces: York's Gauge 1o-)
. . 128
..:'\PPE\"DIX .:\. ~Ietric Conventions.
A.PPE\"\)IX B.
1. ExtrinsicCur\"aturc (llHlthe Etnhedcling Equations of(~auss andCodazzi 129
2. Prouf of Eq. (i.IS) . . . . . . . . . . . . . . . . . 13°
3. :VIOll1entull1 and A.ngular :YIolllcntuln of the l~rasitational Field. 13°
2
4. Relation of Eq. (-1-.16) to Eq. (1.83) . . . . . . . . . . . . . . 13
.............................. 13·+
I<..EFERE:\CES
F()Rl~\V()RIJ
]'his \v'ork is an outgro\vth of a serIes of lectures given by one of us Cf.
1\..) under the auspices of the i\.ccaden1ia Nazionale dei Lincei in 1\.0111e in the
spring of 1974. It is intended to help fill the need for a unified treat111cnt of
Dirac's approach to the canonical Han1iltonian forn1ldation of singular I..a
grangian systen1s. \\7c have attelnpted as far as possible to refer to the original
literature on the subjcct, but there have undoubtedly been S0111C inad\'crtant
0111issions, for \vhich \ve apologize.
\\ie \vish especially to thank Peter Goddard and Giorgio l)onzano for their
essential participation in the for111ulation of the " string lTIodcl " gi\"en here,
and Karel Kuchar for per111itting us to use parts of his unpublished lecture
notes at Princeton in Chapter 7.
C. T'. is grateful to J....~. \\lhceler for much encouragcn1ent, and to the
National Science foundation for support under grant GP 30799X to Prin
ceton lJnivcrsity \vhile i\.. ]. 1-1. thanks the Institute for i~d\'anced Study,
the )Jational Science ~---oundation, and the U. S. i\t0111ic Energy COn1111ission
for their support of various phases of this project.
\\7e are indebted to .i\caclen1ic Press, Inc, the publishers of ...~nnals of
Physics C:\". \T.), for perlnission to usc various sections of I-1anson and l\.eggc
(1974) and Regge and Teitelboin1 (1974) in this ,,"ork. T\\'o of us (...-\. J. H.
and 1'. R.) arc grateful to the ...~ccaden1ia Xazionalc dei L.incci for the congenial
hospitality enjoyed \vhile this \,'ork ,,'as being prepared.
..~. J. I-IA:\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~~XS\TSTElVIS
Constrained canonical systen1s occur \vith ren1arkahlc frequency in phy
sics. :\Iax\vell's theory of electrol11agnctisl11, I~instein's theory of gTavitation,
and nun1crous Inanifestly Lorentz in\·ariant 111echanical S)1'stCIT1S possess
constraints \vhich invalidate the strictly canonical classical systen1s. It is
clear that a correct Ha111iltonian for111ulation of a constrained classical systcln
is interesting in its o-\\'n right, as \\'ell as being quite useful in de\·cloping a
valid canonical quantization procedure for the syste111. Our purpose here is
to introduce the reader to the systclnatic trcatll1cnt of constrained I-Ia1l1iltollian
systelns devel0ped i11itially by I)irac (I950) and to indicatcits rcIati()n to
quantum 111echanics \vhere kno\vn.
T'his chapter \vill deal \vith the for111al aspects of constrained 1-1all1ilto
nian systelns. rrhc ren1aining chapters are devoted to specific applications of
the 11lethods. In particular, \VC \vill exan1ine relativistic spinless particles,
relativistic spinning particles, the relati\,istic string, vector fields \\,ith ~\hclian
and non-...\helian gauge groups, and gravitation.
rrhe reader \vho v\·ishes to acquire a gcncr?l feeling for the applic~lti()n
of these 11l,ethods \vithout getting bogged do\\'n in details is adyiscd to skinl
Chapter I, and then carefully study a falTIiliar systenl. (e.g. ("hapter 2, 4 or 5),
referring back to Chapter I \vhen necessary.
\\Te define a singular Lagrangian L (ql 'Yl") as one for \vhich thc \'cloci
tcs (jz" cannot be expressed uniquely in tern1S of the canonical 11101nenta
pI"= I.:L/2Y1due to the existence of constraints a1110ng the canonical coordinates
o
and 1110n1enta fol1o\ving fro1l1 the for111 of the Lagrangian alone. rrhe pl-Ohlerll
of developing a consistent classical Ha111iltonian dynalnics corresponding to
a singular Lagrangian systen1 \,'as apparently attacked first by Dirac (1950).
Subsequently Dirac (195I), ...r\.nderson and Bergn1ann (1951), Bergll1ann and
Goldberg (1954), Dirac (1958a) and })e\'/itt (1959) refined Dirac's original
n1ethods. ...r\.n expanded treatlnent of the general constrained Hamiltonian
systelTI appears in Dirac's lectures on quantun1 mechanics (1964); sec also
Dirac (1969). IZundt (1966) and Shann1ugadhasan (1963, 1973) rcvic\v
some fine points.
Finally, \ve ll1ention several other approaches to the quantization of sin
gular Lagrangian systelTIs. Schvvinger (1951 (l, I951b, 1953) and l)eierls (1952)
utilize variational techniques; SY1l1anzik (1971) gives an extended treatlnent
(see also the Appendix of rron1boulis (1973)). DeWitt (1967 b), l,"addce\' and
Popov (1967 a, 1967 b) and Faddeev (1969) ll1ake use of Feynlnan path
integrals to undcrstand singular systen1s. No attempt vvill be 111ade to treat
these methods here.
-8-
'1'he forn1ulations of both classical and quantuln Inechanics have under
gone profound changes in recent tilDes through the usc ofmodern mathernatical
language and advanced techniques of functional analysis. \\"'"e think that
ultin1atcly these concepts should be introduced into our treatlTIent to give a
less heuristic vic\v ofthe subject than \,'"C have succeeded in developing so far.
V\T have not pursued this lTIattcr in vic\\'" ofthe practical character ofthepresent
C
notes. Their usefulness, \,'"e feel, is in providing a set of direct guidelines to
setting up a consistent canonical forlnalis111 for an an1azing varietyofphysically
significant systelTIS \vhile avoiding lTIany COn11l10n pitfalls.
...\. f~OR:\IAL IXTRODLCTIO\,
()ur forn1aI discussion of constrained systcnls hegins \\'jth the consider
ation of an action functional
J..b
(1.1) c= ciT L(qi' q,)
\vhere qi('r) is a canonical coordinate and qi = dq//d-: is a crlnonical \-clocity.
We confine ourselyes to I-Jagrangians ",'"ithollt explicit 7-dependcncc. I)cfining
the canonical n10n1enta as
dL
(1.2) pi -- ,
dlo:.
Jl
\ve find the l~ulcr equations
ctpi 8L
(1·3) 0
c1--r Jqi
hy reqUIrIng the variation of the action S to be stationary.
If \,'"e choose for our l)oisson brackets the con\'"entl011
~~~\ 813 2:\. 2B
ciqi 3pi cqi
\ve have
__ 61~
(1.5) .I
\\'here 0;° is the Kronecker delta. Hereafter, repeated indices \vill be sun1111ed
over unless other\yise stated. The canonical I-IalTIiltonian
(1.6)
then forn1ally generates the I-Iamilton equations of motion
q•£ --- ftq£, !-~]c"} -- -~d!jJ!1:~
p.t' == {p't,Hc}== - ~diH~c- ·
9
NO\V \ve suppose that I...(qi , qi) is singular, so that there is no unique
solution qz"(q ,p) expressing the velocities in terms ofthe 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
\veakly zero" and 111eans that 9m 111ay have non\'anishing canonical Poisson
d
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 effective I-Ianliltonian
(1.9) H --~- I-Ic -Lim1! (Ir'jJJlf\q 'LlJ),~ lIe..
II generates lle\\' equations of lTIotion replacing (1.7),
ql' {q/ ,I-I lj ~ L'd~Ip-zI"( 11m l"l''I,Y?m;i
(1.10)
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\·atives of the
constraints (1.8) to be zero, or to be linear cOITIhinations of the constraints so
they are \veakly zero:
Yu 9n , H}~ {91/ , 1-Ic}
:--== {
If I~q. (1.11) is not already true as a consequence of the original prin1ary
constraints (1.8), t\\TO possibilities occur. First, \\·e n1ay find that Eq. (1.JJ
gives no ne\\T inforn1ation, but sin1ply ilTIpOSeS conditions on the forn1 of 11m ,
Second, Eq. (1.11) Inay in1ply a ne\v relation an10ng thep's and q's, independent
of 11m , These are seco/lc!ary constraints and n1ust be adjoined to the original
constraints (1.8). NO\V \ve repeat the process, requiring the '7 deri\Tatives of
the secondary constraints to vanish, and so on, until all independent constraints
and conditions on Um 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 ofall the constraints \vith the equations ofn10tion requires
that there exist a solution for Um as a function of q and p
-10-
so that Hitself 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.I3) { 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 to
linear conl,hinations of first class quantities. :\otethat 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.15b) ~.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 141SC
to a nonsingular :\ XX lnatrix of Poisson brackets \\'hieh \\'C \\'rite
(I.16)
(\\?hcn COlllpllting(~xr~, it is clear that onc 11111st not usc the constraint cquatiollS
until after calculating the l)oisson bracket.) Sincc the dctcrnlinant of an
antisynlnlctric nlatrix vanishes if the dinlension is odd, \\'C conclude that the
nunlbcr N of second class constraints ll1USt be e'veu. Since CY.r~ is nonsing'ular,
its in\'erse (~~~l exists and satisfies
\Ve novv proceedto construct fron1 any dynatnical variable...-\. a lle\V variable
...-\.' \vhich has vanishing brackets 'lvitlz all secoJt(l class constraints. \\Te define
(I.18) A,-- ;.\..J.. - {A· , ~a 1f C-a-~1 q;",
and observe indeed that
(I.19)
-- {..'I.J.., mrYJl--- fl ..t.'l.. , mrYJt.--o .
Note that {A', ~i} IS not necessarily \veakly zero.