An example&: T"&ananelement

TheAPi(p)function9(9'=...,w =defineswithisacoordinate

expressedsystem [19-,-..

92)

as9

nota,

on (W):

denotedT (W)isdenotedP .. --iPn))

isaTTIN)

is

on

on

[d9indPibyPi,depend onisa

simply by 9.

coordinate

systemTV .

ongiven by.

the choicesymplectic formIT" &, w)

isPlot)Go

symplectic form

This doesn.

Pid9 :

function PHP
eThenUCQPETE (U)=.

dimension.
cotangent bundle

itsopen subset

i .

The&-Onpmanifold ofsmoothaonof coordinate system

TPQ.

Thus,

symplectic manifold.

and,The case of cotangent bundle
MT=*27& GLetUs
thew =,dindPi

right action
rightfield Xg

vectoronpg

via

action=g"*p.

G generated by 30 %. Thenon

T & generated
by 3

is

locallyas

X3

D[Pid9"

be the rector field

expressed

U191=For gt

9:(p9t)et3=

Pi(p9 + )and-DiV(9) Pit

p-Tg &,

for:=9 1997) Hence(X39")(p)·=T &8 G-·P==

49 +af9(19 )(zVj(9)=+= 0.

( )ag ) g*P(((ag)==+Sincea (i)* Pi(p) =g+/toPi(p9t)(t==0Lust) aP) d V(2))-:P(97 (i) agt)*-=-0:Us K&:Us (9)9j.

.

Thenixw
Thus< M(p) 3),.

+peTE

190 Vj(9)Psatisfies

momentd(P : Us"(9))=:P(Uz)-the condition (i) ofisfor.

: =,

which

Ie

Vj(9)dPi=&[M 3).

=-ix,Condition (ii) ?

.

mapPg(Ug199)) = PlUgig (9)+,Ilp(gU , (99)Uz(99) istotangent

theage*curveat+=0.

ThusgUg(95) is fangent to the curve aget gat to,=henceie.

isequal to Uggg+ (9) .

gUg(99).

SoYES,=theVg3g+ge+93g7(9).

condition (ii)isalso satisfied.

We conclude :Amomentmap

M : T"Q-< M(k 3),=-Y"P(Vz)

isprovided by.

Suppose & /G&Thenstructure

a&/G-isof a smooth manifold and=>&Cotangent bundle TY /G)the& It is enough to prove this,T & lyYLetFor 3 %,=:iTPU XT"Gas

M(p)=M10In(ppo)Ug(9 9).

=,·==[M(p) 3)0-10=,

be93)(Pu Pa)(0 93),,=

eachonolu

local trivialization

There.

symplectic quotient M(0)/G

theaG-bundle

principalasymplectic manifolds

as,

has

Pa

=UCQ/G

UX6a.

symplectic manifold.

themap.

andhence=Pa(s3)-with.

0ETPU X 6
&O-section

and

MI/6

=&T U.

An

example-Yang-Mills theory with gauge group G
1x Re is equivalent tod-dimensional spacetime IRP

Recall that theon=

Hamiltonian

thewithi,jel -ide,withsystem
azl ,,

EAials Ejb(y))--,=,variables

conjugate

Aia(* )

,Ejo(X)

having Poisson bracket

din G

dij das G(X-Y),

Hamiltonian

H(A E),(d x)Ei + t
Fij

whereandathephase:==diAj-0; Ai + (Ai Aj],=.

D : Ei=&(0 Ei:*bundle T A

one

naturaldirection+(Ai Ei])

=0,.

theory may be identified withof the space(AFR (I* y)/A()(We imposedThe fiberD E

space of thiscotangent*=constraint& (x)TheFij()*=+0boundary condition

corresponds totheasat( + 03.

spatial infinity )field #(x).

.

The group9(g /R":=Al (henceactson

For-E

Lie( e): =,E,isTM
thetoAle,&nothing

but theE

st+01x1 + 0as

T=as

undermoment

mapofthe reduced phase space

and,,(A)/g)all

isisomorphic:.

physical phase spacetheory,Al/g

the action+

theequivalently

invariant

A +A = gAg + g'dIg.

Rol of:cotangent bundle oforg :E(deA)-of the quantumstatesaree1as(x) +=regard thisweTA)) byE.

i .

,E(0)/gIf

onx]-(axG(x) ((x) -)dXE(x) DE(x)

=

Thus

G(g(x)-theareware,physicalwavefunctionals
functionals on

on

which

spatial gauge transformations:

E(AS)

Alternatively,

So

where
theorEg

9.

.
identity component of $thephysical state

condition

is

F(1P]

FLA]

Fge Do

=

Then,

.

equivalently
Je[(A)

thatis ,the=Fy-valued function E(X)

0Gauss law&F (A)=asTMAYgo)=.

is=physical phase spacethemay regard

weEi0)/

F(A]

=0.

,
constraint,