Legendre

LagrangianS( (19 9)d=,↓extremize

Hamiltonian[A B)+=,

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fixmg E(tH)(8) 0

H(P 2) Hamiltonian,fA=AB Aoe[A HY=,&↳
Operator

Path-integral-Transition(ti (i) +,==97,

19 21st=it 59,p)Time9(ti) 9evolution

itf ti Fl
#=e-:

=it

States 74 Vectors in 2,=,,(t) 9f)(09e)((t+ )(5 ]amplitudeU+,-,A reminder

L

Legendre transform:H ((9 9)+given,(9SolveH(p 9)+L( 9=& dLScan=: =also-,q"(p 9)=,

L(9 9 (p 9),,9 for p=,.

9 : giforPigiven,L(9 9)

RHS=:H(p 9)Solveat9)=P " (p 9)

==,

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L+ H
>
-

L= H

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-by extremizing

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Path-integral>-amplitudez(tf 2+i ti (i),,in=operator formalism(9):= - Ei19 :)time.....

tiEtf= (t) t : #-===

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#==NE

#=...
=
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=

[Pj(#19; 7 ====((pj(9j) (1&(E)-+then,+V(9j)= H(1 (j),0(E))(9; ]+=H(p 9j)O(Ez))H(EOC-EH(ESOL,z (t 9) : ti.

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109
=97,

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.

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,

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Absolutely convergent
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-

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well-behaved-

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Symmetry and Ward identity
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ie.

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.

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expression ofan92,ga,

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on

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=

(0(t)

=

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indepottThen ,Effici (0(s) =0=/.

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.

z(t+, Ef : 0, 1) 0 : ti 9.
.

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Correlation function of product of local observables
corresponds to the time ordered product of thecorresponding operators.

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inSuppose-classical mechanics(in Lagrangiansymmetry 99

89

a87(9 9)E(=,Sit.

85=This2E(tf)=8

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EU(9=9),

total derivative->

depend on time

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variational parameter

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theNoether charge

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.

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=

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S E D.
.

.

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692

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=

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=:

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=

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,

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=

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also

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Apply

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,

=

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Ward (G(89eS()U(t)=(03e S(9]( = GS(C) Or+80(to)>[0i, e=(03e S19]( = GS(C) OrJRTEQ=80(to)+ED(S)-EDSt).

z(tf E : 80(o) ; ti 9),,=Ez(t+, 9+;&(5)=(=+x,t (80()) =:Take the limit

Go.

Put E-1;-:

and to xto

5 5=R(t]) Octos Eicti)+st ( QH)

tox to---Q(5)( 0(80):0.

:o== (2 0],

Ward identity in quantum mechanics(in operator formalism)In classical mechanics, a continuous symmetry yields
a conserved charge (Noether charge).
After quantization, the Noether charge generates
the symmetry transformation.

The case oftranslation

time-theWeotheralso=== (H(p 9) ]
,

hand,

,

F defined by#.

- (f=t:= (tfc.

-t( 5]Cf,S(a70(t)=(-p =+.

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E (0(t) = (e=.

know(+ -=:-0Onsymmetry

0Comparison =>

(H ].

=,--H

=H(p 9),.
C-number+&

H is the operator corresponding to Hamiltonian(modulo a c-number shift).

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Apply89()

9-translation (not a symmetry in general).
=E(t)

=E+E
to ,+in0

theintegrand of=(8(09e==,.

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P(t=+Take theThe:part0

=:

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toxto,

toto to 9(t0)=(90p(9 , ]-=tovanishesp0)

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