Computation of transition amplitude and partition functions
Let us compute&operator

ECt 4 iti 4: ),.

path-integral

functionspartition&inboth.

Operator
We choosethesymmetric ordering

The results forand willotherorderings

mentioned

be,,4.)

=/Fle
-

=

=

(0)(1

((((++==:"

tf-ti=4, 4) e44)(e" -+i)*-

giftf-til e+(4+ 4),

after that=:- +)-+

1e+

eiltf-ti ~14+4 )10)::i(tf-til(1 1 -5) W 4i)

2=44 /10)(1 +:-&,14:

eist-tie 14=filtf-tile-,easily foundbecan(4 4).

-z(t 4+ :tiFI:.

(-Tra=Wemayecompute

F&-

(1 s-

=resultwouldinzitf 4, it 4 ).

,:Tre(I) e*=e-orderings
-

jwY4~

Y s4Y
I+=z (4 4)=(y+ y]-,

shift of-,the exponent

giftf-tilsw et-

SWT-is'=2( )tw-,(s- t)tin

theteiltfutil

=

2+2-

ae- --

operator(1 s)w4+ Y=(E)=+function-ofe

symmetry-twisted partition

Tre(I echange--*A=- E=also)

e+FTrje()-)-overall factors~-44:(1I e WT+ix).

Path-integral
ZIt iti 4 ),=:=/ot↑(E) 4

4=Y
where4xYa=

teeaYelt)

Yalt)+4(ti)+,

YY

fee=i: eI,,,

are-=44-w44)

e=

Write

*441t + / + / :

=

*O4i

↑It

=z(t)
solution

0=,0,ofEOM

obeying boundary condition

iwIt-tileiwit-tf

then--i4, (4x(t+) Y(t+))
0,+-I - wIFat) (ta 4))
Fas Init+Erwt

e-(-iTa-w -i -w)Yne⑧,

=--i (4 first-ti +Y(t)::(+ 3) I++4, 4 ::ZStx4r==:/Faltis(d+ (- :in -ti)+sti)ES-w4)+ )-y- w)

t : 4: )e4f4:start!
↳note

eiwstf-tis

4 eicoctf-ti)
e4F(tf-t ),::Constraint

onFCt)

Ext : E(Y)
is:w

If

=--I unitaryobeys it:***
EdF'Y'ZCt Fit 4') e
E(E,,andEts th East = Stast,,,Recall(E /En).

DThenThese,EclY).

1=F (ts -tr) F(t-t1)

i At=definitionbyberegardedof

&F&4

cana

shift ofoverallthe result

theaF(t - t ).

real AeiAltf-tie

=,

Afor

e=Zlt . ;ti 4 : )in**require

F(t)

This

/EdEd4 E CY)"et/Fits

=⑪..

=.

by

shift of

as

an

14:.

eiwstf-ti)()

ambiguity in theIt can also
constantaenergyabsorbed

bewhich result.

(#) match with the Operator result

ambiguity

corresponds

tointhedefinitionofOF & 4operator ordering ambiguity.

whereFor partition function

Li44=Le->-=

first,

Wickdorotation

wF4-(ii -wF4)
at-= =eid4R)+mode expansion

Se=dI

S

4=

2n= x

Letusputw)4Le

4(T+T),=

2 <4 (t)-eillantares=I=I4e: <an -23 /,2niFreIi=+e0584↑(t T)ya=

Let us compute

Za(S1):(2πm++c//Tz) Preiln-*- w) YuEn) :*+&4 &4 :

=in=x

dEnd4

anduse3-fen reg.zalst)=

Enlist whtu-(ne Edten(i - w)+-=

I(I dneeB

⑰A

=

E.

-

3 function regularization (See** xx/P(1 y)=*below) :+a) - E)T)--↑(1 z)4(1 z)-=+)=

24 (1 z)P()-=-zism(E)

=z

*3

-function regularization

Generalizedwhich

continuation

310 a)
,

P(a)

Here

CA

=t- a

property3'( as:

,

=xn+Y

=

we

=0,

withlog2t-P(a +1)=

aP(a)M(asP(1 a),,=exp(Eexp(-(log(n+y)
3 10 /+y),-=AE-log(n a) (n a ..
+find=exp)

.

Si,

=Sat.

53( a)=,

Using these:

analytic

and haslogP(a)=Gamma functionisconvenient

Noteregularis310 a),,for Re(s)>absolutely convergent

is

E. (n+as3(S a) =function

ZetaP11 +y) +---logx))
logX 310 1)·.

log2x + logX)+GC&11-ir)

=2·

at

int =

simple zero=iNT= han-d(nex)atmisin(E)=>-=( 9:za(St)This)/2isin(E)ei+i-I)/zisin)-=matchTre(-)*

A B

with-the*=-=ize

Operator

resultEforF*withethesymmetric ordering

F E (4 4)

8 /(4 4]==,:+,,

3- function regularization
corresponds

tothe

symmetric ordering.

.

Remark(Op)The

cases"I"

twoTre(I) e
*

8

S=

=

and

Thus&,+Indeed,

.

&+ E

for S'=0

S din

ei=()

particular

in

other

ei=

and(1I e WT+ix)e=with eachconsistentis

SWT-is'*-resultthe operatorofF

for &=,

E Ite

orE,

mod 25

does=(-1) e·-·1That&isis+ix+two

also

I -ethe+ acases

forT

wT + i-->ethe,d-This+WT

S'= 0inwhencase&E

S'

twoagenerals'cases

are,

also

=swapped

()

is

*

by

swapped bye**whichTFor

are-ane=(1) *

integer

even

its,

and the& + & T+.

.

.

.

(P I )On the.

.

the

hand

other,

path-integral

ZalS1)==/05o+↑(t T)+=abitunder&underastrange+&Related problem
over+

Howeverbut,45,=" <4(t)i+appears

resultthe=isbeperiodic

periodic

onlyto.

which

supposedis4(T T)+,

Ea(S! ) for &:4(t) F(i)and thisei64()

The definition.

+ 2+eis

-is=

Le

at-

tois

=

IF

antiperiodic

beequaltoistheExa(S1)-E****e)
FileTree)differ byphases

=.

e"Yz

Y+pathintgral

under -- TrTTrele-EB)

the results=resultand.

,At first sightit,

3-function

ACHS

appears

RHS

changes

However

Gto-

beits

whateveratas

periodic

signmustOnethe set ofotheris+underchange

itsparticular& +x+ 25

use

sign

particularly wrong

under& +x+2tchanges its signbackto

itself.

tomodify the definition ofis& F &4

filmttsddFndtn

in.

3-function regularization

measurewhich results

but

about

path-integral=,

the Co-product,CT-X(exin-d)'s goes

way-out of the problem

thethein& + &+ hi

underregularization you

2 ,a - &NothingA

problem is-zisin(t)=andthethatregularization=

Asappears

aswithme,.

za(S1)ThisisIt isetimx _=asperiodic

underZolSt)
Zi(S! )

depends

However,certain

aand=IS=,-m.

indeedE"cheap trick" and it looks strange thatanon

it turns

study anomaly& ->4+ 25 ;STree F

=

sense

-

operator result forTre(F=This looks likeitthethe same+ilmtd.

.

additional parameter

outWe

tobewillame 2"reasonable

seethatwhen.

-solution
weina

systematically