non-perturbative methods in gauge theory

Heller, The path integral is a method of quantization h is equivalent to the operatorformalism.. The h based on path integrals has several advantages o er the operator formalism.. Onthe

Institute of and Experimental Ph w, Russia and

TheNielsBohrInstitute,Copenhagen,Denmark

intralineartensions

J Heller,

These notesarebasedon givenat:

1) AutonomaUniversityofMadrid,WinterSemester of1993;

2) LeipzigUniversity,Winter Semesterof1995;

3) wPh andT Institute,SpringSemesterof1995

Myintentionwastointro graduateandPh.D.studentstothe

meth-odsof temporaryquantum eld theory The term\non-perturbative"in

thetitle means literally \beyond the eof perturbation theory"

There-fore,itisassumedthatthereaderisfamiliarwithquantum aswell

as with the standard methods of perturbative expansion in quantum eld

theoryand,in withthetheoryofrenormalization

The purpose wasto makethe usefulforseniorpeople

(in-thoseworkingin theory),asasurveyofideas,

ter-minologyandmethods, hweredevelopedinquantum eldtheoryin the

seventiesandthebeginningoftheeighties Forthisreason,thesenotesdonot

goverydeeplyintodetails,sothepresentationissometimesabit

Correspondingly, thesub hare usually veredb modern

instringtheory, hasthetwo-dimensional eldtheories,arenot

hed Itisassumedthat ha willfollowthisone

Themainb dyof the notes dealswith gaugetheoriesand

large-N methods These two Chapters are b Chapter 1 h

is devoted to the method of path integrals The path-integral h is

looselyusedinquantum eldtheoryand InChapter1,

I shall pay most attention to asp of the path integrals, h are then

usedinthenexttwoChapters

In hChapter,Iwasgoingtobeas totheoriginalpapers,where

the involved methods were proposed, as possible The list of these papers

resp ely

1 R.P Feynman, An operator having ations in quantum

ele o Phys.Rev 84 (1951)108

2 K.G.Wilson,Con nement ofquarks,Phys.Rev D10 (1974)2445

3 G.'tHooft,Aplanardiagramtheoryforstronginter Phys

B72(1974)461

The werefollowed b seminars wheresomeproblems for deeper

studies had been solved ona kboard They are insertedin the text as

theproblems, h beomittedat rstreading Somemoreinformation

is also added as remarks after the main text Both of them tain some

relevant

usu-allygivenonlytoeither a rstpaper(orpapers)inaseriesorthose

tain-inga p presentation of thematerial With themodern

databaseatqspires(SLAC),alistofsubsequentpapers inmost

beretrievedb downloading ofthe rstpaper

Iwouldliketothankthestudentsfortheirattention, and

ques-tions IamindebtedtoMartinGurtlerforhishelpinpreparingthese

notes

Contents

1.1 Operator 2

1.1.1 Freepropagator 2

1.1.2 formulation 5

1.1.3 Path-orderingofoperators 10

1.1.4 Feynmandisentangling 12

1.1.5 oftheGaussianpathintegral 17

1.1.6 Transitionamplitudes 20

1.1.7 Propagatorsinexternal nite

Æ(d)

(0)will be inthenext

Note thatthein nitesimalversionthetransformation(1.3.7)is a

ular ofthemoregeneralone

h is an analog of the transformation (1.2.25) and leaves the measure

invariant The givenin Eq (1.3.18) isan illustrationof

this

Thegeneraltransformation(1.3.19)leads,whenappliedtothepath

inte-gralinEq.(1.3.1),tothe hwinger{Dysonequations

=Æ

More etransformationsof thetypeof (1.3.7), h are asso

atedwithsymmetriesofthe andresultin ed ts,

leadto some(less e) relationsbetween h are

Wardidentities Thisterminologygoes kto the ftieswhenaproper

re-lation between the 2-and 3-pointGreen was rst derived for the

gaugesymmetryin QED

Thesimplest Wardidentity, h isasso with the hiral

transfor-mation(1.3.7),reads



J5

(0)

i(x)



j(y)

m=0

= iÆ(d)

(x)

5 )

i(0)



j(y)

iÆ(d)

(y)D

i(x)



5

j(0)E

: (1.3.21)

Itis fromthewa Eq.(1.3.21)wasderived,thatitisalwayssatis edas

a ofthequantumequationsof motion(1.3.20)

Problem1.20 Derive Eq.(1.3.21) inthe operator formalismwhenthe a erages

aresubstitutedbythev exp valuesoftheT{pro

Solution UseEq.(1.3.14) h anextra iinMinkowski where

and holdsin the quantum inthe weak sense, i.e whenapplied to a

state) and equal-timean utation relations for and

j(x)

= ÆijÆ(d)

For

thisreasonthedimensionalregularizationisnot in of

the hiralanomaly

Remarkon gauge- xing

Note that we did not add a gauge- xing term to the (1.3.4) It is

harmless to do that the gauge- xing term does not tribute to the

variationofthe underthe hiraltransformation Moreover,allgauge

invariant quantities donot depend onthe gauge xing How to quantize a

gaugetheorywithoutaddingagauge- xingtermwillbeexplainedinthenext

Chapter

1.3.3 Chiral anomaly

Asisalreadymentioned,Eq.(1.3.18)involvesthe ty

Æ(d)

(0), oneneeds [Ver78,Fuj79℄ to regularizethe measure in

thepathintegralo er and



, thisterm fromthe hangeofthe

measureunderthe hiraltransformation

Let us expand the elds and



o er someset of the orthogonal basis

similarly tohowitisdoneinEq.(1.1.78):

i

(x) =

X

ni

n

i

n(x);

i

(x) =

X

n

i

n

iy

n(x); (1.3.25)

where there is no summation o er the spinor index i Here

i

nand i

nare

idi

n1

Y

m=1Y

jj

m

Theideaofregularizingthemeasureisto ourselvestoalargebut

nite numberofthe basis This isanalogous to the

of 1.1.4 Wethereforede netheregularizedmeasure as

(D



)

R(D )

R

=M

Y

n=1Y

idi

nM

Y

m=1Y

jj

R

= (D

0

)

R(D 0

)

Rdet



dd

x

ky

n(x)e

k j

5

j

m(x)



; (1.3.28)

wherethedeterminantis botho erthenandm and o erthespinor

k andj Thisistheregularizedanalogofthenonregularized

expres-sion(1.3.17)

Usingtheorthogonalityofthebasis

Z

dd

xj

n(x)

i

m(x) = Æ

nmÆij

x

ky

n(x)e2i( x)

k j

5

j

m(x)

dd

xy

n(x)(x)

5

n(x); (1.3.30)

wherethespinor are in theusualwa

Itis easyto seehowthisformula versEq.(1.3.18)

1

X

i

n(x)

j

n(y) = Æ

(d)

(x y)Æ

ij

(1.3.31)

inthenonregularized dueto the ofthebasis

In the regularized the sum o er n on the LHS of Eq (1.3.31) is

b M from abo esothat theRHS isnolongerequaltothe

n(x)

j

n(y) = R

ij

withtheRHSbeingthematrixelementofsomeregularizingoperatorR

It be hosenin manyways Weshallworkwithseveralforms:

a2

b

r2

1

1 a2

b

r2

1

1+ab

 Theparameteraistheultraviolet The

disappearsasa!0whenEq.(1.2.57)holds

These regularizations (1.3.33) { (1.3.35) are non-perturbative, and

pre-servegaugeinv theyare fromthe variant

x(x)

J5



= 2iSp

5R

= 2iZ

dd

x(x)sp

5

R(x;x): (1.3.36)

It is worthnotingthat an extra R in Eq (1.3.36) is a of a

moregeneralformula

(y))jl

r1

jyiappearsontheRHS.It

Remarkon regularizationof the measure

The regularization of the measure in the path integral b Eq (1.3.27) is

equivalent to the point-splitting pro where the in the

utatortermissmeared to Eq.(1.2.55)

Toshow this, letus note that thevariationalderivative be

approxi-matedforthe nitenumberofbasis b

Æ

Æ j

R(y)

=M

X

n=1

j

n(y)X

k



k

The sumo erk is in order forthe regularizedvariational

derivativetofeelvariationsofallthespinor onentsof

nwhenthevari-

ationisnotdiagonalin thespinor

Whenappliedto

i

R(x) =

M

X

n=1i

n

i

n

ityields

Æ i

R(x)

Æ j

R(y)

=M

X

n=1

i

n(x)

j

n(y) = R

ij

(x;y); (1.3.40)

or,equivalently,

Æij

Æ(d)

(x y)

Reg

=) Rij

h isa analogofEq.(1.2.55)

Thus, we that the regularization of the measure in the path

integralisequivalentto smearingthe in utatorterms

Remarkon regularized hwinger{Dysonequations

to 1.2.5, thepro from the previousparagraph

re-sultsinthefollowingregularized hwinger{Dysonequations

=Z

dd

dd

yR (x;y)

Æ

Æ (y)

Theseequations areunderstoo again intheweaksense,i.e under thesign

ofa eragingo er



and and obviouslyrepro Eq.(1.3.20)asa!0

Problem1.21 Derive Eq.(1.3.36)usingthe regularized hwinger{Dyson

equa-tion(1.3.42)

Solution The is similarto that ofProblem1.19 forthe

addi-tionalterms hareduetotheRHSofEq.(1.3.42) Onegetsfor m=0

yÆ

Æ (y)

R (x;y

5 (x) i



5Z

dd

Inordertoderivean expressionforthe hiralanomaly,weshould

theRHSofEq.(1.3.36)forsome oftheregularizingoperatorR

Letus hooseRgivenb Eq.(1.3.34) Theoperator

b

r2

in thedenominator

betransformedas

b

r2

2

+1



2ie

0(:::)R

tothe eldstrength

d4

x(x)

e2

16

2F

isthedual ... standard methods of perturbative expansion in quantum eld

theoryand ,in withthetheoryofrenormalization

The purpose wasto makethe usefulforseniorpeople

(in- thoseworkingin theory) ,asasurveyofideas,... theory) ,asasurveyofideas,

ter-minologyandmethods, hweredevelopedinquantumeldtheoryin the

seventiesandthebeginningoftheeighties Forthisreason,thesenotesdonot

goverydeeplyintodetails,sothepresentationissometimesabit... krotationfromMinkowskito

(indi-b the arrows)fora) timeand b)energy Thedotsrepresent

sin-gularitiesofafreepropagatorina) ordinateandb)momentum

The toursofintegrationinMinkowski