The Erwin Schrodinger International Boltzmanngasse

ESI Institute for Mathematical Physics A Wien Austria

Cohomological Analysis

of GaugedFixed Gauge Theories

Glenn Barnich

Marc Henneaux

Tobias Hurth

Kostas Skenderis

Vienna Preprint ESI Octob er

Supp orted by Federal Ministry of Science and Transp ort Austria

Available via httpwwwesiacat

ESI

ULBTH

FTUV

CERNTH

MPIPhT

SPIN

hepthyymmnnn

Cohomological analysis

of gaugedxed gauge theories

ab bc de

Glenn Barnich Marc Henneaux Tobias Hurth

f

and Kostas Skenderis

a

Departament de Fsica Teorica Universitat de Valencia

E Burjassot Valencia Spain

b

Physique Theorique et Mathematique Universite Libre de Bruxelles

Campus Plaine CP B Belgium

c

Centro de Estudios Cientcos de Santiago

Casilla Santiago Chile

d

CERN Theory Division

CH Geneva Switzerland

e

MaxPlanckInstitute for Physics WernerHeisenb ergInstitute

Fohringer Ring D Munich Germany

f

Spinoza Insitute University of Utrecht Leuvenlaan

CE Utrecht The Netherlands

Abstract

The relation b etween the gaugeinvariant lo cal BRST cohomology involving the

antields and the gaugexed BRST cohomology is claried It is shown in particular

that the co cycle conditions b ecome equivalent once it is imp osed on the gaugexed

side that the BRST co cycles should yield deformations that preserve the nilp otency

of the gaugexed BRST dierential This shows that the restrictions imp osed

on lo cal counterterms by the Quantum No ether condition in the EpsteinGlaser

construction of gauge theories are equivalent to the restrictions imp osed by BRST

invariance on lo cal counterterms in the standard Lagrangian approach

gbarnichulbacbe henneauxulbacbe tobiashurthcernch KSkenderisphysuunl

Intro duction

In the BRST approach to p erturbative gauge theories the p ossible counterterms

are restricted by WardSlavnovTaylor identities which have a cohomological

interpretation If one follows the pathintegral approach and takes into account the renor

malization of the BRST symmetrya la ZinnJustin by intro ducing sources coupled to

the BRST variation of the elds and the ghosts it may b e shown that the

counterterms must full the BRST invariance condition

sA

where s is the BRST dierential acting in the space of elds ghosts and asso ciated sources

antields The counterterms are lo cal so A in is given by the integral of a

lo cal nform a in terms of which the BRST invariance condition b ecomes

sa db

for some n form b Following an initial investigation by Joglekar and Lee the

general solution of for YangMills gauge mo dels has b een determined in where

it was shown that up to trivial terms of the form sc de the counterterm a in is

equal to a strictly gaugeinvariant op erator plus ChernSimons terms in o dd spacetime

dimensions in the absence of U factors for which there are further solutions also

dealt with in This guarantees renormalizability of the theory in the mo dern sense

in any numb er of spacetime dimensions and in the standard p owercounting sense in

dimensions

If one follows instead the op erator formalism and the Quantum No ether metho d based

on the gaugexed BRST formulation one nds that the counterterms are con

strained by the condition

g

a db

g

where is the gaugexed BRST dierential acting on the elds and where b oth a and

b involve only the elds no antield The symb ol means equal when the gaugexed

equations of motion hold

The question then arises as to whether and are equivalent It may b e

shown that the antield and gaugexed lo cal cohomologies are equivalent so that

0 g 0

any solution a of sa denes a solution a of a and vice versa This is not true

however for the cohomologies mo dulo d In particular there are solutions of

that have no analogue in the antield cohomology and which therefore do not corresp ond

to an integrated gaugeinvariant op erator An example is given by the CurciFerrari mass

term

a a

A A C C

a

a

which is a solution of in the gauge where the equation of motion for the auxiliary b

b c

eld is b A C f C but which do es not dene an integrated gaugeinvariant

a b

a ac

op erator The prop erties of have b een studied in Thus

and are in general not equivalent

If however the co cycle condition is supplemented by the requirement of nilp o

tency of the deformed BRST dierential which is required if we want the theory to

b e unitary the CurciFerrari mass term is excluded It is the purp ose of this letter to

show that quite generally the gaugedxed co cycle condition supplemented by the

requirement that the deformation generated by the p ermissible counterterms should pre

serve onshell nilp otency of the BRST is equivalent to the antield co cycle

condition which controls the counterterms in the ZinnJustin approach

This letter is organized as follows In the next section we recall some salient prop erties

of the gaugexed action The equivalence of the two co cycle conditions is shown in section

while a discussion of trivial solutions is presented in section In section we review

the analysis of counterterms in the Quantum No ether metho d and show how nilp otency of

the deformed BRST dierential arises in that context Finally in an app endix we present

an analysis of the relation b etween the antield and the weak gaugedxed cohomology

using metho ds of homological algebra

Gaugexed action



The starting p oint is the solution S of the master equation

L L R R

S S S S

S S

 

A A

A A

We use DeWitts condensed notations The solution S is a lo cal functional as are all func

A

tionals without free indices o ccurring b elow The elds include the original elds

the ghosts as well as the auxiliary elds and the antighosts of the nonminimal sector

We assume that the canonical transformation necessary for gaugexing has already b een

p erformed so that the gaugexed action is simply obtained by setting the antields equal

g  g

to zero S S The gaugexed BRST dierential is dened by

R

S

g A



j



A

R L

where right and left derivatives are dened by F F z z z F z

We use the conventions of but the derivations are taken to act from the left so

g

sF S F etc The transformation generated by leaves the gaugexed action

g

invariant b ecause of the master equation As a result the functional derivatives of S

transform into themselves

R R g L R g

S S S

g



A B A

B

1

We keep here the auxiliary beld but similar considerations can b e made if one eliminates the

auxiliary elds since the gaugexed BRST cohomological groups are invariant under such an elimination



here and b elow it is understo o d that is set equal to zero after the second derivatives

have b een computed The gaugexed BRST dierential is weakly nilp otent

R g R L

S S

g A

 

B

B A

Both and are direct consequences of the denition and the master equa

tion

The BRST dierential in the space of the elds and the antields is dened by

sF S F

 g A g A

for any F It is related to as s antielddep endent terms or which

g A A



is the same s j It is strictly nilp otent s It will b e useful in the

 A

sequel to give a sp ecial name to the terms linear in in the expansion of s

A g A g A 

s O

with

g A A

S j 

linear in

L R

S



B

 

B A

The action of s on the antields can also b e expanded in p owers of the antields One

 g g  g  

O where is the Koszul dierential asso ciated with the has s

A A A

gaugexed stationary surface

R g

S

 g 



j S

A A

A

and where

R L

S

g   



S j

A A B

linear in 

A

B

g g g g g g g g g A g A

One easily veries the relations

g g g

from the denitions of the derivations and These relations are actually the rst

ones to arise in the expansion of s in p owers of the antields

The canonical transformation appropriate to gaugexing do es not mo dify the coho

mology of s either in the space of lo cal functions or in the space of lo cal functionals

b ecause it is just a change of variables So in the case of YangMills theory the coho

mology group H s F of the BRST dierential in the space F of lo cal functionals is still

given by the analysis of In H s F the sup erscript is the total ghost numb er

g g g

Note however that the expansion s is not the standard expansion

arising prior to gaugexing since the degree involved here is the total antield numb er

that gives equal weight to each antield irresp ective of its antighost numb er This is

why it is the Koszul resolution asso ciated with the gaugexed stationary surface that

arises in the present analysis and not the KoszulTate resolution asso ciated with the

gaugeinvariant equations of motion

Since the equations of motion following from the gaugexed action have no gauge

invariance by assumption one may invoke the general results of to assert

that

g

H F k

k

where k is the total antield numb er used in the ab ove expansions In words any lo cal

R

g g

functional A F A a that solves A a db and is at least quadratic

g g

in the antields has the form A C a c dm

g

We have of course H L for k in the space L of lo cal functions but

k

we shall need the version valid for lo cal functionals b elow This is a direct consequence

of Theorem or of which states that there can b e no nontrivial higher

order conservation laws for an action having no gauge symmetries This theorem is

also known as the Vinogradov twoline theorem Because higherorder conservation

g g

laws and elements of H F also denoted H jd are in bijection the prop erty

k k

g

follows In general however the homological group H F do es not vanish

g

even though H L in the space of lo cal functions and is related to the global

symmetries of the gaugexed action

Reconstruction Theorem

We now have all the required to ols to show that a lo cal counterterm of the gaugexed

formalism that preserves nilp otency denes a lo cal counterterm of the antield ZinnJustin

approach That is the condition

g

A

R

for the lo cal functional A a which implies for the integrand a together

g

with the fact that the asso ciated deformed BRST symmetry e should remain weakly

nilpotent for the new equations of motion to O e in the deformation parameter e

g 0

e O e



determines a lo cal functional co cycle A of the antield cohomology

sA

0 R

In the symb ol means equal when the deformed equations of motion S

A

eA hold The relationship b etween A and A is



A A A A O

where A resp ectively A is linear resp ectively quadratic in the antields

The ab ove derivation is the deformation of the BRSTsymmetry and is related to

the deformation A of the action as follows When one adds eA to the gaugexed action

g g g g

S S eA one mo dies the gaugexed BRST symmetry as e

g g

in such a way that eS eA O e The existence of is guaranteed by

the co cycle condition which we can rewrite as

g g

A A

A A  A g

A

for some lo cal functional A linear in the antields We have A

A

A

where g is the Grassman parity of

A

 

As shows the relationship b etween A and A is that A starts like

A to zeroth order in the antields Thus the question is whether any lo cal functional

A that fulls b oth and can b e completed by terms of higher orders in the

antields to yield a lo cal functional solution of



The converse statement namely that any lo cal functional A solution of

denes when setting the antields equal to zero a co cycle of the weak cohomology

g

fullling is rather obvious Indeed if sA then A term indep endent of

the antields in sA Furthermore at the next order

g g g

A A A

a relation that is seen to b e equivalent to by rephrasing the condition in terms

of A and A On the one hand direct calculations yield

L R

S

g  B  g A

A

A A



B

A

R L R

S A

g 

A

A

 

B

B A

g A 0

On the other hand if one replaces the weak equality by a strong equality in e

O e one gets in view of

R g R L R

S A S

g A AB

e e e O e

 

B B

B A

AB

for some Thus b ecomes to order e

L R R R L

S A S

g A B

  

B B

A B A

g g

which shows that is indeed equivalent to the statement that A A vanishes

weakly or which is the same

Accordingly to each counterterm of the antield ZinnJustin approach corresp onds a

counterterm of the BRSTNo ether metho d

Conversely given a solution of or which also fulls the question

is whether one can construct a lo cal functional A that starts like A A and is BRST

invariant That or by itself do es not guarantee the existence of A is illustrated

by the CurciFerrari mass term and has b een explained in

The problem arises b ecause the p erturbative construction yielding successively A

A etc given the initial data A and A along the lines of homological p erturbation

theory applied to the antield formalism can b e obstructed in the space of

g

lo cal functionals The obstructions are in the homological groups H F also denoted

k

g

by H jd The p oint is that the equations dening the higherorder terms A A etc

k

take the form

g

A B

k k

where the lo cal functional B involves only the lowerorder terms A i k and can b e

k i

g g

shown to b e closed To infer that B is exact one needs either H F or

k k

g

if H F do es not vanish additional information guaranteeing that B is in the

k k

zero class

g

As we recalled ab ove H F for j Thus the only obstructions may arise

j

for k ie for A If it can b e proven that A exists there cannot b e any further

obstruction at the next orders and A also exists The strategy of the construction of A

from A and A consists then in showing that one avoids the obstruction for A It is

here that the condition is necessary

The equation for A is actually with

g g

B A A

g

We must show that B is exact ie that it vanishes weakly But this is guaranteed

b ecause and have b een shown to b e equivalent so that implies or

Therefore the obstruction for A is avoided as announced

One may understand the equivalence b etween and more directly in terms

of the master equation itself As is known the elements of H s F can b e viewed

0

as consistent rstorder deformations of the master equation S S S eA S S

0 0

S S O e As we have indicated given A the obstruction to the construction

of A can only o ccur for A ie we must verify that the term is zero But this term is

the term linear in the antields in the master equation So the absence of obstruction is

0 0 0 0 A





equivalent to the statement that S S j vanishes or S S j

linear in

This is precisely the statement that the deformed BRST symmetry remains nilp otent as

the Jacobi identity for the antibrackets easily shows

Trivial Solutions

The map b etween the antield cohomology and the gaugedxed cohomology fails to b e

surjective since only classes with representatives fullling the extra condition are

in the image of the map The map fails also to b e injective b ecause there are nontrivial

co cycles of the antield cohomology that are mapp ed on trivial co cycles of the gauged

xed cohomology This is b est seen on a simple example Consider

with a neutral scalar eld and imp ose the gauge condition A through the

equation of motion for the auxiliary beld where is a constant with dimension L

R

With that gauge choice the nontrivial co cycle d x of the gaugeinvariant cohomology

b ecomes trivial in the weak gaugexed cohomology since one has sC A Similar

considerations would apply to any function f in the gauge A f Although

we will not provide a precise argument we note that these mo dd cob oundaries of the

gaugexed cohomology which are present in p eculiar gauges are not exp ected to b e

physically trivial The reason is that correlation functions of gaugeinvariant op erators

do not change in dierent gauges for a pro of within the EG framework see

Note that the gaugexed action has a nontrivial global symmetry acting on the un

physical variables namely the shift C C where is a constant Grassmann o dd

R



parameter corresp onding to the cohomology class d x C This phenomenon is precisely

related in the app endix to the noninjectivity of the ab ove map

Counterterms in the Quantum No ether metho d

We show in this section how the nilp otency condition arises in the Quantum No ether

metho d This metho d is a general metho d for constructing theories with global

symmetries using the EpsteinGlaser EG approach to quantum eld theory In this

approach which was intro duced by Bogoliub ov and Shirkov and develop ed by Epstein

and Glaser the p erturbative Smatrix is directly constructed in the Fo ck space

of asymptotic elds by imp osing causality and Poincare invariance The metho d can b e

regarded as an inverse of the cutting rules one builds np oint functions by appropriately

gluing together mp oint functions m n Moreover this metho d directly yields a

nite p erturbation theory one avoids UV innities altogether by prop er treatment of

np oint functions as op eratorvalued distributions The coupling constants of the theory

e are replaced by temp ered test functions g x ie smo oth functions rapidly decreasing

at innity which switch on the interactions The iterative construction of the Smatrix

starts by giving a numb er of free elds satisfying gaugedxed elds equation so that

there are propagators and the rst term T in the p erturbative expansion of the S

matrix Ultimately one is interested in the theory in which g x b ecomes again constant

g x e This is the socalled adiabatic limit We use the convention to still keep e

explicit in which case the adiabatic limit is g x We work b efore the adiabatic limit

is taken as the latter do es not always exist b ecause of physical infrared singularities

Causality and Poincare invariance completely x the Smatrix up to lo cal terms The

remaining lo cal ambiguity is further constrained by symmetries It is the purp ose of

our analysis to determine the precise restrictions imp osed on these lo cal terms by Ward

identities At tree level the lo cal terms are equal to the Lagrangian of the conventional

approach but new lo cal terms may b e intro duced at each order in p erturbation theory

The lo cal terms at lo op level corresp ond to the counterterms in the Lagrangian approach

although their role is not to subtract innities as the p erturbative expansion is already

nite If the form of these lo cal terms remains the same to all orders in p erturbation

theory then the theory is renormalizable

The Quantum No ether metho d consists of adding a coupling to the No ether current

j that generates the asymptotic and hence linear symmetry in the theory and then

requiring that this current b e conserved inside correlation functions There are a numb er

of equivalent ways to present this condition Here we follow where the

condition was formulated in terms of the interacting No ether current The Ward identity

formula in contains terms that vanish in the naive adiabatic limit g x

Their explicit form which can b e found in is not imp ortant for the present analysis

Here we will schematically denote them by g j Due to these terms the interacting

BRST charge is not conserved b efore the adiabatic limit is taken For a discussion of

the implications of this fact and also of other diculties encountered when attempting

to construct the interacting BRST charge we refer to We note however that

considerations involving only currents are sucient to derive all consequences of nonlinear

symmetries for timeordered pro ducts The Quantum No ether condition reads

x

T j xT x T x g j

n

Working out the consequences of this condition to all orders one recovers the nonlinear

structure in a manner similar to the way the No ether metho d works in classical eld

theory Further consistency requirements on the theory follow by considering

multicurrent correlation functions In particular the twocurrent equation is

x

xj y T x T x g j T j

n

where again we have only schematically included terms that vanish in the naive adiabatic

limit The explicit form of these terms as well as an allorder analysis of will b e

presented in

We are interested in gauge theories In this case the relevant symmetry is BRST

symmetry We now present the analysis of for this case to rst nontrivial

order This is sucient to connect with the analysis of the preceding sections Equation

at rst order yields the following condition on L hiT

g A B

L L e K

AB

R

A g

where denotes collectively all the elds j generates the asymptotic trans

A A

formation rules is dened by eq It was shown in that is the

A

nextorder symmetry transformation rule L is some lo cal function of and its rst

A B

derivative and K are the freeeld equations

AB

To work out the consequences of condition we rst note that since j is the

gaugedxed BRST current it satises

A

g A

j T J K

AB

A

where T is antisymmetric in and J may contain derivatives acting on the free

eld equations Equation guarantees that is satised at n ie no T

involved At n one nds the following condition

L

A A

g B

j j T J K J

AB

A

A

B A

for some T and J also p ossibly containing derivatives acting on K L

AB

A

is the Euler derivative of L and if J contained derivatives in they now act on

A

L j arises as a lo cal normalization term of the correlation function T j x T x

It was shown in that it is the No ether current that generates the symmetry transfor

A

mation rules Combining with we obtain

A A

g A

ej ej T eT J eJ K O e

AB

A

where K are the eld equations that follow from the Lagrangian L eL where L

AB

generates the free eld equations

Conditions and are equivalent to conditions and we analysed in

section

Conclusions

In this letter we have shown that the restrictions imp osed on counterterms by the Quan

tum No ether condition in the EpsteinGlaser construction of gauge theories are equivalent

to those imp osed in the ZinnJustin antield approach to the renormalization of gauge

theories The crucial requirement that guarantees the equivalence of the restrictions on

the counterterms co cycle conditions is the nilp otency of the deformed BRST gener

ator We have also analysed how this requirement arises in the EG approach Similar

considerations apply to anomalies This will b e discussed elsewhere

Acknowledgements

The authors acknowledge the hospitality of the Erwin Schrodinger International Institute

for Mathematical Physics in Vienna where this collab oration has b een started This work

has b een partly supp orted by the Actions de Recherche Concertees of the Direction

de la Recherche Scientique Communaute Francaise de Belgique by I ISN Belgium

convention by Proyectos FONDECYT and Chile TH

was supp orted by the DOE under grant No DEFGER during a visit of the

theory group at CALTECH where part of this work was done KS is supp orted by the

Netherlands Organization for Scientic Research NWO

App endix Antield canonical versus weak gauge

xed BRST cohomology

In this app endix the general relation in the space of lo cal functionals F b etween the an

tield BRST cohomology computed b efore gaugexing and the weak gaugexed version

is analysed by using standard to ols from homological algebra

As mentioned in section the canonical transformation used for gaugexing do es

not mo dify the antield BRST cohomology and we assume that this transformation has

b een done The complete BRST dierential s in canonical form then diers from that

in gaugexed form only in the grading used for the expansion called generically reso

lution degree b elow The grading asso ciated to the canonical form consists in assigning

antighost numb er to the antifelds of the original elds to the antields of the ghosts

to the antields of the ghosts for ghosts etc while in the gaugexed case the grading

consists in assigning antield numb er to all the antields

P

0 0 0

In b oth cases we have an expansion of the form s s in the bigraded

k 

k

g

space V V with g Z the ghost numb er and k N the resolution degree The

k g

k

0 0 0

ghost numb er of s is the resolution degree of s are resp ectively k

k

Let V b e the space containing only terms of resolution degree larger than n A

k n

V if the expansion of A according to the resolution degree is A A A In

k n n n

particular V V

k 

g

For n consider the spaces H s V dened by the co cycle condition sA

k n n

A and the cob oundary condition A A sB B

n n n n n

0 g g

In particular B Consider the maps i H s V H s V dened

n n k n k n

by i A A A A They are well dened b ecause they map

n n n n n

co cycles to co cycles and cob oundaries to cob oundaries Note that the dierence b etween

g

H s V and im i is the cob oundary condition an element A A A

k n n n n

g

with sA is trivial in im i H s V if A sB with B B B

n k n n n

g 0 0

For n consider the spaces H H V The co cycle condition for an element

n

g 0 0 0 0 0

A H H V is A A A for some A and the cob oundary

n n n n n

n

0 0 0 g

condition is A B B with B Consider the maps H s V

n n n n n k n

g 0 0

H H V dened by A A A

n n n n

n

0 0 g g

H V H s V dened by Consider nally the maps m H

k n n

n

m A sA A It is straightforward to check that the maps m are well

n n n n n

dened

We are now in a p osition to prove the decomp osition

g

H s V ker m im i A

k n n n

g

The pro of follows from the isomorphism as real vector spaces H s V im

k n n

ker and by showing that ker im i and im ker m From A it then

n n n n n

follows that

g g

H s V ker m i H s V

k 

g

ker m i ker m i H s V

k 

M

i i ker m A ker m

n n

n

g

Note that the isomorphism H s V im ker used in the pro of is noncanonical

k n n n

in the sense that it involves a choice of supplementary subspace to ker

n

Discussion If V is the space of lo cal functions or of horizontal forms we have

0

H V for n and this b oth in the canonical and the gaugexed form It follows

n

g 0 0 g

that H H V and thus ker m for n Since H s V it also

n k 

n

g g

0 0 g 0 0

follows that m and ker m H H V so that H s V H H V

This result has b een deduced in

If V is the space of lo cal functionals F for the canonical form of the BRST dierential

with the cohomologically trivial pairs of the nonminimal sector eliminated there are

no elds with negative pure ghost numb ers This implies that the antield numb er must

b e larger than or equal to max g K Furthermore if k K the presence of

g g

the ghosts implies that H F This implies for g that H F for

k k

g

g

k hence ker m for n and m Again we get H s F H H F

n

g

For g the only nonvanishing cohomology group is H F This implies

g

g

H H F for n g so that ker m for n g Furthermore

n

n

g

g g

H F H s F so that m Hence H s F i i H

k g g g

g

g g g

Finally H H jd H jd which follows from H F and i

g

g g g

i at ghost numb er g since there are no terms with antield numb er less than

g

g

g so that H s F H F which is the result obtained in

g

As already stated in section in the space of lo cal functionals for the gaugexed form

g g

g g

F for k with H F characterizing the nontrivial global symmetries H

k

of the gaugexed action and their asso ciated No ether currents for the classical elds

the ghost elds and the elds of the gaugexing sector We thus have ker m ker m

and m implying that

g g

g g g g g

H s F ker m H H F i H H F A

g

It follows that the canonical antield BRST cohomology H s F is isomorphic to the

g

g g

direct sum of a subset of the weak gaugexed BRST cohomology H H F

and of a subset of the nontrivial global symmetries of the gaugexed action Since

g

g

H s F H H F the condition that a ker m b ecomes s A A

k 

B B with B This is precisely condition and thus equivalent to

References

C Becchi A Rouet and R Stora Renormalization of the ab elian HiggsKibble

mo del Commun Math Phys Renormalization of gauge theories

Ann Phys NY

IV Tyutin Gauge invariance in eld theory and statistical mechanics Leb edev

preprint FIAN n

JC Ward An identity in Quantum Electro dynamics Phys Rev

AA Slavnov Ward identities in gauge theories Theor Math Phys

JC Taylor Ward identities and charge renormalization of the YangMills eld

Nucl Phys B

J ZinnJustin Renormalisation of gauge theories in Trends in elementary particle

theory Lecture notes in Physics n Springer Berlin Quantum Field Theory

rd

and Critical Phenomena Edition Clarendon Press Oxford

H Klub ergStern and JB Zub er Ward identities and some clues to the renormal

ization of gauge invariant op erators Phys Rev D Renormalization

of nonab elian gauge theories in a background eld gauge Green functions Phys

Rev D Renormalization of nonab elian gauge theories in a back

ground eld gauge Gauge invariant op erators Phys Rev D

JA Dixon Calculation of BRS cohomology with sp ectral sequences Commun

Math Phys

C Itzykson and JB Zub er Quantum Field Theory Mc GrawHill New York

IA Batalin and GA Vilkovisky Gauge algebra and quantization Phys Lett

B Quantization of gauge theories with linearly dep endent genera

tors Phys Rev D erratum Phys Rev D

SD Joglekar and BW Lee General theory of renormalization of gauge invariant

op erators Ann Phys NY

G Barnich and M Henneaux Renormalization of gauge invariant op erators and

anomalies in YangMills theory Phys Rev Lett hepth

G Barnich F Brandt and M Henneaux Lo cal BRST cohomology in the antield

formalism I I Application to YangMills theory Commun Math Phys

hepth

G Bandelloni A Blasi C Becchi and R Collina Nonsemisimple gauge mo dels

Classical theory and the prop erties of ghost states Ann Inst Henri Poincare

Nonsemisimple gauge mo dels Renormalization Ann Inst Henri

Poincare

J Gomis and S Weinb erg Are nonrenormalizable gauge theories renormalizable

Nucl Phys B hepth

T Hurth and K Skenderis Quantum No ether metho d Nucl Phys B

hepth

T Hurth and K Skenderis The Quantum No ether condition in terms of interacting

elds in New Developments in Quantum Field Theory eds P Breitenlohner D

Maison and J Wess Springer Berlin to app ear hepth

M Henneaux On the algebraic structure of the BRST symmetry in NATO

Advanced Summer Institute and Ban Summer School in Theoretical Physics on

Physics Geometry and Topology Ban Canada Aug HC Lee ed

Plenum Press New York M Henneaux and C Teitelb oim Quantization of

Gauge Systems Press Princeton

M Henneaux On the gaugexed BRST cohomology Phys Lett B

hepth

G Curci and R Ferrari On a class of Lagrangian mo dels for massive and massless

YangMills elds Nuovo Cim A The unitarity problem and the

zero mass limit for a mo del of massive YangMills theory Nuovo Cim A

I Ojima Comments on massive and massless YangMills Lagrangians with a quartic

coupling of FaddeevPop ov ghosts Z Phys C

J de Bo er K Skenderis P van Nieuwenhuizen and A Waldron On the renormal

izability and unitarity of the CurciFerrari mo del for massive vector b osons Phys

Lett B hepth

T Hurth Nonab elian gauge symmetry in the causal EpsteinGlaser approach Int

J Mod Phys A hepth

N Dragon T Hurth and P van Nieuwenhuizen Polynomial form of the Stueckel

b erg mo del Nucl Phys Proc Suppl B hepth

T Hurth Higgsfree massive nonab elian gauge theories Helv Phys Act

hepth

F Brandt Deformations of global symmetries in the extended antield formalism

J Math Phys hepth

G Barnich F Brandt and M Henneaux Lo cal BRST cohomology in the anti

eld formalism I General theorems Commun Math Phys hep

th

AM Vinogradov On the algebrageometric foundations of Lagrangian eld the

ory Sov Math Dokl A sp ectral sequence asso ciated with a non

linear dierential equation and algebrageometric foundations of Lagrangian eld

theory with constraints Sov Math Dokl The theory of higher in

nitesimal symmetries of non linear partial dierential equations Sov Math Dokl

RL Bryant and PA Griths Characteristic Cohomology of Dierential Systems

I General Theory Duke University Mathematics Preprint Series volume n

January

JML Fisch and M Henneaux Homological p erturbation theory and the algebraic

structure of the antield antibracket formalism for gauge theories Commun Math

Phys

M Henneaux Lectures on the antieldBRST formalism for gauge theories Nucl

Phys B Proc Suppl A

G Barnich and M Henneaux Consistent couplings b etween elds with a gauge

freedom and deformations of the master equation Phys Lett B

hepth

NN Bogoliub ov and DV Shirkov Introduction to the Theory of Quantized Fields

Interscience New York

H Epstein and V Glaser Le role de la lo calite dans la renormalisation p erturba

tive en theorie quantique des champs in Statistical Mechanics and Quantum Field

Theory Pro ceedings of the Summer Scho ol of Les Houches eds C DeWitt and

R Stora Gordon and Breach New York The role of lo cality in p erturbation

theory Ann Inst Poincare

H Epstein and V Glaser Adiabatic limit in p erturbation theory in G Velo AS

Wightman eds Renormalization Theory D Reidel Publishing Company Dor

drecht p O Piguet and A Rouet Symmetries in Perturbative Quan

tum Field Theory Phys Rep H Epstein V Glaser and R Stora

General prop erties of the np oint functions in lo cal quantum eld theory in J

Bros D Jagolnitzer eds Les Houches Pro ceedings G Popineau and R

Stora A p edagogical remark on the main theorem of p erturbative renormalization

theory unpublished R Stora Dierential Algebras ETHlectures un

published G Scharf Finite Quantum Electro dynamics Text and Monographs in

Physics Springer Berlin T Hurth NonAb elian gauge theories the causal

approach Ann Phys hepth

M Dutsc h and K Fredenhagen A lo cal p erturbative construction of observables

in gauge theories The example of QED Commun Math Phys

hepth

T Hurth and K Skenderis Analysis of anomalies in the Quantum No ether metho d

in preparation

M Henneaux Spacetime lo cality of the BRST formalism Commun Math Phys