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Gribov ambiguity and non-trivial vacuum structure of gauge theories on a cylinder
Physics Letters B 303 (1993) 303-307 North-Holland
PHYSICS LETTERS 13
Gribov ambiguity and non-trivial vacuum structure of gauge theories on a cylinder Edwin L a n g m a n n
i
and G o r d o n W. S e m e n o f f
2
Department of Physics, University of British Columbia, Vancouver, B.C., Canada V6T 1ZI Received 22 December 1992
Using the hamiltonian framework, we analyze the Gribov problem for U (N) and SU (N) gauge theories on a cylinder ( = ( 1 + 1 ) dimensional spacetime with compact space S ~). The space of gauge orbits is found to be an orbifold. We show by explicit construction that a proper treatment of the Gribov ambiguity leads to a highly non-trivial structure of all physical states in these q u a n t u m field theory models. The especially interesting example of massless Q C D is discussed in more detail: There, some of the special static gauge transformations which are responsible for the Gribov ambiguity also lead to a spectral flow, and this implies a chiral condensate in all physical states. We also show that the latter is closely related to the Schwinger term and the chiral anomaly.
1. The elimination of gauge degrees of freedom is essential for understanding and extracting physical information from Yang-Mills (YM) gauge theories. This is usually done by "fixing a gauge", i.e. requiring that the field configurations obey some gauge condition such as the Coulomb or the Landau gauge (e.g. in the path integral formalism by means of the Faddeev-Popov trick). Though adequate for perturbative calculations, it is well-known that such a procedure is not sufficient for a deeper understanding of non-abelian YM theories: It has been pointed out already by Gribov [1] that there are many gauge equivalent configurations obeying the Coulomb or the Landau gauge condition, and that the existence of these should play a crucial role for non-perturbative features of these theories such as confinement. Later on it was shown by Singer [2] that in 4 (compact) spacetime dimensions such a Gribov ambiguity arises for any reasonable gauge fixing condition. It has also been found that the lack of a full understanding of Gribov ambiguities and their proper treatment is one major obstacle to a rigorous non-perturbative construction of gauge theories in 4 dimensions [ 3 ]. (For Supported by the " F o n d s zur FSrderung der wissenschaftlichen Forschung'" under the contract Nr. J0789-PHY. 2 Supported in part by the Natural Sciences and Engineering Research Council of Canada.
a recent discussion of the Gribov problem see ref. [4].) In 2 spacetime dimensions gauge theories are much simpler. Pure YM theory on a plane BqXR is trivial with no propagating degrees of freedom. However, on a manifold with non-trivial topology such as a Riemann surface [5,6] or (in the hamiltonian approach) a cylinder (spacetime S ~X BR) [ 7-10 ], it has a finite number of physical degrees of freedom and several non-trivial features which makes it an interesting toy model. Moreover, YM gauge theories with matter (fermions or bosons) on a cylinder allow for a (essentially) rigorous construction on an operator level by means of the theory of quasi-free representations of fermion and boson field algebras [ 11,12 ] (we are planning to report on that in more detail in future publications; for an alternative approach see ref. [9]). This corresponds to the well-known fact that on a cylinder, normal ordering of the bilinears of the matter fields with respect to the free vacuum is sufficient to eliminate all divergences in the fully interacting gauge theory. In this paper we shall examine the Gribov problem for U (N) and SU (N) YM gauge theories with matter on a cylinder and discuss (some of) its physical consequences. Though the Gribov ambiguity does not allow a gauge fixing, we argue that it is possible to maximally reduce the gauge freedom. One is left with
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
303
Volume 303, number 3,4
PHYSICS LETTERSB
an infinite discrete non-abelian group ~' of residual static gauge transformations. It is possible to determine the (topologically highly non-trivial) space of all gauge orbits explicitly and to give a general construction of all physical states which are invariant under ~'. We discuss some of the implications of this structure for massless QCD( 1 + 1 ) (i.e. YM theory with massless fermions on a cylinder). The latter is expecially interesting due to the interplay between the Gribov ambiguity and the anomaly structure [ 13 ]. We note that the existence of a non-trivial vacuum structure in massless QCD ( 1 + 1 ) has already been found in a one-loop calculation by Hetrick and Hosotani [ 8]. Our construction is without approximation and applies to all physical states (not only the vacuum). Moreover, it is a special case of a more general construction valid for any YM gauge theory on a cylinder. 2. The following discussion is in the hamiltonian framework. The gauge group is G = S U ( N ) or U ( N ) and assumed in the fundamental representation. As mentioned above, a YM-field on a cylinder has a finite number of physical degrees of freedom. A simple argument identifying these is as follows: the temporal component Ao of the YM-field is not dynamical but only the Lagrange multiplier enforcing Gauss' law (=invariance under all static gauge transformations), and the only gauge invariant quantities one can construct at some fixed time from the spatial component A1 of the YM-field are the eigenvalues of the the Wilson loop over the whole space ~1, 2~
W[&J=Pexp(-ifdXAl(X))~G.
(2) with real Y( This suggests that the physical YM degrees of freedom can be represented by these Y', and that it should be possible to "fix" the gauge to Ao(x)=0,
A , ( x ) = Z Y~Hi,
(3)
i
which is essentially the Coulomb gauge OA~/Ox=O. Indeed, one can systematically eliminate the gauge degrees of freedom by explicitly solving Gauss' law and end up with the same effective hamiltonian one obtains by imposing the gauge condition (3) [ 10 ] (see also ref. [8] ). After that there are still gauge transformations left which act non-trivially on the Y~ [9,10 ], so (3) is rather a reducing than a fixing of the gauge. Firstly, there are gauge transformations permuting the eigenvalues of the diagonalized Wilson loop D: The gauge transformations pq = exp [ l i n ( e o eiz+ eji e -ix) ]
(4)
(ZeN arbitrary) interchange the ith and thejth eigenvalues and provide a representation n ~ ~ of the permutation group S~,
-iyj,
VrceSN.
(5)
J
(Representing the N eigenvalues of W[Al ] as e iz' one has (z~Z)i_-Z ~-'(o, hence by writing Yi=Y~ja}Z j and Z~= Ej fl} YJ one can easily work out the matrices i k ~ j = ~ k c%(k)fl~ ~2. ) Secondly, the functions
(1)
o
W[A~ ] can always be written as h -~Dh with h~G and D in the Cartan subgroup of G (note that this representation is not unique). It is convenient to introduce the basis {Hi}~__-~l in the Cartan subalgebra of the Lie algebra g of G = S U ( N ) with Hi- e , - ei+ 1.i+t where e u is the N × Nmatrix with the matrix elements (eu)kt=~afijl. For G = U ( N ) we have to add to these H o = 1 (the N × N u n i t matrix). Then we can write D
h(x)=exp(ix~vJHj),
#~ P is the p a t h - o r d e r i n g s y m b o l , x e [ - 7t, ~) the spatial v a r i a b l e .
uJ~Z,
(6)
are gauge transformations [i.e. obey h(0) = h ( 2 g ) ]. ~2 E.g. f o r G = S U ( 3 )
onegets
representing the $3 elements
as
304
15 A p r i l 1993
respectively.
Volume 303, number 3,4
Though they leave D invariant, they act non-trivially on the Yg: yi___~ y i q - l ~ i '
lliEz.
15 April 1993
PHYSICS LETTERS B
(7)
Obviously this provides a representation ~,~ ~ of the abelian group Z ~ with r=N-1 for G = S U ( N ) and
r=NforG=U(N). It is easy to see that the group f#' generated by all these gauge transformations (5) and (7) is a semidirect product o f 7/~ by Su [ 9 ]. This residual gauge group gives a full characterization of the Gribov ambiguity on a cylinder and allows us to determine the space of gauge orbits explicitly: the set of all YM fields A~(x) obeying our gauge condition (3) can be identified with ~r ( r as above), and f#' is the group of all gauge transformations compatable with (3). Therefore ~ generates a//Gribov copies of a given YM field configurations obeying (3), and the space of gauge orbits is N'/ff. It is interesting to note that (for G # U ( 1 ) ) this is not a manifold but only an orbifold [14]. From this we can also easily see that (for G # U ( 1 )) there is no gauge fixing on the cylinder avoiding the Gribov ambiguity: the space ~ / o f all YM fields A~ (x) is contractible but ~ / f f is not. Hence the fibre bundle ~/__,Rr/~, is non-trivial and does not allow for a global cross section. 3. To discuss the implications of this in gauge theories with matter, we refer to the construction of these models mentioned above. By means of the theory of quasifree representations of fermion and boson field algebras [ 11,12 ] one can construct a Hilbert space ,~ so that the (Fourier components of the) YM field A~ (x), the matter fields and the matter currents are represented by (closeable) operators on ,~ (with a common, dense, invariant domain [ 12 ] ), the hamiltonian H is a hermitian operator on Yt~, and there is a unitary representation F o f t h e group c# o f all static gauge transformations on ~ so that F(U)H=
HF( U) V U~ ~. The physical Hilbert space of the model is of course 3%hy~= { Te o ~ I F ( U ) T = ~ V Ue .~}, and it can be explicitly constructed in two steps: After imposing the gauge condition (3), ~X(is reduced to a subspace ,~,hy~. ~ p h y s is a tensor product o f the Hilbert space L2(N ~) of the YM variables yi and a HilO~-tM of the matter fields. On ffk~ we have bert space ,~"
a unitary represenation FM of the residual gauge group ~', and eqs. (5) and (7) provide a representation of f#' on L 2 ( ~ ) . The subspace "~l~hys of~-~phys invariant under the gauge transformations (6) is obviously spanned by the states ~ FIexp[iOi(yi+ui)]FM(hi) "~g"
~(0)=
(8)
ViEZ i
[0eRr; strictly speaking, we should "smear these out" by appropriate test functions f(O)], where hi(x)exp(ixHD and ~ e . T ~ . On "~phys w e have then a unitary representation/~ofthe SN subgroup of ~',
/'(~) ~u(0) =F~(lr) t'icZ
V~SN
I-I exp [i0i(2( y)i + ui) ]FM(hi)"'~P, i
(9)
.
Hence the states invariant under the whole residual gauge group ~', thus spanning ,~phys, are given by 1
n! ~
rtESN
F(~z)~g(0),
5Ue,~'M,0E~ ~.
(10)
We can write ~¢/ephys--S~¢"--phys where S=(l/n!)× ~,,sN/~(Tr) is an orthogonal projection. It is worth pointing out that for this construction to work the semidirect product structure of ~' is crucial, i.e. that ~ - I/)~E 7/r for all v E U and ~ze SN. 4. As an illustration and an especially interesting example, we discuss in more detail massless Q C D ( 1 + 1 ). Here the very gauge transformations h (x) (6) also cause a spectral flow of the fermions. Due to this, the YM variables yi can combine with fermionic degrees of freedom Q~ resulting in variables invariant under the h(x), and this leads to an interesting additional structure of the physical states. To show this, we refer to a (essentially) rigorous construction of massless Q C D ( 1 + 1 ) by means of the non-trivial quasi-free representation of the fermion field operators ~3 ~ut.~ (x) =- q/~) (x) naturally associated with the free fermion hamiltonian 2n
Ho= f dx :~,, ( x ) y ~1x ~0 U ( x ) : , 0 ~3 a, a'e { 1, 2} and A, Be { 1, ..., N} are the spin and color indices,
respectively. 305
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PHYSICS LETTERS B
with 7 5 = ( ~ , 5 ) ~ , = ( ~ o ) [11] and : : normal ordering with respect to theJ~ee fermion vacuum £2 (see below). This representation is on the fermion Fock space 0% generated from a state £2 ( = " f r e e Dirac sea" ) obeying (h.~ (n)~2= ( / ~ ( - n - 1 )~2= 0,
VA, n>~O,
(11)
where (&~ (n) = f~" dx exp ( - inx) ~uo~( x ) / x / ~ and neT/. After reducing the gauge to (3), the remnant o f Gauss' law is fii(0) - rE, dx : ~t(x)Hi~u(x) : -~ 0 (see ref. [ 1 0 ] ) , with the r N × N matrices H ~ obeying tr(H~Hj)=3~ I ~. Hence the reduced Hilbert space Off,h : of the model is a tensor product of the space ~ = { ~U~~ I/~ (0) 7 ' = 0, Vi) and the Hilbert space L2(~ r) for the YM variables Y', and there is a unitary representation of the residual gauge group if' on ~:ghy~- Especially, the special gauge transformations h(x) (6) are implemented by unitary operators Fv(h) on ~ [ 15,1 1 ], and they act non-trivially on the axial "charges" Q~5-~f~" dx :~u*(x)75Hi~u(x) : (the H ~as above):
Q~-,Q~ +2u ~ vi.
(12)
Proof. We first consider the operators Q ~ = f~" dx :~/*(x)?~e~a~u(x) : (with e~B as above) which can be written as
QS.As= ~ ((t'~(n)q/,m(n) n>~O k
-~l.s( - l -n)(/TA(- 1 Obviously
under
the
_.)_(kn'--,- 1- n ]B] " transformation
~u(x)
exp(ixeii)~(x) we have ~ ( ~ ] ( n ) ~ ( ~ ( n - O ~ ) , hence by the canonical anticommutator relations of the fermion field operators,
Qs~B ~ Qs~4B+ 20ABt~iA = Q5~4~+ 2 ( eii)AB . From this we readily deduce that under ~ ' ( x ) ~ h(x)~u(x), Q~n-oQs~B+ 2~j ui(Hfl~m hence as
Qi = ~'A,B( H i ) BA Qs~B, Qi
15 April 1993
From this proof it is clear that the implementers FF(Hg) create particle-antiparticle pairs in the fermion Fock space (this can also be seen from the explicit formulas for these derived in ref. [ 15 ] ), hence all physical states ~v(0) (10) (including, of course, the g r o u n d s t a t e = v a c u u m of the model) contain a condensate of fermion-anti-fermion pairs. Moreover, on ~phys the YM-degrees o f freedom yi can be combined with the axial "charges" Q~ to the gauge invariant 0-variables
Oi- i SYi'
pi=
i SOi
_ vi
l ~i
--~5.
(13)
This can be understood also as a result of the anomalies in this model [ 13 ]: The naive fermion currents **5pa(x ) = :~*(x)Ta~(x): and y ( x ) = • ~*(X)ysTa~(x) : are constructed by means of normal ordering : : with respect to the free fermion vacuum £2. Due to the well-known Schwinger term in the [ p , j ] - c o m m u t a t o r [ 16,11 ] the currentsja(x) do not transform covariantly under static gauge transformations. However, there is a freedom in the normal ordering prescription [ 11 ] which can be used to fix this problem: one can show that the (essentially unique) gauge covariant currents are given by ~6 p ( x ) = p ( x ) and f ~ ( x ) = j T ( x ) - A ~ ( x ) / n (one can think of this as gauge-covariant normal ordering). The gauge covariant axial "charges" are therefore Os=f~dxf(x)=Qs-f2~dxAl(x)/rc. By direct calculation one can show that these give covariantly conserved vector currents"7 f . = ( ~ ( x ) , f ( x ) ) : #s D d'" (x) = 0, whereas the gauge covariant axial current f~(x) = (j-'(x), p ( x ) ) has an anomaly; D . f ~ ( x ) = - E ( x ) / r c (with ~9 E=Fol=OoAi--OiAo +i[Ao, At] the YM field strength as usual). In the gauge (3) this implies O.~(x) = - Z~OoYiHff~r and gauge invariance (under the residual gauge transformations in (q') and non-conservation of the axial charges Q~ identical with - 2 p ~ (p~cf. eq. (13) ). 5. Usually (especially in path integral approaches)
Q~+2 ~ ( H ' ) n . ~ uJ(Hj)AB, A,B
j
identical with eq. ( 12 ). ~4 I.e. Hi=Ej(b-l)°Hj with (b-l) ° the inverse matrix to
bij=tr{HiHfl. 306
,5 The T ° are the generators of the Lie algebra g of G. #6 We write p(x) ~ Zapa(x) T aand similarly for j, A~etc, ~7 u~{0, 1} is a spacetime index. ~8 Dr is the covariant derivative in the adjoint representation. ~90v=O/Ox",x°=t (time) and x I =x.
Volume 303, number 3,4
PHYSICS LETTERS B
G r i b o v ambiguities are accounted for by restricting the YM-field to one f u n d a m e n t a l d o m a i n [ 1,4 ]. In our case this would correspond to restricting the y i ( 3 ) to some appropriate compact subset F D of ~r and forgetting about the special gauge transformations (5), (7). [E.g. for G = U ( N ) , N>~2, such an appropriate F D would be given by
O<~ZI<~Z2<~...<~ZN
15 April 1993
4 d i m e n s i o n s which has not been achieved yet [4]. One of us (E.L.) would like to thank M. Bergeron, A. Carey, D. Eliezer, H. Grosse, A. K o v n e r and M. Salmhofer for helpful discussions.
Note added. After submission of this paper we received ref. [ 18 ] where also the space of gauge orbits for SU (N) Yang-Mills fields on a cylinder is found.
References [ 1] V.N. Gribov, Nucl. Phys. B 139 (1978) 1. [2] 1.M. Singer, Commun. Math. Phys. 60 (1978) 7. [ 3 ] V. Rivasseau, From perturbative to constructive regularization,Princeton Series in Physics (Princeton U.P., Princeton, NJ, 1991). [4] P. van Baal, Nucl. Phys. B 369 (1992) 259. [5] E. Winch, Commun. Math. Phys. 141 (1991) 153. [6 ] M. Blauand G. Thompson,Intern. J. Mod. Phys. A 7 (1992) 3781. [7] S.G. Rajeev, Phys. Lett. B 212 (1988) 203. [8] J.E. Hetrick and Y. Hosotani, Phys. Lett. B 230 (1989) 88. [9] M. Mickelsson,Phys. Lett. B 242 (1990) 217. [ 10] E. Langmannand G.W. Semenoff,Phys. Lett. B 296 (1992) 117. [ 11 ] A.L. Carey and S.N.M. Ruijsenaars, Acta Appl. Mat. 10 (1987) 1. [ 12] H. Grosse and E. Langmann,J. Math. Phys. 33 (1992) 1032. [13] R. Jackiw, in: Relativity, groups and topology II (Les Houches, 1983), eds. B.S. DeWitt and R. Stora (NorthHolland, Amsterdam, 1984). [ 14] L. Dixon, J. Harvey, C. Vafa and E. Winen, Nucl. Phys. B 274 (1986) 285. [ 15 ] S.N.M. Ruijsenaars, J. Math. Phys. 18 ( 1977 ) 517. [ 16] P. Goddard and D. Olive, Intern. J. Mod. Phys. A 1 (1986) 303. [17] O. Braneli and D.W. Robinson, Operator algebras and quantum statistical mechanics, Vol. 2 (Springer, Berlin, 1981). [ 18 ] J.E. Hetrick, Gauge fixingand Gribov copies in pure YangMills on a circle, preprint hep-lat/9212005.
307
PHYSICS LETTERS 13
Gribov ambiguity and non-trivial vacuum structure of gauge theories on a cylinder Edwin L a n g m a n n
i
and G o r d o n W. S e m e n o f f
2
Department of Physics, University of British Columbia, Vancouver, B.C., Canada V6T 1ZI Received 22 December 1992
Using the hamiltonian framework, we analyze the Gribov problem for U (N) and SU (N) gauge theories on a cylinder ( = ( 1 + 1 ) dimensional spacetime with compact space S ~). The space of gauge orbits is found to be an orbifold. We show by explicit construction that a proper treatment of the Gribov ambiguity leads to a highly non-trivial structure of all physical states in these q u a n t u m field theory models. The especially interesting example of massless Q C D is discussed in more detail: There, some of the special static gauge transformations which are responsible for the Gribov ambiguity also lead to a spectral flow, and this implies a chiral condensate in all physical states. We also show that the latter is closely related to the Schwinger term and the chiral anomaly.
1. The elimination of gauge degrees of freedom is essential for understanding and extracting physical information from Yang-Mills (YM) gauge theories. This is usually done by "fixing a gauge", i.e. requiring that the field configurations obey some gauge condition such as the Coulomb or the Landau gauge (e.g. in the path integral formalism by means of the Faddeev-Popov trick). Though adequate for perturbative calculations, it is well-known that such a procedure is not sufficient for a deeper understanding of non-abelian YM theories: It has been pointed out already by Gribov [1] that there are many gauge equivalent configurations obeying the Coulomb or the Landau gauge condition, and that the existence of these should play a crucial role for non-perturbative features of these theories such as confinement. Later on it was shown by Singer [2] that in 4 (compact) spacetime dimensions such a Gribov ambiguity arises for any reasonable gauge fixing condition. It has also been found that the lack of a full understanding of Gribov ambiguities and their proper treatment is one major obstacle to a rigorous non-perturbative construction of gauge theories in 4 dimensions [ 3 ]. (For Supported by the " F o n d s zur FSrderung der wissenschaftlichen Forschung'" under the contract Nr. J0789-PHY. 2 Supported in part by the Natural Sciences and Engineering Research Council of Canada.
a recent discussion of the Gribov problem see ref. [4].) In 2 spacetime dimensions gauge theories are much simpler. Pure YM theory on a plane BqXR is trivial with no propagating degrees of freedom. However, on a manifold with non-trivial topology such as a Riemann surface [5,6] or (in the hamiltonian approach) a cylinder (spacetime S ~X BR) [ 7-10 ], it has a finite number of physical degrees of freedom and several non-trivial features which makes it an interesting toy model. Moreover, YM gauge theories with matter (fermions or bosons) on a cylinder allow for a (essentially) rigorous construction on an operator level by means of the theory of quasi-free representations of fermion and boson field algebras [ 11,12 ] (we are planning to report on that in more detail in future publications; for an alternative approach see ref. [9]). This corresponds to the well-known fact that on a cylinder, normal ordering of the bilinears of the matter fields with respect to the free vacuum is sufficient to eliminate all divergences in the fully interacting gauge theory. In this paper we shall examine the Gribov problem for U (N) and SU (N) YM gauge theories with matter on a cylinder and discuss (some of) its physical consequences. Though the Gribov ambiguity does not allow a gauge fixing, we argue that it is possible to maximally reduce the gauge freedom. One is left with
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
303
Volume 303, number 3,4
PHYSICS LETTERSB
an infinite discrete non-abelian group ~' of residual static gauge transformations. It is possible to determine the (topologically highly non-trivial) space of all gauge orbits explicitly and to give a general construction of all physical states which are invariant under ~'. We discuss some of the implications of this structure for massless QCD( 1 + 1 ) (i.e. YM theory with massless fermions on a cylinder). The latter is expecially interesting due to the interplay between the Gribov ambiguity and the anomaly structure [ 13 ]. We note that the existence of a non-trivial vacuum structure in massless QCD ( 1 + 1 ) has already been found in a one-loop calculation by Hetrick and Hosotani [ 8]. Our construction is without approximation and applies to all physical states (not only the vacuum). Moreover, it is a special case of a more general construction valid for any YM gauge theory on a cylinder. 2. The following discussion is in the hamiltonian framework. The gauge group is G = S U ( N ) or U ( N ) and assumed in the fundamental representation. As mentioned above, a YM-field on a cylinder has a finite number of physical degrees of freedom. A simple argument identifying these is as follows: the temporal component Ao of the YM-field is not dynamical but only the Lagrange multiplier enforcing Gauss' law (=invariance under all static gauge transformations), and the only gauge invariant quantities one can construct at some fixed time from the spatial component A1 of the YM-field are the eigenvalues of the the Wilson loop over the whole space ~1, 2~
W[&J=Pexp(-ifdXAl(X))~G.
(2) with real Y( This suggests that the physical YM degrees of freedom can be represented by these Y', and that it should be possible to "fix" the gauge to Ao(x)=0,
A , ( x ) = Z Y~Hi,
(3)
i
which is essentially the Coulomb gauge OA~/Ox=O. Indeed, one can systematically eliminate the gauge degrees of freedom by explicitly solving Gauss' law and end up with the same effective hamiltonian one obtains by imposing the gauge condition (3) [ 10 ] (see also ref. [8] ). After that there are still gauge transformations left which act non-trivially on the Y~ [9,10 ], so (3) is rather a reducing than a fixing of the gauge. Firstly, there are gauge transformations permuting the eigenvalues of the diagonalized Wilson loop D: The gauge transformations pq = exp [ l i n ( e o eiz+ eji e -ix) ]
(4)
(ZeN arbitrary) interchange the ith and thejth eigenvalues and provide a representation n ~ ~ of the permutation group S~,
-iyj,
VrceSN.
(5)
J
(Representing the N eigenvalues of W[Al ] as e iz' one has (z~Z)i_-Z ~-'(o, hence by writing Yi=Y~ja}Z j and Z~= Ej fl} YJ one can easily work out the matrices i k ~ j = ~ k c%(k)fl~ ~2. ) Secondly, the functions
(1)
o
W[A~ ] can always be written as h -~Dh with h~G and D in the Cartan subgroup of G (note that this representation is not unique). It is convenient to introduce the basis {Hi}~__-~l in the Cartan subalgebra of the Lie algebra g of G = S U ( N ) with Hi- e , - ei+ 1.i+t where e u is the N × Nmatrix with the matrix elements (eu)kt=~afijl. For G = U ( N ) we have to add to these H o = 1 (the N × N u n i t matrix). Then we can write D
h(x)=exp(ix~vJHj),
#~ P is the p a t h - o r d e r i n g s y m b o l , x e [ - 7t, ~) the spatial v a r i a b l e .
uJ~Z,
(6)
are gauge transformations [i.e. obey h(0) = h ( 2 g ) ]. ~2 E.g. f o r G = S U ( 3 )
onegets
representing the $3 elements
as
304
15 A p r i l 1993
respectively.
Volume 303, number 3,4
Though they leave D invariant, they act non-trivially on the Yg: yi___~ y i q - l ~ i '
lliEz.
15 April 1993
PHYSICS LETTERS B
(7)
Obviously this provides a representation ~,~ ~ of the abelian group Z ~ with r=N-1 for G = S U ( N ) and
r=NforG=U(N). It is easy to see that the group f#' generated by all these gauge transformations (5) and (7) is a semidirect product o f 7/~ by Su [ 9 ]. This residual gauge group gives a full characterization of the Gribov ambiguity on a cylinder and allows us to determine the space of gauge orbits explicitly: the set of all YM fields A~(x) obeying our gauge condition (3) can be identified with ~r ( r as above), and f#' is the group of all gauge transformations compatable with (3). Therefore ~ generates a//Gribov copies of a given YM field configurations obeying (3), and the space of gauge orbits is N'/ff. It is interesting to note that (for G # U ( 1 ) ) this is not a manifold but only an orbifold [14]. From this we can also easily see that (for G # U ( 1 )) there is no gauge fixing on the cylinder avoiding the Gribov ambiguity: the space ~ / o f all YM fields A~ (x) is contractible but ~ / f f is not. Hence the fibre bundle ~/__,Rr/~, is non-trivial and does not allow for a global cross section. 3. To discuss the implications of this in gauge theories with matter, we refer to the construction of these models mentioned above. By means of the theory of quasifree representations of fermion and boson field algebras [ 11,12 ] one can construct a Hilbert space ,~ so that the (Fourier components of the) YM field A~ (x), the matter fields and the matter currents are represented by (closeable) operators on ,~ (with a common, dense, invariant domain [ 12 ] ), the hamiltonian H is a hermitian operator on Yt~, and there is a unitary representation F o f t h e group c# o f all static gauge transformations on ~ so that F(U)H=
HF( U) V U~ ~. The physical Hilbert space of the model is of course 3%hy~= { Te o ~ I F ( U ) T = ~ V Ue .~}, and it can be explicitly constructed in two steps: After imposing the gauge condition (3), ~X(is reduced to a subspace ,~,hy~. ~ p h y s is a tensor product o f the Hilbert space L2(N ~) of the YM variables yi and a HilO~-tM of the matter fields. On ffk~ we have bert space ,~"
a unitary represenation FM of the residual gauge group ~', and eqs. (5) and (7) provide a representation of f#' on L 2 ( ~ ) . The subspace "~l~hys of~-~phys invariant under the gauge transformations (6) is obviously spanned by the states ~ FIexp[iOi(yi+ui)]FM(hi) "~g"
~(0)=
(8)
ViEZ i
[0eRr; strictly speaking, we should "smear these out" by appropriate test functions f(O)], where hi(x)exp(ixHD and ~ e . T ~ . On "~phys w e have then a unitary representation/~ofthe SN subgroup of ~',
/'(~) ~u(0) =F~(lr) t'icZ
V~SN
I-I exp [i0i(2( y)i + ui) ]FM(hi)"'~P, i
(9)
.
Hence the states invariant under the whole residual gauge group ~', thus spanning ,~phys, are given by 1
n! ~
rtESN
F(~z)~g(0),
5Ue,~'M,0E~ ~.
(10)
We can write ~¢/ephys--S~¢"--phys where S=(l/n!)× ~,,sN/~(Tr) is an orthogonal projection. It is worth pointing out that for this construction to work the semidirect product structure of ~' is crucial, i.e. that ~ - I/)~E 7/r for all v E U and ~ze SN. 4. As an illustration and an especially interesting example, we discuss in more detail massless Q C D ( 1 + 1 ). Here the very gauge transformations h (x) (6) also cause a spectral flow of the fermions. Due to this, the YM variables yi can combine with fermionic degrees of freedom Q~ resulting in variables invariant under the h(x), and this leads to an interesting additional structure of the physical states. To show this, we refer to a (essentially) rigorous construction of massless Q C D ( 1 + 1 ) by means of the non-trivial quasi-free representation of the fermion field operators ~3 ~ut.~ (x) =- q/~) (x) naturally associated with the free fermion hamiltonian 2n
Ho= f dx :~,, ( x ) y ~1x ~0 U ( x ) : , 0 ~3 a, a'e { 1, 2} and A, Be { 1, ..., N} are the spin and color indices,
respectively. 305
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with 7 5 = ( ~ , 5 ) ~ , = ( ~ o ) [11] and : : normal ordering with respect to theJ~ee fermion vacuum £2 (see below). This representation is on the fermion Fock space 0% generated from a state £2 ( = " f r e e Dirac sea" ) obeying (h.~ (n)~2= ( / ~ ( - n - 1 )~2= 0,
VA, n>~O,
(11)
where (&~ (n) = f~" dx exp ( - inx) ~uo~( x ) / x / ~ and neT/. After reducing the gauge to (3), the remnant o f Gauss' law is fii(0) - rE, dx : ~t(x)Hi~u(x) : -~ 0 (see ref. [ 1 0 ] ) , with the r N × N matrices H ~ obeying tr(H~Hj)=3~ I ~. Hence the reduced Hilbert space Off,h : of the model is a tensor product of the space ~ = { ~U~~ I/~ (0) 7 ' = 0, Vi) and the Hilbert space L2(~ r) for the YM variables Y', and there is a unitary representation of the residual gauge group if' on ~:ghy~- Especially, the special gauge transformations h(x) (6) are implemented by unitary operators Fv(h) on ~ [ 15,1 1 ], and they act non-trivially on the axial "charges" Q~5-~f~" dx :~u*(x)75Hi~u(x) : (the H ~as above):
Q~-,Q~ +2u ~ vi.
(12)
Proof. We first consider the operators Q ~ = f~" dx :~/*(x)?~e~a~u(x) : (with e~B as above) which can be written as
QS.As= ~ ((t'~(n)q/,m(n) n>~O k
-~l.s( - l -n)(/TA(- 1 Obviously
under
the
_.)_(kn'--,- 1- n ]B] " transformation
~u(x)
exp(ixeii)~(x) we have ~ ( ~ ] ( n ) ~ ( ~ ( n - O ~ ) , hence by the canonical anticommutator relations of the fermion field operators,
Qs~B ~ Qs~4B+ 20ABt~iA = Q5~4~+ 2 ( eii)AB . From this we readily deduce that under ~ ' ( x ) ~ h(x)~u(x), Q~n-oQs~B+ 2~j ui(Hfl~m hence as
Qi = ~'A,B( H i ) BA Qs~B, Qi
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From this proof it is clear that the implementers FF(Hg) create particle-antiparticle pairs in the fermion Fock space (this can also be seen from the explicit formulas for these derived in ref. [ 15 ] ), hence all physical states ~v(0) (10) (including, of course, the g r o u n d s t a t e = v a c u u m of the model) contain a condensate of fermion-anti-fermion pairs. Moreover, on ~phys the YM-degrees o f freedom yi can be combined with the axial "charges" Q~ to the gauge invariant 0-variables
Oi- i SYi'
pi=
i SOi
_ vi
l ~i
--~5.
(13)
This can be understood also as a result of the anomalies in this model [ 13 ]: The naive fermion currents **5pa(x ) = :~*(x)Ta~(x): and y ( x ) = • ~*(X)ysTa~(x) : are constructed by means of normal ordering : : with respect to the free fermion vacuum £2. Due to the well-known Schwinger term in the [ p , j ] - c o m m u t a t o r [ 16,11 ] the currentsja(x) do not transform covariantly under static gauge transformations. However, there is a freedom in the normal ordering prescription [ 11 ] which can be used to fix this problem: one can show that the (essentially unique) gauge covariant currents are given by ~6 p ( x ) = p ( x ) and f ~ ( x ) = j T ( x ) - A ~ ( x ) / n (one can think of this as gauge-covariant normal ordering). The gauge covariant axial "charges" are therefore Os=f~dxf(x)=Qs-f2~dxAl(x)/rc. By direct calculation one can show that these give covariantly conserved vector currents"7 f . = ( ~ ( x ) , f ( x ) ) : #s D d'" (x) = 0, whereas the gauge covariant axial current f~(x) = (j-'(x), p ( x ) ) has an anomaly; D . f ~ ( x ) = - E ( x ) / r c (with ~9 E=Fol=OoAi--OiAo +i[Ao, At] the YM field strength as usual). In the gauge (3) this implies O.~(x) = - Z~OoYiHff~r and gauge invariance (under the residual gauge transformations in (q') and non-conservation of the axial charges Q~ identical with - 2 p ~ (p~cf. eq. (13) ). 5. Usually (especially in path integral approaches)
Q~+2 ~ ( H ' ) n . ~ uJ(Hj)AB, A,B
j
identical with eq. ( 12 ). ~4 I.e. Hi=Ej(b-l)°Hj with (b-l) ° the inverse matrix to
bij=tr{HiHfl. 306
,5 The T ° are the generators of the Lie algebra g of G. #6 We write p(x) ~ Zapa(x) T aand similarly for j, A~etc, ~7 u~{0, 1} is a spacetime index. ~8 Dr is the covariant derivative in the adjoint representation. ~90v=O/Ox",x°=t (time) and x I =x.
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G r i b o v ambiguities are accounted for by restricting the YM-field to one f u n d a m e n t a l d o m a i n [ 1,4 ]. In our case this would correspond to restricting the y i ( 3 ) to some appropriate compact subset F D of ~r and forgetting about the special gauge transformations (5), (7). [E.g. for G = U ( N ) , N>~2, such an appropriate F D would be given by
O<~ZI<~Z2<~...<~ZN
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4 d i m e n s i o n s which has not been achieved yet [4]. One of us (E.L.) would like to thank M. Bergeron, A. Carey, D. Eliezer, H. Grosse, A. K o v n e r and M. Salmhofer for helpful discussions.
Note added. After submission of this paper we received ref. [ 18 ] where also the space of gauge orbits for SU (N) Yang-Mills fields on a cylinder is found.
References [ 1] V.N. Gribov, Nucl. Phys. B 139 (1978) 1. [2] 1.M. Singer, Commun. Math. Phys. 60 (1978) 7. [ 3 ] V. Rivasseau, From perturbative to constructive regularization,Princeton Series in Physics (Princeton U.P., Princeton, NJ, 1991). [4] P. van Baal, Nucl. Phys. B 369 (1992) 259. [5] E. Winch, Commun. Math. Phys. 141 (1991) 153. [6 ] M. Blauand G. Thompson,Intern. J. Mod. Phys. A 7 (1992) 3781. [7] S.G. Rajeev, Phys. Lett. B 212 (1988) 203. [8] J.E. Hetrick and Y. Hosotani, Phys. Lett. B 230 (1989) 88. [9] M. Mickelsson,Phys. Lett. B 242 (1990) 217. [ 10] E. Langmannand G.W. Semenoff,Phys. Lett. B 296 (1992) 117. [ 11 ] A.L. Carey and S.N.M. Ruijsenaars, Acta Appl. Mat. 10 (1987) 1. [ 12] H. Grosse and E. Langmann,J. Math. Phys. 33 (1992) 1032. [13] R. Jackiw, in: Relativity, groups and topology II (Les Houches, 1983), eds. B.S. DeWitt and R. Stora (NorthHolland, Amsterdam, 1984). [ 14] L. Dixon, J. Harvey, C. Vafa and E. Winen, Nucl. Phys. B 274 (1986) 285. [ 15 ] S.N.M. Ruijsenaars, J. Math. Phys. 18 ( 1977 ) 517. [ 16] P. Goddard and D. Olive, Intern. J. Mod. Phys. A 1 (1986) 303. [17] O. Braneli and D.W. Robinson, Operator algebras and quantum statistical mechanics, Vol. 2 (Springer, Berlin, 1981). [ 18 ] J.E. Hetrick, Gauge fixingand Gribov copies in pure YangMills on a circle, preprint hep-lat/9212005.
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