Origin of the Gribov ambiguity

Volume 78B, number 2,3

PHYSICS LETTERS

25 September 1978

ORIGIN OF THE GRIBOV AMBIGUITY Y. TAKAHASHI 1 and M. KOBAYASHI a, 2 Department of Physics, University of Connecticut, Storrs, CT06268, USA a and Theoretical Physics Institute, DepartmentofPhysics, University of Alberta, Edmonton, Alberta, Canada T6G 2J1 Received 12 June 1978

It is pointed out that the Gribov ambiguity for the non-abelian transverse gauge field has the same origin as the JohnsonSudarshan problem for the spin-3/2 field as well as the propagation problem discovered by Velo and Zwanziger.

The purpose of this note is to demonstrate that the Gribov ambiguity for the non-abelian transverse gauge field [1 ] has the same origin as the Johnson-Sudarshan problem for the spin-3/2 field [2] as well as the propagation problem discovered by Velo and Zwanziger [3]. The Gribov ambiguity states that three dimensional transverse fields Bi(x ) in non-abelian gauge theories cannot be determined uniquely. This proof starts with the transverse gauge (the Coulomb gauge): aiB i = O.

(1)

Following Gribov, we prove the non-uniqueness of the transverse fields by showing the existence o f nontrivial solutions for U such that

B'--i Ut BiU + U*OiU"

(2)

Suppose 6B i = B ' i - B i be infinitesimal, then eq. (2) becomes - i 6 B k = [Bk,X] + ak x.

(3)

[Bi, ai×] + V2X = 0.

(4)

Unless eq. (4) has a solution such that g is identically zero, B i cannot be determined uniquely even though the gauge is fixed. In the abelian gauge theory, eq. (4) turns out to be

v2x = o.

(5)

Therefore, the only solution satisfying the boundary condition X ~ 0 at Ixl ~ ~ is X = 0. While in the nonabelian theory eq. (4) yields [V2 6ac + ieabCB b ( x ) ~k ] Xc(x) = O,

(6)

by substituting the following decompositions into eq. (4):

B i -- ~1 7 a--a z~i ,

X = ½ "tax a.

(7)

Let A = det[V26 ac + ieabeBb(x) Ok] ,

(8)

By virtue of the condition (1), eq. (3) leads to

1 Permanent address: Theoretical Physics Institute, Depart,' ment of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2J1. 2 Permanent address: Department of Physics, Gifu University, Gifu, Japan 502.

then A is singular, as is shown by Gribov. Therefore, a nontrivial solution o f xC(x) exists. This creates the disaster that the transverse fields are not determined uniquely in the non-abelian gauge theory. The Gribov ambiguity originates from the fact that the constraint equation has a zero point. This situation occurs in the Rarita-Schwinger field interacting with 241

Volume 78B, number 2,3

PHYSICS LETTERS

electromagnetic fields. The primary and secondary constraint equations are ( m - ~ 2 ,,[kDk ) 41/2 + Dk43/2 = O,

(9)

and

('Yu - i(e/3m2)3'o3'u~/oFoa ) 4u = 0,

(10)

respectively, where

41/2(x) = ~4t(x),

(11)

4~/2(x) = (~k~ - ~ %~l) 4~(x),

(lZ)

25 September 1978

A2(n0 )2 = Inl2,

(16)

where the four-vectors n u are the normals to the characteristic surface. It is worthwhile to notice that the same structure as eq. (14) appears in eq. (16). Therefore the normals cannot be determined uniquely. The analysis just mentioned above is applied to massive spin-2 fields interacting with the electromagnetic fields. The problem in question arises when deriving the tenth constraint equation from the secondary constraints [5]. The constraint is obtained under the assumption that the determinant A3 is nonsingular, where A3 is defined by ( 3 e ]2 IHI2"

2x3 = ~ 6 det [m25 kl + -} ieekln Hn ] = 1 - \ 2m 2]

and

(17)

D k = Ok - ieAk(X ).

(13)

In the canonical formalism the constraint variables have to be expressed in terms of the canonical variables through the constraint equations. This cannot be performed for the Rarita-Schwinger field, as is discussed by Jenkins [4]. Because the determinant of the coefficient of 44 in eq. (10) which is denoted to A{ and expressed by e 3m 2

A22=det[l÷i--Tk~tlfkl?=[

I

- t 2e ] 2 , , l ~ l

,~m2!

~3

=

(14) is singular so that the unique inverse of the determinant does not exist. For the coefficient of 41/2 we get its determinant A 1 from eq. (9): A1 = det [m - } 7kDk] ,

(15)

which is also singular. Therefore, 44 and 41/2 are not determined uniquely through the constraint equations (9) and (10). This should be compared with eq. (8) in the non-abelian gauge theory. The invertibility of constraint equations is closely related to the wave propagation of higher-spin fields. To investigate the nature of the propagation one calculates the characteristic determinant D(n) and examines its characteristic roots, D(n) = 0. For the RaritaSchwinger field, the characteristic roots lead to the form

242.

Then, the characteristic roots are calculated as A3(n0 )2 = In[2,

(18)

Eq. (17) states, however, that A 3 is a singular determinant. The same structure can be found for a spin-1 theory coupled via an electric quadrupole to an external field [6]. The problem of secondary constraints still remains unsolved since Johnson-Sudarshan's work in 1961. In conclusion the pathologies among the Gribov ambiguity, the Johnson-Sudarshan problem and the propagation problem discovered by Velo and Zwanziger originate from the lack of invertibility of the constraint equations. One of the authors (M.K.) would like to thank Dr. A.Z. Capri for enlightening discussions on the characteristic determinant. This work was supported in part by the National Research Council of Canada. [1] V.N. Gribov, Instability of non-abelian gauge theories and impossibility of choice of Coulomb gauge, preprint SLACTRANS-176 (1977) (lecture at the 12th Winter School of the Leningrad Nuclear PhysicsInstitute). [2] K. Johnson and E.C.G. Sudarshan, Ann. Phys. (NY) 13 (1961) 126. [3] G. Velo and D. Zwanziger,Phys. Rev, 186 (1969) 1337. [4] J.D. Jenkins, J. Phys. A7 (1974) 1129. [5] M. Kobayashiand A. Shamaly, Phys. Rev. D17 (1978) 2179. [6] A.Z. Capri and A. Shamaly, Can. J. Phys. 54 (1976) 1089.