- Email: [email protected]
Remarks on the “Homogeneous” gauge
Volume 101 B, number 6
PHYSICS LETTERS
28 May 1981
REMARKS ON THE "HOMOGENEOUS" GAUGE
M. AZAM
Tara Institute of Fundamental Research. Bombay 400005. India Received 10 November 1980
It is noted that some recently claimed advantages for the gauge xvA u = 0 are not in general true.
The gauge condition [ 1,2]
x'A - xv.A u = 0
(1)
has recently been the object of much interest. A number of advantages have been claimed [ 2 - 4 ] for nonabelian gauge theories in this gauge: (i) an inversion formula l
Au(x ) = f ds sxvFv(sx) 0
(2)
holds (A u and Fur take values in the Lie algebra of the gauge group); (ii) potentials satisfying (1) are free from the gaugefixing ambiguity [5l. These claims are clearly of great interest; in particular, Shifman [3] has used eq. (2) in a rather detailed study of the Wilson loop. Unfortunately, there are configurations of the gauge potentials for which formula (2) breaks down and gauge fixing ambiguities are present. We consider euclidean space-time with the origin removed, so that the gauge condition (1) is defined everywhere. It is easy to see then (e.g. ref. [2] ) that the orbit of a given A~, under the gauge transformations
{A g = g - l A u g - e - l g - l O
g}
always intersects x ' A = 0 and that in the gauge given by (1), gauge transformations are homogeneous functions of x (hence the name homogeneous gauge): g(sx) = g(x).
(3)
Now, the Coulomb-like configuration of the potentials satisfying
(4)
A (sx) = s - l A (x),
is invariant under homogeneous gauge transformations: =
Clearly, for these configurations the inversion formula (2) fails (it is equally easy to see where the derivation of (2) given in refs. [2,3] breaks down ~ t ). As far as the second claim is concerned, note that eqs. (3) and (4) enable a compactification of s p a c e time from R 4 - {0) to the three-dimensional sphere S 3. The gauge functions satisfying (3) are trivially defined on S 3 and the Coulomb-like configurations can also be mapped on S 3 by scaling by the radial distance. In the spherical coordinate system the homogeneous gauge condition (1) reduces to
A r = 0.
(5)
Non-vanishing components of the potentials are
Ai(r, Ol,O2,03)-Ai(r,O ),
i=01,02,03,
(6)
and eq. (4) becomes
rAi(r, 0) = Ai(I, 0),
(7)
(as r ~ 0% Ag(r, 0 ) ~ O, independent of 0). This equation establishes a one-one correspondence between the Coulomb-like potentials on R 4 _ (0} and a general poI In ref. [21 only those potentials which are (72 everywhere in space-time are considered. Clearly this excludes the example we construct.
0 031 9163/81/0000-0000/$ 02.50 © North-Holland Publishing Company
401
Volume I01B, number 6
PHYSICS I.ETTERS
tential on S 3. We may now use the result of Singer [6] and Narasimhan and Ramadas [7] (that gauge-fixing is impossible in gauge theories on S 3) to conclude that the Coulomb-like configuration is not free from gauge ambiguity. It may now be asked whether these troublesome configurations are in some sense negligible in the computation of gauge-invariant ("observable") quantities. We consider the Wilson loop integral as an example. All closed curves on I:14 - (0) project onto closed curves on S 3. The relation between Coulomb-like configurations on euclidean s p a c e - t i m e and general configurations on S 3, established above, then means that the euclidean space loop integral
W(C)=tr P exp(-e f Au dxU) C cannot be trivial for Coulomb-like configurations, un-
402
28 May 1981
less the corresponding quantity for general potentials on S 3 is trivial, which is o f course not true. My warmest thanks are due to P.P. Divakaran for encouragement and several suggestions, l am also grateful to the members of the Theoretical Physics Group of T I F R for many fruitful discussions.
References [ 1 ] M.S. Dubovikov and A.V. Smilga, in: Proc. Intern. Seminar Group methods in field theory (Moscow, 1979). [21 C Cronstrom, Phys. Lett. 90B (1980) 267. [3l M.A. Shifman, preprint ITEP-12 (Moscow, 1980). [4] A. N iemi, 1telsinki preprint I-1U-TI.r-80-16. [5 ] V.N. Gribov, Lecture 12th Winter School of the t,eningrad Nuclear Physics Institute (1977). [6] I.M. Singer, Commun. Math. Phys. 60 (1978) 7. [7] M.S. Narasimhan and T.R. Ramadas, (ommun. Math. Phys 67 (1979) 121.
PHYSICS LETTERS
28 May 1981
REMARKS ON THE "HOMOGENEOUS" GAUGE
M. AZAM
Tara Institute of Fundamental Research. Bombay 400005. India Received 10 November 1980
It is noted that some recently claimed advantages for the gauge xvA u = 0 are not in general true.
The gauge condition [ 1,2]
x'A - xv.A u = 0
(1)
has recently been the object of much interest. A number of advantages have been claimed [ 2 - 4 ] for nonabelian gauge theories in this gauge: (i) an inversion formula l
Au(x ) = f ds sxvFv(sx) 0
(2)
holds (A u and Fur take values in the Lie algebra of the gauge group); (ii) potentials satisfying (1) are free from the gaugefixing ambiguity [5l. These claims are clearly of great interest; in particular, Shifman [3] has used eq. (2) in a rather detailed study of the Wilson loop. Unfortunately, there are configurations of the gauge potentials for which formula (2) breaks down and gauge fixing ambiguities are present. We consider euclidean space-time with the origin removed, so that the gauge condition (1) is defined everywhere. It is easy to see then (e.g. ref. [2] ) that the orbit of a given A~, under the gauge transformations
{A g = g - l A u g - e - l g - l O
g}
always intersects x ' A = 0 and that in the gauge given by (1), gauge transformations are homogeneous functions of x (hence the name homogeneous gauge): g(sx) = g(x).
(3)
Now, the Coulomb-like configuration of the potentials satisfying
(4)
A (sx) = s - l A (x),
is invariant under homogeneous gauge transformations: =
Clearly, for these configurations the inversion formula (2) fails (it is equally easy to see where the derivation of (2) given in refs. [2,3] breaks down ~ t ). As far as the second claim is concerned, note that eqs. (3) and (4) enable a compactification of s p a c e time from R 4 - {0) to the three-dimensional sphere S 3. The gauge functions satisfying (3) are trivially defined on S 3 and the Coulomb-like configurations can also be mapped on S 3 by scaling by the radial distance. In the spherical coordinate system the homogeneous gauge condition (1) reduces to
A r = 0.
(5)
Non-vanishing components of the potentials are
Ai(r, Ol,O2,03)-Ai(r,O ),
i=01,02,03,
(6)
and eq. (4) becomes
rAi(r, 0) = Ai(I, 0),
(7)
(as r ~ 0% Ag(r, 0 ) ~ O, independent of 0). This equation establishes a one-one correspondence between the Coulomb-like potentials on R 4 _ (0} and a general poI In ref. [21 only those potentials which are (72 everywhere in space-time are considered. Clearly this excludes the example we construct.
0 031 9163/81/0000-0000/$ 02.50 © North-Holland Publishing Company
401
Volume I01B, number 6
PHYSICS I.ETTERS
tential on S 3. We may now use the result of Singer [6] and Narasimhan and Ramadas [7] (that gauge-fixing is impossible in gauge theories on S 3) to conclude that the Coulomb-like configuration is not free from gauge ambiguity. It may now be asked whether these troublesome configurations are in some sense negligible in the computation of gauge-invariant ("observable") quantities. We consider the Wilson loop integral as an example. All closed curves on I:14 - (0) project onto closed curves on S 3. The relation between Coulomb-like configurations on euclidean s p a c e - t i m e and general configurations on S 3, established above, then means that the euclidean space loop integral
W(C)=tr P exp(-e f Au dxU) C cannot be trivial for Coulomb-like configurations, un-
402
28 May 1981
less the corresponding quantity for general potentials on S 3 is trivial, which is o f course not true. My warmest thanks are due to P.P. Divakaran for encouragement and several suggestions, l am also grateful to the members of the Theoretical Physics Group of T I F R for many fruitful discussions.
References [ 1 ] M.S. Dubovikov and A.V. Smilga, in: Proc. Intern. Seminar Group methods in field theory (Moscow, 1979). [21 C Cronstrom, Phys. Lett. 90B (1980) 267. [3l M.A. Shifman, preprint ITEP-12 (Moscow, 1980). [4] A. N iemi, 1telsinki preprint I-1U-TI.r-80-16. [5 ] V.N. Gribov, Lecture 12th Winter School of the t,eningrad Nuclear Physics Institute (1977). [6] I.M. Singer, Commun. Math. Phys. 60 (1978) 7. [7] M.S. Narasimhan and T.R. Ramadas, (ommun. Math. Phys 67 (1979) 121.