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Gauge choices and physical variables in QED
Physics Letters B 316 (1993) 172-174 North-Holland
PHYSICS LETTERS B
Gauge choices and physical variables in QED M a r t i n Lavelle a,1 a n d D a v i d M c M u l l a n b,2 a Institut J~r Physik, Johannes Gutenberg-Universit?it, 14:-55099Mainz, FRG b Dublin Institute for Advanced Studies, School of Theoretical Physics, Burlington Road, Dublin 4, Ireland Received 21 July 1993 Editor: P.V. Landshoff In a gauge theory like QED not all components of the gauge potentials are physical. Generally it proves necessary to remove some of the unphysical fields by a gauge-fixing. Clearly this should not, however, be done in such a way that physical degrees of freedom are removed. In this note it shall be heuristically shown that some widely used gauges can potentially do just that. In QED the physical degrees o f freedom [ 1] are the two transverse photon components, A/L(k), (kiA~-(k) = 0). After gauge-fixing, the gaugeinvariance of the theory is broken and the relevant symmetry is BRST-invariance. The above physical degrees of freedom are indeed BRST-invariant. The A0 component has zero canonical m o m e n t u m and is, in fact, a Lagrange multiplier for Gauss' law rather than a dynamical field. The longitudinal component, A~, is not BRST-invariant. The usual fermion is also gauge and BRST-dependent, but there is also a physical electron field as suggested by Dirac [2]. We have recently shown that the physical Green functions are gauge invariant and infrared finite [3 ]. In this short note we want to ask if all "gauges" are consistent with this physical content o f QED. Firstly, however, let us argue that there are indeed some "good" (physical) gauges in QED. In Coulomb gauge the longitudinal component is set to zero and two dynamical components remain. Looked at in such a light it is clear that this is a "good" gauge. In Landau gauge, O~A~ = 0, we see that we are actually identifying the longitudinal part of the spatial components with the time derivative of the multiplier. In this way we remove the unphysical, dynamical field, Af. The other Lorentz gauges may be built up in a similar manner. l E-mail address: [email protected]. 2 E-mail address: [email protected].
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Although, due to their mathematical simplicity, Lorentz gauges are very widely used, algebraic gauges (for an introduction see ref. [4] ) have acquired some popularity over the years. In perturbation theory their use is claimed to lead to some simplifications since some degrees of freedom vanish; note, however, that the gauge boson propagator is then a source of trouble. Outside of perturbation theory, these gauges are claimed to yield in Yang-Mills theory a description free of the Gribov ambiguity. We will now discuss some of these gauges in the above spirit. The temporal gauge is fixed by demanding A0 = 0. This means that the multiplier is set to zero and so Gauss' law is not automatically imposed. The unphysical field has not been removed and there is said to be a residual gauge freedom (of time independent gauge transformations). The perturbative propagator is not well defined (there are singularities o f the l/k0 form). Gauss' law must be reimposed in this formulation o f the theory, this is often done by fixing Coulomb gauge on an arbitrary time slice [5 ]. In a spatial axial gauge, say A3 = 0, there is again a residual gauge freedom. However, there is another, potentially more serious, problem: the theory appears to be over-constrained for certain momenta. If the photon three-momentum is orthogonal to the direction of the gauge fixing, then, without loss of generality, we may assume that the m o m e n t u m is in the " l " direction, thus A~ is then the longitudinal, unphysical photon. The A0 component ~s still not dynamical and Elsevier Science Publishers B.V.
Volume 316, number 1
PHYSICS LETTERS B
A3 is zero by the gauge choice. The only photon component left is A2 - and we see that the gauge choice has removed a physical degree of freedom. This represents a major difference between the temporal and the spatial axial gauge: in the former Gauss' law is missing and needs to be imposed, in the latter the condition A3 = 0 is too strong for all momenta such that k3 = 0. This is reminiscent o f results [6 ] from light-cone quantisation of gauge theories in a box, where it has also been shown that the light cone equivalent of the spatial axial gauge could not be employed for all momenta. The problem in this gauge appears only if the momentum k3 vanishes, one might ask if this is so serious? There is, however, a problem in perturbation theory for just k3 = 0, since the propagator diverges there. Many regularisations [4 ] o f this divergence attempt to retain the gauge condition, A3 = 0, and the theory would appear to be over-constrained in these approaches. A similar gauge where the problem discussed here does not appear is defined by a3A3 = 0. However, there is still a residual gauge freedom in that gauge, in fact there is no gauge fixing for vanishing k3, and the perturbative divergence is worse than in the usual axial gauge for inhomogeneous (gauge parameter non-zero) versions of 03A3 gauge. Similar simple arguments can be followed through for other gauges. In light-cone gauge (in equal-time quantisation!) we demand, say, A0 = A3. In other words we set the field A3 equal to the multiplier. If k has now zero component in the "3"-direction, then there is again only one physical component and the theory is over-constrained. The perturbative propagator has a divergence, at k0 = k3, which one can generally understand as following from the residual gauge freedom. Only for ko = 0 would these two problems coincide. In Fock-Schwinger gauge, one requires x , A ~ ( x ) = 0. The arguments above may be followed through and one finds that if xiA L vanishes there is only one component that one can identify with a physical photon. The perturbative propagator in this gauge is the subject of some debate [7] #1
#1 Note that in section V of Modanese [7] time ordering is not taken properly into account.
14 October 1993
For the various "flow gauges", i.e., gauges which via some parameter can "flow" between two different gauge fixings, similar considerations may be taken into account. It is not completely clear what the physical consequences of the above problems are. However, although these gauges seem to yield gauge invariant results at tree level, they evince difficulties when loops are taken into account and little is known of their non-perturbative validity. In fact the status of these gauges is uncertain and so some caution would perhaps not be amiss. The above analysis may not be directly carried over to Yang-Mills theories, where the physical degrees of freedom may not be so straightforwardly identified as in QED. (This is related to the Gribov ambiguity.) However, it seems clear that similar difficulties, or worse, will appear in non-abelian gauge theories. We conclude by remarking that many algebraic gauges appear to be unphysical already in QED. Note added. After completion of this note we received
a preprint by P.V. Landshoff and P. van Nieuwenhuizen [8]. Here the propagator in a class o f gauges, Ao d- ~A3 = 0, is considered. They obtain a nonstandard result for the regularisation of the pole such that for k3 = 0 they obtain the Feynman gauge propagator. This is fully in accord with the considerations above. It is not clear to us however, if the treatment in that paper is complete: are ghosts (which are usually present in Feynman gauge) required for k3 = 0 and would they yield non-vanishing contributions at higher loops? M.J.L. thanks the Dublin Institute for Advanced Studies for their warm hospitality and the Graduierten Kolleg of Mainz University for support. References
[1] N. Nakanishi and I. Ojima, Covariant Operator Formalism of Gauge Theories and Quantum Gravity (World Scientific, Singapore, 1990). [2] P.A.M. Dirac, Principles of Quantum Mechanics (OUP, Oxford, 1958), p. 302. [ 3 ] M. Lavelle and D. McMullan, Phys. Lett. B 312 ( 1993 ) 211. [4] P. Gaigg et al. (eds.), Physical and Nonstandard Gauges (Springer, Berlin, 1990); 173
Volume 316, number 1
PHYSICS LETTERS B
M. Lavelle and D. McMuUan, The Radiation Class: A Set of Temporal Gauges, Mainz/Dublin preprint, MZTH/92-29, DIAS-STP-92-13, Z. Phys. C, to appear. [5 ] See M. LaveUe, M.Schaden and A. Vladikas, Phys. Lett. B 203 (1988) 441, and references therein. [6] T. Heinzl, S. Krusehe and E. Werner, Phys. Lett. B 256 (1991) 55.
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[7] W. Kummer and J. Weiser, Z. Phys. C 31 (1986) 105; G. Modanese, J. Math. Phys. 33 (1992) 1523; R. Delbourgo and Triyanta, Intern. J. Mod. Phys. A 7 (1992) 5833. [8] P.V. Landshoff and P. van Nieuwenhuizen, Canonical Quantisation in n • A = 0 gauges, preprint DAMTP 93-33, ITP-SB-93-38.
PHYSICS LETTERS B
Gauge choices and physical variables in QED M a r t i n Lavelle a,1 a n d D a v i d M c M u l l a n b,2 a Institut J~r Physik, Johannes Gutenberg-Universit?it, 14:-55099Mainz, FRG b Dublin Institute for Advanced Studies, School of Theoretical Physics, Burlington Road, Dublin 4, Ireland Received 21 July 1993 Editor: P.V. Landshoff In a gauge theory like QED not all components of the gauge potentials are physical. Generally it proves necessary to remove some of the unphysical fields by a gauge-fixing. Clearly this should not, however, be done in such a way that physical degrees of freedom are removed. In this note it shall be heuristically shown that some widely used gauges can potentially do just that. In QED the physical degrees o f freedom [ 1] are the two transverse photon components, A/L(k), (kiA~-(k) = 0). After gauge-fixing, the gaugeinvariance of the theory is broken and the relevant symmetry is BRST-invariance. The above physical degrees of freedom are indeed BRST-invariant. The A0 component has zero canonical m o m e n t u m and is, in fact, a Lagrange multiplier for Gauss' law rather than a dynamical field. The longitudinal component, A~, is not BRST-invariant. The usual fermion is also gauge and BRST-dependent, but there is also a physical electron field as suggested by Dirac [2]. We have recently shown that the physical Green functions are gauge invariant and infrared finite [3 ]. In this short note we want to ask if all "gauges" are consistent with this physical content o f QED. Firstly, however, let us argue that there are indeed some "good" (physical) gauges in QED. In Coulomb gauge the longitudinal component is set to zero and two dynamical components remain. Looked at in such a light it is clear that this is a "good" gauge. In Landau gauge, O~A~ = 0, we see that we are actually identifying the longitudinal part of the spatial components with the time derivative of the multiplier. In this way we remove the unphysical, dynamical field, Af. The other Lorentz gauges may be built up in a similar manner. l E-mail address: [email protected]. 2 E-mail address: [email protected].
172
Although, due to their mathematical simplicity, Lorentz gauges are very widely used, algebraic gauges (for an introduction see ref. [4] ) have acquired some popularity over the years. In perturbation theory their use is claimed to lead to some simplifications since some degrees of freedom vanish; note, however, that the gauge boson propagator is then a source of trouble. Outside of perturbation theory, these gauges are claimed to yield in Yang-Mills theory a description free of the Gribov ambiguity. We will now discuss some of these gauges in the above spirit. The temporal gauge is fixed by demanding A0 = 0. This means that the multiplier is set to zero and so Gauss' law is not automatically imposed. The unphysical field has not been removed and there is said to be a residual gauge freedom (of time independent gauge transformations). The perturbative propagator is not well defined (there are singularities o f the l/k0 form). Gauss' law must be reimposed in this formulation o f the theory, this is often done by fixing Coulomb gauge on an arbitrary time slice [5 ]. In a spatial axial gauge, say A3 = 0, there is again a residual gauge freedom. However, there is another, potentially more serious, problem: the theory appears to be over-constrained for certain momenta. If the photon three-momentum is orthogonal to the direction of the gauge fixing, then, without loss of generality, we may assume that the m o m e n t u m is in the " l " direction, thus A~ is then the longitudinal, unphysical photon. The A0 component ~s still not dynamical and Elsevier Science Publishers B.V.
Volume 316, number 1
PHYSICS LETTERS B
A3 is zero by the gauge choice. The only photon component left is A2 - and we see that the gauge choice has removed a physical degree of freedom. This represents a major difference between the temporal and the spatial axial gauge: in the former Gauss' law is missing and needs to be imposed, in the latter the condition A3 = 0 is too strong for all momenta such that k3 = 0. This is reminiscent o f results [6 ] from light-cone quantisation of gauge theories in a box, where it has also been shown that the light cone equivalent of the spatial axial gauge could not be employed for all momenta. The problem in this gauge appears only if the momentum k3 vanishes, one might ask if this is so serious? There is, however, a problem in perturbation theory for just k3 = 0, since the propagator diverges there. Many regularisations [4 ] o f this divergence attempt to retain the gauge condition, A3 = 0, and the theory would appear to be over-constrained in these approaches. A similar gauge where the problem discussed here does not appear is defined by a3A3 = 0. However, there is still a residual gauge freedom in that gauge, in fact there is no gauge fixing for vanishing k3, and the perturbative divergence is worse than in the usual axial gauge for inhomogeneous (gauge parameter non-zero) versions of 03A3 gauge. Similar simple arguments can be followed through for other gauges. In light-cone gauge (in equal-time quantisation!) we demand, say, A0 = A3. In other words we set the field A3 equal to the multiplier. If k has now zero component in the "3"-direction, then there is again only one physical component and the theory is over-constrained. The perturbative propagator has a divergence, at k0 = k3, which one can generally understand as following from the residual gauge freedom. Only for ko = 0 would these two problems coincide. In Fock-Schwinger gauge, one requires x , A ~ ( x ) = 0. The arguments above may be followed through and one finds that if xiA L vanishes there is only one component that one can identify with a physical photon. The perturbative propagator in this gauge is the subject of some debate [7] #1
#1 Note that in section V of Modanese [7] time ordering is not taken properly into account.
14 October 1993
For the various "flow gauges", i.e., gauges which via some parameter can "flow" between two different gauge fixings, similar considerations may be taken into account. It is not completely clear what the physical consequences of the above problems are. However, although these gauges seem to yield gauge invariant results at tree level, they evince difficulties when loops are taken into account and little is known of their non-perturbative validity. In fact the status of these gauges is uncertain and so some caution would perhaps not be amiss. The above analysis may not be directly carried over to Yang-Mills theories, where the physical degrees of freedom may not be so straightforwardly identified as in QED. (This is related to the Gribov ambiguity.) However, it seems clear that similar difficulties, or worse, will appear in non-abelian gauge theories. We conclude by remarking that many algebraic gauges appear to be unphysical already in QED. Note added. After completion of this note we received
a preprint by P.V. Landshoff and P. van Nieuwenhuizen [8]. Here the propagator in a class o f gauges, Ao d- ~A3 = 0, is considered. They obtain a nonstandard result for the regularisation of the pole such that for k3 = 0 they obtain the Feynman gauge propagator. This is fully in accord with the considerations above. It is not clear to us however, if the treatment in that paper is complete: are ghosts (which are usually present in Feynman gauge) required for k3 = 0 and would they yield non-vanishing contributions at higher loops? M.J.L. thanks the Dublin Institute for Advanced Studies for their warm hospitality and the Graduierten Kolleg of Mainz University for support. References
[1] N. Nakanishi and I. Ojima, Covariant Operator Formalism of Gauge Theories and Quantum Gravity (World Scientific, Singapore, 1990). [2] P.A.M. Dirac, Principles of Quantum Mechanics (OUP, Oxford, 1958), p. 302. [ 3 ] M. Lavelle and D. McMullan, Phys. Lett. B 312 ( 1993 ) 211. [4] P. Gaigg et al. (eds.), Physical and Nonstandard Gauges (Springer, Berlin, 1990); 173
Volume 316, number 1
PHYSICS LETTERS B
M. Lavelle and D. McMuUan, The Radiation Class: A Set of Temporal Gauges, Mainz/Dublin preprint, MZTH/92-29, DIAS-STP-92-13, Z. Phys. C, to appear. [5 ] See M. LaveUe, M.Schaden and A. Vladikas, Phys. Lett. B 203 (1988) 441, and references therein. [6] T. Heinzl, S. Krusehe and E. Werner, Phys. Lett. B 256 (1991) 55.
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14 October 1993
[7] W. Kummer and J. Weiser, Z. Phys. C 31 (1986) 105; G. Modanese, J. Math. Phys. 33 (1992) 1523; R. Delbourgo and Triyanta, Intern. J. Mod. Phys. A 7 (1992) 5833. [8] P.V. Landshoff and P. van Nieuwenhuizen, Canonical Quantisation in n • A = 0 gauges, preprint DAMTP 93-33, ITP-SB-93-38.