VOLUME 73, NUMBER 7 PH YS ICAL REVIEW LETTERS 15 AUrUsT 1994

Lavelle and McMullan Reply: In his Comment Kieu [I]makes two distinct points. He suggests that transformingfrom the usual fermion fields to the physical electrons,Pvh„, = exp[ —ig(B;A'/V2)]lit(x), may slightly simplifythe bookkeeping involved in perturbatively calculatingthe physical, gauge-invariant and infrared finite Green'sfunctions first examined in Ref. [2]. Second he claimsthat the new symmetry of QED found in Ref. [3] cannotbe used to help single out physical fields, contrary to ourprevious claims. In this Reply we wish to point out theerror in this latter reasoning.

Kieu claims that the fields ti,.„(x) = exp [ i g-x (8 A" /i)2)]t/I(x), and their barred equivalents, arephysical because they are invariant under gauge, BRST,and anti-BRST transformations. He then points out thatthey are not invariant under the noncovariant symmetryfound in Ref. [3] and concludes that the symmetry cannotbe used to distinguish between physical and nonphysicalfields as was done in [3]. Other people with whom wehave discussed our recent work have also asked why

P„„ is not to be preferred to ttvh„, and they were alsointroduced in Ref. [4]. We would now like to clear thematter up.

First, it must be stressed that the description of aphysical electron given in Ref. [3] is of a static electron.In particular it has a Coulomb electric field. As expectedthis electron is not Lorentz covariant. The covariant,gauge invariant fields considered by Kieu do not havea Coulomb field and would not lead to infrared-finiteGreen's functions.

Finding the physical degrees of freedom is a moresubtle affair than looking for mere invariance under gaugetransformations. After all this would seem to imply thatthe three transverse components of AT are physical and,despite Kieu's Comment, we know that in fact only thetwo A; are. To show this last is a subtle affair andis especially problematic when fermions are taken intoaccount. This was pointed out in Ref. [3]. The symmetrypresented there correctly singles out A; and not AT. Thatthe time component of the AT field can be related tothe charge density in fact demonstrates that it is not anindependent physical variable.

The fields P„„are moreover nonlocal in time, in

contrast to the Dirac electron which is merely spatiallynonlocal. This renders in our opinion a Hamiltonian for-malism based around lit„„ impossible. (As a minor asideit would also render time ordering rather difficult. ) Indeedthe nonlocality in time would prevent us from decompos-ing the fields in terms of positive and negative frequencycomponents. Thus the identification of the Dirac sea,the one-particle states, and the like is incompatible withthe use of these fields. Kieu's Lagrangian (8) still has agauge symmetry and he is forced to introduce the Landaugauge constraint. The physical fields in Landau gauge areconsidered in great detail in Ref. [5] and are shown to be

just the A; .A second possible description of the electron is

based upon a stringlike picture. Consider titr (x):=-

exp[ig I „A„(z)dz"]P(z), where I is any contour fromthe point x to —~. Although this is, by construction,gauge invariant, it cannot describe a physical field. Itis again nonlocal in time and is furthermore dependentupon the arbitrary line I . A physical electron cannothave either of these properties. If we demand that thecontour does not introduce any temporal nonlocality wemust have fr (x) —= exp[ig P „A;(z)dz']P(x). We notethat the change in the electric field due to such an ansatzis determined by the commutator of the electric field and

Since this vanishes everywhere except along theintegration contour tetr only generates an electric fieldalong the contour. The physical electron, by contrast,generates the correct classical Coulomb field that oneassociates with an electron and this was Dirac's initialmotivation for introducing it. If we now decompose thespatial components into the physical, transverse compo-nents, A;, and the unphysical, longitudinal component,A; =- 8; i),A, / 7 we find pr (x) = Nr (x) A~h&s (~), where

Nr(x) = exp[ig I"„A; (z)dz']. In other words we havefactorized the contour dependence in a gauge invariantway. Removing this unphysical dependence from theansatz leaves us with Dirac's physical electron and theCoulomb electric field.

We conclude by noting that Dirac's electron seemsto offer the only satisfactory description of the particleobserved in experiments and that our interpretation of thesymmetry observed in Ref. [3] holds.

M. J. Lavelle' and D. McMullan2~Institut fiir Physik (ThEP) der Universitat Mainz

D-55099 Mainz, Federal Republic of Germany"-School of Mathematics and Statistics

University of Plymouth

Drake Circus, Plymouth

Devon, PL4 8AA, United Kingdom

Received 28 March 1994PACS numbers: 12.20.—m, 11.30.—j

[1] T. D. Kieu, preceding Comment, Phys. Rev. Lett. 73,1047 (1994).

[2] M. J. Lavelle and D. McMullan, Phys. Lett. B 312, 211(1993).

[3] M. J. Lavelle and D. McMullan, Phys. Rev. Lett. 71, 3758(1993).

[4] T. Kashiwa and Y. Takahashi, Kyushu and Alberta ReportNo. KYUSHU-HET-14 (to be published).

[5] N. Nakanishi and I. Ojima, Covariant Operator Formalism of Gauge Theories and Quantum Gravity (World Sci-entific, Singapore 1990).

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