Abstract
The covariant quantization of QED and QCD requires the introduction of subsidiary gauge-fixing and ghost fields and it is crucial to understand the role of these in the energy-momentum tensor, and in the momentum and angular momentum operators. These issues were studied in \cite{Leader:2011za}, and key results from this study were utilized in the major review of angular momentum in \cite{Leader:2013jra}. Damski \cite{Damski:2021feu} has rightly criticized as incorrect the derivation of certain equations in \cite{Leader:2011za}. We show, however, that the key results of \cite{Leader:2011za} which are utilized in \cite{Leader:2013jra} are unaffected by Damski's criticism.
Highlights
It is well known that the covariant quantization of QED and QCD, i.e., in which the photon vector potential AμðxÞ and the gluon vector potential Aμa transform as genuine Lorentz 4-vectors, is a nontrivial task [1–4] involving the introduction of a scalar gauge-fixing field (Gf) in QED and both a gauge-fixing field and Faddeev-Popov ghosts fields (Gf þ Gh) in QCD
Damski [8] has pointed out that this “proof,” for both the canonical and Bellinfante cases in QED, is wrong, because it assumes that the physPical states alone form a complete set and uses 1 1⁄4 jΦihΦj, which is an incorrect “resolution of the identity” because it leaves out the states of negative norm. He does not comment on QCD, Damski’s argument shows that the “proof” that hΦ0jtμcaνnðGf þ GhÞjΦi 1⁄4 0 in [7] is incorrect. This criticism seems to imply that the conclusion reached in [7], that the subsidiary fields do not contribute to the physical matrix elements of the Canonical and Bellinfante versions of the momentum and angular momentum in QED and QCD, is false, but as will be shown, this implication is wrong
(c) Despite Eq, (1) it turns out that the subsidiary fields do not contribute to the physical matrix elements of the canonical version of the momentum or angular momentum
Summary
It is well known that the covariant quantization of QED and QCD, i.e., in which the photon vector potential AμðxÞ and the gluon vector potential Aμa transform as genuine Lorentz 4-vectors, is a nontrivial task [1–4] involving the introduction of a scalar gauge-fixing field (Gf) in QED and both a gauge-fixing field and Faddeev-Popov ghosts fields (Gf þ Gh) in QCD. In all phenomenological papers dealing with the QCD case it is assumed, without comment, that the contribution from the subsidiary fields is zero That the latter is true for the Bellinfante case in QCD follows from the proof given in Joglekar and Lee and discussed by Collins in the above-cited works, where it is shown that the physical matrix elements of a Becchi, Rouet, Stora, Tyutin (BRST)-exact operator vanish. He does not comment on QCD, Damski’s argument shows that the “proof” that hΦ0jtμcaνnðGf þ GhÞjΦi 1⁄4 0 in [7] is incorrect This criticism seems to imply that the conclusion reached in [7], that the subsidiary fields do not contribute to the physical matrix elements of the Canonical and Bellinfante versions of the momentum and angular momentum in QED and QCD, is false, but as will be shown, this implication is wrong. We shall comment briefly on the important difference between gauge invariance and gauge independence, which was inadequately explained in [7]
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