Operator Gauge Transformation Research Articles

Gauge invariance and renormalization

The renormalization of Abelian and non-Abelian local gauge theories is discussed. It is recalled that whereas Abelian gauge theories are invariant to local c-number gauge transformations δA μ ( x) = ∂ μ ,…, with □ Λ = 0, and to the operator gauge transformation δA μ ( x) = ∂ μ φ( x), …, δφ( x) = α −1 ∂· A( x), with □ φ = 0, non-Abelian gauge theories are invariant only to the operator gauge transformations δA μ(x) ∼ ▪ μC(x) , …, introduced by Becchi, Rouet and Stora, where ▪ μ is the covariant derivative matrix and C is the vector of ghost fields. The renormalization of these gauge transformation is discussed in a formal way, assuming that a gauge-invariant regularization is present. The naive renormalized local non-Abelian c-number gauge transformation δA μ(x) = (Z 1/Z 3)gA μ(x) × Λ(x)+∂ μ Λ(x) , …, is never a symmetry transformation and is never finite in perturbation theory. Only for Λ(x) = (Z 3/Z 1)L with L finite constants or for Λ(x) = Ω z ̃ 3C(x) with Ω a finite constant does it become a finite symmetry transformation, where z ̃ 3 is the ghost field renormalization constant. The renormalized non-Abelian Ward-Takahashi (Slavnov-Taylor) identities are consequences of the invariance of the renormalized gauge theory to this formation. It is also shown how the symmetry generators are renormalized, how photons appear as Goldstone bosons, how the (non-multiplicatively renormalizable) composite operator A μ × C is renormalized, and how an Abelian c-number gauge symmetry may be reinstated in the exact solution of many asymptotically fr ee non-Abelian gauge theories.

The fermion propagator of quantum electrodynamics in an arbitrary gauge

The fermion propagator equation derived in the preceding paper has been solved using the following approximations: (i) the free proton propagator in an arbitrary gauge is used: (ii) for the vertex a functional of the fermion propagator which approximately obeys the Ward-Takahashi identity is taken; (iii) in the resulting equation a small term is neglected. Closed expressions for the fermion propagator and the renormalization function are obtained in an arbitrary gauge. The gauge dependence found for the fermion propagator is in agreement with the one which follows directly from the gauge transformation of the time ordered product defining the propagator. One recovers the well-known result that Z1 = Z2 (=Z(0)) can only be a finite non-zero number in one special gauge. From a discussion of the operator gauge transformation it seems likely that this finite gauge is non-physical and that in all physical gauges the electron propagator is more singular than the free one and that Z(0) = 0. In contraadistinction to the usual situation this result causes in the present formulation no difficulty and may be essential for the finiteness of the vacuum polarization.

Operator Gauge Transformations and Gauge Transformations of the Third Kind

A connection between gauge transformations of the third kind and operator gauge transformations of the first and second kinds is shown to hold for the electron propagator and for the vertex function of quantum electrodynamics as well as for the photon propagator. A method for calculating the gauge change of any time ordered n-point function is indicated.

Operator and generalized gauge transformations in quantum electrodynamics

Previous studies on the relations between Green’s functions are generalized. In the majority of the published literature dealing with this problem, gauge transformations are defined through the change of a generating functional, or equivalently, through certain specified changes of the Feynman rules. The connection between this definition, and the definition of defining a gauge transformation by the addition of the gradient of an operator-valued gauge function to the electromagnetic potential (operator gauge transformation), is studied in this paper. It is found that each operator gauge transformation from the radiation gauge may be defined through certain modifications of the Feynman rules in that gauge. However, usually it consists of more modifications than those assumed before. These modifications due to an operator gauge transformation are all of a special type. Modifications of other type may be included, which serves to define generalized gauge transformations. An example of a generalized gauge transformation is given in which the interacting electron propagator coincides with the free one outside any prespecified space-time domain. This example serves to illustrate the many other possibilities of gauge transformations that have not been considered before. A theorem giving a relation between the electron Green’s functions in different gauges is also proved.

Commutation relations for electromagnetic potentials

Given the known (gauge-independent) value of [Fαβ(x),Fγδ(y)], we derive the most general expression for the commutator [Aα(x),Aβ(y)]. The latter contains four arbitrary functionsNα(x,y), which, if suitably chosen, may yield the Coulomb gauge, the Feynman gauge, the Landau gauge, etc. It is however impossible to go over from one value ofNα(x,y) to another by means of an operator gauge transformation, unless the difference between both values ofNα satisfies some differential relations.

Gauge invariance, the vertex function, and the magnitude of the renormalization constants of quantum electrodynamics

Using the Gupta-Bleuler formalism for quantum electrodynamics, we have proven that the regularized second-order vertex function, in the limit of high momentum-transferP2, on the mass shell, is given by γΜC logP2 rather than γΜ. This proves that the assumption ofG. KAllen that the renormalized vertex function tends to Z1γΜ in the above limit, is not valid to second order in the Feynman gauge. Since the KAllen formalism is the proper renormalization of the Feynman gauge, this indicates that KAllen’s assumption is not valid in the KAllen gauge. The failure of this assumption would invalidate KAllen’s proof that a completely finite theory of quantum electrodynamics cannot be formulated in a consistent way. Our examination of the asymptotic behavior of the vertex function was motivated by the research ofB. Zumino andK. Johnson, who objected to the lack of gauge invariance in KAllen’s proof. Zumino, in particular, has pointed out that the assumption of KAllen concerning the high momentum-transfer behavior of the renormalized vertex function was not gauge-invariant, and that the assumption was therefore not likely to be true in the KAllen gauge. The gauge transformations that produce changes in the propagators, and therefore produce changes in the magnitude of the renormalization constants, were first given byLandau andKhalatnikov. Their derivation of the transformation formulas for the propagators from an operator gauge-transformation, was considered dubious byZumino, who derived the same formulas using his functional formulation for quantum electrodynamics. We have formulated a valid operator gauge-transformation that reproduces the Landau transformations for the propagators. Our operator gauge-transformation introduces a set of noninteracting scalar fields in a Hubert space which has an indefinite metric. In theories which do not have a Ward identity, the high-momentum limit of the renormalized vertex function has sometimes been taken as the definition of Z1. We have shown that this is not correct for the vector meson, but that it may be valid for the pseudoscalar meson, when these mesons interact with a spinor field.