Some remarks on local operators in quantum electrodynamics

Abstract

We assume the existence of a conserved current which generates locally gauge transformations of first kind. We are working in a local quantum Field Theory, where the fields are defined on a vector space where indefinite metric is allowed.We show that the Maxwell equations are not consistent with the above assumptions and the vectors obtained by applying local charged operators on the vacuum cannot describe physical states.Moreover we show that, if charged fields have non-trivial expectation value on the physical states, the vector space must contain vectors with negative norm.We discuss the relation between the local formulation of QED and a formulation in terms of physical states. As an example we study the transition from Gupta-Bleuler free QED to the Coulomb-gauge formulation.

Gauss’ Law and string-localized quantum field theory

“…Thus, it "inherits" the string-localization of the interaction density. This is in accord with the NoGo result of [9] mentioned in the introduction. If e is chosen spacelike, this also ensures sufficient spacelike separability for the needs of scattering theory.…”

“…commutes with all fields relatively local w.r.t. F µν , it follows that the interacting Dirac field cannot be relatively local [1,9].…”

Collision theory for massless Bosons

“…It is well known that locality of the charge carrying fields is in general not compatible with positivity of the metric in the state space. In quantum electrodynamics for example, one has either to abandon locality of the Fermi fields (as in the Coulomb-gauge) or one looses positivity of the metric (as in the Gupta-Bleuler gauge) [5]. For this reason gauge theories like quantum electrodynamics do not fit completely into the framework which is used in this paper.…”

Euclidean formulation of quantum field theory without positivity

“…Wl (Temperedness): The rc-point Wightman function W n (x l9 ...,x n ) are distributions, for simplicity taken of tempered type. 1 They also satisfy the standard hermiticity condition. W2 (Poincare covariance): For any n, W n (x i ,...,n n )=W(Λx 1 -a 9 ... 9 Λx n -a) 9…”

“…With the advent of gauge theories it became clear that it was natural (if not necessary) to consider QFT in which not all the Wightman axioms are satisfied. In particular, it appeared that the introduction of "charged" fields was in conflict with either locality or positivity [1][2][3][4]. On the other hand, the success of (perturbative) renormalization theory (also for gauge theories) [5,6] and the usefulness of keeping a relation with the wisdom gathered from conventional perturbation theory made clear that it could be better, at least at a technical level, to keep locality rather than positivity [3,4].…”