Gauge invariant fields
Quantum field theory is extended to include purely virtual “cloud sectors”, which allow us to define point-dependent observables, including a gauge invariant metric and gauge invariant matter fields, and calculate their off-shell correlation functions perturbatively in quantum gravity. Each extra sector is made of a cloud field, its anticommuting partner, a cloud function and a cloud Faddeev-Popov determinant. Thanks to certain cloud symmetries, the ordinary correlation functions and S matrix elements are unmodified. The clouds are rendered purely virtual, to ensure that they do not propagate unwanted degrees of freedom. So doing, the off-shell, diagrammatic version of the optical theorem holds and the extended theory is unitary. Every insertion in a correlation function can be dressed with its own cloud. The one-loop two-point functions of dressed scalars, vectors and gravitons are calculated. Their absorptive parts are positive, cloud independent and gauge independent, while they are unphysical if non purely virtual clouds are used. Renormalizability is proved to all orders by means of an extended Batalin-Vilkovisky formalism and its Zinn-Justin master equations. The purely virtual approach is compared to other approaches available in the literature.
We extend quantum field theory by including purely virtual “cloud” sectors, which allow us to define physical off-shell correlation functions of gauge invariant quark and gluon fields. Thanks to certain “cloud symmetries”, the new sectors do not change the fundamental physics. In particular, the ordinary correlation functions and the S matrix amplitudes remain the same. Each cloud sector is made of a cloud field, its anticommuting partner, a cloud function and a cloud Faddeev-Popov determinant. Every field insertion in a correlation function can be made gauge invariant by dressing it with an independent cloud. The cloud sectors are rendered purely virtual, to ensure that they do not propagate extra degrees of freedom. The off-shell, diagrammatic version of the optical theorem holds, and the extended theory is unitary. The one-loop two-point functions of the dressed quarks and gluons are calculated. Their absorptive parts are gauge independent, cloud independent and positive (while they are cloud dependent and possibly negative, if the clouds are defined by means of the Feynman prescription). A gauge/cloud duality simplifies the computations and shows that the gauge choice is just a particular cloud. Renormalizability is proved to all orders by means of an extended Batalin-Vilkovisky formalism and its Zinn-Justin master equations. We compare the purely virtual approach with the Coulomb nonlocal dressing of Dirac for QED, and the one of Lavelle and McMullan for non-Abelian gauge theories. We also comment on the use of Wilson lines and ‘t Hooft composite fields.