It has been shown that by using a Lagrange multiplier field to ensure that the classical equations of motion are satisfied, radiative effects beyond one-loop order are eliminated. It has also been ...shown that through the contribution of some additional ghost fields, the effective action becomes form invariant under a redefinition of field variables, and furthermore, the usual one-loop results coincide with the quantum corrections obtained from this effective action. In this paper, we consider the consequences of a gauge invariance being present in the classical action. The resulting gauge transformations for the Lagrange multiplier field as well as for the additional ghost fields are found. These gauge transformations result in a set of Faddeev–Popov ghost fields arising in the effective action. If the gauge algebra is closed, we find the Becci–Rouet–Stora–Tyutin (BRST) transformations that leave the effective action invariant.
Using the background field method, we study in a general covariant gauge the renormalization of the six-dimensional Yang-Mills theory. This requires background gauge invariant counterterms, some of ...which do not vanish on shell. Such counterterms occur, even off shell, with gauge-independent coefficients. The analysis is done at one-loop order and the extension to higher orders is discussed by means of the Becchi-Rouet-Stora-Tyutin identities. We examine the behavior of the beta function, which implies that this theory is not asymptotically free.
We examine the self-consistency of the first-order formulation of the Yang-Mills theory. By comparing the generating functional Z before and after integrating out the additional field Fμνa, we derive ...a set of structural identities that must be satisfied by the Green's functions at all orders. These identities, which hold in any dimension, are distinct from the usual Ward identities and are necessary for the internal consistency of the first-order formalism. They relate the Green's functions involving the fields Fμνa, to Green's functions in the second-order formulation which contain the gluon strength tensor fμνa. In particular, such identities may provide a simple physical interpretation of the additional field Fμνa.
We study the Becchi-Rouet-Stora-Tyutin (BRST) renormalization of an alternative formulation of the Yang-Mills theory, where the matrix-propagator of the gluon and the complementary fields is ...diagonal. This procedure involves scalings as well as nonlinear mixings of the fields and sources. We show, in the Landau gauge, that the BRST identities implement a recursive proof of renormalizability to all orders.
We study the self-consistency of the first-order formulation of quantum gravity, which may be attained by introducing, apart from the graviton field, another auxiliary quantum field. By comparing the ...forms of the generating functional Z before and after integrating out the additional field, we derive a set of structural identities, which must be satisfied by the Green's functions at all orders. These are distinct from the usual Ward identities, being necessary for the self-consistency of the first-order formalism. They relate the Green's functions involving the additional quantum field to those containing a certain composite graviton field, which corresponds to its classical value. Thereby, the structural identities lead to a simple interpretation of the auxiliary field.
By using the Faddeev-Popov quantization procedure, we demonstrate that the radiative effects computed using the first-order and second-order Einstein-Hilbert action for general relativity are the ...same, provided one can discard tadpoles. In addition, we show that the first-order form of this action can be used to obtain a set of Feynman rules that involves just two propagating fields and three three-point vertices; using these rules is considerably simpler than employing the infinite number of vertices that occur in the second-order form. We demonstrate this by computing the one-loop, two-point function.
We consider the first order form of the Einstein-Hilbert action and quantize it using the path integral. Two gauge fixing conditions are imposed so that the graviton propagator is both traceless and ...transverse. It is shown that these two gauge conditions result in two complex fermionic vector ghost fields and one real bosonic vector ghost field. All Feynman diagrams to any order in perturbation theory can be constructed from two real bosonic fields, two fermionic ghost fields and one real bosonic ghost field that propagate. These five fields interact through just five three point vertices and one four point vertex.
The problem of eliminating divergences arising in quantum gravity is generally addressed by modifying the classical Einstein-Hilbert action. These modifications might involve the introduction of a ...local supersymmetry, the addition of terms that are higher order in the curvature to the action, or invoking compactification of superstring theory from ten to four dimensions. An alternative to these approaches is to introduce a Lagrange multiplier field that restricts the path integral to field configurations that satisfy the classical equations of motion; this has the effect of doubling the usual one-loop contributions and of eliminating all effects beyond one loop. We show how this reduction of loop contributions occurs and find the gauge invariances present when such a Lagrange multiplier is introduced into the Yang-Mills and Einstein-Hilbert actions. Moreover, we quantize using the path integral, discuss the renormalization, and then show how Becchi-Rouet-Stora-Tyutin (BRST) invariance can be used to both demonstrate that unitarity is retained and to find BRST relations between Green's functions. In the Appendices we show how background field quantization can be implemented, consider the use of a Lagrange multiplier field to restrict higher-order contributions in supersymmetric theories, and derive the BRST equations satisfied by the generating functional.