Abstract
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 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 Greens functions. In the Appendices, we show how the 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.
Highlights
The removal of ultraviolet divergences that arise in quantum field theory has been a long-standing problem
It involves introducing a term into the action in which a Lagrange multiplier (LM) field is used to ensure that the classical equations of motion are satisfied
As SYM þ SLM þ Sgf þ Sghost has the structure of Eq (2.12), we see that there are no diagrams beyond one-loop order, with the tree level diagrams being those that occur in normal YM theory and the one-loop diagrams being doubled
Summary
The removal of ultraviolet divergences that arise in quantum field theory has been a long-standing problem. We propose a relatively uncomplicated way of quantizing the EH action that makes it possible to remove divergences induced by quantum effects without losing unitarity and that retains GR in the classical limit It involves introducing a term into the action in which a Lagrange multiplier (LM) field is used to ensure that the classical equations of motion are satisfied. If one quantizes a classical Lagrangian for a field φi that has been supplemented by such a term by using the path integral formalism, it can be shown that the one-loop radiative corrections to the classical action are twice those that occur if the term involving the LM were absent, and that all radiative effects beyond one-loop order are absent The introduction of such a LM field has been considered in YM theory [26] in the Proca model [27] and with the EH Lagrangian [28].
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