Use of Lagrange multiplier fields to eliminate multiloop corrections

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 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.

Thermal gauge theories with Lagrange multiplier fields

“…Such a feature is reminiscent of that observed in the case of the strange particles, which are produced by strong interactions only pairwise. Since the Lagrange multiplier theory reduces to Einstein's theory in the classical limit at zero temperature [11][12][13][14][15], it is compatible with general relativity in the low energy domain.…”

“…Recently, it has been proposed that there may be an alternative way of quantizing the EH action that removes such divergences in a simpler manner, preserves unitarity and produces the expected tree level effects [11][12][13][14][15]. This involves the introduction of a Lagrangian multiplier (LM) field that restricts the path integral used to quantize the theory to paths that satisfy the classical Euler-Lagrange equations.…”

On the breakdown of space-time in General Relativity

“…In [14] and [15], the constraints derived for linear 1EH action and HGR action are shown to be different from those derived from the full theory action, and the resulting path integral is shown to be covariant as a result.Recovery of covariance in the weak limit of the metric tensor is an important finding, whereby this symmetry is restored and field path integral quantization is indeed then made possible. Recently, McKeon and collaborators et al have used background field theory approach in the weak limit of the gravitational field to render the 1EH and 2EH actions renormalizable, finite and restricted it to one-loop order using background and lagrange multplier fields [18][19][20][21][22][23][24][25]. These are also consistent with EFT results whereby quantum corrections are possible at 1-loop order in the low-energy limit of 4-dimensional GR, however there is no indication of breakdown in covariance in this approach, except that renormalizability beyond oneloop in this formalism is not possible [28,29].…”

“…We then deduce that this breakdown is actually a non-perturbative property of d-dimensional GR theory (d ≥ 2) itself from the recovery of covariance and renormalizability from linearized versions (which have different constraint structures for 1EH and HGR actions [14,15]). This is recent work focusing on 1EH and second order (2EH) actions using background and Lagrange multiplier fields and the path integral approach [18][19][20][21][22][23][24][25]. These findings are consistent with existing work on space-time breakdown in GR by Penrose [26,27], whereby this breakdown in GR happens in the strong limit of the gravitational field of a black hole, and also with Effective Field Theory (EFT) results which hold at one loop order in the low-energy limit of the theory [28,29].…”

Thermal gauge theories with Lagrange multiplier fields

“…Recently, it has been proposed that there may be an alternative way of quantizing the EH action that removes such divergences in a simpler manner, preserves unitarity and produces the expected tree level effects [11][12][13][14][15]. This involves the introduction of a Lagrangian multiplier (LM) field that restricts the path integral used to quantize the theory to paths that satisfy the classical Euler-Lagrange equations.…”