Lagrangian constraints and renormalization of 4D gravity

I Y Park

doi:10.1007/jhep04(2015)053

Abstract

It has been proposed in \cite{Park:2014tia} that 4D Einstein gravity becomes effectively reduced to 3D after solving the Lagrangian analogues of the Hamiltonian and momentum constraints of the Hamiltonian quantization. The analysis in \cite{Park:2014tia} was carried out at the classical/operator level. We review the proposal and make a transition to the path integral account. We then set the stage for explicitly carrying out the two-loop renormalization procedure of the resulting 3D action. We also address a potentially subtle issue in the gravity context concerning whether renormalizability does not depend on the background around which the original action is expanded.

Highlights

It has been proposed in [21] that 4D Einstein gravity becomes effectively reduced to 3D after solving the Lagrangian analogues of the Hamiltonian and momentum constraints of the Hamiltonian quantization
Instead of remaining entirely in the Hamiltonian quantization followed by the Wheeler-DeWitt equation, the present method employs the Lagrangian method
Gauge-fixing of these non-dynamical fields introduces constraints; 4D Einstein gravity reduces to 3D as a result of solving these constraints, and thereby a possibility for renormalizability of 4D Einstein gravity opens

Summary

Constraints and quantization

We review the quantization of 4D Einstein gravity proposed in [21] with a more detailed and refined account. If is often stated in the general gauge theory context in literature that one may take the path integral, instead of the canonical analysis, as the starting point ( the title of this subsection) We adopt this viewpoint in this subsection and push quantization as far as possible. Let us turn to the “independent” path integral It is rather obvious from the beginning that staying entirely within the path integral formulation (i.e., without the close guidance from the canonical operator formalism) should be impossible since, for one thing, 6Although δn has a highly non-linear and complicated metric dependence, it becomes simple once the gauge conditions n = 1, Nm = 0 are imposed:. As often stated in the gauge theory context, one may take, as the starting point, the more covariant form of the path integral in terms of the usual metric component measure dgμν (· · · )ei d4x R (2.45). The measure in (2.36) can be viewed as a gauge-fixed version of the 4D diffeomorphism invariant measure

Effectively 3D perturbative analysis

Discussions

Ghost sector

Graviton sector