Bimetric truncations for quantum Einstein gravity and asymptotic safety
Section snippets
Introduction and motivation
Unifying the principles of quantum mechanics and general relativity is perhaps still the most challenging open problem in fundamental physics [1]. While the various approaches that are currently being developed, such as string theory, loop quantum gravity [2], [3], [4], or asymptotic safety [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], for instance, are based upon rather
Conformally reduced gravity as a theoretical laboratory
Our toy model is inspired by the observation that the (Euclidean) Einstein–Hilbert action,when evaluated for metrics , assumes the form of a standard action:Here is a “reference metric” which is fixed once and for all; in the following we usually assume it flat, whence . The corresponding classical equation of motion reads thenIn [25], [26] the scalar-like theory defined by (2.2) was
Generalized local potential approximation
In the following, we employ a generalized truncation ansatz which will allow us to disentangle the ϕ- and -dependencies of the EAA. We no longer identify the dynamical metric with the background metric as in (2.17). The ansatz has a nontrivial extra dependence now. It reads10
Comparison with standard scalar field theory on a rigid background
The gravitational average action is a functional of two metrics, and the full information is available only if is kept arbitrary. In particular this is necessary for setting up an FRGE. Likewise, in the conformally reduced case, is defined for an arbitrary , in accordance with the requirement of “background independence”. Nevertheless, should contain also the information about the beta functions one computes in the standard rigid-background approach.
To see
Summary and conclusion
Our discussion started from the principle of “background independence” which any satisfactory theory of quantum gravity should respect. This requirement can be met in two complementary ways: either one constructs the theory without using a background at all, or one does introduce some background, as a technical tool and for mathematical convenience, but shows that no prediction of the theory depends on it. Aiming at the quantization of gravity along the lines of asymptotic safety we employed
Acknowledgements
We thank J. Pawlowski, R. Percacci, and O. Rosten for helpful discussions.
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