Elsevier

Annals of Physics

Volume 325, Issue 4, April 2010, Pages 785-815
Annals of Physics

Bimetric truncations for quantum Einstein gravity and asymptotic safety

https://doi.org/10.1016/j.aop.2009.11.009Get rights and content

Abstract

In the average action approach to the quantization of gravity the fundamental requirement of “background independence” is met by actually introducing a background metric but leaving it completely arbitrary. The associated Wilsonian renormalization group defines a coarse graining flow on a theory space of functionals which, besides the dynamical metric, depend explicitly on the background metric. All solutions to the truncated flow equations known to date have a trivial background field dependence only, namely via the classical gauge fixing term. In this paper, we analyze a number of conceptual issues related to the bimetric character of the gravitational average action and explore a first nontrivial bimetric truncation in the simplified setting of conformally reduced gravity. Possible implications for the Asymptotic Safety program and the cosmological constant problem are discussed in detail.

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,SEH[gμν]=-116πGd4xg(R(g)-2Λ),when evaluated for metrics gμν=ϕ2gˆμν, assumes the form of a standard ϕ4 action:SEH[ϕ]=-34πGd4xgˆ12gˆμνμϕνϕ+112R^ϕ2-16Λϕ4Here gˆμν is a “reference metric” which is fixed once and for all; in the following we usually assume it flat, whence R^R(gˆ)=0. The corresponding classical equation of motion reads then^ϕ+23Λϕ3=0In [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 χB-dependencies of the EAA. We no longer identify the dynamical metric gμνϕ2gˆμν with the background metric g¯μνχB2gˆμν as in (2.17). The ansatz has a nontrivial extra χB dependence now. It reads10Γk[ϕ,χB]=-34πd4xgˆ12Gkϕ-^ϕ+12G

Comparison with standard scalar field theory on a rigid background

The gravitational average action Γk[gμν,g¯μν] is a functional of two metrics, and the full information is available only if g¯μν is kept arbitrary. In particular this is necessary for setting up an FRGE. Likewise, in the conformally reduced case, Γk[ϕ,χB] is defined for an arbitrary χB, in accordance with the requirement of “background independence”. Nevertheless, Γk[ϕ,χB] 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.

References (65)

  • E. Manrique, M. Reuter, F. Saueressig, in...
  • M. Reuter et al.
  • A. Bonanno et al.

    Gen. Rel. Grav.

    (2003)
    A. Bonanno et al.

    Class. Quant. Grav.

    (2004)
    A. Bonanno et al.

    Class. Quant. Grav.

    (2006)
    A. Bonanno et al.

    Class. Quant. Grav.

    (2007)
  • M. Reuter et al.

    Phys. Rev. D

    (2004)
  • M. Reuter et al.

    Phys. Rev. D

    (2004)
  • M. Reuter et al.

    JCAP

    (2004)
  • F. Girelli et al.

    Class. Quant. Grav.

    (2007)
  • O. Lauscher et al.

    Phys. Rev. D

    (2000)
  • D. Litim

    Phys. Lett. B

    (2000)
    D. Litim

    Phys. Rev. D

    (2001)
    D. Litim

    Int. J. Mod. Phys. A

    (2001)
  • C. Kiefer

    Quantum Gravity

    (2007)
    H. Hamber

    Quantum Gravitation

    (2008)
  • A. Ashtekar

    Lectures on Non-Perturbative Canonical Gravity

    (1991)
    A. Ashtekar et al.

    Class. Quant. Grav.

    (2004)
  • C. Rovelli

    Quantum Gravity

    (2004)
  • Th. Thiemann

    Modern Canonical Quantum General Relativity

    (2007)
  • S. WeinbergS. Weinberg,...
  • M. Reuter

    Phys. Rev. D

    (1998)
  • D. Dou et al.

    Class. Quant. Grav.

    (1998)
  • O. Lauscher et al.

    Phys. Rev. D

    (2002)
  • M. Reuter et al.

    Phys. Rev. D

    (2002)
  • O. Lauscher et al.

    Phys. Rev. D

    (2002)
  • O. Lauscher et al.

    Class. Quant. Grav.

    (2002)
  • O. Lauscher et al.

    Int. J. Mod. Phys. A

    (2002)
  • W. Souma

    Prog. Theor. Phys.

    (1999)
  • M. Reuter et al.

    Phys. Rev. D

    (2002)
    Fortschr. Phys. 52 (2004) 650,...
  • A. Bonanno et al.

    JHEP

    (2005)
  • For reviews, see: M. Reuter, F. Saueressig,...O. Lauscher et al.O. Lauscher et al.
  • R. Percacci et al.

    Phys. Rev. D

    (2003)
    R. Percacci et al.

    Phys. Rev. D

    (2003)
    R. Percacci et al.

    Class. Quant. Grav.

    (2004)
  • A. Codello et al.

    Phys. Rev. Lett.

    (2006)
    A. Codello et al.

    Int. J. Mod. Phys. A

    (2008)
  • D. Litim

    Phys. Rev. Lett.

    (2004)
    P. Fischer et al.

    AIP Conf. Proc.

    (2006)
    P. Fischer et al.

    Phys. Lett. B

    (2006)
    P. Fischer et al.

    AIP Conf. Proc.

    (2006)
  • P. Machado et al.

    Phys. Rev. D

    (2008)
  • D. Benedetti, P. Machado, F. Saueressig, hep-th/0901.2984 and...
  • O. Lauscher et al.

    JHEP

    (2005)
  • M. Reuter et al.

    JHEP

    (2006)
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